Electrical Engineering

 ‫א‬‫א‬‫א‬ ã¹]...

Author: General Management for Creating and Developing Courses

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 ‫א‬‫א‬‫א‬ ã¹]
 ٢ J

١٤١

‫א‬

‫ א‬١٤١

‫א‬

٢-



 W،،‫א‬‫א‬،‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬ ‫א‬  ،‫א‬   ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K ‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬        ،‫א‬ ‫א‬‫א‬   ‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫א‬    ،   ‫א‬،‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬      ‫א‬  ‫א‬   ،‫א‬     K‫א‬‫א‬،‫א‬‫א‬    ?  ? ٢J   ? ‫א‬ ‫א‬    K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬?  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ،‫א‬،‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬  ،‫א‬‫א‬‫א‬‫א‬  K‫א‬

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‫א‬‫א‬‫א‬‫א‬















‫ א‬١٤١

‫א‬

٢-



 ،KKK،‫א‬‫א‬‫א‬‫א‬  ‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬‫א‬‫א‬ ‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٢J  ‫א‬‫א‬  K‫א‬

‫א‬‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬،‫א‬ ،‫א‬‫א‬

،  ‫א‬‫א‬‫א‬‫א‬،

‫א‬،‫א‬‫א‬‫א‬‫א‬   K‫א‬‫א‬

 KKKKKK،‫א‬  ‫א‬



 ‫א‬‫א‬‫א‬ ã¹]
٢ J   ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬

١

‫א‬‫א‬

‫ א‬١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



 ‫א‬‫א‬‫א‬‫א‬



 W‫א‬‫א‬،‫א‬‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬ •

-١-

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



 ١J ١ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬   ‫א‬ ‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬  

 K‫א‬‫א‬‫א‬‫א‬‫א‬



Energy Conversion‫א‬٢J ١

 W‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬ K١ KEF‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W K٢ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K‫א‬

 K‫א‬‫א‬‫א‬

Primary Energy Resources‫א‬‫א‬‫א‬

١J ٢ J ١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬

‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬

 W‫א‬‫א‬

 W‫א‬‫א‬

 (Fossil Energy Resources)‫א‬‫א‬

 (Renewable Energy sources)‫א‬‫א‬ 

(Nuclear Energy sources) ‫א‬‫א‬

E١F

E٢F

E٣F

 ‫א‬  Fossil Energy Sources‫א‬‫א‬

١J ١J ٢J ١

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬ -٢-

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



‫א‬‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬

 (Petroleum Oil)‫א‬

EIF

 (Natural Gas)‫א‬‫א‬ EIIF (Coal)‫א‬ EIIIF

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬    ‫א‬    ‫א‬ ‫א‬      

   ‫א‬ ‫א‬      ‫א‬‫א‬     ‫א‬ ‫א‬

   ‫א‬   ‫א‬‫א‬  ‫א‬‫א‬   ‫א‬ ‫א‬  

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 ‫א‬   ‫א‬  ‫א‬    ‫א‬       

‫א‬ ‫א‬  ‫א‬    ‫א‬‫א‬ ‫א‬  ‫א‬‫א‬   ‫א‬ ‫א‬

‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬K  ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬ ،‫א‬‫א‬ ‫א‬‫א‬

 ‫א‬ ‫א‬   ‫א‬    ‫א‬    

‫א‬‫א‬‫א‬

 W‫א‬‫א‬ E‫א‬‫א‬F‫א‬

 ‫א‬‫א‬

 ٤٠

 ‫א‬

 ٦٠

 ‫א‬‫א‬

 ٢٥٠

 ‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬E١J ١F

-٣-



‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬

 Nuclear Energy‫א‬‫ א‬٢J ١J ٢J ١ ‫א‬‫א‬ ‫א‬E‫א‬F‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K(Radioactive Material)‫א‬

  ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬–‫א‬‫א‬J ‫א‬‫א‬‫א‬  K‫א‬

‫א‬‫א‬EfissionF‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ (Fusion)  K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬E  ‫א‬F‫א‬‫א‬‫א‬E  ‫א‬F‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬ K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬  Renewable Energy‫א‬‫ א‬٣J ١J ٢J ١   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬ ‫א‬‫א‬ K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ -٤-

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬

 W‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

EIF

EIIF

‫א‬F‫(א‬Wind Energy)  KE‫א‬

‫א‬‫א‬(Water falls)  ‫א‬‫א‬‫א‬‫א‬

 KE‫א‬‫א‬F

‫א‬‫א‬(Sea waves)‫א‬‫א‬‫א‬  K ‫א‬‫א‬‫א‬

‫א‬(Geophysics)‫א‬‫א‬‫א‬‫א‬‫א‬

EIIIF EIVF EVF

 KE‫א‬‫א‬F‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬(Ocean Energy)

EVIF

 K

 KE‫א‬‫א‬‫א‬F(Solar Lakes)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ EVIIF   ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬ EVIIIF  K(Bio gas)‫א‬

‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ ،‫א‬‫א‬‫א‬‫א‬ ‫א‬     ‫א‬     ‫א‬     

 ‫א‬  ‫א‬  ‫א‬ ‫א‬ ‫א‬   ،‫א‬  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬

‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬

-٥-

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬  K‫א‬ 

Generation of Electrical Energy‫א‬‫א‬٣J ١

، ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬K    K‫א‬‫א‬

 W‫א‬‫א‬ Hydro- Power  ‫א‬‫א‬‫א‬‫א‬ ١J ٣J ١

 Station E ‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬K 

‫א‬‫א‬  KE١J ١F 

-٦-



‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



 ‫א‬



‫א‬‫א‬

 

T

 

G



 ‫א‬‫א‬‫א‬

 

 ‫א‬‫א‬E١J ١F



‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬KE‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬E ‫א‬F‫א‬‫א‬  KE٢J ١F

 ‫א‬‫א‬

  ‫א‬‫א‬

  T  

G

 ‫א‬‫א‬‫א‬E٢J ١F -٧-



‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬E‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬K ‫א‬‫א‬ ،‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬E‫א‬‫א‬‫א‬ ‫א‬‫א‬

‫א‬‫א‬F‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬

 KE٣FE 



 ‫א‬‫א‬

 



P





G  



‫א‬



 ‫א‬‫א‬‫א‬‫א‬‫א‬E٣J ١F  Thermal Power Station‫א‬‫א‬‫א‬‫ א‬٢J ٣J ١

‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ -٨-

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬     E٤J ١FK  ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬E‫א‬F‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬  

 







 

 T

‫א‬‫א‬‫א‬

   

 P



 ‫א‬‫א‬‫א‬‫א‬E٤J ١F

 



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ،‫א‬    F ‫א‬ ‫א‬ ‫א‬‫א‬ ‫א‬    ‫א‬‫א‬E‫א‬،‫א‬

،  ‫א‬ ‫א‬     E٥J ١F   K   ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

-٩-

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-









 ‫א‬

‫א‬ 











‫א‬‫א‬‫א‬





 



 

‫ﻣﻀﺨﺔ‬



‫ﻣﻜﺜﻒ‬



 ‫א‬‫א‬E٥J ١F Gas Turbine ‫א‬‫א‬‫א‬‫א‬ ٣J ٣J ١

 Power Station ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬E‫א‬‫א‬‫א‬F‫א‬‫א‬  



KE٦J ١F،

 

‫א‬‫א‬   









‫א‬‫א‬‫א‬  ‫א‬‫א‬E٦J ١F

،‫א‬‫א‬  K‫א‬

- ١٠ -

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-

 

Diesel Power ‫א‬‫א‬‫א‬ ٤J ٣J ١ Station

‫א‬‫א‬‫א‬‫א‬ ‫א‬

K ‫א‬‫א‬،‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬  K‫א‬

 W‫א‬E٧J ١F ‫א‬







 

‫א‬‫א‬‫א‬

 

 ‫א‬E٧J ١F 

 W‫א‬‫א‬‫א‬ ٥J ٣J ١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

E‫א‬٠٠٠٠٠٠J ‫א‬‫א‬J ‫א‬J J J F J  ‫א‬F‫א‬E ‫א‬‫א‬‫א‬F

‫א‬‫א‬‫א‬،K E ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬ 

- ١١ -

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬ 





  ‫א‬





 







 





 





 ‫א‬‫א‬‫א‬‫א‬‫א‬E٨J ١F  



  

 

+











-



 ‫א‬‫א‬‫א‬‫א‬E٩J ١F

- ١٢ -

‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



  

 

 



 ‫א‬‫א‬E١٠J ١F

- ١٣ -

 



‫א‬‫א‬

 ١٤١

‫א‬

‫א‬‫א‬‫א‬

٢-



 ‫א‬‫א‬ 

K‫א‬‫א‬‫א‬‫ א‬K١ K‫א‬‫א‬‫א‬‫ א‬K٢ K‫א‬‫א‬‫א‬‫ א‬K٣

K‫א‬‫א‬‫ א‬K٤ K‫א‬‫א‬‫א‬‫ א‬K٥

K‫א‬‫א‬‫א‬‫ א‬K٦ K‫א‬‫א‬‫א‬‫א‬‫ א‬K٧ K‫א‬‫א‬‫א‬‫ א‬K٨

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٩ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٠ K‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١١

K‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٢ K‫א‬‫א‬‫א‬‫א‬‫ א‬K١٣

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٤

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٥  K‫א‬‫א‬‫א‬‫א‬‫ א‬K١٦ 

- ١٤ -

 ‫א‬‫א‬‫א‬ ã¹]
٢ J

 ‫א‬‫א‬

‫א‬‫א‬

٢

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬‫א‬‫א‬



 W‫א‬‫א‬،‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬‫• א‬ K‫א‬‫א‬ • K‫א‬‫א‬ •

K ‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬ •

 K ‫א‬‫א‬‫א‬ •

- ١٥ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



Introduction١J ٢

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K Magnetic Effects of Electrical Current ‫א‬‫א‬‫א‬٢J ٢

Generation & Concentration of Magnetic Field‫א‬‫א‬١J ٢J ٢

‫א‬‫א‬‫א‬،‫א‬‫א‬



‫א‬F‫א‬‫א‬‫א‬(Magnetic Field)‫א‬

‫א‬‫א‬‫א‬،Eφ،E  ‫א‬‫א‬F‫א‬  KE١J ٢F I ϕ

 K‫א‬‫א‬‫א‬E١J ٢F ،‫א‬‫א‬‫א‬‫א‬‫א‬



‫א‬‫א‬‫א‬ W  ‫א‬‫א‬ ‫א‬،‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬

‫א‬F‫א‬‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬،E  ‫א‬F‫א‬‫א‬،Eφ‫א‬

 KE٢J ٢F‫א‬‫א‬‫א‬‫א‬،‫א‬ - ١٦ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



 K‫א‬E٢J ٢F



 W‫א‬ ‫א‬‫א‬‫א‬    ‫א‬‫א‬‫א‬ E١F

F‫א‬‫א‬‫א‬، (•) ‫א‬ (×)  ‫א‬‫א‬‫א‬،E ‫א‬‫א‬‫א‬

 KE٢J ٢F

،‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬ E٢F ‫א‬،‫א‬‫א‬ ‫א‬،

 K‫א‬،‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ E٣F K‫א‬،‫א‬

J ٢F‫א‬‫א‬ ‫א‬‫א‬ E٤F KE٣

ϕ١



 K‫א‬‫א‬‫א‬E٣J ٢F



‫א‬،‫א‬EφF‫א‬‫א‬ ‫א‬،‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ - ١٧ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬،E  F‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 

A

B

 K‫א‬‫א‬‫א‬‫א‬E٤J ٢F ‫א‬‫א‬،‫א‬‫א‬‫א‬ E  ‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،‫א‬  ‫א‬‫א‬‫א‬  ‫א‬‫א‬ ‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬Eφ١F‫א‬‫א‬(φ٢)‫א‬‫א‬‫א‬

 KE٤J ٢F‫א‬ ‫א‬،‫א‬‫א‬‫א‬

 KE٥J ٢F‫א‬‫א‬‫א‬‫א‬ ϕ

 KE٥J ٢F

 W

 ϕ١<ϕ٢<ϕ٣ ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬ - ١٨ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



 Faraday's Law‫א‬‫א‬٢J ٢J ٢

N ‫א‬‫א‬‫א‬

FE‫א‬،E٦J ٢F‫א‬

‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬E

،  ‫א‬‫א‬N E = −N

dφ dt

 WE١J ٢F‫א‬ (١-٢) ϕ = f(t)

E = - N dϕ/dt

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٦J ٢F Basic Concepts of Electrical Generator‫א‬‫א‬٣J ٢

‫א‬‫א‬‫א‬‫א‬، E٥J ٢F



‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬،

 KE٧J ٢F،‫א‬ N

S  K‫א‬‫א‬‫א‬‫א‬‫א‬E٧J ٢F - ١٩ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬W    ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬ E = −N

dϕ dt

(٢-٢)  WB‫א‬‫א‬‫א‬

ϕ =B A

(٣-٢)  W

ϕ = BA

(٤-٢)

dϕ = BdA

(٥-٢)

،ϕ‫א‬‫א‬‫א‬‫א‬ l ‫א‬

‫א‬‫א‬W  E‫א‬‫א‬

 W، dϕ = ‫א‬‫א‬‫א‬‫א‬،dx

dϕ = BdA

(٦-٢)

dA = ldx

(٧-٢)

dϕ = Bldx

(٨-٢)  W،N=١W،E٢J ٢F‫א‬،‫א‬‫א‬

E = Bl

dx = Blν dt

(٩-٢)  W

 E‫א‬‫א‬‫א‬F‫א‬‫א‬ZB  ‫א‬Z l

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬Z ν

- ٢٠ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬،EWF‫א‬E‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬، ν  ϕ W

 W،‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬

Generation of Sine Wave‫א‬‫א‬١J ٣J ٢

‫א‬‫א‬‫א‬‫א‬‫א‬

،E٨J ٢F،‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬EF‫א‬ N B C ω

A x

φ

D

S

 K‫א‬‫א‬E٨J ٢F

 

‫א‬‫א‬،E٨J ٢F‫א‬‫א‬‫א‬

 W‫א‬،ADCBA

‫א‬‫א‬‫א‬A‫א‬

E١F

‫א‬‫א‬‫א‬‫א‬B‫א‬

E٢F

 K‫א‬

‫א‬‫א‬E  ‫א‬‫א‬F‫א‬‫א‬ KE٨J ٢F‫א‬

- ٢١ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



K‫א‬‫א‬‫א‬‫א‬C‫א‬

‫א‬ ‫א‬‫א‬‫א‬ D‫א‬

E٣F

(٤)

‫א‬‫א‬‫א‬E  ‫א‬‫א‬F‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K‫א‬ ‫א‬‫א‬ ‫א‬،‫א‬‫א‬‫א‬ ‫א‬‫א‬L‫א‬

KE٩J ٢F 1 05 0 05 1

0

90

180

270

360

‫א‬‫א‬

 K‫א‬‫א‬E٩J ٢F



Mathematical Analysis of Sine Wave‫א‬‫א‬‫א‬٢J ٣J ٢

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W(V)‫א‬‫א‬

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W(ω)‫א‬‫א‬‫א‬

 radians‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W(f)‫א‬ ‫א‬‫א‬‫א‬d‫א‬،‫א‬ d =2 π D 2

ϑ=

(2 π D 2) =2π (D 2)

 WD‫א‬ (١٠-٢)  Wθ‫א‬‫א‬‫א‬‫א‬ (١١-٢)

- ٢٢ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



V‫א‬،‫א‬‫א‬f‫א‬‫א‬‫א‬‫א‬‫א‬،

 W (١٢-٢)  Wω‫א‬‫א‬

V = (2 π D 2 ) ⋅ f ω= 2 πf

(١٣-٢)  WωV‫א‬

V = ω ⋅ (D 2)

(١٤-٢)  W‫א‬t

d=V⋅t

(١٥-٢)  W‫א‬

θ= ω⋅ t

(١٦-٢) θ‫א‬‫א‬‫א‬‫א‬‫א‬

 W،E١٠J ٢F،V  θ ‫א‬

N ‫اﻟﺴﺮﻋﺔ‬ ‫اﻟﺨﻄﻴﺔ‬

θ

‫א‬

V

θ θ

‫اﻟﺴﺮﻋﺔ‬ ‫اﻟﻌﻤﻮدﻳﺔ‬

ω

S ‫א‬‫א‬E١٠J ٢F υ = V sinθ

(١٧-٢)

 WKϕ‫א‬‫א‬‫א‬‫א‬υ

E = B l ν = B l V sin θ = B l V sin (ωt )

- ٢٣ -

(١٨-٢)

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

∴ E = B l V sin (ωt ) = E max sin (ωt )

(١٩-٢) (٢٠-٢)

Q E max = B l V = B l ω (D 2 ) ⎛Bl ωD⎞ ∴E = ⎜ ⎟ sin (ωt ) 2 ⎠ ⎝

(٢١-٢) K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Evaluation of Sine Wave‫א‬‫א‬‫א‬٣J ٣J ٢

‫א‬



0

90

180

270

360

‫א‬‫א‬

  ‫א‬‫א‬E١١J ٢F ،E١١J ٢F ‫א‬‫א‬‫א‬‫א‬



‫א‬W ‫א‬‫א‬‫א‬،‫א‬

 ‫؟‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬W  ‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Average value of sine wave‫א‬‫א‬‫א‬١J ٣J ٣J ٢

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬W  ‫א‬‫א‬

‫א‬،(T)‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ T/٢‫א‬‫א‬‫א‬KZ‫א‬ T/٢‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ - ٢٤ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

،E١٢J ٢F‫א‬  ‫א‬‫א‬

‫א‬‫א‬‫א‬ T/٢‫א‬‫א‬‫א‬‫א‬W‫א‬  K‫א‬‫א‬

π

π I av = ∫ I m sin θdθ = I m [cos (π ) − cos(0)] = 2I m

(٢٢-٢)

0

I av =

2 Im π

(٢٣-٢) 

 Z‫א‬‫א‬ I av = (

1 ) ∫ I m sin θ = I m [cos(0) − cos(2π)] = 0 2π

(٢٤-٢) 

Im 1 0.75

Iav

0.5 0.25 0 0

30

60

90

120

150

180

‫א‬‫א‬‫א‬‫א‬E١٢J ٢F 

Effective value of sine wave‫א‬‫א‬‫א‬٢J ٣J ٣J ٢

 ‫א‬،  ‫א‬‫א‬‫א‬  ‫א‬W ‫א‬‫א‬

‫א‬‫א‬،‫א‬ ‫א‬‫א‬‫א‬،   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ root ‫א‬r.m.s ‫א‬K‫א‬

- ٢٥ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬mean square  W i = I m sin θ W‫א‬‫א‬

،(square) i 2 = (I m sin θ) 2 W ‫א‬،‫א‬ KE١٣J ٢F

E١F

I 2m

Im

π

٢π

‫א‬‫א‬E١٣J ٢F ‫א‬‫א‬‫א‬‫א‬‫א‬ E٢F :(mean) ‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

(

)

⎛ I 2m 1 2 2 ∫ I m − sin θ = ⎜⎜ π π ⎝

⎞ (1 − cos 2θ) I 2m ⎟∫ = ⎟ 2 2 ⎠

(٢٥-٢)  W(root)‫א‬‫א‬‫א‬‫א‬ E٣F

I eff

I eff =

I 2m I m = = 2 2 ⎛ ⎜ ⎝

⎛1⎞ ⎜ ⎟ ⎝π⎠

2

(٢٦-٢)  W(rms)‫א‬‫א‬(root mean square) ‫א‬‫א‬

∫ I m sin

2

⎞ θ⎟ ⎠

(٢٧-٢)

- ٢٦ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

Phase angle‫א‬‫א‬٤J ٣J ٢

‫א‬‫א‬‫א‬‫א‬E١٠J ٢F

‫א‬‫א‬‫א‬‫א‬

‫א‬E١٤J ٢F‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬   

N ‫ﻗﻄﺐ ﺷﻤﺎﻟﻲ‬



φ

  

S ‫ﻗﻄﺐ ﺟﻨﻮﺑﻲ‬



 KE١٤J ٢F



،α & ψW ‫א‬‫א‬‫א‬  W‫א‬،E١٥J ٢F 

N

 ψ

 α

θ = ωt

  

C Aِ



B

 S

E١٥J ٢F - ٢٧ -





 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

 WA‫א‬‫א‬‫א‬‫א‬‫א‬ eA = Em sin ωt

J

(٢٨-٢)   W‫א‬‫א‬‫א‬‫א‬B‫א‬

eB = Em sin (ωt+ψ)

J

(٢٩-٢)

 W‫א‬‫א‬‫א‬‫א‬C‫א‬ J eC = Em sin (ωt - α)

(٣٠-٢)

B‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬A‫א‬ψ‫א‬‫א‬‫א‬ KE١٦J ٢F‫א‬C‫א‬α‫א‬ B A C

ψ α

K‫א‬‫א‬‫א‬‫א‬E١٦J ٢F



‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ،‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬(reference)‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬C،B‫א‬‫א‬A‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ - ٢٨ -

 ‫א‬‫א‬ ‫א‬‫א‬

e A = E m sin(ωt + 0) e B = E m sin(ωt + ψ) e C = E m sin(ωt − α)

 ١٤١

‫א‬

٢-



WZ‫א‬‫א‬،‫א‬A‫א‬ ♦ (٣١-٢) W‫א‬‫א‬‫א‬‫א‬ ♦ (٣٢-٢) W‫א‬‫א‬‫א‬‫א‬ ♦ (٣٣-٢)

Representation of Sine Wave‫א‬‫א‬‫א‬٥J ٣J ٢

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ω‫א‬Em‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ωt‫א‬‫א‬

 KE١٧J ٢F

1

0.5

ωt

0

ωt

π

٢π

-0.5

-1

 K‫א‬‫א‬‫א‬E١٧J ٢F Mathematical Representation of Vectors‫א‬‫א‬١J ٥J ٣J ٢

 W‫א‬‫א‬

 K‫א‬‫א‬‫א‬ J ١

K‫א‬‫ א‬J ٢

K‫א‬‫ א‬J ٣

K‫א‬‫ א‬J ٤ - ٢٩ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

 W  W‫א‬‫א‬‫א‬

E١F

A



 W

A = a 1 + ja 2

(٣٤-٢)  W‫א‬j

  − 1 Z‫א‬‫א‬‫א‬‫א‬Wj  W‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬

 KE١٨J ٢F

to - ∞

٠

to ∞

K‫א‬‫א‬E١٨J ٢F

  



j‫א‬‫א‬،X‫א‬   X ⋅ j ⋅ j = −X 

 EE–FF‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬،j‫א‬‫א‬

‫א‬‫א‬K ٩٠‫א‬‫א‬‫א‬‫א‬j‫א‬،٩٠

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  KE١٩J ٢F‫א‬‫א‬‫א‬ 

     - ٣٠ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

 

j a٢

   ⎢ ⎟A 

θ 

a١

  K‫א‬‫א‬‫א‬‫א‬E١٩J ٢F A = (a 12 + a 22 )

⎛a θ = tan −1 ⎜⎜ 2 ⎝ a1

(٣٥-٢)

⎞ ⎟⎟ ⎠

(٣٦-٢)  W‫א‬‫א‬

E٢F

 W‫א‬‫א‬‫א‬

a 1 = A cos θ

(٣٧-٢) (٣٨-٢)  W A = a 1 + ja 2 W‫א‬

a 2 = A sin θ

A = A cos θ + jA sin θ = A(cos θ + jsin θ)

(٣٩-٢)  W‫א‬‫א‬

E٣F

 W‫א‬‫א‬‫א‬

e jθ = cos θ + j sin θ

(٤٠-٢)  W‫א‬ A = A (cos θ + j sin θ ) W‫א‬‫א‬

A

= A e

jθ

(٤١-٢)  W‫א‬‫א‬

E٤F

 W‫א‬‫א‬‫א‬

A = A ∠θ

(٤٢-٢) - ٣١ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



Basic Calculations of Vectors‫א‬‫א‬‫א‬٢J ٥J ٣J ٢

 ‫א‬

E١F

 W‫א‬ D ‫א‬ C ‫א‬ A, B ‫א‬ A = a 1 + ja 2

(٤٣-٢) (٤٤-٢)

B = b1 + jb 2

C = c1 + jc 2

d2 = a 2 − b2

(٤٥-٢) (٤٦-٢) (٤٧-٢) (٤٨-٢) (٤٩-٢) (٥٠-٢) (٥١-٢) (٥٢-٢) (٥٣-٢) (٥٤-٢)

C = [(a 1 + b1 ) 2 + (a 2 + b 2 ) 2 ]

(٥٥-٢)

D = d 1 + jd 2 C=A+B D=A−B c1 + jc 2 = a 1 + ja 2 + b1 + jb 2 = (a 1 + b1 ) + j(a 2 + b 2 )

d1 + jd 2 = (a 1 + ja 2 ) − (b1 + jb 2 ) = (a 1 − b1 ) + j(a 2 − b 2 ) c1 = a 1 + b 1

c2 = a 2 + b2 d 1 = a 1 − b1

θ C = tan −1

(a 2 + b 2 ) (a 1 + b1 )

(٥٦-٢)

D = [(a 1 − b1 ) 2 + (a 2 − b 2 ) 2 ]

θ d = tan −1

(٥٧-٢)

(a 2 − b 2 ) (a 1 − b1 )

(٥٨-٢)

- ٣٢ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

 ‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

E٢F

 K‫א‬‫א‬‫א‬‫א‬

  : B ⋅ A C

C = A ⋅ B = (a 1 + ja 2 ) ⋅ (b1 + jb 2 )

(٥٩-٢) (٦٠-٢)

= (a 1 b1 − a 2 b 2 ) + j(a 1 b 2 + a 2 b1 )

C = [(a 1 b1 − a 2 b 2 )2 + (a 1 b 2 + a 2 b1 )2 ]

θ C = tan −1

(a 1 b 2 + a 2 b1 ) (a 1 b1 − a 2 b 2 )

(٦٢-٢)  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

C = A ⋅ B = A ∠θ A ⋅ B ∠θ B = C ∠θ C C=A⋅B

∠θ C = ∠θ a + ∠θ b C = A e jθ a ⋅ B e jθ b = A ⋅ B e j(θ a +θ b ) = C e jθ c

D= D=

(٦١-٢)

A B A

(٦٣-٢) (٦٤-٢) (٦٥-٢)  W‫א‬ (٦٦-٢)  W‫א‬ (٦٧-٢)

B

(٦٨-٢) (٦٩-٢)

∠θ d = ∠θ a − ∠θ b

- ٣٣ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬

 ‫א‬‫א‬ 

 W‫א‬W 

‫א‬‫א‬‫א‬‫א‬‫א‬

EF

‫א‬‫א‬‫א‬‫ א‬E F

‫א‬‫א‬‫א‬‫ א‬E F ‫א‬‫א‬ E F

Kα‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

EF

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬EF ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

EF

‫א‬‫א‬‫א‬‫א‬

EF

K ‫א‬‫א‬‫א‬‫א‬

EF

 W‫א‬W 

W‫א‬‫א‬‫ א‬K١ EF  A = 6 − j 8, B = 8 − j 6 EF E F A = 6 − j 8, B = 8 + j 6 E F A = 6 + j 8, B = 8 − j 6 A = 6 + j 8, B = 8 + j 6 E F K = 6 − j 8, L = − 8 + j 6 EF K = − 6 − j 8, L = − 8 + j 6 EF K = − 6 − j 8, L = − 8 − j 6 EF EF K = − 8 + j 6, L = − 8 + j 6 EF K = − 6 − j 8, L = 8 + j 6

- ٣٤ -

W A − B ‫א‬‫א‬ K٢ A = 6 − j 8, B = 8 − j 6 EF E F A = 6 − j 8, B = 8 + j 6 A = 6 + j 8, B = 8 − j 6 E F E F A = 6 + j 8, B = 8 + j 6

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



A = 6 − j 8, B = − 8 + j 6

A = − 6 − j 8, B = − 8 + j 6 A = − 6 − j 8, B = − 8 − j 6 A = − 8 + j 6, B = − 8 + j 6 A = − 6 − j 8, B = 8 + j 6

EF EF EF EF EF

W‫א‬‫א‬‫א‬‫א‬ K٣ A = 6− j8 E F E F B =8+ j6 B = 8 − j6 A = 6+ j8 E F EF A = − 6− j8 B = −8− j6 EF EF A = −8+ j6 EF A = − 6 + j 8, W‫א‬‫א‬‫א‬‫א‬‫א‬ K٤ A = 10 ∠30° EF E F A = 10 ∠ − 30° A = 10 ∠60° E F E F A = 10 ∠ − 60° EF A = 10 ∠36.87° EF A = 10 ∠ − 36.87° A = 10 ∠53.13° EF EF A = 10 ∠ − 53.13° W‫א‬‫א‬‫ א‬K٥ A = 10 ∠30°, B = 10 ∠ − 30° EF E F A = 10 ∠30°, B = 10 ∠60° E F A = 10 ∠30°, B = 10 ∠0° A = 10 ∠0°, B = 10 ∠0° E F A = 10 ∠120°, B = 10 ∠ − 60° EF - ٣٥ -

 ‫א‬‫א‬ ‫א‬‫א‬

 ١٤١

‫א‬

٢-



A ‫א‬‫א‬ K٦ B EF A = 10 ∠30°, B = 10 ∠ − 30° E F A = 10 ∠30°, B = 10 ∠60° A = 10 ∠30°, B = 10 ∠0° E F

W

A = 10 ∠0°, B = 10 ∠0° A = 10 ∠120°, B = 10 ∠ − 60°

E F EF

B ‫א‬،‫א‬ K٧ A EF A = 10 ∠30°, B = 10 ∠ − 30°

W

A = 10 ∠30°, B = 10 ∠60° A = 10 ∠30°, B = 10 ∠0°

A = 10 ∠0°, B = 10 ∠0°

A = 10 ∠120°, B = 10 ∠ − 60°

- ٣٦ -

E F E F E F EF

 ‫א‬‫א‬‫א‬ ã¹]
٢ J

 ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬

٣

- ١٤ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬‫א‬‫א‬



 W‫א‬‫א‬،‫א‬‫א‬‫א‬ K‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫• א‬ K‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫• א‬ K‫א‬‫א‬‫א‬ • K‫א‬‫א‬ •

Kj‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫• א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬ •

- ٣٧ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Introduction١J ٣

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬  K‫א‬‫א‬ Basic Elements of A.C Circuits ‫א‬‫א‬٢J ٣ The Resistance‫א‬‫א‬١J ٢J ٣

i



 E

 

R

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬E١J ٣F ،E١J ٣F‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬

E = E m sin ωt

(١-٣)  W

E = E m sin ωt = i ⋅ R

(٢-٣)

⎛E ⎞ i = ⎜ m ⎟ sin ωt = I m sin ωt ⎝ R ⎠ i = I m sin ωt

(٣-٣) (٤-٣) ‫א‬‫א‬‫א‬E٤J ٣FE١J ٣F‫א‬  K‫א‬‫א‬‫א‬،‫א‬‫א‬    - ٣٨ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



The Inductive Reactance‫א‬‫א‬٢J ٢J ٣

،E٢J ٣F،‫א‬

Φ   ‫א‬‫א‬‫א‬‫א‬Φ  W،i‫א‬N ‫א‬ 

Φ

  

i



N

 K‫א‬‫א‬‫א‬‫א‬E٢J ٣F Nφ ∝ i Nφ = Li

(٥-٣) (٦-٣)  K(Henry)‫א‬L‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬

dφ dt Q Nφ = Li E = −N

(٧-٣) (٨-٣)

dφ di =L dt dt dφ di = −L ∴ E = −N dt dt ∴N

E=−L

(٩-٣) (١٠-٣)  i = I m sin(ωt ) W‫א‬‫א‬‫א‬‫א‬

d ⋅ (I ⋅ sin (ωt )) dt m

E = − L ⋅ I m ⋅ ω ⋅ cos ( ωt

)

E = − Im ⋅ ω⋅ L ⋅ sin (ωt + 90 ) = Em ⋅ sin (ωt + 90 )

- ٣٩ -

(١١-٣) (١٢-٣) (١٣-٣)

 W

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Em = ω⋅ L Im

(١٤-٣) E F‫א‬‫א‬‫א‬‫א‬

 ،(Ω)‫א‬،(Inductive Reactance)‫א‬‫א‬‫א‬ωL   ωL = 2 π f L W

‫א‬i‫א‬E‫א‬،‫א‬‫א‬‫א‬

 K٩٠

 W‫א‬‫א‬‫א‬ (١٥-٣)

ωL = 2πf ⋅ L

The Capacitive Reactance‫א‬‫א‬٣J ٢J ٣

،‫א‬‫א‬‫א‬

،E٣J ٣F‫א‬ Q∝V Q = CV

 WK‫א‬‫א‬،‫א‬

(١٦-٣) (١٧-٣) ‫א‬‫א‬،‫א‬C‫א‬‫א‬‫א‬  

E

 K Farad‫א‬‫א‬

i



++++ Q



----

   K‫א‬‫א‬‫א‬‫א‬E٣J ٣F

 

- ٤٠ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ‫א‬١J ٣J ٢J ٣

،R‫א‬‫א‬C‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،E٤J ٣F ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،‫א‬،‫א‬ ‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬K‫א‬‫א‬ R 

C

ER

EC

  

E

 

I

  ‫א‬‫א‬E٤J ٣F ‫א‬‫א‬ E١٧J ٣F‫א‬  W‫א‬

dV dQ =C C =I dt dt

(١٨-٣) ،E١J  ‫א‬F‫א‬‫א‬‫א‬ VC = E − I ⋅ R = E − C ∴ VC + RC

dV dVC ⋅ R = E − RC C dt dt

 W‫א‬

(١٩-٣)

dVC =E dt

(٢٠-٣) ‫א‬‫א‬،‫א‬‫א‬،E٢٠J ٣F‫א‬  W‫א‬‫א‬ - ٤١ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬ −t ⎛ ⎞ ⎜ VC = E ⋅ ⎜1 − e RC ⎟⎟ ⎝ ⎠

IC =

(E − VC )

R E ⎛ − tτ ∴ I C = ⎜⎜ e R⎝

(٢١-٣)  W‫א‬،E٤J ٣F

−t ⎛ ⎛ ⎞⎞ ⎜ E − E ⋅ ⎜1 − e RC ⎟ ⎟ ⎜ ⎟⎟ ⎜ ⎝ ⎠⎠ ⎝ = R ⎞ ⎟⎟ ⎠

(٢٢-٣) (٢٣-٣)  W  ‫א‬

 ‫א‬



WI C

WRC=τ

 ‫א‬٢J ٣J ٢J ٣ ،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬،E٥J ٣F

،‫א‬‫א‬‫א‬‫א‬K ‫א‬‫א‬ K ‫א‬‫א‬‫א‬ R

C

ER

 W‫א‬‫א‬

+ −



EC



 

I



 ‫א‬‫א‬E٥J ٣F

- ٤٢ -

 

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

VC + I d ⋅ R = 0

(٢٤-٣)  W‫א‬‫א‬،‫א‬Id‫א‬

Id = − C

dVC dt

VC − RC

dVC =0 dt

(٢٥-٣)  W‫א‬،E٢٤J ٣F‫א‬

(٢٦-٣) ‫א‬‫א‬،‫א‬‫א‬،E٢٦J ٣F‫א‬ ⎛ −t ⎞ VC = VCi ⋅ ⎜⎜ e RC ⎟⎟ ⎝ ⎠

 W‫א‬‫א‬

(٢٧-٣)  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬VCi

 WId‫א‬E٢٤J ٣F‫א‬ V V I d = − C = − Ci R R

⎛ − t RC ⎞ ⎟⎟ ⋅ ⎜⎜ e ⎝ ⎠

(٢٨-٣)  K‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬،E٦J ٣F

‫א‬

 KE‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬،E‫א‬

‫א‬

  ‫א‬‫א‬‫א‬‫א‬‫א‬E٦J ٣F  E١J ٣F

‫א‬١٢ V‫א‬‫א‬،٥٠ µF‫א‬

‫א‬K٢٠ msec‫א‬‫א‬،τ‫א‬‫א‬W‫א‬،٢ kΩ‫א‬‫א‬ - ٤٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬  K‫א‬١٠ msec‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

 ‫א‬

 W،τ‫א‬‫א‬

τ = R ⋅ C = 0.1 sec

 W،٢٠ msec‫א‬

−t − 0.020 ⎛ ⎞ ⎛ 0.1 ⎞ = 2.175 V VC = E ⋅ ⎜⎜1 − e RC ⎟⎟ = 12 V ⋅ ⎜1 − e ⎟ ⎝ ⎠ ⎝ ⎠

E ⎛ − tτ ∴ I C = ⎜⎜ e R⎝

W،٢٠ msec‫א‬

⎞ 12 V ⎛ −0.020 0.1 ⎞ ⎟⎟ = ⎜e ⎟ = 4.91 mA ⎠ ⎠ 2000 Ω ⎝

 W،‫א‬‫א‬١٠ msec‫א‬

⎛ −t ⎞ ⎛ − 0.010 0.1 ⎞ ⎟⎟ = 1.97 V VC = VCi ⋅ ⎜⎜ e RC ⎟⎟ = 2.175 V ⋅ ⎜⎜ e ⎝ ⎠ ⎝ ⎠ V I d = − Ci R

⎛ − t RC ⎞ 2.175 V ⎛ − 0.010 0.1 ⎞ ⎜ ⎟ ⎟⎟ = − 0.98 mA ⋅⎜ e ⎟ = − 2000 ⋅ ⎜⎜ e ⎝ ⎠ ⎝ ⎠

 ‫א‬‫א‬‫א‬‫א‬٣J ٣J ٢J ٣

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ i=

 W‫א‬‫א‬

dQ dV =C dt dt

(٢٩-٣) ‫א‬‫א‬، v = Vm sin ωt W   ‫א‬‫א‬‫א‬‫א‬‫א‬ dv d (sin ωt ) = C ⋅ Vm ⋅ dt dt i = C ⋅ Vm ⋅ ω ⋅ cos ωt = ω ⋅ C ⋅ Vm ⋅ sin (ωt + 90 ) i =C

- ٤٤ -

 W‫א‬

(٣٠-٣) (٣١-٣)

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 W‫א‬‫א‬

i = I m ⋅ sin (ωt + 90)

(٣٢-٣)  W

I m = Vm ⋅ ωC

(٣٣-٣)  W

Vm 1 = I m ωC

(٣٤-٣)

‫א‬ 1 1 = ωC 2 πf C

1 ،E F‫א‬ ωC  W،(Ω)‫א‬،(Capacitive Reactance)‫א‬

(٣٥-٣)  K٩٠°‫א‬‫א‬‫א‬ ‫א‬‫א‬  ‫א‬‫א‬‫א‬٣J ٣

j Operator in Electrical Circuits ‫א‬‫א‬‫א‬j‫א‬‫א‬‫א‬١J ٣J ٣

L‫א‬‫א‬‫א‬Ri‫א‬‫א‬

 KE٧J ٣F،C

R ER E

L   

EL

C EC



 i  K‫א‬‫א‬‫א‬ C، L،RE٧J ٣F  W  K(volt) ‫א‬، E r = i ⋅ R WR‫א‬

 (volt) ‫א‬، E L = i ⋅ ωL WL ‫א‬ - ٤٥ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 (volt)‫א‬، E C =

i WC‫א‬ ωC

، E L = i(ωL) W ،E ‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ j‫א‬‫א‬‫א‬،E٨J ٣Fi‫א‬٩٠°‫א‬  W‫א‬



EL

 ER 

i





EC

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٨J ٣F E L = jωL ⋅ i

(٣٦-٣) i‫א‬EL٩٠°‫א‬‫א‬‫א‬jωLi‫א‬  KEC

 ٩٠°‫א‬‫א‬‫א‬–j/ωC

 W‫א‬‫א‬‫א‬‫א‬ R + jω L −

j 1 ⎞ ⎛ = R + j⎜ ωL − ⎟ ωC ωC ⎠ ⎝

(٣٧-٣) Z‫א‬  KRX

Z = R + jX

(٣٨-٣) WZ‫א‬‫א‬‫א‬

Z = (R 2 + X 2 ) θ Z = tan −1

(٣٩-٣)

X R

(٤٠-٣)  

- ٤٦ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬‫א‬٢J ٣J ٣

 ‫א‬‫א‬‫א‬ ١J ٢J ٣J ٣ ،‫א‬‫א‬E٩J ٣F‫א‬‫א‬‫א‬

 WXL‫א‬‫א‬

L٢

L١

E L1

L

≡

E L2

E

i

 

E



i



 ‫א‬‫א‬‫א‬‫א‬E٩J ٣F X L = jω L = j ω L 1 + jω L 2 = j ω ( L 1 + L 2 )

∴ L = (L1 + L 2 )

(٤١-٣) (٤٢-٣)

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬  E٢J ٣F

٥ mH‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K٧ mH‫א‬  ‫א‬

 W‫א‬‫א‬‫א‬‫א‬،E٤٢J ٣F ∴ L = (L1 + L 2 ) = 5 mH + 7 mH = 12 mH

 ‫א‬‫א‬‫א‬ ٢J ٢J ٣J ٣

،‫א‬‫א‬E١٠J ٣F‫א‬‫א‬‫א‬

 WXC‫א‬‫א‬ - ٤٧ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

C١

C٢





C٢



E i



≡

E C2

E C1

 

E



i



 ‫א‬‫א‬‫א‬‫א‬E١٠J ٣F 1 1 ⎞ ⎛ ⎛ 1 ⎞ ⎛ ⎟⎟ + ⎜⎜ − j XC = − j ⎜ ⎟ = ⎜⎜ − j ⎝ ωC ⎠ ⎝ ωC 1 ⎠ ⎝ ωC 2

⎛ 1⎞ ⎜ ⎟= ⎝ C⎠

⎞ ⎛ 1 1 ⎟⎟ = − j ⎜⎜ + ⎠ ⎝ ωC 1 ωC 2

⎛ 1 1 ⎞ ⎜⎜ ⎟⎟ + ⎝ C1 C 2 ⎠

⎛ C ⋅C C = ⎜⎜ 1 2 ⎝ C1 + C 2

⎞ ⎟⎟ ⎠

(٤٣-٣)  W (٤٤-٣)

⎞ ⎟⎟ ⎠

(٤٥-٣)  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٤٥J ٣F‫א‬  E٣J ٣F ‫א‬١٠٠ µF‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K٢٥ µF  ‫א‬

 W‫א‬‫א‬،E٤٥J ٣F ⎛ C ⋅C C = ⎜⎜ 1 2 ⎝ C1 + C 2

⎞ 100 µF ⋅ 25 µF ⎟⎟ = = 20 µF ⎠ 100 µF + 25 µF

- ٤٨ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

RL Series Circuit ‫א‬‫א‬

٣J ٢J ٣J ٣

،‫א‬‫א‬E١١J ٣F‫א‬‫א‬‫א‬  WZ‫א‬‫א‬ R

L

ER

EL

E



  







i



 K‫א‬‫א‬‫א‬E١١J ٣F

Z = R + jωL

(٤٦-٣)

Z = (R 2 + (ωL) 2 ) ∠θ Z = tan −1

(٤٧-٣)

ωL R

(٤٨-٣)  W i ‫א‬K‫א‬ θ Z ‫א‬‫ א‬E٤٨J ٣F‫א‬ i=

i=

E E ∠θ E = = i ∠θ i Z Z ∠θ Z E Z

=

(٤٩-٣)  W

E

(R 2 + ω 2 L2 ) , ∠θ i = ∠θ E − ∠θ Z E = E ∠0

(٥٠-٣) (٥١-٣)  W‫א‬‫א‬EF‫א‬ (٥٢-٣)  W‫א‬‫א‬

∠θ i = ∠0 − ∠θ Z = ∠ − θ Z = ∠ϕ

(٥٣-٣)

- ٤٩ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ϕ K ‫א‬ ‫א‬،‫א‬‫א‬،E 

 KE١٢J ٣F‫א‬‫א‬‫א‬  

E = i ⋅ Z

E L = i ⋅ jω L



θi= ϕ



 i

ER = i ⋅ R

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٢J ٣F  W

E = ER + EL E = i ⋅ R + i ⋅ jωL = i ⋅ (R + jωL ) = i ⋅ Z

(٥٤-٣) (٥٥-٣)  E٤J ٣F

،L = ٠{٠١ Henry‫א‬  ‫א‬‫א‬٥ Ω‫א‬  Kf = ٥٠ HzW e = 200 volts W

 W‫א‬

 K‫א‬‫א‬

 K‫א‬

E١F

E٢F

 ‫א‬

ω = 2πf 

 Wω‫א‬‫א‬‫א‬W‫א‬E١F

ω = 2 × 3.14 × 50 = 314 rad / sec

 WXL‫א‬‫א‬

XL = ω× L

X L = 314 × 0.01 = 3.14 Ω - ٥٠ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

Z =

 WZ‫א‬‫א‬

5 2 + (3.14 )2 = 5.9 Ω

⎛ 3.14 ⎞ θ Z = tan −1 ⎜ ⎟ = 32.13° ⎝ 5 ⎠

i=

 Wi‫א‬

e Z

‫א‬‫א‬‫א‬٠° ‫א‬‫א‬،EF‫א‬ i =

 W

200 V∠0° = 33.9 A ∠ − 32.13° 5.9 Ω ∠32.13°

٠°‫א‬‫א‬،٣٢{١٣°‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬، ٣٢{١٣° ‫א‬‫א‬  W‫א‬

E٣F

 W‫א‬‫א‬

♦

 W‫א‬‫א‬

♦

E R = i ⋅ R = 33.9 A ∠ − 32.13° ⋅ 5 Ω = 169.5 V ∠ − 32.13°

E L = i ⋅ X L = 33.9 A ∠ − 32.13° ⋅ 3.14 Ω ∠90° = 106.45 V ∠57.87°

RC series Circuit‫א‬‫א‬

٤J ٢J ٣J ٣

 W،‫א‬‫א‬‫א‬E١٣J ٣F

- ٥١ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬



R



C

 

EC

ER

 

E



i

 

 K‫א‬‫א‬‫א‬E١٣J ٣F Z=R− j

1 ωC

(٥٦-٣)

⎛ 1 ⎞ Z = R +⎜ ⎟ ⎝ ωC ⎠ 2

2

⎛ −1 ⎞ ∠θ Z = tan −1 ⎜ ⎟ ⎝ ωCR ⎠

(٥٧-٣) ⇒ ‫א‬

(٥٨-٣)  W i ‫א‬

i=

i=

E ∠θ E E E ∠θ E = = = i ∠θ i 2 ⎞ Z Z ∠θ Z ⎛ ⎜ R 2 + ⎛ 1 ⎞ ⎟ ∠θ ⎜ ⎟ Z ⎜ ⎝ ωC ⎠ ⎟ ⎝ ⎠ E

=

(E )

2 ⎛ ⎜ R2 + ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎝ ωC ⎠ ⎝ θi = θ E − θ Z

Z

θ i = 0° − θ Z = ϕ

(٥٩-٣)  W

⎞ ⎟ ⎟ ⎠ ,

(٦٠-٣) (٦١-٣)  W،٠° ‫א‬E‫א‬F‫א‬‫א‬‫א‬ (٦٢-٣) - ٥٢ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

K ‫א‬‫א‬‫א‬،‫א‬‫א‬، θ Z  ‫א‬‫א‬

 KE١٤J ٣F‫א‬‫א‬ ER

i

ϕ

E

EC



 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٤J ٣F

 W

E = ER + EC

(٦٣-٣)

ER = i ⋅ R

(٦٤-٣)

−j EC = i ⋅ ( ) ωC

(٦٥-٣)

j ) ωC 1 ⎞ ⎛ E = i ⋅ ⎜R − j ⎟=i⋅Z ωC ⎠ ⎝

E = i ⋅R − i ⋅(

(٦٦-٣) (٦٧-٣)

E =i•Z

(٦٨-٣)

⎛ 2 ⎛ 1 ⎞2 ⎞ E = i • ⎜R + ⎜ ⎟ ⎟ ⎜ C ω ⎝ ⎠ ⎟⎠ ⎝ 2 ⎛ 2 2 2 ⎛ 1 ⎞ ⎞ ⎜ E = i R + i •⎜ ⎟ ⎟ = ⎜ ω C ⎝ ⎠ ⎟⎠ ⎝

(٦٩-٣) ER

2

+ EC

2

(٧٠-٣)

- ٥٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E٥J ٣F ‫א‬٥٠ Hz١٢٥ volts‫א‬ ٢{٢ Amp.‫א‬‫א‬‫א‬‫א‬‫א‬،E١٥J ٣F،‫א‬‫א‬ ‫א‬‫א‬ ‫א‬‫א‬،٩٦{٨ Watt‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬ 

C

P = 96.8 Watts

 ١٢٥ volts ٥٠ Hz

2 .2 A

 

P = i2R P i2

=

96.8 2 .2 2

= 20 Ω

E R = iR = 2.2 × 20 = 44 volts EC =

E 2 − E 2R =

(125)2 − (44)2

= 117 volts

EC = i • XC XC =

117 = 53.2 Ω 2 .2

X C = 53.2 =





 KE٥J ٣F‫א‬‫א‬‫א‬E١٥J ٣F

R =



1 1 1 = = ωC 2πf ⋅ C 2 × 3.14 × 50 × C

C = 60 × 10 −6 Farad

- ٥٤ -

 ‫א‬

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



K‫א‬‫א‬‫א‬E١٦J ٣F E R = 44 volts

ϕ

i

 

E C = 117 volts

E = 125 volts 



 KE٥J ٣F‫א‬‫א‬‫א‬E١٦J ٣F  ‫א‬‫א‬‫א‬‫א‬ ٥J ٢J ٣J ٣

RLC Series Circuit R‫א‬‫א‬‫א‬‫א‬‫א‬

  C  ‫א‬‫א‬L‫א‬  ‫א‬‫א‬

E‫א‬F‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬i،E١٧J ٣F

F‫א‬E  ‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬  W Z ‫א‬‫א‬‫א‬KE L

R



C

 

ER

EL



EC

E

 

i



 K‫א‬‫א‬‫א‬ C، L،RE١٧J ٣F 1 ⎞ 1 ⎞ ⎛ ⎛ Z = ⎜ R + jωL − j ⎟ = R + j⎜ ωL − ⎟ ωC ⎠ ωC ⎠ ⎝ ⎝

Z=

(R )2 + ⎛⎜ ωL − ⎝

1 ⎞ ⎟ ωC ⎠

(٧١-٣)

2

(٧٢-٣) - ٥٥ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

1 ⎞⎞ ⎛⎛ ⎜ ⎜ ωL − ⎟⎟ 2 LC − 1 ⎞⎟ ω ω C ⎝ ⎠⎟ −1 ⎜ −1 ⎛ ∠θ Z = tan = tan ⎜⎜ ⎟ ⎜ ⎟ R ⎝ ωCR ⎠ ⎜ ⎟ ⎝ ⎠

(٧٣-٣)  W‫א‬

i=

E ∠0° E = ∠ − θZ ° Z ∠θ Z Z

(٧٤-٣)

⎛ ω 2 LC − 1 ⎞ ⎟ ∠ − tan −1 ⎜⎜ ⎟ 2 CR ω ⎠ ⎝ 1 ⎞ ⎛ + ⎜ ωL − ⎟ ωC ⎠ ⎝ E

i=

(R )2

(٧٥-٣)

 W‫א‬‫א‬‫א‬ ER =

E ⋅R

(R )2 + ⎛⎜ ωL − ⎝

1 ⎞ ⎟ ωC ⎠

2

∠ − tan

−1

⎛ ω 2 LC − 1 ⎞ ⎟ ⎜ ⎜ ωCR ⎟ ⎠ ⎝

(٧٦-٣)

E L = i ⋅ jωL = i ⋅ X L

(٧٧-٣)

⎛ ω LC − 1 ⎞ ⎟ ∠90° − tan −1 ⎜⎜ ⎟ 2 CR ω ⎠ ⎝ 1 ⎞ ⎛ + ⎜ ωL − ⎟ ωC ⎠ ⎝ 2

E ⋅ ωL

EL =

(R )2 E C = −i ⋅ j

1 = −i ⋅ X C ωC

(٧٩-٣)

(E ωC )

EC =

(R )2

(٧٨-٣)

⎛ ω LC − 1 ⎞ ⎟ ∠ − 90° − tan −1 ⎜⎜ ⎟ 2 CR ω ⎝ ⎠ 1 ⎞ ⎛ + ⎜ ωL − ⎟ ωC ⎠ ⎝ 2

- ٥٦ -

(٨٠-٣)

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 (XC < XL) (XL < XC)‫א‬‫א‬‫א‬E١٨J ٣F 

E

θi

EL

i

EL



ER

i

ER

 



 



E C

θi EC



E

(XC < XL) WEJ ١٨J ٣F





 (XL < XC)WEJ ١٨J ٣F E٦J ٣F

‫א‬   ٢٠ Ω‫א‬‫א‬‫א‬١٠٠ µF

‫א‬‫א‬‫א‬‫א‬‫א‬KE١٩J ٣F، L = 0.05 H ‫א‬‫א‬K٢٠٠ volts‫א‬‫א‬،٥٠ Hz‫א‬

 K‫א‬  ‫א‬

0.05 H

20 Ω

200 V

100 µF

i



50 Hz

  KE٦J ٣F‫א‬‫א‬‫א‬E١٩J ٣F 

ω = 2πf = 2 × 3.14 × 50 = 314 rad / sec 

X L = ωL = 314 × 0.05 = 15.7 Ω

XC =

1 1 = = 31.84 Ω −6 ωC 314 × 100 × 10

(

)

- ٥٧ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

Z = 20 + j(15.7 − 31.84 ) = 20 − j16.14

Z =

(20)2 + (16.14)2

= 25.7 Ω

⎛ − 16.14 ⎞ θ Z = tan −1 ⎜ ⎟ = − 38.9° ⎝ 20 ⎠

i =

200 V∠0° E = = 7.78 ∠38.9° Amp. Z 25.7 Ω ∠ − 38.9°

E R = i ⋅ R = 7.78 A ∠38.9° • 20 Ω = 155.6 V ∠38.9

E L = i ⋅ X L = 7.78 A ∠38.9 • 15.7 Ω ∠90 = 122.146 V ∠128.9° E C = i ⋅ X C = 7.78 A ∠38.9 • 31.84 Ω ∠ − 90 = 247.72 V ∠ − 51.1°

 ‫א‬‫א‬‫א‬E٢٠J ٣F

i⋅XL= ١٢٢٫١٤٦ V



١٢٨٫٩°

i⋅R = ١٥٫٦ V

-٥١٫١°

i = ٧٫٧٨ A

i⋅XC= ٢٤٧٫٧٢

٣٨٫٩° 

 

E = ٢٠٠ V

 E٦J ٣F‫א‬‫א‬E٢٠J ٣F

- ٥٨ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 Parallel Circuits ‫א‬‫א‬‫א‬٣J ٣J ٣  ‫א‬‫א‬‫א‬١J ٣J ٣J ٣

،‫א‬‫א‬E٢١J ٣F‫א‬‫א‬‫א‬

 WXL‫א‬‫א‬  

L L١

E i

≡

L٢

i1

i2



E



i



 ‫א‬‫א‬‫א‬‫א‬E٢١J ٣F  W‫א‬،‫א‬‫א‬‫א‬‫א‬ i=

E j ωL

i = i1 + i 2 =

(٨١-٣)  W‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬ E E + =E⋅ j ωL 1 j ωL 2

⎛ 1 1 ⎜⎜ + ⎝ j ωL 1 j ωL 2

⎞ ⎟⎟ ⎠

(٨٢-٣)

 W‫א‬،‫א‬‫א‬ E = jωL i E E XL = = i ⎛ 1 1 + E ⋅ ⎜⎜ ⎝ j ωL 1 j ωL 2

XL =

(٨٣-٣) ⎞ ⎟⎟ ⎠

=

1 ⎛ 1 1 ⎜⎜ + ⎝ jω L 1 j ω L 2

⎞ ⎟⎟ ⎠

(٨٤-٣) 

- ٥٩ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 W‫א‬،‫א‬‫א‬‫א‬ 1 1 1 = + L L1 L 2 ∴L = (

(٨٥-٣)

L1 ⋅ L 2 ) L1 + L 2

(٨٦-٣)

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬

 E٧J ٣F ٥ mH ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K٧ mH‫א‬  ‫א‬

 W،‫א‬‫א‬‫א‬‫א‬،E٨٦J ٣F ∴L = (

L1 ⋅ L 2 5 mH × 7 mH )= = 2.92 mH L1 + L 2 5 mH + 7 mH  ‫א‬‫א‬‫א‬٢J ٣J ٣J ٣

،‫א‬‫א‬E٢٢J ٣F‫א‬‫א‬‫א‬

 WXC‫א‬‫א‬   C١

E i

i1

C٢ 

C٢ 

≡

E

i2

 i 

 ‫א‬‫א‬E٢٢J ٣F - ٦٠ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 W‫א‬،‫א‬‫א‬‫א‬‫א‬ i=

E = E ⋅ j ωC (1 jωC)

i = i1 + i 2 =

(٨٧-٣)  W‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

E E + = E ⋅ j ω C 1 + E ⋅ jω C 2 (1 jωC1 ) (1 jωC 2 )

(٨٨-٣)  W‫א‬،‫א‬‫א‬‫א‬

i = E ⋅ jω C 1 + E ⋅ j ω C 2 = E ⋅ j ω C

(٨٩-٣)  W‫א‬، j ω E ‫א‬

C = C1 + C 2

(٩٠-٣)  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٩٠J ٣F‫א‬  E٨J ٣F

‫א‬١٠٠ µF‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K٢٥ µF  ‫א‬  W‫א‬‫א‬،E٩٠J ٣F C eq = C1 + C 2 = 100 µF + 25 µF = 125 µF

RL Parallel Circuit‫א‬‫א‬٣J ٣J ٣J ٣

،‫א‬‫א‬E٢٣J ٣F‫א‬‫א‬‫א‬

 WZ‫א‬‫א‬ iL

iR V

E

L

iT

 ‫א‬‫א‬E٢٣J ٣F - ٦١ -

    

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



(

 WZ‫א‬‫א‬

R ⋅ jω L R ⋅ jωL ⋅ (R − j ωL ) Rω 2 L2 + j R 2 ωL = = Z= R + j ωL (R + j ωL ) ⋅ (R − j ωL ) R 2 + ω 2 L2 Z =

Z =

(Rω L ) 2 2

(R

(R

+j

2

ωL

)

) (R + ω L ) ⎛ ( R ωL ) ⎞ ⎛ (Rω L ) ⎞ ⎟ ⎜ ⎟ +⎜ ⎜ (R + ω L )⎟ ⎜ (R + ω L )⎟ ⎝ ⎠ ⎝ ⎠ 2

+ ω 2 L2

2 2

2

2

⎛ R ωL ∠θ Z = tan −1 ⎜⎜ 2 2 ⎝ Rω L 2

2

2

E ∠0° E = ∠0° R ∠0° R

iL =

E ∠0° E = ∠ − 90° ωL ∠90° ωL

2

2 2

(٩٣-٣) (٩٤-٣)  W‫א‬‫א‬ (٩٥-٣) (٩٦-٣)

E E ∠0° + ∠ − 90° R ωL E E (0 − j1) = E − j E = E⎛⎜ 1 − j 1 ⎞⎟ i T = (1 + j 0 ) + R ωL R ωL ωL ⎠ ⎝R iT = iR + iL =

iT

(٩١-٣) (٩٢-٣)

⎞ ⎟ = tan −1 ⎛⎜ R ⎞⎟ ⎟ ⎝ ωL ⎠ ⎠

iR =

)

2 2

2

2 2

(

)

⎛⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎞ = E ⋅ ⎜⎜ ⎟ + ⎜ ⎟ ⎟ ⎜⎝ R ⎠ L ω ⎝ ⎠ ⎟⎠ ⎝

(٩٧-٣) (٩٨-٣) (٩٩-٣)

⎛ R ⎞ θ i = tan −1 ⎜ − ⎟ ⎝ ωL ⎠ 

(١٠٠-٣)  KE٢٤J ٣F‫א‬‫א‬

- ٦٢ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬



θi

iR

E

 

iT



iL



 K‫א‬‫א‬‫א‬‫א‬E٢٤J ٣F

 E٩J ٣F ‫א‬  ‫א‬‫א‬٥ Ω‫א‬E٢٥J ٣F

 Kf = ٥٠ HzW e = 200 volts W،L = ٠{٠١ Henry  K‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ 

E = 200 V

iL

iR

R =5Ω

V

L = 0.01 H

f = 50 Hz iT

 KE٩J ٣F‫א‬‫א‬‫א‬E٢٥J ٣F

     ‫א‬

iR =

E ∠0° E 200 V ∠0° = ∠0° = = 40 A ∠0° R ∠0° R 5 Ω ∠0°

iL =

E E ∠ − 90° = ∠ − 90° ωL 2πf L

iL =

200 V 2 π ⋅ 50 ⋅ 1 × 10 − 2 Ω

∠ − 90° = 63.7 A ∠ − 90°

- ٦٣ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

iT

⎛⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎞ = E ⋅ ⎜⎜ ⎟ + ⎜ ⎟ ⎟ = 200 × ⎜⎝ R ⎠ ω L ⎝ ⎠ ⎟⎠ ⎝

⎛⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎞ ⎜⎜ ⎟ + ⎜ ⎟ ⎟ = 75.2 A ⎜⎝ 5 ⎠ 3 . 14 ⎝ ⎠ ⎟⎠ ⎝

5 ⎞ ⎛ R ⎞ −1 ⎛ θ i = tan −1 ⎜ − ⎟ = tan ⎜ − ⎟ = − 57.9° ⎝ ωL ⎠ ⎝ 3.14 ⎠

RC Parallel Circuit‫א‬‫א‬ ٤J ٣J ٣J ٣

،‫א‬‫א‬E٢٦J ٣F‫א‬‫א‬‫א‬  WZ‫א‬‫א‬  iC

iR V

R

E

C

   

iT



 K‫א‬‫א‬‫א‬‫א‬‫א‬E٢٦J ٣F ⎛ ⎛ 1 ⎞⎞ R ⋅ ⎜ − j⎜ ⎟⎟ ⎝ ⎝ ωC ⎠ ⎠ Z= = ⎛ ⎛ 1 ⎞⎞ ⎛ ⎜ R − j⎜ ⎟⎟ ⎜ ⎝ ωC ⎠ ⎠ ⎝ ⎝

R ⎞ ⎛ ⎜− j ⎟ ωC ⎠ ⎝ (RωC − j) ⎞ ⎟ ωC ⎠

- ٣)

(١٠١ Z=

− jR − j R ⋅ (RωC + j) = (RωC − j) (RωC − j) ⋅ (RωC + j)

- ٣)

(١٠٢

( (

) )

− j R ⋅ (RωC + j) R − jR 2 ωC Z = = (RωC − j) ⋅ (RωC + j) R 2 ω 2 C 2 + 1

- ٦٤ -

(١٠٣-٣)

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

Z=

(R

R 2

−j

) (R

ω2 C 2 + 1

⎛ R Z= ⎜ 2 2 2 ⎜ R ω C +1 ⎝ ∠θ Z

R 2 ωC 2

(١٠٤-٣)

)

ω2 C 2 + 1

2

⎛ ⎞ R 2 ωC ⎟ +⎜ ⎜ R 2 ω2 C 2 + 1 ⎟ ⎝ ⎠

⎞ ⎟ ⎟ ⎠

2

( ) ( ) ( − R ωC (R ω C + 1)) = tan ⎛⎜ − R ωC ⎞⎟ = ⎜ ⎟ R (R (R ω C + 1)) ⎝ ⎠ 2

2

2

2

2

2

−1

2

2

(١٠٥ -٣)

(١٠٦-٣)

 W‫א‬‫א‬ iR = iC =

E ∠0° E = ∠0° R ∠0° R E ∠0° 1 ∠ − 90° ωC

iT = iR + iC =

E١٠٧J ٣F

= EωC ∠90° E١٠٨J ٣F

E ∠0° + EωC ∠90° R

E١٠٩J ٣F

⎛E ⎞ ⎛E ⎞ i T = ⎜ + j 0 ⎟ + (0 + j EωC ) = ⎜ + j EωC ⎟ E١١٠J ٣F ⎝R ⎠ ⎝R ⎠ 2

iT



⎛E⎞ = ⎜ ⎟ + (EωC )2 E١١١J ٣F ⎝R⎠

⎛ ⎛ E ⎞⎞ ∠θ i = tan −1 ⎜ EωC ⎜ ⎟ ⎟ = tan −1 (ωCR ) ⎝ R ⎠⎠ ⎝

E١١٢J ٣F

 KE٢٧J ٣F‫א‬‫א‬ 

- ٦٥ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬





iT

θi



iC

 

iR



E

 K‫א‬‫א‬‫א‬‫א‬E٢٧J ٣F

 E١٠J ٣F

،C = ٠{٠١ µF‫א‬‫א‬٥ Ω‫א‬

‫א‬‫א‬‫א‬KE٢٨J ٣Ff = ٥٠ HzW e = 200 volts W  K‫א‬‫א‬‫א‬



iC

iR E = 200 V

R =5Ω

V

 

C = 0.01 µF



f = 50 Hz



iT



 KE١٠J ٣F‫א‬‫א‬‫א‬E٢٨J ٣F

 ‫א‬ iR =

200 V ∠0° E ∠0° E = ∠0° = = 40 A ∠0° 5 Ω ∠0° R ∠0° R

i C = EωC ∠90° = 200 × 2 π × 50 × 0.01 × 10 −6 ∠90° = 62.8 × 10 −3 A ∠90° 2

iT

2

2

(

⎛E⎞ ⎛1⎞ ⎛1⎞ = ⎜ ⎟ + (EωC )2 = E ⎜ ⎟ + (ωC )2 = 200 × ⎜ ⎟ + 3.14 × 10 −6 ⎝R⎠ ⎝R⎠ ⎝5⎠

- ٦٦ -

)

2

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

2

iT

(

⎛1⎞ = 200 × ⎜ ⎟ + 3.14 × 10 −6 ⎝5⎠

)

2

= 40 A

(

)

∠θ i = tan −1 (ωCR ) = tan −1 2 × π × 50 × 0.01 × 10 −6 × 5 = 0°

 ‫א‬‫א‬‫א‬‫א‬٥J ٣J ٣J ٣ RLC Parallel Circuit ‫א‬‫א‬L‫א‬  ‫א‬‫א‬R

J ٣FE‫א‬‫א‬‫א‬C  KE٢٩ 

iT iL

iC C

E





L

  



iR



R

 

 K‫א‬‫א‬‫א‬‫א‬‫א‬E٢٩J ٣F (iL)‫א‬‫א‬‫א‬H(iR)‫א‬‫א‬‫א‬Wi‫א‬‫א‬‫א‬

 K(iC)‫א‬‫א‬‫א‬H

(١١٣-٣)

i = i R + i L + iC

،‫א‬‫א‬‫א‬‫א‬E‫א‬  EZ‫א‬F‫א‬ i =

E E E + + j ⎞ R jωL ⎛ ⎜− ⎟ ⎝ ωC ⎠

(١١٤-٣) - ٦٧ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

⎛1 1 ωC ⎞ 1 ⎞⎞ ⎛1 ⎛ ⎟⎟ = E ⋅ ⎜ + j ⎜ ωC − i = E ⋅ ⎜⎜ + − ⎟⎟ ωL ⎠ ⎠ j ⎠ ⎝ ⎝R ⎝ R j ωL 2

1 ⎞ ⎛1⎞ ⎛ i = E ⋅ ⎜ ⎟ + ⎜ ωC − ⎟ ωL ⎠ ⎝R⎠ ⎝

(١١٥-٣)

2

(١١٦-٣)

⎛ ω 2 LC − 1 ⎞ ⎞ 1 ⎞ ⎛ 1 ⎞⎞ ⎛⎛ −1 ⎜ ⎛ ⎟⎟ ∠θ i = tan −1 ⎜ ⎜ ωC − = tan R ⎜⎜ ⎟ ⎟ ⎜ ⎟ ⎟⎟ ⎜ ω ω L R L ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎠⎠ ⎝ ⎝

(١١٧-٣)

E٣٠J ٣F‫א‬‫א‬ iC

iC E

iR

 

iL

i iL

 

 K‫א‬‫א‬‫א‬‫א‬E٣٠J ٣F

- ٦٨ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 E١١J ٣F  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٣١J ٣F‫א‬‫א‬‫א‬

 ‫א‬ 

iT



iL

iC



E = ٥٠ V f = ١ kHz

C= ٠٫٠١ µf

 L = ٢٫٥٣ H 

iR R = ١٠٠ Ω

  

 KE١١J ٣F‫א‬‫א‬‫א‬E٣١J ٣F iR =

50 V ∠0° = 0.5 A ∠0° 100 Ω ∠0°

iL =

50 V ∠0° = 3.145 mA ∠ − 90° (2 × π × 1000 × 2.53) Ω ∠90°

iC =

50 V ∠0°

(1 2 × π × 1000 × 0.01 × 10 ) Ω ∠ − 90° −6

= 3.145 mA ∠90°

i T = i R + i L + i C = 0.5 A ∠0° + 3.145 mA ∠ − 90° + 3.145 mA ∠90° i T = 0.5 A ∠0°

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬،‫א‬

- ٦٩ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 E١٢J ٣F W   ،‫א‬‫א‬ ،E٣٢J ٣F‫א‬‫א‬‫א‬ K(f =٦٠ Hz) E = 100 volts ‫א‬‫א‬‫א‬، Z1 = 6 + j 8 & Z 2 = 8 - j 6

 K‫א‬‫א‬‫א‬  ‫א‬ i١

Z١

i٢

Z٢

i ١٠٠ V

 KE١٢J ٣F‫א‬‫א‬‫א‬E٣٢J ٣F Z1 = 6 + j 8 = 10 Ω ∠51.3° Z 2 = 8 − j 6 = 10 Ω ∠ − 36.9°

E = 100 V ∠0° i1 =

100∠0° = 10 A ∠ − 51.3° 10∠51.3°

i2 =

100∠0° = 10 A ∠36.9° 10∠ − 36.9°

Zt =

Z1 Z 2 (6 + j8)(8 − j6) = (6 + j8)(8 − j6) Z1 + Z 2

Zt =

i =

96 + j28 100∠16.26° = = 7.1 Ω ∠8.13° 14 + j2 14.14∠8.13°

E 100∠0 = = 14.1 A ∠ − 8.13° Zt 7.1∠8.13 - ٧٠ -

    

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

Power in A.C Circuits ‫א‬‫א‬‫א‬٤J ٣ Power Triangle‫א‬١J ٤J ٣

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

(Voltage lead)‫א‬‫א‬‫א‬،

‫א‬‫א‬ (Current lead)‫א‬‫א‬ ‫א‬‫א‬،(in phase)‫א‬‫א‬‫א‬‫א‬

 KE٣٣J ٣Fϕ‫א‬‫א‬‫א‬

ϕ



V





 

i

 K‫א‬‫א‬‫א‬‫א‬E٣٣J ٣F 

 I ⋅ cos ϕ =‫א‬‫א‬W I‫א‬  KE٣٤J ٣F I ⋅ sin ϕ = ‫א‬‫א‬ 

  ϕ I ⋅ cos

V



ϕ



I



 I ⋅ sin ϕ



 K‫א‬‫א‬‫א‬‫א‬E٣٤J ٣F

- ٧١ -



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬V‫א‬‫א‬

 KE٣٥J ٣F

P = V ⋅ I ⋅ cos ϕ ϕ

 

S=V⋅I



V



Q = V ⋅ I ⋅ sin ϕ

 

 ‫א‬E٣٥J ٣F ‫א‬،‫א‬W‫א‬‫א‬

 W‫א‬‫א‬

 WApparent PowerW(S)‫א‬‫א‬E١F (١١٨-٣)  WActive PowerW(P)‫א‬‫א‬E٢F

S=V⋅I P = V ⋅ I ⋅ cos ϕ

(١١٩-٣)  WReactive PowerW(Q)‫א‬‫א‬E٣F

Q = V ⋅ I ⋅ sin ϕ

(١٢٠-٣)  Power Factor‫א‬٢J ٤J ٣

، cos ϕ  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ cos (ϕ) =

 K‫א‬‫א‬‫א‬‫א‬ϕ‫א‬‫א‬

P S

(١٢١-٣)

- ٧٢ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Power Factor Correction‫א‬٣J ٤J ٣

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،E٣٦J ٣F(Current lag.) ‫א‬K‫א‬‫א‬P‫א‬‫א‬‫א‬‫א‬S‫א‬

PS‫א‬‫א‬،Q‫א‬ ‫א‬K ‫א‬‫א‬،‫א‬‫א‬ 

‫א‬‫א‬ cos ϕ ϕ‫א‬‫א‬

 K‫א‬‫א‬‫א‬PSK‫א‬ 

I1 ‫ﻣﺼﺪر اﻟﺘﻐﺬﻳﺔ‬ V

V

ϕ1 ‫א‬

  



I1

  K‫א‬‫א‬‫א‬‫א‬E٣٦J ٣F



،E٣٦J ٣F،‫א‬‫א‬‫א‬‫א‬‫א‬

،ϕ١‫א‬‫א‬I١‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬KP.F١ = cos ϕ١‫א‬ ،‫א‬‫א‬‫א‬E ‫א‬‫א‬‫א‬‫א‬F  K‫א‬ϕ‫א‬‫א‬،E٣٧J ٣F

- ٧٣ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 IC



V

I1

ϕ1

IC



I2



V



ϕ2

IC

 

I1



 K‫א‬‫א‬E٣٧J ٣F  E١٣J ٣F ‫א‬‫א‬،‫א‬‫א‬‫א‬E٣٨J ٣F  K‫א‬‫א‬‫א‬

I1

I2



I1

  

IC V = 380 V∠0

ZL = 6 + j 8

V

ZL

C

   

 KE١٣J ٣F‫א‬‫א‬‫א‬E٣٨J ٣F  ‫א‬

ZL =

(6)2 + (8)2

θ ZL = tan −1

 W‫א‬ZL‫א‬ = 10 Ω

 WθZL‫א‬‫א‬

8 = 51.3° 6 - ٧٤ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 W I1 ‫א‬ I1 =

380 ∠0° = 38 A ∠ − 51.3° 10∠51.3°

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬  W‫א‬‫א‬Kϕ٢‫א‬‫א‬ I C = I1 ⋅ sin ϕ1 = 38 ⋅ sin (−51.3°) = − 29.66 A

 W θ C = 90° ‫א‬‫א‬‫א‬ ∴ I C = − 29.66 A ∠90°

 W‫א‬‫א‬ IC = C=

V ∠0 V ∠0 = = VωC ∠90 − j (1 ωC) (1 ωC) ∠ − 90

− 29.66 = − 248.42 µF 380 × 2π × 50

 KE٣٩J ٣F‫א‬‫א‬  I1 ⋅ cos ϕ1 = I 2 ⋅ cos ϕ 2 = I 2 ϕ1

V

IC

 

   I 1 

I1 ⋅ sin ϕ1

 K‫א‬‫א‬E٣٩J ٣F - ٧٥ -

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬  W‫א‬W  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K١ K‫א‬°٩٠‫א‬‫א‬‫א‬‫א‬ K٢ K‫א‬°٩٠‫א‬‫א‬‫א‬‫א‬ K٣ .XL‫א‬‫א‬‫ א‬K٤ .XC‫א‬‫א‬‫ א‬K٥

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٦ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٧ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٨ K‫א‬ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٩ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٠ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١١  K‫א‬

 W‫א‬W  ‫א‬(٥ mH)‫א‬‫א‬(٥Ω) ‫א‬‫א‬‫א‬‫ א‬K١ K‫א‬

(١µf)   ‫א‬‫א‬(١٥Ω)‫א‬‫א‬‫א‬‫ א‬K٢ K‫א‬‫א‬

(٥ mH)  ‫א‬‫א‬(٢٠Ω)  ‫א‬‫א‬‫א‬‫ א‬K٣ K‫א‬‫א‬(١µf)‫א‬‫א‬ - ٧٦ -

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

‫א‬(٥ mH)‫א‬‫א‬(٥Ω) ‫א‬‫א‬‫א‬‫ א‬K٤ K‫א‬

(١µf)   ‫א‬‫א‬(١٥Ω)‫א‬‫א‬‫א‬‫ א‬K٥ K‫א‬‫א‬ (٥ mH)  ‫א‬‫א‬(٢٠Ω)  ‫א‬‫א‬‫א‬‫ א‬K٦ K‫א‬‫א‬(١µf)‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٧ K٧٠ mH‫א‬٥٠ mH

‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫ א‬K٨ K٢٠ mH‫א‬٣٠ mH ١٠٠ µF‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٩ K٢٥ µF‫א‬ ١٠٠ µF‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٠ K٢٥ µF‫א‬

W،‫א‬‫א‬ ،‫א‬‫א‬‫א‬‫א‬ K١١ K(f =٥٠ Hz) E = 200 V ‫א‬‫א‬‫א‬، Z1 = 8 + j 6 & Z 2 = 6 - j 8

 K‫א‬‫א‬‫א‬ i١

Z١

i٢ i

Z٢



٢٠٠ V

 - ٧٧ -

 

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫ א‬K١٢ K‫א‬‫א‬‫א‬

I1

I2



I1



IC V = 380 V∠0

ZL = 6 + j 8

V

C

ZL

  

١٢ V‫א‬‫א‬،١٠ µF‫א‬ K١٣ ٥٠ ‫א‬‫א‬، τ‫א‬‫א‬W  ‫א‬،١ kΩ‫א‬‫א‬‫א‬

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬Kmsec ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬٢ kΩ  K‫א‬٣٠ msec 

- ٧٨ -

 ‫א‬‫א‬‫א‬ ã¹]
‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

٢ J ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

٤

- ٣٦ -

 ‫א‬‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ٢-

‫א‬ 

 ‫א‬‫א‬‫א‬‫א‬‫א‬



 W‫א‬‫א‬،‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •  K‫א‬‫א‬‫א‬‫א‬‫א‬ •          

- ٧٩ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Introduction١J ٤

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Basic Theorems of A.C Circuits ‫א‬‫א‬‫א‬‫א‬‫א‬٢J ٤

 Ohm's Law ١J ٢J ٤ ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬Z‫א‬I‫א‬V‫א‬‫א‬‫א‬‫א‬ E‫א‬F‫א‬،

 W‫א‬

V , Z (b) V = i • Z , V , (c) Z = i (a) i =

V ∠θ V Z ∠θ Z V ∠θ V = i ∠θ i • Z ∠θ Z V ∠θ V Z ∠θ Z = i ∠θ i

(١-٤)

i ∠θ i =

(٢-٤) (٣-٤)  E١J ٤F  W‫א‬‫א‬  K Z ‫א‬‫א‬E١F  K i ‫א‬‫א‬E٢F 

i



١٠٠ V ٥٠ Hz

Z = 6 + j8

  

 KE١J ٤F‫א‬‫א‬‫א‬E١J ٤F

- ٨٠ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬  W، Z ‫א‬

E١F

 K 6 Ω ‫א‬٦‫א‬W 

  ω ⋅ L = 8 Ω E‫א‬‫א‬F‫א‬‫א‬W  L=

 W

8Ω 8Ω 8Ω = = = 25.465 m H ω 2 π f 100 π R =6Ω

L = 25.465 m H

 KE١J ٤F Z ‫א‬E٢J ٤F  W، Z ‫א‬، i ‫א‬E٢F Z=

i=

6 2 + 8 2 ∠ tan −1

8 = 10 Ω ∠ 53.13°  6

 W، i ‫א‬

100 V ∠0° V = = 10 A ∠ − 53.13° Z 10 Ω ∠53.13°

 E٢J ٤F  W‫א‬‫א‬

 Z ‫א‬‫א‬

E١F



E٢F

i = 20 A ∠30°

  

١٠٠ V

Z

 

 KE٢J ٤F‫א‬‫א‬‫א‬E٣J ٤F - ٨١ -



 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬ V i

Z=

100 V∠0° = 5 Ω ∠ − 30° 20 A∠30°

Z ∠θ Z =

E١F

W

E٢F

V ∠θ V i ∠θ i

Z=

,

W‫א‬ Z ‫א‬

Z = 5 Ω {cos (−30) + j sin (−30)}

Z = 5 Ω (0.866 − j 0.5) = 4.33 − j 2.4

 W‫א‬‫א‬  W Z ‫א‬  4.33 Ω ‫א‬٤{٣٣‫א‬W 

 

1 = 2.4 Ω E‫א‬‫א‬F‫א‬‫א‬W  ω⋅C

 W C=

1 1 = = 1.326 mF 2.4 ω 2.4 × 100 × π

C = 1.326 mF

R = 4.33 Ω

KE٢J ٤F Z ‫א‬E٤J ٤F

- ٨٢ -



 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 Kirchhoff's Laws  ٢J ٢J ٤ Kirchhoff's Current Law

١J ٢J ٢J ٤

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬?

 W،‫א‬

k =n

∑ ik = 0

(٤-٤)

k =1

‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬ n ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬E٥J ٤FK‫א‬  

in

i1

 

i2



i3



 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E٥J ٤F  E٣J ٤F  KIT‫א‬‫א‬‫א‬،E٦J ٤F‫א‬‫א‬ IT

A

 

Z١ I١= ١٠ A ∠٣٠°

Z٢ I٢=٢٠ A ∠٢ °

B

 KE٣J ٤F‫א‬‫א‬‫א‬E٦J ٤F - ٨٣ -

    

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬  WBA‫א‬‫א‬، IT‫א‬‫א‬ WA

•

 EITF=‫א‬‫א‬‫א‬‫א‬‫א‬

EF

EI١ + I٢F=‫א‬‫א‬‫א‬‫א‬

E F  W

(IT) = (I١ + I٢)

WB

•

 EI١ + I٢F=‫א‬‫א‬‫א‬‫א‬‫א‬

EF

EITF=‫א‬‫א‬‫א‬‫א‬

E F

(I١ + I٢) =(IT)

 K‫א‬BA‫א‬‫א‬

 W

 W

I T = (I1 + I 2 ) = 10 A∠30° + 20 A∠ − 20° = 

I T = (8.66 + j 5) + (18.8 − j 6.84) = 27.46 − j1.84 = 27.5 A ∠ − 3.84°

 E٤J ٤F

 W‫א‬‫א‬،Z١, Z٢‫א‬‫א‬E٣J ٤F E=١١٠ V, f=٦٠HZ IT

A

  

E=١١٠ V f = ٦٠ HZ

Z١ I١= ١٠ A∠٣٠°

Z٢ I٢=٢٠ A∠-٢٠°

   

B - ٨٤ -



 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 KE٤J ٤F‫א‬‫א‬‫א‬E٧J ٤F  ‫א‬

 WW‫א‬ Z1 =

110 V∠0° = 11 Ω∠ − 30° 10 A∠30°

Z2 =

110 V∠0° = 5.5 Ω∠20° 20 A∠ − 20°

 W‫א‬ Z1 = 11 Ω∠ − 30° = 11⋅ cos (− 30°) + j 11 ⋅ sin (− 30°) = 9.52 − j 5.5

‫א‬‫א‬R١= ٩{٥٢ Ω٩{٥٢‫א‬  WC١‫א‬،XC= ٥{٥ Ω،‫א‬‫א‬‫א‬

XC = C1 =

1 ω⋅C

1 1 = = 482.28 µF ω ⋅ X C 2 π × 60 × 5.5

 W Z 2 = 5.5 Ω∠20° = 5.5 cos (20°) + j 5.5 sin (20°) = 5.17 + j 1.88 

‫א‬‫א‬R٢= ٥{١٧ Ω٥{١٧‫א‬  WL‫א‬‫א‬‫א‬،XL= ١{٨٨ Ω،‫א‬‫א‬‫א‬ XL = ω ⋅ L2 L2 =

X L 1.88 Ω = = 4.98 mH ω 2 π × 60

 - ٨٥ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



C1 = 482.28 µF

R 1 = 9.52 Ω

L 2 = 4.98 mH

R 2 = 5.17 Ω

 

Z٢

Z١



 KE٤J ٤F Z1 , Z 2 ‫א‬E٨J ٤F



 E٥J ٤F  KI١‫א‬‫א‬‫א‬،E٩J ٤F‫א‬‫א‬ 

IT =٢٠ A∠٢ °



A

 

Z١ I٢= ١٠ A∠٣٠°

I١



Z٢

   

B

 KE٥J ٤F‫א‬‫א‬‫א‬E٩J ٤F  ‫א‬

 WA‫א‬‫א‬،I١‫א‬‫א‬ WA

•

EI١ + I٢F‫א‬‫א‬‫א‬‫א‬=EITF‫א‬‫א‬‫א‬‫א‬‫א‬

 W

(I T ) = (I1 + I 2 )

(I1 ) = (I T − I 2 ) I1 = (I T − I 2 ) = 20 A∠20° − 10∠30° = 18.8 + j 6.84 − (8.66 + j 5)

I1 = 10.14 + j 1.84 = 10.3 A ∠10.28° - ٨٦ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 K‫א‬‫א‬‫א‬‫א‬ Parallel Connection‫א‬‫א‬‫א‬

١J ١J ٢J ٢J ٤

Z١, Z٢‫א‬،E٤J ٤FE٧J ٤F

 ‫א‬‫א‬‫א‬،E١J ‫א‬F‫א‬‫א‬



‫א‬‫א‬ ‫א‬‫א‬،‫א‬

 KK‫א‬‫א‬ V ‫א‬

 Z‫א‬‫א‬‫א‬‫א‬‫א‬

 W،E١٠J ٤F

IT

IT

A

Z١ I١

(I T ) = (I1 + I 2 )

 B

 ‫א‬‫א‬‫א‬E١٠J ٤F

 

IT



V = I1 Z1 = I 2 Z 2 = I T Z

Z

V

I٢

B

 

≡

Z٢

V

A



 

 W‫א‬‫א‬‫א‬ (٥-٤)  W

(٦-٤)  W،EF‫א‬I٢

V ⎞ ⎛V⎞ ⎛ V ⎟⎟ + ⎜⎜ ⎟⎟ = ⎜⎜ Z Z Z ⎝ ⎠ ⎝ 1 2 ⎠

(٧-٤)  W‫א‬،Z‫א‬

1 ⎞ ⎛1⎞ ⎛ 1 ⎟⎟ + ⎜ ⎟ = ⎜⎜ ⎝ Z ⎠ ⎝ Z1 Z 2 ⎠

(٨-٤) - ٨٧ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

⎛ 1 ⎞ ⎛ Z1 + Z 2 ⎜ ⎟ = ⎜⎜ ⎝ Z ⎠ ⎝ Z1 Z 2

(Z ) = ⎛⎜⎜

Z1 Z 2 ⎝ Z1 + Z 2

 ١٤١

‫א‬

٢-



⎞ ⎟⎟ ⎠

(٩-٤) 

⎞ ⎟⎟ ⎠

(١٠-٤) ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬  E٦J ٤F  KZ١ & Z٢‫א‬‫א‬‫א‬E٤J ٤F IT

IT

A

Z٢ = ٥٫٥ Ω ∠٢٠°

Z١ = ١١ Ω ∠-٣٠°

≡

A

 

V

Z

B

  

IT

I٢

I١



B





 E٦J ٤F‫א‬‫א‬‫א‬E١١J ٤F  ‫א‬  W‫א‬،E٤J ٤F‫א‬Z١ & Z٢‫א‬ Z1 = 11 Ω∠ − 30° Z 2 = 5.5 Ω∠20°

 WE١٠J ٤F‫א‬Z‫א‬‫א‬

(Z ) = ⎛⎜⎜

Z1 Z 2 ⎝ Z1 + Z 2

⎞ 11 Ω ∠ − 30° × 5.5 Ω ∠20° 11 Ω × 5.5 Ω ∠ − 30° + 20° ⎟⎟ = = ⎠ 11 Ω ∠ − 30° + 5.5 Ω ∠20° (9.52 − j 5.5) + (5.17 + j 1.88) - ٨٨ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



60.5 ∠ − 10° 60.5 Ω 2 ∠ − 10° (Z ) = = (14.69 − j 3.62) 15.13 Ω ∠ − 13.84° 60.5 Ω 2 (Z ) = ∠ (− 10° + 13.84°) = 4 Ω ∠3.84° 15.13 Ω

J ٤FF‫א‬‫א‬‫א‬،‫א‬

 W،EE٣

IT =

110 V∠0° = 27.5 A ∠ − 3.84°  4 Ω ∠3.84°

 KE٣J ٤F‫א‬‫א‬‫א‬ ٢J ١J ٢J ٢J ٤

Current Divider‫א‬

 KI١, I٢IT‫א‬‫א‬E١٢J ٤F IT



A

 Z1 I١

V



Z2



I٢

  

B

  Z1 , Z 2 ‫א‬‫א‬E١٢J ٤F



 WI١, I٢‫א‬‫א‬‫א‬IT‫א‬‫א‬‫א‬‫א‬

Q (I T ) = (I1 + I 2 )

(١١-٤) (١٢-٤) (١٣-٤) (١٤-٤) (١٥-٤) (١٦-٤)

∴ (I 2 ) = (I T − I1 )

Q V = I1 Z1 = I 2 Z 2 = I T Z ∴ V = I1 Z1 = (I T − I1 ) Z 2 I1 Z1 + I1 Z 2 = I T Z 2

I1 (Z1 + Z 2 ) = I T Z 2

- ٨٩ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

∴ I1 = I T

Z2 (Z1 + Z 2 )

∴ I2 = IT

Z1 (Z1 + Z 2 )

 ١٤١

‫א‬

٢-



(١٧-٤)  W

(١٨-٤) ‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬،‫א‬‫א‬‫א‬  E٧J ٤F

 K i1 , i 2 ‫א‬‫א‬‫א‬‫א‬،E١٣J ٤F‫א‬‫א‬ i T = ٢٧٫٥ A ∠-



A

 Z1 = ١١ Ω ∠-٣٠°

i1

Z1 = ٥٫٥ Ω i2

    

B

 E٧J ٤F‫א‬‫א‬‫א‬E١٣J ٤F



 ‫א‬

I١, I٢‫א‬‫א‬‫א‬،E١٨J ٤FE١٧J ٤F‫א‬

 W i1 = i T

Z2 ⎛ ⎞ 5.5 Ω ∠20° ⎟⎟ = (27.5 A ∠ − 3.84°) × ⎜⎜ (Z1 + Z 2 ) ⎝ 11 Ω ∠ − 30° + 5.5 Ω ∠20° ⎠

⎛ ⎞ 5.5 Ω ∠20° ⎟⎟ i1 = (27.5 A ∠ − 3.84°) × ⎜⎜ ⎝ (9.53 − j 5.5) + (5.17 + j 1.88) ⎠ ⎛ 5.5 Ω ∠20° i1 = (27.5 A ∠ − 3.84°)× ⎜⎜ ⎝ 14.7 − j 3.62 - ٩٠ -

⎞ ⎟⎟ ⎠

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



⎛ ⎞ 5.5 Ω ∠20° ⎟⎟ i1 = (27.5 A ∠ − 3.84°) × ⎜⎜ ⎝ 15.14 Ω ∠ − 13.84° ⎠ i1 = (27.5 A ∠ − 3.84°) × (0.363 ∠33.84°) = 10 A ∠30° i2 = iT

Z1 ⎛ ⎞ 11 Ω ∠ − 30° ⎟⎟ = (27.5 A ∠ − 3.84 ) × ⎜⎜ (Z1 + Z 2 ) 11 Ω ∠ − 30 ° + 5 . 5 Ω ∠ 20 ° ⎝ ⎠

⎛ 11 Ω ∠ − 30° ⎞ ⎟⎟ = 20 A ∠ − 20° i 2 = (27.5 A ∠ − 3.84 ) × ⎜⎜ 15 . 14 13 . 84 Ω ∠ − ° ⎝ ⎠

E٣J ٤F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬ Kirchhoff's Voltage Law

٢J ٢J ٢J ٤

‫א‬‫א‬‫א‬?

 W،‫א‬ ‫א‬‫א‬‫א‬‫א‬ k =n

∑

k =1

(١٩-٤)

vk = 0

‫א‬‫א‬ ‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬n

‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬E١٤J ٤FK‫א‬

- ٩١ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



  V1



Z1



 

i 

E

i

i

Z2

 V2



i



Z3



 

 V3



 ‫א‬‫א‬‫א‬E١٤J ٤F



 W‫א‬‫א‬

(٢٠-٤)

E − V1 − V2 − V3 = 0

 W

(٢١-٤)

E = V1 + V2 + V3

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

- ٩٢ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E٨J ٤F  KA, B‫א‬‫א‬‫א‬E١٥J ٤F Z1 = 5 Ω ∠ − 20°

A

i = 10 A ∠ − 30°



 

Z 2 = 7 Ω ∠ − 10°

 

Z 3 = 8 Ω ∠15°

B

 E٨J ٤F‫א‬‫א‬‫א‬E١٥J ٤F

    ‫א‬

 W،‫א‬‫א‬ VAB = V1 + V2 + V3

 W‫א‬‫א‬ V1 = i ⋅ Z1 = (10 A ∠ − 30°) ⋅ (5 Ω ∠ − 20°) = 50 V∠ − 50° V2 = i ⋅ Z 2 = (10 A ∠ − 30°) ⋅ (7 Ω ∠ − 10°) = 70 V∠ − 40° V3 = i ⋅ Z 3 = (10 A ∠ − 30°) ⋅ (8 Ω ∠15°) = 80 V∠ − 15°

 W‫א‬،VAB‫א‬ VA − B = V1 + V2 + V3 = 50 V∠ − 50° + 70 V∠ − 40° + 80 V∠ − 15° VAB = (32.14 − j38.3) + (53.62 − j45) + (77.27 − j20.7 ) =

VAB = 163 − j104 = 193.4 V∠ − 32.54°

 K‫א‬‫א‬‫א‬‫א‬

- ٩٣ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 Series Connection‫א‬‫א‬‫א‬

١J ٢J ٢J ٢J ٤

 Z1 , Z 2 , Z 3 ‫א‬،E٨J ٤FE١٥J ٤F

‫א‬ Z1 ‫א‬،E١J ‫א‬F‫א‬‫א‬

‫א‬ Z 3 ‫א‬‫א‬ Z 2 ‫א‬ Z 2 ‫א‬  i ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬  Z ‫א‬‫א‬‫א‬‫א‬‫א‬

 W،E١٦J ٤F 

V1 A

Z1

i

V3 B

V2

Z2

V



A

Z3

≡



i



V

Z

 

B



 ‫א‬‫א‬‫א‬E١٦J ٤F  W‫א‬‫א‬ V ‫א‬ V = V1 + V2 + V3

 W،‫א‬A, B‫א‬‫א‬‫א‬

(٢٢-٤)

V = i Z1 + i Z 2 + i Z 3 = i Z

 W‫א‬، i ‫א‬‫א‬ ∴ Z1 + Z 2 + Z 3 = Z

(٢٣-٤)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬

- ٩٤ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E٩J ٤F  K‫א‬‫א‬‫א‬‫א‬E٨J ٤F‫א‬ A i



Z1 = 5 Ω ∠ − 20° Z 2 = 7 Ω ∠ − 10°

B



A

i

≡

 Z



B

Z 3 = 8 Ω ∠15°

 

 E٩J ٤F‫א‬‫א‬‫א‬E١٧J ٤F

 ‫א‬

Z٢‫א‬‫א‬Z١‫א‬E١٧J ٤F  ‫א‬‫א‬Z٣‫א‬‫א‬Z٢‫א‬  K‫א‬‫א‬i‫א‬‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬ Z eq = Z1 + Z 2 + Z 3 = 5 Ω ∠ − 20° + 7 Ω ∠ − 10° + 8 Ω ∠15°

Z eq = (4.7 − j 1.71) Ω + (6.9 − j 1.22) Ω + (7.7 + j 2.1) Ω Z eq = (19.3 − j 0.83) = 19.32 Ω ∠ − 2.46°

 Potential Divider ‫א‬

٢J ٢J ٢J ٢J ٤

Z١, Z٢& Z٣‫א‬‫א‬‫א‬‫א‬‫א‬E١٨J ٤F

،‫א‬‫א‬V‫א‬‫א‬،E‫א‬  W

- ٩٥ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 

V1 A



Z1



i

E

Z2

V3 B



V2

 

Z3



 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٨J ٤F  W V1 = i ⋅ Z1

(٢٤-٤)

V2 = i ⋅ Z 2

(٢٥-٤)

V3 = i ⋅ Z 3

(٢٦-٤)  W

E = V1 + V2 + V3 = i ⋅ (Z1 + Z 2 + Z 3 )

(٢٧-٤)  W، E  V1 

V1 Z1 = E (Z1 + Z 2 + Z 3 )

(٢٨-٤)  W، E ‫א‬

⎛ ⎞ Z1 ⎟⎟ V1 = E ⋅ ⎜⎜ ⎝ (Z1 + Z 2 + Z 3 ) ⎠

(٢٩-٤)

- ٩٦ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 W، E  V2 ‫א‬ V2 Z2 = E (Z1 + Z 2 + Z 3 )

(٣٠-٤)  W، E ×‫א‬

⎛ ⎞ Z2 ⎟⎟ V2 = E ⋅ ⎜⎜ ( ) + + Z Z Z 2 3 ⎠ ⎝ 1

(٣١-٤)  W، E  V3 ‫א‬

V3 Z3 = E (Z1 + Z 2 + Z 3 )

(٣٢-٤)  W، E ‫א‬

⎛ ⎞ Z3 ⎟⎟ V3 = E ⋅ ⎜⎜ ( ) + + Z Z Z 2 3 ⎠ ⎝ 1

(٣٣-٤)  W‫א‬

⎛ Z Vx = E ⋅ ⎜⎜ x ⎝ (Z T

⎞ ⎟ ) ⎟⎠

 E١٠J ٤F ‫א‬‫א‬‫א‬‫א‬‫א‬E١٩J ٤F

 K‫א‬

- ٩٧ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Z1 = 5 Ω ∠ − 20°

A E= ١١٠ V f = ٦٠ Hz

Z 2 = 7 Ω ∠ − 10°

    

Z 3 = 8 Ω ∠15°

B

 

 E١٠J ٤F‫א‬‫א‬‫א‬E١٩J ٤F  ‫א‬ KEF‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬  W‫א‬‫א‬‫א‬ ⎛ ⎞ Z1 ⎟⎟ V1 = E ⋅ ⎜⎜ ⎝ (Z1 + Z 2 + Z 3 ) ⎠ ⎛ ⎞ 5 Ω ∠ − 20° ⎟⎟ V1 = (110 V∠0°) ⋅ ⎜⎜ ⎝ ((5 Ω ∠ − 20°) + (7 Ω ∠ − 10°) + (8 Ω ∠15°)) ⎠ ⎛ ⎞ 5 Ω ∠ − 20° ⎟⎟ V1 = (110 V∠0°) ⋅ ⎜⎜ ⎝ ((5 Ω ∠ − 20°) + (7 Ω ∠ − 10°) + (8 Ω ∠15°)) ⎠

⎛ ⎞ 5 Ω ∠ − 20° ⎟⎟ V1 = (110 V∠0°) ⋅ ⎜⎜ ⎝ ((4.7 − j 1.71) Ω + (6.9 − j 1.22) Ω + (7.7 + j 2.1) Ω ) ⎠ ⎛ ⎞ 5 Ω ∠ − 20° ⎟⎟ = 28.47 V∠ − 17.54° V1 = (110 V∠0°) ⋅ ⎜⎜ ⎝ ( 19.32 Ω ∠ − 2.46°) ⎠ ⎛ ⎞ Z2 ⎛ ⎞ 7 Ω ∠ − 10° ⎟⎟ = (110 V∠0°) ⋅ ⎜⎜ ⎟⎟ V2 = E ⋅ ⎜⎜ ( ) ( ) Z + Z + Z 19 . 32 Ω ∠ − 2 . 46 ° ⎝ ⎠ 2 3 ⎠ ⎝ 1 V2 = 39.86 V ∠ − 7.54°

- ٩٨ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



⎛ ⎞ Z3 ⎛ ⎞ 8 Ω ∠15° ⎟⎟ = (110 V∠0°) ⋅ ⎜⎜ ⎟⎟ V3 = E ⋅ ⎜⎜ ⎝ ( 19.32 Ω ∠ − 2.46°) ⎠ ⎝ (Z1 + Z 2 + Z 3 ) ⎠ V3 = 45.55 V ∠17.46°

 WV١, V٢, V٣‫א‬‫א‬ E = V1 + V2 + V3 = 28.47 ∠ − 17.54° + 39.86 ∠ − 7.54° + 45.55 ∠17.46° E = (27.1 − j 8.5) V + (39.5 − j 5.2 ) + (43.4 + j 13.7 ) = (110 − j 0 ) = 110 V ∠0°

 Compound Circuits‫א‬‫א‬‫ א‬٣J ٢J ٤

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬



‫א‬‫א‬‫א‬‫א‬،E٢٠J ٤F ‫א‬F‫א‬‫א‬Z٢, Z٣ ‫א‬،‫א‬‫א‬ ‫א‬‫א‬

‫א‬F‫א‬‫א‬‫א‬Z١, Z٤‫א‬،E‫א‬‫א‬

 KE‫א‬‫א‬

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ A

Z1

Z2 B

Z4



i3

i2

i1 V

 K‫א‬

C

V١



Z3

  

D

 ‫א‬E٢٠J ٤F  Simplification Method‫א‬‫א‬

١J ٣J ٢J ٤

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬،‫א‬

 K‫א‬‫א‬‫א‬

- ٩٩ -



 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E١١J ٤F K i1 , i 2 & i 3 ‫א‬‫א‬W‫א‬E٢١J ٤F i1

A

C

Z1 = 5Ω ∠10°

Z 2 = 5 Ω ∠10°

f = 60 Hz

Z 4 = 10 Ω ∠ − 10°

B

i3

i2

E = 110 V



E Z 3 = 10 Ω ∠ − 10°

  

F

D





 E١١J ٤F‫א‬‫א‬‫א‬E٢١J ٤F  W‫א‬ ‫א‬‫א‬‫א‬‫א‬  K

 ‫א‬W 

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬K ‫א‬‫א‬  W‫א‬

i1 = i 2 + i 3

 W‫א‬C‫א‬ •

 WACDB‫א‬‫א‬ •

E = i1 Z1 + i 2 Z 2 + i1 Z 4 = i1 (Z1 + Z 4 ) + i 2 Z 2 E = i1 (Z1 + Z 4 ) + (i1 − i 3 ) Z 2

 WCEFD‫א‬‫א‬ • i 2 Z 2 = i3 Z 3 

 E‫א‬‫א‬‫א‬‫א‬F‫א‬‫א‬  W i 3 ‫א‬

i3 =

i2 Z2 Z3 - ١٠٠ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

i1 = i 2 +

(

Z 3 + Z2 i2 Z2 = i2 Z3 Z3

 ١٤١

‫א‬

٢-



 W‫א‬‫א‬

)

 W i 2 ‫א‬‫א‬ i 2 = i1

Z3

(Z 3 + Z 2 )  W‫א‬‫א‬

⎛ Z 3Z 2 E = i1 (Z1 + Z 4 ) + i1 ⎜ ⎜ Z 3 + Z2 ⎝

(

)

⎛ ⎞ Z 3Z 2 ⎟ = i1 ⎜ Z1 + Z 4 + ⎜ ⎟ Z 3 + Z2 ⎝ ⎠

(

)

⎞ ⎟ ⎟ ⎠

 W i ‫א‬‫א‬ i1 =

E ⎛ Z 3Z 2 ⎜ Z1 + Z 4 + ⎜ Z 3 + Z2 ⎝

(

i1 =

i1 =

i1 =

i1 =

)

⎞ ⎟ ⎟ ⎠

110 V ∠0°

⎛ (10 Ω ∠ − 10°) ⋅ (5 Ω ∠10°) ⎞⎟ ⎜⎜ 5 Ω ∠10° + 10 Ω ∠ − 10° + (10 Ω ∠ − 10°) + (5 Ω ∠10°) ⎟⎠ ⎝ 110 V ∠0°

(

)

⎛ ⎞ 50 Ω 2 ∠0° ⎜ 5 Ω ∠10° + 10 Ω ∠ − 10° + ⎟ ⎜ ⎟ ( ) ( ) 9 . 85 j 1 . 74 4 . 92 j 0 . 87 − Ω + + Ω ⎝ ⎠

110 V ∠0°

(

)

(

)

⎛ 50 Ω 2 ∠0° ⎜ 5 Ω ∠10° + 10 Ω ∠ − 10° + ⎜ (14.77 − j 0.87 ) Ω ⎝ 110 V ∠0°

⎛ 50 Ω 2 ∠0° ⎜ 5 Ω ∠10° + 10 Ω ∠ − 10° + ⎜ (14.8 Ω ∠ − 3.37° ⎝

- ١٠١ -

⎞ ⎟ ⎟ ⎠

)

⎞ ⎟ ⎟ ⎠

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



i1 =

110 V ∠0° (5 Ω ∠10° + 10 Ω ∠ − 10° + 3.38 Ω ∠3.37°)

i1 =

110 V ∠0° ((4.92 + j 0.87 ) Ω + (9.85 − j 1.74) Ω + (3.37 + j 0.2) Ω )

i1 =

110 V ∠0° 110 V ∠0° = = 6.05 A ∠2.12° (18.14 − j 0.67 ) Ω 18.15 Ω ∠ − 2.12°

 W i 2 ‫א‬ i 2 = i1

Z3

(Z 3 + Z 2 )

⎛ ⎞ (10 Ω ∠ − 10°) ⎟⎟ = (6.05 A ∠2.12°) ⋅ ⎜⎜ ( ( ) ( ) ) 10 10 5 10 Ω ∠ − ° + Ω ∠ ° ⎝ ⎠

⎛ ⎞ (10 Ω ∠ − 10°) ⎟⎟ = i 2 = (6.05 A ∠2.12°) ⋅ ⎜⎜ ( ( ) ( ) ) 9 . 85 − j 1 . 74 Ω + 4 . 92 + j 0 . 87 Ω ⎝ ⎠ ⎛ (10 Ω ∠ − 10°) ⎞ ⎟⎟ = i2 = (6.05 A ∠2.12°) ⋅ ⎜⎜ ⎝ (14.77 − j 0.87 ) ⎠ ⎛ 10 Ω ∠ − 10° ⎞ ⎟⎟ = 4.09 A ∠ − 4.5° i 2 = (6.05 A ∠2.12°) ⋅ ⎜⎜ ⎝ 14.8 Ω ∠ − 3.4° ⎠

 W i 3 ‫א‬ i3 =

i2 Z2 (4.09 A ∠ − 4.5°) ⋅ (5 Ω ∠10°) = = 2.05 A ∠15.5° Z3 10 Ω ∠ − 10°

- ١٠٢ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬W A

i1

C

Z1

i3

Z2

Z3

D

F

Z4

B

E

i2

E = 110 V f = 60 Hz





A

≡

i1

E, f B

 

Z



 ‫א‬‫א‬E٢٢J ٤F ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،E٢٢J ٤F

‫א‬Z١, Z٤‫א‬‫א‬‫א‬‫א‬‫א‬ Z٢, Z٣  WZ‫א‬‫א‬ Z = Z1 + Z 4 +

Z 2 Z3 Z 2 + Z3

 W‫א‬، i1 ‫א‬‫א‬‫א‬‫א‬‫א‬ i1 =

E ⎛ Z 3Z 2 ⎜ Z1 + Z 4 + ⎜ Z 3 + Z2 ⎝

(

)

⎞ ⎟ ⎟ ⎠

،‫א‬ i1 ‫א‬‫א‬

 W، i1 ‫א‬

i1 = 6.05 A ∠2.12°

 W‫א‬، i 2 ‫א‬ i 2 = i1

Z3

(Z 3 + Z 2 ) - ١٠٣ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



،‫א‬ i 2 ‫א‬‫א‬  W، i 2 ‫א‬ i 2 = 4.09 A ∠ − 4.5°

 W‫א‬‫א‬، i 3 ‫א‬ W‫א‬ • i 3 = i1

(Z

Z 2

2

+ Z3

)

i 3 = (6 . 05 A ∠ 2 . 12 )

5 Ω ∠10 ° = 2 . 05 A ∠15 . 5 ° (14 . 8 Ω ∠ − 3 . 4 ° )

 ‫א‬ Z 2 + Z 3 ‫א‬‫א‬ W‫א‬ • Q i1 = i 2 + i 3

∴ i 3 = i1 − i 2 ∴ i 3 = ( 6.05 A ∠2.12 ) − (4.09 A ∠ − 4.5° ) = ∴ i3 = (6.05 + j0.22 ) − (4.08 − j 0.32) = 1.97 + j0.54 = 2.05 A ∠15.5°

W‫א‬‫א‬‫א‬‫א‬‫א‬ • i3 Z 3 = i 2 Z 2

 W، Z 3 ‫א‬ ∴ i3 =

i2 Z 2 Z3

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ∴ i3 = 2.05 A ∠15.5°

- ١٠٤ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 K i 2 ‫א‬‫א‬‫א‬‫א‬  Star-Delta, Delta-Star ConversionJ ‫א‬J ‫א‬

٢J ٣J ٢J ٤

‫א‬‫א‬‫א‬،E٢٣J ٤F‫א‬

K ‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬‫א‬



‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬ 

a

 iT

i1

E, f

Z1

i2 i5

b i3

Z3 d

Z5 i4

Z2



c



Z4



  



 ‫א‬‫א‬E٢٣J ٤F



‫א‬‫א‬‫א‬‫א‬J ‫א‬J  ‫א‬

A, B, ‫א‬‫א‬E٢٤J ٤F،‫א‬‫א‬K‫א‬‫א‬

(∆)‫א‬‫א‬‫א‬‫א‬K ‫א‬‫א‬‫א‬  ‫א‬C K(Y)‫א‬‫א‬‫א‬‫א‬  K‫א‬

- ١٠٥ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-





A

A ZA



≡

O ZB



ZCA

ZAB

 

ZC

B

C

ZBC

B

C

 

 ‫א‬‫א‬E٢٤J ٤F ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬K‫א‬‫א‬‫א‬‫א‬

 ‫א‬

‫א‬A, B‫א‬‫א‬،E٢٥J ٤F

‫א‬‫א‬∆A, B‫א‬‫א‬

 W‫א‬KYA, B‫א‬ A



A

 

ZA ZAB

ZCA

O ZB B

 

ZC C

B

ZBC

C

A, B‫א‬‫א‬E٢٥J ٤F

(Z A + Z B ) = ⎛⎜⎜ Z AB ⋅ (Z BC + Z CA ) ⎞⎟⎟ ⎝ Z AB + (Z BC + Z CA ) ⎠

- ١٠٦ -

(٣٤-٤)

 

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



(Z A + Z B ) = ⎛⎜⎜ Z AB ⋅ (Z BC + Z CA ) ⎞⎟⎟ = ⎛⎜⎜ Z AB Z BC + Z AB Z CA ⎞⎟⎟ ⎝ Z AB + (Z BC + Z CA ) ⎠ ⎝ (Z AB + Z BC + Z CA ) ⎠

(٣٥-٤)

B, C ‫א‬‫א‬،E٢٦J ٤F

‫א‬‫א‬∆B, C‫א‬‫א‬‫א‬

 W‫א‬KYB, C‫א‬

(Z B + Z C ) = ⎛⎜⎜ Z BC ⋅ (Z AB + Z CA ) ⎞⎟⎟ ⎝ Z BC + (Z AB + Z CA ) ⎠

(٣٦-٤)

(Z B + Z C ) = ⎛⎜⎜ Z BC ⋅ (Z AB + Z CA ) ⎞⎟⎟ = Z BC Z AB + Z BC Z CA ⎝ Z BC + (Z AB + Z CA ) ⎠ (Z AB + Z BC + Z CA )

(٣٧-٤)

A

A

ZA ZAB

ZCA

O ZB B

ZC C

B

ZBC

C

 B, C‫א‬‫א‬E٢٦J ٤F A, ‫א‬‫א‬، E٢٧J ٤F

‫א‬∆A, C‫א‬‫א‬‫א‬C

 W‫א‬KYA, C‫א‬‫א‬  

- ١٠٧ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

A



ZA



 ١٤١

‫א‬

٢-



A

 

O ZB

ZAB

ZCA



ZC

B

C

 B 

ZBC

C

 C, A‫א‬‫א‬E٢٧J ٤F

(Z C + Z A ) = ⎛⎜⎜ Z CA ⋅ (Z AB + Z BC ) ⎞⎟⎟ ⎝ Z CA + (Z AB + Z BC ) ⎠ (Z C + Z A ) = ⎛⎜⎜ Z CA Z AB + Z CA Z BC ⎞⎟⎟ ⎝ Z CA + Z AB + Z BC ⎠

(٣٨-٤)

(٣٩-٤)

 W‫א‬E٣٩J ٤FE٣٧J ٤FE٣٥J ٤F‫א‬‫א‬ ⎛ Z CA ⋅ Z AB Z A = ⎜⎜ ⎝ Z CA + Z AB + Z BC

⎞ ⎟⎟ ⎠

(٤٠-٤)

⎛ Z BC ⋅ Z AB Z B = ⎜⎜ ⎝ Z AB + Z BC + Z CA

⎞ ⎟⎟ ⎠

(٤١-٤)

⎛ Z CA ⋅ Z AB Z C = ⎜⎜ ⎝ Z CA + Z AB + Z BC

⎞ ⎟⎟ ⎠

(٤٢-٤)  W‫א‬

Z AB = Z A + Z B +

ZA ⋅ ZB ZC

(٤٣-٤)

Z BC = Z B + Z C +

ZB ⋅ ZC ZA

(٤٤-٤) - ١٠٨ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

Z CA = Z C + Z A +

 ١٤١

‫א‬

٢-



ZC ⋅ ZA ZB

(٤٥-٤)  E١٢J ٤F

‫א‬‫א‬‫א‬‫א‬‫א‬،E٢٨J ٤F‫א‬‫א‬‫א‬  K(Y-∆)(∆-Y)

a i1 iT E=٢٢٠ V f= ٥٠ Hz

Z1 = 10 Ω ∠ − 53.13°

i5

b



i2 Z 2 = 10 Ω ∠36.87°

Z 5 = 20 Ω ∠0°

Z 3 = 10 Ω ∠53.13°

 

c



i4

i3



Z 4 = 10 Ω ∠ − 36.87°

  

d



 E١٢J ٤F‫א‬‫א‬‫א‬E٢٨J ٤F

 ‫א‬

a, ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

،∆b, c, d‫א‬،∆b, c   ‫א‬،bYa, c, d ‫א‬

‫א‬ ‫א‬‫א‬KcYa, b, d ‫א‬∆‫א‬‫א‬K ‫א‬‫א‬‫א‬

 K‫א‬Ya, b, c‫א‬

،E٤٢J ٤FE٤٠J ٤F‫א‬،‫א‬‫א‬

 W

⎞ ⎛ ⎛ Z1 ⋅ Z 2 ⎞ (10 Ω ∠ − 53.13°) ⋅ (10 Ω ∠36.87°) ⎟⎟ = ⎜⎜ ⎟⎟ Z A = ⎜⎜ Ω ∠ − ° + Ω ∠ + Ω ∠ ° Z + Z + Z 10 53 . 13 10 36 . 87 20 0 ⎠ 2 5 ⎠ ⎝ ⎝ 1 - ١٠٩ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 a

a Z 2 = 10 Ω ∠36.87°

Z1 = 10 Ω ∠ − 53.13° b

ZA o

c

Z 5 = 20 Ω ∠0°

ZB

ZC

b



 

c

 E٢٩J ٤F

(

)

(

⎛ ⎞ 10 0 Ω 2 ∠ − 16.26° 10 0 Ω 2 ∠ − 16.26° ⎟= ZA = ⎜ ⎜ (6 − j 8) Ω + (8 + j 6 ) Ω + (20 + j 0 ) Ω ⎟ (34 − j2) Ω ⎝ ⎠ ZA

(10 0 Ω =

)

)

2

∠ − 16.26° = 2.94 Ω ∠ − 12.86° (34 Ω ∠ − 3.4°)

⎛ Z1 ⋅ Z 5 ⎞ (10 Ω ∠ − 53.13°) ⋅ (20 Ω ∠0°) ⎟⎟ = Z B = ⎜⎜ (34 Ω ∠ − 3.4°) Z + Z + Z 2 5 ⎠ ⎝ 1

ZB

(200 Ω =

2

)

∠ − 53.13° = 5.88 Ω ∠ − 49.73° (34 Ω ∠ − 3.4°)

⎞ (10 Ω ∠36.87°) ⋅ (20 Ω ∠0°) ⎛ Z 2 ⋅ Z5 ⎟⎟ = Z C = ⎜⎜ (34 Ω ∠ − 3.4°) Z + Z + Z 2 5 ⎠ ⎝ 1 ZC =

200 Ω ∠36.87° = 5.88 Ω ∠40.27° 34 Ω ∠ − 3.4°

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 KE٣٠J ٤F  

- ١١٠ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 

a

 

Z A = 2.94 Ω ∠ − 12.86° iT E=٢٢٠ V f= ٥٠ Hz



o Z B = 5.88 Ω ∠ − 49.73°

b

Z C = 5.88 Ω ∠40.27°

i b

Z 3 = 10 Ω ∠53.13°



ic

 

c

Z 4 = 10 Ω ∠ − 36.87°

d 

 E٣٠J ٤F ،‫א‬‫א‬Z٣, ZB  ‫א‬E٣٠J ٤F   ‫א‬،‫א‬‫א‬Z٤, ZC‫א‬

‫א‬KZA‫א‬‫א‬‫א‬‫א‬‫א‬  W،Zeq‫א‬‫א‬‫א‬‫א‬‫א‬

⎛ (Z + Z B ) ⋅ (Z 4 + Z C ) ⎞ ⎟⎟ + Z A Z eq = ⎜⎜ 3 ⎝ (Z 3 + Z B ) + (Z 4 + Z C ) ⎠

(Z 3 + Z B ) = ((10 Ω ∠53.13°) + (5.88 Ω ∠ − 49.73°)) = (6 + j 8) + (3.8 − j 4.5)

(Z 3 + Z B ) = 9.8 + j 3.5 = 10.41 Ω ∠19.65° (Z 4 + Z C ) = ((10 Ω ∠ − 36.87°) + (5.88 Ω ∠40.27°)) = (8 − j 6) + (4.5 + j 3.8)

(Z 4 + Z C ) = 12.5 − j 2.2 = 12.7 Ω ∠ − 10° (Z 3 + Z B ) + (Z 4 + Z C ) = (9.8 + j 3.5) + (12.5 − j 2.2) (Z 3 + Z B ) + (Z 4 + Z C ) = 22.3 + j 1.3 = 22.34 Ω ∠3.34° ⎛ (10.41 Ω ∠19.65°) ⋅ (12.7 Ω ∠ − 10°) ⎞ ⎟⎟ + (2.94 Ω ∠ − 12.86°) Z eq = ⎜⎜ (22.34 Ω ∠3.34 °) ⎝ ⎠ - ١١١ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Z eq = (5.92 Ω ∠6.31°) + (2.94 Ω ∠ − 12.86°) Z eq = (5.88 + j 0.65) Ω + (2.87 − j 0.65) Ω = (8.75 + j 0 ) Ω = 8.75 Ω ∠0°

‫א‬ ‫א‬‫א‬،Zeq‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬K iT =

220 V ∠0° = 25.1 A ∠0° 8.75 Ω ∠0°

 E١٣J ٤F  K(Y-∆)(∆-Y)‫א‬ i 5 ‫א‬‫א‬،E١٢J ٤F‫א‬‫א‬

 ‫א‬

b, c‫א‬‫א‬ i 5 ‫א‬،E٢٨J ٤F  K i 5 ‫א‬‫א‬ Z 5 ‫א‬Vbc‫א‬‫א‬

b, c‫א‬Vbc‫א‬E٣٠J ٤F

 WE٣١J ٤Fobc‫א‬‫א‬

Vob + Vbc + Vco = 0

‫א‬o, b‫א‬‫א‬‫א‬obc‫א‬‫א‬‫א‬

‫א‬Vbc  ‫א‬‫א‬،‫א‬Vco‫א‬،Vob‫א‬،o, c‫א‬  W‫א‬،Vob‫א‬‫א‬‫א‬ Vbc − Vco + Vob = 0

 W‫א‬‫א‬ Vbc = Vco − Vob Vbc = i c ⋅ Z C − i b ⋅ Z B 

- ١١٢ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



iT

o

Z B = 5.88 Ω ∠ − 49.73° b

Z C = 5.88 Ω ∠40.27° ic

ib

c

Z 4 = 10 Ω ∠ − 36.87°

Z 3 = 10 Ω ∠53.13°

 

d



 E٣١J ٤F



 W‫א‬، i b , i c ‫א‬‫א‬

(Z C + Z 4 ) ⎞ ⎛ ⎟⎟ i b = i T ⋅ ⎜⎜ ⎝ (Z C + Z 4 ) + (Z B + Z 3 ) ⎠ ⎛ (12.7 Ω ∠ − 10°) ⎞ ⎟⎟ = 14.29 A ∠ − 13.34° i b = (25.14 A ∠0°) ⋅ ⎜⎜ ⎝ (22.34 Ω ∠3.34°) ⎠

 W

(Z B + Z 3 ) ⎞ ⎛ ⎟⎟ i c = i T ⋅ ⎜⎜ ( ) ( ) Z Z Z Z + + + 4 B 3 ⎠ ⎝ C ⎛ (10.41 Ω ∠19.65°) ⎞ ⎟⎟ = 11.72 A ∠16.31° i c = (25.14 A ∠0°) ⋅ ⎜⎜ ⎝ (22.34 Ω ∠3.34°) ⎠

 W Vbc ‫א‬ Vbc = i c ⋅ Z C − i b ⋅ Z B Vbc = (11.72A∠16.31°) ⋅ (5.88Ω ∠40.27°) − (14.29A∠ − 13.34°) ⋅ (5.88Ω∠ − 49.73°) - ١١٣ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Vbc = (68.91 V ∠56.58°) − (84.03 V ∠ − 63.1°) Vbc = (38 + j 57.5) − V (38 − j 75) V − = 0 + j 132.5 V = 132.5 V ∠90° i5 =

Vbc 132.5 V ∠90° = = 6.63 A ∠90° Z5 20 Ω ∠0°

 Thevinen's Theorem

٣J ٣J ٢J ٤

K ‫א‬‫א‬‫א‬‫א‬

F‫א‬‫א‬‫א‬ Z X  ‫א‬



‫א‬‫א‬ i X ‫א‬‫א‬‫א‬‫א‬‫א‬،EE٣٢J ٤F  Z eq ‫א‬‫א‬  V1 ‫א‬‫א‬‫א‬‫א‬‫א‬  KE‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬ 

Z1

E

Z eq

iX Z2

V1

ZX

V1

i X

ZX 

 

 E٣٢J ٤F ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬‫א‬K‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،EJ ٣٣J ٤F .EJ ٣٣J ٤F   ‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬ • .EJ ٣٣J ٤F VTh ‫א‬‫א‬(a, b)‫א‬‫א‬

F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • ،EJ ٣٣J ٤FE‫א‬‫א‬ - ١١٤ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬(a, b)‫א‬‫א‬‫א‬‫א‬‫א‬ • K Z Th

‫א‬‫א‬،EJ ٣٣J ٤F‫א‬‫א‬‫א‬ i X ‫א‬‫א‬ •

 Z Th ‫א‬ VTh ‫א‬‫א‬‫א‬ Z X ‫א‬‫א‬ W

iX =

VTh (Z Th + Z X )

(٤٦-٤)

a

Z1

Z1

Z2

E

a

ZX

VTh

b

 EF Z Th VTh

 

ZX

Z2

E



iX

 

b

iX











a



 Z1

EF a

ZX

Z2

b

b

 EF



EF

≡









Z Th 

 

 E٣٣J ٤F  E١٤J ٤F

r  K‫א‬ i5 ‫א‬‫א‬E١٢J ٤F

 ‫א‬

r  W i5 ‫א‬،E٢٨J ٤F

- ١١٥ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



r

‫א‬‫א‬،‫א‬‫א‬‫א‬ i5 ‫א‬W  • KE٣٤J ٤F



 

a

 i T1

Z 2 = 10 Ω ∠36.87°

Z1 = 10 Ω ∠ − 53.13° i11

b

E=٢٢٠ V f= ٥٠ Hz

Z 3 = 10 Ω ∠53.13°





i 21 VTh

c

  

Z 4 = 10 Ω ∠ − 36.87°

d



 E٣٤J ٤F ،E٣٤J ٤F VTh ‫א‬W  • W

Vab + VTh + Vca = 0 Vab = i11 ⋅ Z1 Vca = − i 21 ⋅ Z 2

VTh + i11 ⋅ Z1 − i 21 ⋅ Z 2 = 0 VTh = i 21 ⋅ Z 2 − i11 ⋅ Z1

- ١١٦ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 W i11 , i 21 ‫א‬‫א‬ i11 =

220 V ∠0° E = (Z1 + Z 3 ) (10 Ω ∠ − 53.13°) + (10 Ω ∠53.13°)

i11 =

220 V ∠0° 220 V ∠0° 220 V ∠0° = = = 18.33 A ∠0° (6 − j 8) Ω + (6 + j 8) Ω (12 + j 0) Ω 12 Ω ∠0°

i 21 =

220 V ∠0° E = (Z 2 + Z 4 ) (10 Ω ∠36.87°) + (10 Ω ∠ − 36.87°)

i 21 =

220 V ∠0° 220 V ∠0° 220 V ∠0° = = = 13.75 A ∠0° (8 + j 6) Ω + (8 − j 6) Ω (16 + j 0) Ω 16 Ω ∠0°

VTh = (13.75 A ∠0°) ⋅ (10 Ω ∠36.87°) − (18.33 A ∠0°) ⋅ (10 Ω ∠ − 53.13°)

VTh = (137.5 V ∠36.87°) − (183.3 V ∠ − 53.13°) VTh = (110 + j 82.5) V − (110 − j 146.64 ) V VTh = (0 + j 229.14 ) V = 229.14 V ∠90°

‫א‬‫א‬‫א‬‫א‬ Z Th ‫א‬‫א‬ • ،b, c‫א‬‫א‬‫א‬‫א‬E‫א‬‫א‬F‫א‬ E٣٧J ٤FE٣٦J ٤FKE٣٥J ٤F

 Z1 , Z 3 ‫א‬a, d‫א‬‫א‬‫א‬‫א‬‫א‬  Z 2 , Z 4 ‫א‬،‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬

- ١١٧ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

a

 ١٤١

‫א‬

٢-



  Z 2 = 10 Ω ∠36.87°

Z1 = 10 Ω ∠ − 53.13°

i 21

i11

b

 

c

 

Z 4 = 10 Ω ∠ − 36.87° 

Z 3 = 10 Ω ∠53.13°

 

d a

 E٣٥J ٤F

 Z 2 = 10 Ω ∠36.87°

Z1 = 10 Ω ∠ − 53.13°

i 21

i11 b

c

Z 4 = 10 Ω ∠ − 36.87°

Z 3 = 10 Ω ∠53.13°

      

d

  E٣٦J ٤F

- ١١٨ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

a

 ١٤١

‫א‬

٢-



 Z 2 = 10 Ω ∠36.87°

Z1 = 10 Ω ∠ − 53.13°

i 21

i11 b

c

Z 4 = 10 Ω ∠ − 36.87°

Z 3 = 10 Ω ∠53.13° d

 E٣٧J ٤F

      

 W Z Th ‫א‬‫א‬ ⎛ Z ⋅Z ⎞ ⎛ Z ⋅ Z Z Th = ⎜⎜ 1 3 ⎟⎟ + ⎜⎜ 2 4 ⎝ Z1 + Z 3 ⎠ ⎝ Z 2 + Z 4

⎞ ⎟⎟ = ⎠

⎛ (10 Ω ∠ − 53.13°) ⋅ (10 ∠53.13°) ⎞ ⎛ (10 Ω ∠36.87°) ⋅ (10 ∠ − 36.87°) ⎞ ⎟⎟ + ⎜⎜ ⎟⎟ Z Th = ⎜⎜ ⎝ (10 Ω ∠ − 53.13°) + (10 ∠53.13°) ⎠ ⎝ (10 Ω ∠36.87°) + (10 ∠ − 36.87°) ⎠ ⎞ ⎛ ⎞ ⎛ 100 Ω 2 ∠0° 100 Ω 2 ∠0° ⎟+⎜ ⎟ Z Th = ⎜ ⎜ (6 − j 8) Ω + (6 + j 8) Ω ⎟ ⎜ (8 + j 6 ) Ω + (8 − j 6 ) Ω ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ 100 Ω 2 ∠0° ⎞ ⎛ 100 Ω 2 ∠0° ⎞ ⎛ 100 Ω 2 ∠0° ⎞ ⎛ 100 Ω 2 ∠0° ⎞ ⎟+⎜ ⎟=⎜ ⎟+⎜ ⎟ Z Th = ⎜ ⎜ (12 − j 0 ) Ω ⎟ ⎜ (16 + j 0 ) Ω ⎟ ⎜ (12 Ω ∠0°) ⎟ ⎜ (16 Ω ∠0°) ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Z Th = (8.33 Ω ∠0°) Ω + (6.25 Ω ∠0°) Ω Z Th = (8.33 + j 0 ) Ω + (6.25 + j 0 ) Ω = 14.583 Ω ∠0°

r

W i5 ‫א‬‫א‬ • i5 =

VTh 229.14 V ∠90° 229.14 V ∠90° = = Z Th + Z 5 (14.583 Ω ∠0°) + (20 Ω ∠0°) (14.583 + j 0 ) Ω + (20 + j 0 ) Ω - ١١٩ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

i5 =

 ١٤١

‫א‬

٢-



229.14 V ∠90° 229.14 V ∠90° = = 6.63 A ∠90° (34.583 + j 0) Ω 34.583 Ω ∠0°

r

‫א‬‫א‬‫א‬ i5 ‫א‬

 K‫א‬‫א‬(Y-∆)(∆-Y)

 Closed Loop (Mesh) method EF‫א‬‫א‬

٤J ٣J ٢J ٤

،‫א‬‫א‬‫א‬



K  ‫א‬‫א‬

‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬

a

d

Z1



E, f

i1

Z2

Z4

c

 E٣٨J ٤F

i2

Z3 



e

 

‫א‬‫א‬Fabc‫א‬‫א‬‫א‬‫א‬ i1 ‫א‬‫א‬‫א‬‫א‬E٣٨J ٤F

KE٢‫א‬‫א‬Fbdec‫א‬‫א‬‫א‬‫א‬ i 2 ‫א‬‫א‬E١  Z 3 ‫א‬‫א‬‫א‬ i1 ‫א‬ Z1 , Z 4 ‫א‬‫א‬‫א‬

 W‫א‬ i1 − i 2 ‫א‬ Z 2 ‫א‬ i 2 ‫א‬ WE١‫א‬‫א‬Fabc‫א‬ • E = i1 ⋅ (Z1 + Z 4 ) + (i1 − i 2 ) ⋅ Z 2

(٤٧-٤)

E = i1 ⋅ (Z1 + Z 4 + Z 2 ) − i 2 ⋅ Z 2

(٤٨-٤) WE٢‫א‬‫א‬Fbdec‫א‬ •

0 = i 2 ⋅ Z 3 − (i1 − i 2 ) ⋅ Z 2

(٤٩-٤) - ١٢٠ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

0 = − i 1 ⋅ Z 2 + i 2 ⋅ (Z 2 + Z 3

 ١٤١

‫א‬

٢-



)

(٥٠-٤)

W‫א‬‫א‬‫א‬ • ⎡(Z1 + Z 4 + Z 2 ) ⎢ − Z2 ⎣

− Z 2 ⎤ ⎡ i1 ⎤ ⎡ E ⎤ = (Z 2 + Z 3 )⎥⎦ ⎢⎣i 2 ⎥⎦ ⎢⎣ 0 ⎥⎦

(٥١-٤)  W‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K١ ‫א‬‫א‬ [i ]‫א‬ [Z] [V ][Z]= [E ] ‫א‬

K‫א‬ [V ] 

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ [i ] ‫א‬‫א‬ K٢ K‫א‬‫א‬‫א‬‫א‬k‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬ K٣ K ‫א‬  ‫א‬(k × k)

‫א‬‫א‬ Z kk KKKK، Z 22  Z11 ‫א‬‫א‬ E٣٨J ٤F‫א‬،kKKKKKKK٢١‫א‬

W Z11 = (Z1 + Z 4 + Z 2 )

(٥٢-٤)

Z 22 = (Z 2 + Z 3 )

(٥٣-٤)  K‫א‬k = ٢

 Z mn ‫א‬‫א‬‫א‬‫א‬‫א‬  Z nm = Z mn ،n‫א‬‫א‬m‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬m‫א‬‫א‬n‫א‬‫א‬‫א‬‫א‬  Kn‫א‬‫א‬m‫א‬‫א‬

 Vx = V1 , V2 ,....Vk ‫א‬‫א‬ [V ] ‫א‬ K٤  V1 W،x‫א‬‫א‬‫א‬ Vx 

٢‫א‬‫א‬‫א‬‫א‬ V2 ،١‫א‬‫א‬‫א‬‫א‬ - ١٢١ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 Vx ‫א‬‫א‬Kk‫א‬‫א‬‫א‬‫א‬ Vk K i x ‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬ K٥

‫א‬، [i ] = [Z]

−1

[V ]W‫א‬‫א‬‫א‬،‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬K [Z]  [Z]‫א‬ −1

[Z]

W‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬

= Z  [Z] ‫א‬‫א‬ •

 Z x  [Z] x‫א‬ • K [V ] ‫א‬x‫א‬‫א‬‫א‬‫א‬

Wx‫א‬ •

ix =

Z

x

Z

(٥٤-٤)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K‫א‬‫א‬‫א‬‫א‬‫א‬

 E١٥J ٤F  KEF‫א‬‫א‬ab‫א‬‫א‬‫א‬E٣٩J ٤F‫א‬‫א‬‫א‬ 

iT

E = 110 V f = 60 Hz

iΙ

a

Z1 = 10 Ω ∠30°





i ΙΙ

Z 2 = 20 Ω ∠ − 20° i2

i1 Z 3 = 20 Ω ∠0°



b

- ١٢٢ -

  

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E٣٩J ٤F  ‫א‬

، ‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬KE٣٩J ٤F i Ι & i ΙΙ ‫א‬  K‫א‬

 W‫א‬‫א‬

W(٢×٢)‫א‬

[Z]

⎡(Z + Z 3 ) =⎢ 1 ⎣ (− Z1 )

•

(− Z1 )

⎤ (Z1 + Z 2 )⎥⎦

 W‫א‬

(Z1 + Z3 ) = (10 Ω ∠30°) + (20 Ω ∠0°) = 28.66 + j 5 = 29.09 Ω ∠9.9°

(Z1 + Z 2 ) = (10 Ω ∠30°) + 20 Ω ∠ − 20° = 27.45 − j 1.84 = 27.51 Ω ∠ − 3.84° − 10 ∠30° ⎡29.09 ∠9.9° ⎤ ∴Z =⎢ 27.51 Ω ∠ − 3.84°⎥⎦ ⎣ − 10 ∠30

[i ]

[V ]

W‫א‬‫א‬

•

W‫א‬

•

⎡i ⎤ =⎢ Ι⎥ ⎣ i ΙΙ ⎦

⎡110 ∠0°⎤ =⎢ ⎥ ⎣ 0 ∠0° ⎦

 W،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ − 10 ∠30° ⎡29.09 ∠9.9° ⎤ ⎡ i Ι ⎤ ⎡110 ∠0°⎤ ⎢ − 10 ∠30 ⎥ ⎢ ⎥ = ⎢ 0 ∠0° ⎥ 27 . 51 3 . 84 Ω ∠ − ° ⎣ ⎦ ⎣ i ΙΙ ⎦ ⎣ ⎦ - ١٢٣ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ − 10 ∠30° ⎡ i Ι ⎤ ⎡29.09 ∠9.9° ⎤ ⎢ ⎥=⎢ 27.51 Ω ∠ − 3.84°⎥⎦ ⎣ i ΙΙ ⎦ ⎣ − 10 ∠30

−1

⎡110 ∠0°⎤ ⎢ 0 ∠0° ⎥ ⎣ ⎦

W‫א‬‫א‬ Z=

•

29.09 ∠9.9° − 10 ∠30° − 10 ∠30 27.51 Ω ∠ − 3.84°

= ((29.09 ∠9.9°) ⋅ (27.51 Ω ∠ − 3.84°)) − ((− 10 ∠30) ⋅ (− 10 ∠30)) = 800.27 ∠6.06° − 100 ∠60° = 745.8 − j 2.12 = 745.8 ∠0.2°

W‫א‬‫א‬‫א‬ Z1 =

110 ∠0°

− 10 ∠30°

0 ∠0 °

27.51 ∠ − 3.84°

•

= (110 ∠0°) ⋅ (27.51 ∠ − 3.84°) − 0 ∠0°

= 3026.1 ∠ − 3.84° 29.09 ∠9.9° 110 ∠0° = − (110 ∠0°) ⋅ (− 10 ∠30°) − 0∠0° = 1100 ∠30° − 10 ∠30° 0 ∠0°

Z2 =

W،‫א‬‫א‬‫א‬‫א‬ iΙ =

i ΙΙ =

Z

1

Z Z Z

2

=

•

3026.1 ∠ − 3.84° = 4.06 A ∠ − 4.04° 745.8 ∠0.2°

=

1100 ∠30° = 1.475 A ∠29.8° 745.8 ∠0.2°

W i1 ‫א‬ i1 = (i Ι − i ΙΙ ) = (4.06 A ∠ − 4.04°) − (1.475 A ∠29.8°) = 2.77 − j 1.02 = 2.95 A ∠ − 20.2°

- ١٢٤ -

•

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E١٦J ٤F

r  KEF‫א‬‫א‬‫א‬ i5 ‫א‬‫א‬E١٢J ٤F

 ‫א‬

،‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬KE٤٠J ٤F i Ι , i ΙΙ , i ΙΙΙ  ‫א‬‫א‬  K‫א‬‫א‬‫א‬

a

i2

i1 iT E=٢٢٠ V f= ٥٠ Hz

Z1 = 10 Ω ∠ − 53.13°

iΙ

i5

b



i ΙΙ

Z 2 = 10 Ω ∠36.87°

Z 5 = 20 Ω ∠0°

Z 3 = 10 Ω ∠53.13°

i ΙΙΙ

 

c



i4

i3



Z 4 = 10 Ω ∠ − 36.87°

  

d



 E٤٠J ٤F

 W  i T = i Ι W‫א‬

 i1 = i Ι − i ΙΙ W‫א‬

 i 2 = i ΙΙ W‫א‬

 i 3 = i Ι − i ΙΙΙ W‫א‬  i 4 = i ΙΙΙ W‫א‬

 i 5 = i ΙΙΙ − i ΙΙ W‫א‬

 K‫א‬‫א‬،‫א‬‫א‬ ،k = ٣‫א‬‫א‬(٣ × ٣)  ،‫א‬  W‫א‬

Z11 = Z1 + Z 3 = (12 Ω ∠0°) - ١٢٥ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Z 22 = Z1 + Z 2 + Z 5 = (34 Ω ∠ − 3.4°) Z 33 = Z 3 + Z 4 + Z 5 = (10 Ω ∠53.13°) + (10 Ω ∠ − 36.87°) + (20 Ω ∠0°) Z 33 = (34 + j 2 ) Ω = 34.06 Ω ∠3.8°

 W‫א‬

[Z]

[i ]

[V ]

•

12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13°⎤ ⎡ − 20 ∠0° ⎥ = ⎢− 10 ∠ − 53.13° 34.06 ∠ − 3.4° ⎢ ⎥ ⎢⎣ − 10 ∠53.13° − 20 ∠0° 34.06 ∠3.4° ⎥⎦

 W‫א‬‫א‬

•

 W‫א‬

•

⎡ iΙ ⎤ ⎢ ⎥ = ⎢ i ΙΙ ⎥  ⎢ i ΙΙΙ ⎥ ⎣ ⎦

⎡220 ∠0°⎤ = ⎢ 0 ∠0° ⎥ ⎢ ⎥ ⎢⎣ 0 ∠0° ⎥⎦

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 •

 W 12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13°⎤ ⎡ ⎢− 10 ∠ − 53.13° 34.06 ∠ − 3.4° − 20 ∠0° ⎥ ⎥ ⎢ ⎢⎣ − 10 ∠53.13° 34.06 ∠3.4° ⎥⎦ − 20 ∠0°

⎡ iΙ ⎤ ⎢ ⎥ ⎢ i ΙΙ ⎥ = ⎢ i ΙΙΙ ⎥ ⎣ ⎦

⎡220 ∠0°⎤ ⎢ 0 ∠0° ⎥  ⎥ ⎢ ⎢⎣ 0 ∠0° ⎥⎦

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

- ١٢٦ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



⎡ iΙ ⎤ ⎡ 12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13°⎤ ⎢ ⎥ ⎢ − 20 ∠0° ⎥ ⎢ i ΙΙ ⎥ = ⎢− 10 ∠ − 53.13° 34.06 ∠ − 3.4° ⎥ ⎢ i ΙΙΙ ⎥ ⎢⎣ − 10 ∠53.13° ⎥⎦ 20 0 34 . 06 3 . 4 − ∠ ° ∠ ° ⎣ ⎦

−1

⎡220 ∠0°⎤ ⎢ 0 ∠0° ⎥ ⎢ ⎥ ⎢⎣ 0 ∠0° ⎥⎦

 W،‫א‬‫א‬‫א‬‫א‬‫א‬ 220 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13° ⎛ ⎞ ⎜ ⎟ 0 0 34 . 06 3 . 4 20 0 ∠ ° ∠ − ° − ∠ ° ⎜ ⎟ ⎜ ⎟ 0 ∠0° 34.06 ∠3.4° − 20 ∠0° ⎜ ⎟ iΙ = ⎜ 12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13° ⎟ ⎜ ⎟ − 20 ∠0° ⎟ ⎜ − 10 ∠ − 53.13° 34.06 ∠ − 3.4° ⎜ − 10 ∠53.13° 34.06 ∠3.4° ⎟⎠ − 20 ∠0° ⎝ 12 ∠0° 220 ∠0° − 10 ∠53.13° ⎛ ⎞ ⎜ ⎟ − 10 ∠ − 53.13° 0 ∠0° − 20 ∠0° ⎜ ⎟ ⎜ ⎟ 0 ∠0° 34.06 ∠3.4° − 10 ∠53.13° ⎜ ⎟ i ΙΙ = ⎜ 12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13° ⎟ ⎜ ⎟ 10 53 . 13 34 . 06 3 . 4 20 0 − ∠ − ° ∠ − ° − ∠ ° ⎜ ⎟ ⎜ − 10 ∠53.13° ⎟ 20 0 34 . 06 3 . 4 − ∠ ° ∠ ° ⎝ ⎠

i ΙΙΙ

12 ∠0° − 10 ∠ − 53.13° 220 ∠0° ⎞ ⎛ ⎜ ⎟ 0 ∠0° ⎜ − 10 ∠ − 53.13° 34.06 ∠ − 3.4° ⎟ ⎜ ⎟ 0 ∠0° − 10 ∠53.13° − 20 ∠0° ⎜ ⎟ = ⎜ 12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13° ⎟ ⎜ ⎟ − 20 ∠0° ⎟ ⎜ − 10 ∠ − 53.13° 34.06 ∠ − 3.4° ⎜ − 10 ∠53.13° 34.06 ∠3.4° ⎟⎠ − 20 ∠0° ⎝

 W،‫א‬‫א‬ • W‫א‬‫ א‬Ki 12 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13° Z = − 10 ∠ − 53.13° 34.06 ∠ − 3.4° − 20 ∠0° 34.06 ∠3.4° − 10 ∠53.13° − 20 ∠0° - ١٢٧ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Z = (12 ∠0°) ((34.06 ∠ − 3.4°)(34.06 ∠3.4°) − (− 20 ∠0°) (− 20 ∠0°)) + (10 ∠ − 53.13°) ((− 10 ∠ − 53.13°) (34.06 ∠3.4°) − (10 ∠53.13°) (20 ∠0°)) + (− 10 ∠53.13°) ((10 ∠ − 53.13°) (20 ∠0°) + (10 ∠53.13°)(34.06 ∠ − 3.4°)) Z = (12 ∠0°) ((1160 ∠0°) − (400 ∠0°)) + (10 ∠ − 53.13°) ((− 340.6 ∠ − 49.73°) − (200 ∠53.13°)) − (10 ∠53.13°) ((200 ∠ − 53.13°) + (340.6 ∠49.73°))

= (12 ∠0°) (760 + j 0) + (10 ∠ − 53.13°) (− 340.16 + j 99.88) − (10 ∠53.13°) (340.16 + j 99.88) = (12∠0°) (760∠0 °) + (10∠ − 53.13°) (354.52∠163.63°) − (10∠53.13°) (354.5∠16.4°)

= (9120 ∠0°) + (3545.2 ∠110.5°) − (3545 ∠69.53°) = 6638.7 − j 0.5 = 6638.7 ∠0°

W‫א‬‫ א‬Kii 220 ∠0° − 10 ∠ − 53.13° − 10 ∠53.13° Z = 0 ∠0° 34.06 ∠ − 3.4° − 20 ∠0° 1 0 ∠0° 34.06 ∠3.4° − 20 ∠0°

 W‫א‬‫א‬‫א‬‫א‬‫א‬ Z = (220 ∠0°) ((34.06 ∠ − 3.4°) (34.06 ∠3.4°) − (− 20 ∠0°) (− 20 ∠0°)) 1

= (220 ∠0°) ((1160 ∠0°) − (400 ∠0°)) = (220 ∠0°) (760 + j 0) = (220 ∠0°) (760 ∠0°) = 167200 ∠0°

W‫א‬‫ א‬Kiii 12 ∠0° 220 ∠0° − 10 ∠53.13° Z = − 10 ∠ − 53.13° 0 ∠0° − 20 ∠0° 2 0 ∠0° 34.06 ∠3.4° − 10 ∠53.13°

 W‫א‬‫א‬‫א‬‫א‬‫א‬ - ١٢٨ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



Z = (− 220 ∠0°) ((− 10 ∠ − 53.13°) (34.06 ∠3.4°) − (− 10 ∠53.13°) (− 20 ∠0°)) 2

= (− 220 ∠0°) ((− 340.6 ∠ − 49.73°) − (200 ∠53.13°)) = (− 220 ∠0°) (− 340.16 + j 99.88) = (− 220 ∠0°) (340.16 − j 99.88)

= (− 220 ∠0°) (354.52 ∠163.64°) = − 77994.53 ∠163.64° = 77994.53 ∠ − 163.64°

W‫א‬‫ א‬Kiv 12 ∠0° − 10 ∠ − 53.13° 220 ∠0° Z = − 10 ∠ − 53.13° 34.06 ∠ − 3.4° 0 ∠0° 3 0 ∠0° − 10 ∠53.13° − 20 ∠0°

 W‫א‬‫א‬‫א‬‫א‬‫א‬ Z = (220 ∠0°) ((− 10 ∠ − 53.13°) (− 20 ∠0°) − (− 10 ∠53.13°) (34.06 ∠ − 3.4°)) 3

= (220 ∠0°) ((200 ∠ − 53.13°) + (340.6 ∠49.73°)) = (220 ∠0°) (340.16 + j 99.88) = (220 ∠0°) (354.5 ∠16.36°) = 77994.55 ∠16.36°

 W‫א‬‫א‬ iΙ =

i ΙΙ =

i ΙΙΙ =

Z1 Z Z2 Z

Z3 Z

=

167200 ∠0° = 25.16 A ∠0° 6638.7 ∠0°

=

77994.53 ∠ − 163.64° = 11.75 A ∠ − 163.64° 6638.7 ∠0°

=

77994.55 ∠16.4° = 11.75 A ∠16.4° 6638.7 ∠0°

•

 W‫א‬‫א‬‫א‬ • i T = i Ι = 25.16 A ∠0

i1 = i Ι − i ΙΙ = (25.16 ∠0°) − (11.75 ∠ − 163.64°) = 13.89 + j 3.31 = 14.3 A ∠13.4° i 2 = i ΙΙ = 11.75 A ∠ − 163.64° i 3 = i Ι − i ΙΙΙ = 25.16 A ∠0° − 11.75 A ∠16.4° = 13.89 − j 3.32 = 14.3 A ∠ − 13.4° - ١٢٩ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



i 4 = i ΙΙΙ = 11.75 A ∠16.4° i 5 = i ΙΙΙ − i ΙΙ = 11.75 A ∠16.4° − (11.75 A ∠ − 163.64°)

i 5 = 11.75 A ∠16.4° + 11.75 A ∠163.64° = 0 + j 6.63 = 6.63 A ∠90°

E١٢J ٤F‫א‬‫א‬‫א‬‫א‬ i T ‫א‬‫א‬  KE١٤J ٤FE١٣J ٤F‫א‬‫א‬ i 5 ‫א‬‫א‬ Superposition Theorem ‫א‬

٥J ٣J ٢J ٤

‫א‬‫א‬‫א‬، ١–

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬

‫א‬‫א‬E٤١J ٤F‫א‬‫א‬‫א‬،   E1 , E 2 ‫א‬‫א‬ i1 , i 2 , i 3 ‫א‬‫א‬، E1 , E 2

 KE٤٣J ٤FE٤٢J ٤F

- ١٣٠ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



  i1

a

Z1



b



E١, f

Z2

i3



i2

d E ٢, f

Z3

  

c

 E1 , E 2 ‫א‬‫א‬E٤١J ٤F a

i11

Z1

  

E١, f

b i 31

Z2

i 21

d

Z3

  

c

  E1 ‫א‬E٤٢J ٤F a

i12

 Z1

 b

i32

 Z3 

Z2

i 22

d E ٢, f

 c   E 2 ‫א‬E٤٣J ٤F  W i1 = i11 − i12

(٥٥-٤)

i 2 = i 22 − i 21

(٥٦-٤)

i 3 = i 31 + i 32

(٥٧-٤) - ١٣١ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 E١٧J ٤F  KE٤٤J ٤F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ a

i1



Z1 = 10 Ω ∠ − 30° b



Z 2 = 10 Ω ∠30°

i2

d



Z 3 = 15 ∠60°



E١ = ٢٢٠ V ∠٠°

 i3 

E٢= ٢٢٠ V∠٣٠ c

 E١٧J ٤F‫א‬‫א‬‫א‬E٤٣J ٤F

 ‫א‬

‫א‬‫א‬‫א‬E٤٣J ٤F‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬ ،E٤٥J ٤FE٤٤J ٤F‫א‬  W‫א‬‫א‬،‫א‬ i11 =

E1 ⋅ (Z 2 + Z 3 ) E1 = (Z1 Z 2 + Z1 Z 3 + Z 2 Z 3 ) Z Z Z1 + 2 3 Z 2 + Z3 a

i11



Z1 = 10 Ω ∠ − 30° b



Z 2 = 10 Ω ∠30°

Z 3 = 15 ∠60°

E١ = ٢٢٠ V ∠٠°

i 31 c

 E١E٤٤J ٤F 

- ١٣٢ -

i 21

d

   

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 i12

a



Z1 = 10 Ω ∠ − 30° b  Z 2 = 10 Ω ∠30°

Z 3 = 15 ∠60°

i 22

d



E٢= ٢٢٠ V∠٣٠°



i 32



c

 E٢E٤٥J ٤F 

i11 =

220 V ∠0° ⋅ (10 Ω ∠30° + 15 Ω∠60°) (10 Ω ∠ − 30° ⋅ 10 Ω ∠30° + 10 Ω ∠ − 30° ⋅ 15 Ω ∠60° + 10 Ω ∠30° ⋅ 15 Ω ∠60°)

i11 =

(100 Ω

i11 =

220 V ∠0° ⋅ (23.15 + j 18) Ω 2

∠0° + 150 Ω 2 ∠30° + 150 Ω 2 ∠90°

220 V ∠0° ⋅ 29.32 Ω ∠37.85° 321.75 Ω 2 ∠44.4°

)

=

220 V ∠0° ⋅ 29.32 Ω ∠37.85° 230 + j 225

= 20.05 A ∠ − 6.55°

 W‫א‬‫א‬ i 21 = i11

Z2 10 Ω ∠30° = (20.05 A ∠ − 6.55°) ⋅ = 8.84 A ∠ − 14.4° Z 2 + Z3 29.32 Ω ∠37.85°

i31 = i11

15 Ω ∠60° Z3 = 10.26 A ∠ − 15.6° = (20.05 A ∠ − 6.55°) ⋅ 29.32 Ω ∠37.85° Z2 + Z3

i 22 =

i2 2 = i2 2 =

E 2 ⋅ (Z1 + Z 3 ) E2 = (Z1 Z 2 + Z1 Z 3 + Z 2 Z 3 ) Z Z Z2 + 1 3 Z1 + Z 3

220 V ∠30° ⋅ (10 Ω ∠ − 30° + 15 Ω∠60°) 321.75 Ω 2 ∠44.4°

(220 V ∠30°) ⋅ (18.03 Ω ∠26.31°) 321.75 Ω 2 ∠44.4°

=

220 V ∠30° ⋅ (16.16 + j 8) 321.75 Ω 2 ∠44.4°

= 12.33 A ∠12°

- ١٣٣ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



i12 = i 22

Z3 (12.33 A ∠12°) ⋅ (15 Ω ∠60°) = ⋅ = 10.25 A ∠45.6° Z1 + Z 3 18.03 Ω ∠26.31°

i 32 = i 22

Z1 (12.33 A ∠12°) ⋅ (10 Ω ∠ − 30°) = ⋅ = 6.84 A ∠ − 44.4° Z1 + Z 3 18.03 Ω ∠26.31°

 W‫א‬‫א‬ i1 = i11 − i12 = (20.05 A ∠ − 6.55°) − (10.25 A ∠45.6°) = 12.75 − j 9.6 i1 = 15.97 A ∠ − 37° i 2 = i 22 − i 21 = (12.33 A ∠12°) − (8.84 A ∠ − 14.4°) = 2.18 − j 0.217

i 2 = 2.2 A ∠ − 5.7° i 3 = i 31 + i 32 = (10.26 A ∠ − 15.6°) + (6.84 A ∠ − 44.4°) = 14.77 − j 2.03 i 3 = 14.9 A ∠ − 7.8°

          

- ١٣٤ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 ‫א‬‫א‬‫א‬  K Z ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬ 

K١

i



١٠٠ V ٥٠ Hz



Z =8− j6

 

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬

K٢

K٦٠ Hz‫א‬، E = 110 V ∠0° R =6Ω





L = 25.465 m H

  

E

 K‫א‬‫א‬ Z ‫א‬‫א‬  A ∠30° i = 20 

١٠٠ V

      - ١٣٥ -

Z

K٣

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



‫א‬‫א‬، i ‫א‬ ‫א‬‫א‬‫א‬،‫א‬

K٤

K٥٠ Hz‫א‬، E = 110 V ∠0°



C = 1.326 mF

R = 4.33 Ω

 

i

 

E



KIT‫א‬‫א‬‫א‬،‫א‬ IT

K٥ 

A



Z١ I١= ٥ A∠-٣٠°



Z٢



I٢=١٠ A∠٢٠°

 

B



 W‫א‬‫א‬،Z١, Z٢‫א‬‫א‬،‫א‬

E=١١٠ V, f=٦٠HZ

IT

K٦  

A



E=١١٠ V f = ٦٠ HZ

Z١ I١= ١٠ A ∠٣٠°

Z٢ I٢=١٠ A ∠٣ °

B - ١٣٦ -

   

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



  KI١‫א‬‫א‬‫א‬،‫א‬‫א‬ IT = ٤٠ A ∠-٦٠°



K٧



A

 

Z١



Z٢



I٢ = ١٠ A ∠-٣٠°

I١

 

B K‫א‬‫א‬‫א‬‫א‬ IT

K٨

A





Z٢ = ٥ Ω ∠-٢٠°

Z١ = ٧ Ω ∠-٣٠°

 

I٢

I١ B

K‫א‬‫א‬‫א‬،E٦F‫א‬

K٩



K‫א‬‫א‬،I١, I٢‫א‬‫א‬‫א‬‫א‬،E١٣J ٤F‫א‬‫א‬ K١٠ IT = ١٥ A ∠-٤٥°



A

Z١ = ٢٠ Ω ∠I١

I٢ B

- ١٣٧ -

Z٢ = ١٠ Ω ∠٣٠°

   

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



  K‫א‬‫א‬‫א‬،A, B‫א‬‫א‬‫א‬،‫א‬‫א‬ K١١ Z1 = 7 Ω ∠30°

A





i = 8 A ∠ − 30°

 Z 2 = 8 Ω ∠ − 10°

  

Z 3 = 15 Ω ∠15°

B

 

‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬،‫א‬‫א‬ K١٢ 

A ١١٠ V ٦٠ Hz

 K‫א‬‫א‬ Z1 = 15 Ω ∠30°

 

Z 2 = 17 Ω ∠10°

V

 

Z 3 = 8 Ω ∠15°

B

  

K‫א‬‫א‬‫א‬، i1 , i 2 & i 3 ‫א‬‫א‬‫א‬،‫א‬ K١٣ A

i1

Z1 = 10 Ω ∠10°

C

E

Z 2 = 5 Ω ∠10°

f = 60 Hz B

Z 4 = 10 Ω ∠ − 10°

i3

i2

E = 110 V

D



Z 3 = 10 Ω ∠ − 10°

F

     

- ١٣٨ -

 ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬

٢-



 K‫א‬ i 2 ‫א‬‫א‬،E١٣F‫א‬‫א‬ K١٤

‫א‬‫א‬‫א‬‫א‬‫א‬،E٢٨J ٤F‫א‬‫א‬‫א‬ K١٥  K(Y-∆)(∆-Y)

a

i2

i1

Z 2 = 10 Ω ∠ − 36.87°

Z1 = 20 Ω ∠ − 53.13°

iT

i5

b

E=٢٢٠ V f= ٥٠ Hz





i4

i3 Z 3 = 10 Ω ∠53.13°



Z 4 = 20 Ω ∠36.87°

 

d



K‫א‬، i 5 ‫א‬‫א‬،E١٥F‫א‬‫א‬ K١٦ K‫א‬‫א‬‫א‬، i 5 ‫א‬‫א‬،E١٥F‫א‬‫א‬ K١٧ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬ K١٨ a

i1

Z1 = 20 Ω ∠ − 60° b

Z 2 = 20 Ω ∠60°

i2

 d

 

Z 3 = 10 ∠0° E١ = ٢٢٠ V ∠٠°

 

c

Z 5 = 20 Ω ∠0°



i3 c

 E٢= ٢٢٠ V∠٣٠

K(∆-Y)‫א‬i٣‫א‬‫א‬،E١٨F‫א‬‫א‬ K١٩ - ١٣٩ -

 ‫א‬‫א‬‫א‬ ã¹]
٢ J  ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬

٥

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬ 

 ٢-

 ‫א‬‫א‬‫א‬‫א‬



 W‫א‬‫א‬،‫א‬‫א‬‫א‬ K‫א‬‫א‬‫• א‬

K‫א‬‫א‬‫א‬‫א‬‫א‬ •

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬ K‫א‬‫א‬

K‫א‬‫א‬ • K‫א‬‫א‬ •

K‫א‬‫א‬‫א‬‫א‬‫א‬‫• א‬ K‫א‬‫א‬ •

 K‫א‬‫א‬ • 

- ١٤٠ -

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

 ٢-

‫א‬ 

 Introduction١J ٥ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬،‫א‬‫א‬



‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬K  

‫א‬K  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Magnetic Induction‫א‬‫ א‬٢J ٥

 Self Inductance of a Coil‫א‬‫א‬‫ א‬١J ٢J ٥ (L)‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬



 K‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬

 K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬١J ١J ٢J ٥

Factors that Affect the Self Inductance

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ K١

‫א‬،‫א‬‫א‬N‫א‬F‫א‬ K٢ KEN‫א‬

KE‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K٣ K‫א‬ K٤ 

-١٤١-

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١

‫א‬ 

 ٢-

 Calculation of Self Inductance‫א‬‫א‬‫א‬

٢J ١J ٢J ٥

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬E١٠J ٣F

L‫א‬‫א‬‫א‬i‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬، E = −N

di dφ = −L dt dt



di ‫א‬ dt

(١-٥)  WL‫א‬‫א‬‫א‬

L=

−E (di dt )

(٢-٥)

dφ di

(٣-٥)

L=N

 W‫א‬‫א‬‫א‬  (L)‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬E  ‫א‬‫א‬F‫א‬‫א‬

 K

 KEL.F،(Henry)‫א‬‫א‬‫א‬  ‫א‬ ‫א‬   ‫א‬     ‫א‬  ‫א‬‫א‬ ‫א‬   ‫א‬ 

 ‫א‬  ‫א‬ ‫א‬   ‫א‬‫א‬     ‫א‬  ‫א‬

 K‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،E٢J ٥F‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬‫א‬‫א‬ -١٤٢-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬ 

 ٢-

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ٣J ١J ٢J ٥ Effect of Self Inductance on the Relation between induced emf & Current  

ϕ١

 

‫א‬‫א‬ 

‫א‬



 

 ‫א‬‫א‬‫א‬‫א‬‫א‬E١J ٥F



‫א‬‫א‬‫א‬  F‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬ ‫א‬‫א‬‫א‬EE١J ٥F

 K‫א‬‫א‬

 Mutual Inductance‫א‬‫ א‬٢J ٢J ٥ ،‫א‬E‫א‬‫א‬‫א‬F‫א‬

‫א‬‫א‬،E‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬



‫א‬‫א‬‫א‬‫א‬،  ‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬  KE٢J ٥F  K‫א‬‫א‬

 W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬  

-١٤٣-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

 ٢-

‫א‬ 



ϕ١

  

I1

  

I2

 

 ϕ٢ 

 ‫א‬‫א‬E٢J ٥F   Mutual Inductance Coefficient‫א‬‫א‬ ١J ٢J ٢J ٥  ‫א‬‫א‬‫א‬I ♦ ∆I 1 ∆ϕ α ∆t ∆t

W،‫א‬‫א‬

(٤-٥)

ξ٢‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ ♦ W،‫א‬‫א‬‫א‬‫א‬‫א‬

ξ2 α

∆ϕ ∆t

ξ2 α

∆I 1 ∆t

ξ2 = − M

(٥-٥) WE٥J ٥F‫א‬E٤J ٥F‫א‬ ♦

∆I 1 ∆t

(٦-٥) W‫א‬E٦J ٥F‫א‬ ♦ (٧-٥) -١٤٤-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬ 

 ٢-

K‫א‬‫א‬‫א‬E٧J ٥F‫א‬M‫א‬ ♦ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ♦ K‫א‬‫א‬‫א‬‫א‬

 ‫א‬‫א‬

 W‫א‬‫א‬M‫א‬‫א‬E٧J ٥F‫א‬ −M=

ξ2 ∆t = ξ2 ∆I1 ⎛ ∆I1 ⎞ ⎟ ⎜ ⎝ ∆t ⎠

(٨-٥) ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬  K‫א‬١‫א‬‫א‬‫א‬

 ‫א‬‫א‬

 KE÷×F‫א‬‫א‬E٨J ٥F‫א‬

 KEL.F،(Henry)‫א‬‫א‬  E١J ٥F

٥mA/sec‫א‬‫א‬،‫א‬‫א‬‫א‬  ٠{٥ V‫א‬‫א‬

 ‫א‬

 W‫א‬،E٨J ٥F‫א‬

−M=

ξ2 0 .5 V = = 100 V ⋅ sec/ A ⎛ ∆I1 ⎞ 5 × 10 −3 A / sec ⎟ ⎜ ⎝ ∆t ⎠

-١٤٥-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬ 

 ٢-

 Transformer‫א‬‫ א‬٣J ٥  Function of Transformer‫א‬‫א‬ ١J ٣J ٥  K‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬  Transformer Applications‫א‬‫א‬ ٢J ٣J ٥ K‫א‬‫א‬‫א‬‫א‬ ♦

K‫א‬‫א‬ ♦

E  ‫א‬‫א‬F‫א‬‫א‬‫א‬ ♦ K‫א‬‫א‬‫א‬

 Construction of Transformer‫א‬ ٣J ٣J ٥ ‫א‬‫א‬E٥J ٢F‫א‬‫א‬‫א‬

‫א‬‫א‬ ‫א‬‫א‬  ‫א‬‫א‬‫א‬،

‫א‬‫א‬  ،‫א‬‫א‬ E1 ‫א‬  KE٣J ٥F E 2 ‫א‬‫א‬‫א‬ 

ϕ

  ‫اﻟﻤﻠﻒ‬ ‫ اﻟﺜﺎﻧﻮي‬E 2

‫اﻟﻤﻠﻒ‬ E1 ‫اﻻﺑﺘﺪاﺋﻲ‬

    

 ‫א‬‫א‬‫א‬E٣J ٥F



‫א‬‫א‬‫ א‬ ‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬   -١٤٦-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬

 ٢-



 Transformer Operation‫א‬ ٤J ٣J ٥  W‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K١

K‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬

K٢

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K٣

K‫א‬‫א‬‫א‬‫א‬‫א‬

K‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K٤ K٥

K‫א‬‫א‬ emf Relationship‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

٥J ٣J ٥

 W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ .N٢ =‫א‬‫א‬N١ = ‫א‬‫א‬‫א‬ K١

‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K٢ W E 2 ‫א‬ E2 = N2

∆ϕ ∆t

(٩-٥)

،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K٣ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ E1 = N 1

∆ϕ ∆t

(١٠-٥)

W‫א‬E١٠J ٥F‫א‬E٩J ٥F‫א‬ K٤

-١٤٧-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

⎛ ⎜N E2 ⎝ 2 = E1 ⎛ ⎜ N1 ⎝

 ٢-

∆ϕ ⎞ ⎟ ∆t ⎠ ∆ϕ ⎞ ⎟ ∆t ⎠

‫א‬ 

(١١-٥)

E2 N = 2 E1 N1

(١٢-٥)

‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬E١٢J ٥F‫א‬ K٥ W E 2 = E1 ⋅

N2 N1

(١٣-٥)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K٦ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 Currents Relationship‫א‬‫א‬‫ א‬٦J ٣J ٥ ‫א‬‫א‬‫א‬،‫א‬‫א‬

 W‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬

P1 = P2

(١٤-٥)  W‫א‬

E1 ⋅ i1 = E 2 ⋅ i 2

(١٥-٥)  W‫א‬

i1 E 2 = i 2 E1

(١٦-٥)  W‫א‬،E١٢J ٥F‫א‬

i1 E 2 N = = 2 N1 i 2 E1

(١٧-٥)  W‫א‬‫א‬ -١٤٨-

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

i 2 = i1 ⋅

 ٢-

N1 N2

‫א‬ 

(١٨-٥)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬  E٢J ٥F ‫א‬‫א‬،

1 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 20

‫א‬٥٠ V‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K١ A‫א‬  ‫א‬

 W‫א‬‫א‬‫א‬‫א‬،E١٣J ٥F‫א‬ E 2 = E1 ⋅

N2 = 50 V ⋅ (20 ) = 1000 V N1

 W‫א‬‫א‬‫א‬،E١٨J ٥F‫א‬ i 2 = i1 ⋅

N1 ⎛ 1 ⎞ = 1 A ⋅ ⎜ ⎟ = 0.05 A = 50 mA N2 ⎝ 20 ⎠

 Transformer Efficiency‫א‬‫א‬ ٧J ٣J ٥ ،‫א‬‫א‬‫א‬‫א‬‫א‬

 W‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬

 Wη‫א‬ η=

V2 ⋅ i 2 V1 ⋅ i1

(١٩-٥) -١٤٩-



 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬



 ٢-

 ‫א‬‫א‬

L1

R1

‫א‬

L2

R2

i1

 

i2

V1

E1

 

E2

V2



    ‫א‬‫א‬‫א‬E٤J ٥F



‫א‬‫א‬KE٤J ٥F‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬R١‫א‬Losses١ ‫א‬‫א‬LossesT ‫א‬‫א‬

 W‫א‬‫א‬KR٢‫א‬Losses٢ Losses1 = i12 ⋅ R 1

(٢٠-٥)

Losses 2 = i 22 ⋅ R 2

(٢١-٥)

Losses T = Losses1 + Losses 2 = i12 ⋅ R 1 + i 22 ⋅ R 2

(٢٢-٥)

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

 K‫א‬

 Transformers in Power Networks ‫א‬‫א‬‫א‬‫א‬‫ א‬٨J ٣J ٥ ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬‫א‬ E ‫א‬‫א‬‫א‬‫א‬F‫א‬‫א‬‫א‬ K ‫א‬‫א‬‫א‬‫א‬‫א‬

 KE٥J ٥F،‫א‬‫א‬‫א‬‫א‬  -١٥٠-

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬

 ١٤١  ٢-

‫א‬



‫א‬ 



 



‫א‬





‫א‬

 

 ‫א‬‫א‬‫א‬‫א‬E٥J ٥F 

 E٣J ٥F ‫א‬‫א‬‫א‬‫א‬،٣ kV‫א‬‫א‬‫א‬‫א‬ 

N1 ‫א‬‫א‬،٦٦ kV‫א‬‫א‬٢٢٠ V N2

 K‫א‬‫א‬

 ‫א‬

‫א‬



‫א‬





 E١=٣ kV



E٢=٦٦ kV



E٣=٢٢٠ V ‫اﻟﻤﺴﺘﻬﻠﻚ‬

   E٣J ٥F‫א‬‫א‬E٦J ٥F

-١٥١-

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١  ٢-

‫א‬ 

 ‫א‬

 WE١٢J ٥F‫א‬،‫א‬

N1 ‫א‬ N2

E2 N 66000 = 12 = = 22 E1 N 11 3000

N 11 1 =  N 12 22

 W‫א‬‫א‬،‫א‬ E3 N 220 V 1 = 22 = = E2 N 12 66000 V 300

-١٥٢-

N1 ‫א‬ N2

 ‫א‬‫א‬

 ١٤١

‫א‬‫א‬‫א‬‫א‬‫א‬

‫א‬ 

 ٢-

 ‫א‬‫א‬  K‫א‬‫א‬‫א‬ K١

K‫א‬ K٢

K(L)‫א‬‫א‬‫א‬ K٣ K‫א‬ K٤

K‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬‫ א‬K٥  K‫א‬‫א‬ K٦ K‫א‬‫א‬ K٧

K‫א‬‫א‬ K٨ K‫א‬‫א‬‫ א‬K٩

K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٠ K‫א‬‫א‬‫א‬‫ א‬K١١

K‫א‬‫א‬‫א‬‫א‬‫א‬‫ א‬K١٢ ١mA/sec‫א‬‫א‬‫א‬،‫א‬‫א‬‫ א‬K١٣ K٠{٨ V‫א‬‫א‬

٣mA/sec‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ K١٤  K١{٥ V‫א‬‫א‬

1 ‫א‬، 200 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K١٥

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K١ A‫א‬‫א‬١٠٠ V‫א‬‫א‬

‫א‬، 200 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K١٦

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬

K١ A‫א‬‫א‬١٠٠ V‫א‬‫א‬ ‫א‬، 20 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K١٧

‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K٠{١ A‫א‬‫א‬١٠٠ V‫א‬‫א‬

-١٥٣-

 ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬

 ١٤١  ٢-

‫א‬ 

‫א‬‫א‬‫א‬،١ kV‫א‬‫א‬‫א‬‫א‬ K١٨ ‫א‬‫א‬،٣٣ kV‫א‬‫א‬١١٠ V‫א‬ K‫א‬‫א‬ N 2 N 1

‫א‬‫א‬‫א‬،٣ kV‫א‬‫א‬‫א‬‫א‬ K١٩

‫א‬‫א‬،٦٦ kV‫א‬‫א‬٢٢٠ V‫א‬  K‫א‬‫א‬ N 2 N 1

-١٥٤-

 ‫א‬‫א‬

 ١٤١  ٢-

‫א‬ 

(References) ‫א‬‫א‬



[١] Thomas L. Floyed, Electrical Engineering Fundamentals, Prentice, Inc, sixth edition, ٢٠٠٠. [٢] B. L. Theraja, A. K. Theraja, " Electrical Technology", published by Ninja Construction development Co. Ltd. Ram Nagar, New Delhi, ١١٠٠٥٥, ١٩٩٥. [٣] M. A. PAI, "Introduction to Electric Circuits and Machines", Affiliated east west presses private limited, ١٩٧٥.

- ١٥٥ -

 ‫א‬

 ١٤١

‫א‬

 ٢-



 ‫א‬ ‫א‬ 

 ‫א‬  

  ١

‫א‬‫א‬‫א‬W‫א‬‫א‬

 ٢

١J ١

 ٢

 ‫א‬‫א‬‫א‬‫א‬  ‫א‬٢J ١

 ٢

 ‫א‬‫א‬‫א‬١J ٢J ١

 ٢

 ‫א‬‫א‬١J ١J ٢J ١

 ٤

‫א‬‫א‬٣J ١J ٢J ١

 ٦

 ‫א‬‫א‬٣J ١ ‫א‬‫א‬‫א‬‫א‬١J ٣J ١

 ٨

‫א‬‫א‬‫א‬‫א‬٢J ٣J ١

 ٤  ٦

‫א‬‫א‬٢J ١J ٢J ١

 ١٠

‫א‬‫א‬‫א‬‫א‬٣J ٣J ١

 ١١

‫א‬‫א‬‫א‬٤J ٣J ١

 ١١

‫א‬‫א‬‫א‬٥J ٣J ١

 ١٤   ١٥  ١٦  ١٦

 ‫א‬‫א‬  ‫א‬‫א‬W‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬  ١J ٢

 ١٦

 ‫א‬‫א‬‫א‬٢J ٢ ‫א‬‫א‬١J ٢J ٢

 ١٩

‫א‬‫א‬٢J ٢J ٢

 ٢١

 ‫א‬‫א‬٣J ٢ ‫א‬‫א‬١J ٣J ٢

 ١٩

‫א‬ 

‫ א‬١٤١

‫א‬

٢-



 ٢٢

‫א‬‫א‬‫א‬٢J ٣J ٢

 ٢٤

 ‫א‬‫א‬‫א‬١J ٣J ٣J ٢

 ٢٧

 ‫א‬‫א‬٤J ٣J ٢

 ٢٩

‫א‬‫א‬‫א‬٥J ٣J ٢

 ٣٢

‫א‬‫א‬‫א‬٢J ٥J ٣J ٢

 ٢٤  ٢٥

 ٢٩  ٣٤   ٣٧

 ٣٨  ٣٨

‫א‬‫א‬‫א‬٣J ٣J ٢

 ‫א‬‫א‬‫א‬٢J ٣J ٣J ٢

W‫א‬‫א‬١J ٥J ٣J ٢  ‫א‬‫א‬

 ‫א‬‫א‬‫א‬W‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬  ١J ٣

 ٣٨

 ‫א‬‫א‬٢J ٣ ‫א‬‫א‬١J ٢J ٣

 ٣٩

‫א‬‫א‬٢J ٢J ٣

 ٤٠

‫א‬‫א‬٣J ٢J ٣

 ٤١

‫א‬١J ٣J ٢J ٣

 ٤٢

‫א‬٢J ٣J ٢J ٣

 ٤٤

‫א‬‫א‬‫א‬‫א‬٣J ٣J ٢J ٣

 ٤٥  ٤٥

 ‫א‬‫א‬‫א‬٣J ٣ ‫א‬‫א‬‫א‬j‫א‬‫א‬‫א‬١J ٣J ٣

 ٤٧

‫א‬‫א‬‫א‬١J ٢J ٣J ٣

 ٤٧

‫א‬‫א‬‫א‬٢J ٣J ٣

 ٤٧

‫א‬‫א‬‫א‬٢J ٢J ٣J ٣

 ٤٩

‫א‬‫א‬٣J ٢J ٣J ٣

 ٥١

‫א‬‫א‬٤J ٢J ٣J ٣

 ٥٥

‫א‬‫א‬‫א‬‫א‬٥J ٢J ٣J ٣

‫א‬ 

‫ א‬١٤١

‫א‬

٢-



 

 ٥٩

 ‫א‬‫א‬‫א‬٣J ٣J ٣

 ٥٩

 ‫א‬‫א‬‫א‬١J ٣J ٣J ٣

 ٦٠

 ‫א‬‫א‬‫א‬٢J ٣J ٣J ٣

 ٦١

 ‫א‬‫א‬٣J ٣J ٣J ٣

 ٦٤

 ‫א‬‫א‬٤J ٣J ٣J ٣

 ٦٧  ٧١

‫א‬‫א‬‫א‬‫א‬٥J ٣J ٣J ٣  

 ٧١

 ‫א‬‫א‬‫א‬٤J ٣  ‫א‬١J ٤J ٣

 ٧٢

 ‫א‬٢J ٤J ٣

 ٧٣

 ‫א‬٣J ٤J ٣

 ٧٦

 ‫א‬‫א‬

  ٧٩  ٨٠  ٨٠

 ‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬  ١J ٤

 ٨٠

 ‫א‬‫א‬‫א‬‫א‬‫א‬٢J ٤  ١J ٢J ٤

 ٨٣

 ١J ٢J ٢J ٤

 ٨٧

 ‫א‬‫א‬‫א‬١J ١J ٢J ٢J ٤

 ٩١

 ٢J ٢J ٢J ٤

 ٨٣

 ٨٩

 ٢J ٢J ٤

 ‫א‬٢J ١J ٢J ٢J ٤

 ٩٤

 ‫א‬‫א‬‫א‬١J ٢J ٢J ٢J ٤

 ٩٥

 ‫א‬٢J ٢J ٢J ٢J ٤

 ٩٩

 ‫א‬‫א‬١J ٣J ٢J ٤

 ٩٩

 ‫א‬‫א‬‫א‬٣J ٢J ٤

‫א‬ 

‫ א‬١٤١

‫א‬

٢-



 ١٠٥

 J ‫א‬J ‫א‬٢J ٣J ٢J ٤

 ١٢٠

 EF‫א‬‫א‬٤J ٣J ٢J ٤

 ١٣٥

 ‫א‬‫א‬‫א‬

 ١١٤  ١٣٠   ١٤٠  ١٤١  ١٤١

 ٣J ٣J ٢J ٤

 ‫א‬٥J ٣J ٢J ٤

 ‫א‬‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬  ١J ٥

 ١٤١

 ‫א‬‫א‬٢J ٥  ‫א‬‫א‬‫א‬١J ٢J ٥

 ١٤١

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬١J ١J ٢J ٥

 ١٤٣

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٣J ١J ٢J ٥

 ١٤٤

 ‫א‬‫א‬١J ٢J ٢J ٥

 ١٤٦

 ‫א‬‫א‬٣J ٥  ‫א‬‫א‬١J ٣J ٥

 ١٤٦

 ‫א‬‫א‬٢J ٣J ٥

 ١٤٧

 ‫א‬٤J ٣J ٥

 ١٤٧

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٥J ٣J ٥

 ١٤٩

 ‫א‬‫א‬٧J ٣J ٥

 ١٤٢

 ١٤٣  ١٤٦

 ١٤٦

 ١٤٨

 ‫א‬‫א‬‫א‬٢J ١J ٢J ٥  ‫א‬‫א‬٢J ٢J ٥

 ‫א‬٣J ٣J ٥

 ‫א‬‫א‬‫א‬٦J ٣J ٥

 ١٥٠

 ‫א‬‫א‬‫א‬‫א‬‫א‬٨J ٣J ٥

 ١٥٣

 ‫א‬‫א‬  ‫א‬‫א‬

 ١٥٥

      

 ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬E‫א‬F‫א‬‫א‬

GOTEVOT appreciates the financial support provided by BAE SYSTEMS