A Computer Simulation Of An Induction Heating System - Magnetics, Ieee Transactions On

4343

IEEE TRANSACTIONS ON MAGNETICS. VOL. 21, NO. 5 . SEPTEMBER 1991

A Computer Simulation of an Induction Heating System Lawrence R. Egan and Edward P. Furlani

Abstract-A method is presented for the design and analysis of induction heating systems. The method entails the simulation of system performance using an equivalent circuit approach. Equivalent circuit models are obtained for the three parts of an induction heating system: the power source, the impedance matching circuit, and the load. These models are combined into a system model which is analyzed using the Advanced Continuous Simulation Language (ACSL). This approach is applied to an existing system, and the predicted performance is in close agreement with measured data.

INTRODUCTION

I

NDUCTION heating has been used to heat electrically conductive materials since the early 1990's. Industrial applications of the technology include metal melting and heat treating, crystal growing, semiconductor wafer processing, high-speed sealing and packaging, and curing of organic coatings [ 11, [2]. The advantages of this technology include fast heating rate (6000"F/s in foils) instant stadstop (no warm up required for each cycle) precise heat pattern (heating concentrated where needed) noncontact heating (the heat applicator does not physically contact the part that is heated) An induction heating system usually consists of three distinct parts: the power source (oscillator), an impedance matching circuit, and a load (Fig. 1 shows a block diagram of a typical system). The load consists of a coil of wire (work coil) in close proximity to the heated material (work piece). Copper tubing is often used for the work coil in order to allow for water cooling during continuous operation, During normal operation, the oscillator supplies timevarying current to the work coil, which produces an alManuscript received September 10, 1990; revised May 8, 1991. L. R. Egan is with the Automatic Machine Systems Division, Manufacturing Research and Engineering Organization, Eastman Kodak Company, Rochester, NY 14652. E. P. Furlani is with the Engineering Research Center, Eastman Kodak Company, Rochester, NY 14653. IEEE Log Number 9101771,

temating magnetic field across the work piece. Since the work piece is electrically conductive, an eddy current is induced within it, and heat is generated from the resistance to the eddy current (shown in Fig. 2). Heat is also generated in magnetic materials from alternating magnetization and hysteresis, but this phenomenon is not addressed in this work. Each new application usually requires a unique work coil design. When designing a coil, there are two criteria to consider. 1) The impedance of the work coil and work piece (the load) should be such as to allow for an efficient flow of energy from the source. 2) The coils must be of the appropriate configuration or shape so that the induced heat will be concentrated in the areas of interest. These two criteria are related in the sense that the load impedance is, to some extent, a function of the coil configuration. This paper addresses the first criterion explicitly. Specifically, a method is given for analyzing the steady-state performance of an induction heating system relative to variations in the load. The term steady state, in this context, implies that the electrical and thermal transients have subsided. The approach taken is to first reduce the system to an equivalent circuit, and then simulate the behavior of the circuit as a function of changes in load impedance. This technique was applied to a Cycle-Dyne Model A50 system which consists of a vacuum tube oscillator (source), air core transformer (impedance matching circuit), and a given load. The oscillator is powered by an Amperex Neotron type 8867 vacuum tube. The load consists of a 0.00035-in-thick aluminum foil (work piece) fixed at a given height above a two-turn rectangular work coil having a 3 in side length (Fig. 3). This work was originally undertaken in order to understand the relation between the heat generated in the foil and its distance from the coil. In preliminary experiments, it was observed that the heating ranged from intense to nonexistent, depending on whether the foil was above or below a specific optimum distance. The approach outlined above accurately predicted this phenomenon. It is described in full in the following sections.

0018-9464/91$01.00 0 1991 IEEE

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---

-- -

Line A.C.

POWER OSCILLATOR SOURCE

-

-

MATCHING NETWORK

*

LOAD

Feedback

Fig. 1 . Block diagram of typical induction heating system.

OSCILLATOR WORK COIL MAGNETIC FIELD

>

CURRENT

(EDDY CURRENT)

Fig. 2. Illustration of induction heating from induced eddy currents

Work Piece (0.00035Inch Thick Foil)

[Aluminum

L

Source Current Applied To These Terminals

[

% y:d;+? n ):t

L W o r k Coil Assembly

Fig. 3. Work coil-work piece system

INDUCTION HEATINGSYSTEMANALYSIS Vacuum Tube Oscillator Analysis

The high-frequency power supply investigated in this work consisted of a class-C oscillator with grid-leak bias (Cycle-Dyne Model A50). The operation of the generalized circuit has been described in numerous texts [3]. The high-frequency power generating component was an Amperex Neotron type 8867 vacuum tube (manufacturer’s specifications are given in Appendix I). Fig. 4 shows the vacuum tube oscillator circuit. The plate current is supplied to the resonant tank circuit made up of C, and L,..Power is extracted through coupling to L,, and the feedback voltage (to sustain oscillation) is generated through Lf.The gate, plate, and cathode in Fig. 4 refer to the vacuum tube terminals. The first step in the simulation of this circuit was the specification of the state variables which were taken to be the inductor currents Z2, Z3, Z4, Z, and capacitor voltages Vcp, Vcfl, V c m l ,Vcg, Vcm2.Explicit expressions for these variables were generated by solving nodal and mesh equations using the MACSYMA program [4]. Appendix I1 contains a printout of a MACSYMA session, during which unique solutions were generated for the three coupled currents: Z2, Z3,and Z4.

IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 5, SEPTEMBER 1991

Since the vacuum tube was operating in a class-C mode, it could not be represented by a simple linear equivalent circuit with a dependent current source. Instead, data tables were created from its characteristic operating curves. These tables were used to obtain the terminal characteristics for the plate and gate (shown in Appendix 111). The Advanced Continuous Simulation Language (ACSL) was used to implement the simulation [ 5 ] . In running the ACSL program, the gate and plate voltages were calculated and used as input for the data tables. The program generated the vacuum tube terminal currents from the calculated terminal voltages. In general, any multiport device (linear or nonlinear) for which terminal voltage/ current characteristics are known or measured can be represented by a data table within an ACSL program. Load conditions for which oscillations stopped were determined, and the results of the simulations were verified with experimental measurements. Some of the circuit parameters shown in Fig. 4 were either measured directly with an impedance analyzer or inferred from measured data. In particular, the load impedance was measured with an impedance analyzer and compared to predictions from finite element analysis (FEA). Similarly, the transformer inductance and coupling coefficients were inferred from measured data using linear transformer theory. The analysis of both of these components is described in the following two sections. Impedance Matching Circuit From ac circuit theory, maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. For induction heating applications, maximum power transfer is achieved by first determining the equivalent power source impedance Z g , and then choosing an appropriate impedance matching circuit and equivalent load impedance to yield a complex conjugate match at the operating frequency. The source impedance is typically much higher than the load impedance. It is not uncommon for there to be three orders of magnitude difference between the two. For example, a load consisting of a two-turn work coil coupled to an aluminum foil work piece (described later) had an impedance of 0.84 L! at 384 kHz, while the tube impedance was roughly 1200 Q . One technique that is used to raise the apparent load impedance to a value comparable to that of the vacuum tube is to employ an air-core (and in some cases ferritecore) transformer. The equipment used in this investigation (Cycle-Dyne Model A50) employed an air-core transformer with 35 turns connected to the oscillator circuit and two turns connected to the load. In this study, the air-core transformer was modeled in terms of an equivalent circuit, as shown in Fig. 5. The equivalent circuit parameters R,, L,, R,, L,, L,, RL, RL and Ri were measured at the desired operating frequencies using a Hewlett-Packard model 4192A impedance analyzer. The other equivalent circuit parameters k,,, M,,,, R:, and L j were computed using (1)-(5), which are de-

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EGAN AND FURLANI: COMPUTER SIMULATION OF INDUCTION HEATING SYSTEM

RI LL

Fig. 4. Vacuum tube class-C oscillator circuit.

rived from linear transformer theory [6]:

L; = Lp -

Ri

w2MiS(Ls+ LL) (R, + R ~ +) w ~~ ( L + , L~)'

= Rp i-

(R,

u2M;,(R,

+ RL)

+ RL)2 + w2(Ls + LJ2

R; = Ri

+ w2LI2

L; = L;

+

(2) (3)

R'"

(4) EQUIVALENT CIRCUITS

In particular, the mutual coupling coefficients kps and Mps were computed using (1) and (2), respectively. The same procedure was used to calculate the coupling coefficient ( M p f )between the primary coil of the transformer and the feedback coil. Once all of the circuit parameters are known and the tube data are entered into ACSL data tables, one can proceed with the simulation. Predicted voltage waveforms are illustrated and discussed later in this paper. Prediction of Load Impedance In the previous section, an empirical approach is described for obtaining the equivalent circuit parameters for both the impedance matching transformer and the load. While this approach is useful for existing hardware, it is of limited used in the design and optimization of new hardware. Often, it is desirable to be able to simulate the performance of an induction heating system for a variety of load configurations without having to fabricate or modify hardware. This can be done if one can predict the impedance of the load from first principles. A method for doing this is demonstrated in this section. In induction heating applications, the load usually consists of a work coil in close proximity to a work piece. The coupling between these two elements in shown in Fig. 2. The work coil, which typically consists of a single turn (or multiple turns) of copper wire or tubing, usually carries an alternating current (60 Hz-10 MHz) of several amps to hundreds of amps. This time-varying current in-

Fig. 5 . Linear transformer, load (work coil-work piece), and equivalent circuits.

duces a time-varying electromagnetic field that cuts across the work piece. The impedance associated with this phenomenon, as measured at the terminals of the work coil, has both a resistive (RL)and inductive component (LL). The impedance is a function of several variables, including the configuration or shape of the work coil, the spacing between the work coil and work piece, the thickness of the work piece, the frequency, and the conductivity arid permeability (which are a function of temperature). A general method for computing the impedance entails the use of electromagnetic field theory. The method, as described here, is essentially that of Cendes and Konrad [7]. It entails the use of finite element analysis (FEA), and applies to loads that have an axisymmetric geometry or are long enough in one direction so as to justify a two-dimensional (2D) analysis. A brief description of the technique and an example of its application are given in the remainder of this section. For eddy current problems in which displacement currents and free charges are negligible and in which all materials are linear, the steady-state time-harmonic fields are described by V x H = J

(6)

V x E = -jwB

(7)

V * B = O

(8)

V - D = O

(9)

IEEE TRANSACTIONS ON MAGNETICS, VOL. 21, NO. 5, SEPTEMBER 1991

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where H is the magnetic field strength (A/m), B is the magnetic field intensity (W/m2), D is the electric displacement field (C/m2), and E is the electric field intensity (V /m) , J is the current density (A /m2), and j is the complex number defined by j 2 = - 1 . It is understood that the fields, as defined by (6)-through (9), are complex, vector-valued functions and that their physical counterparts are computed by taking the real part. For example,

E(x, Y , t )

=

Re

[JW, y) exp ( j u t ) ]

where E(x, y, t ) represents the physically measurable field. The Re [ ] and time-exponential phasor notation is suppressed throughout most of this section; however, it is reintroduced as needed. Also, it is assumed that the following constitutive relations for linear and isotropic materials apply: J = uE

(10)

B

=

pH

(1 1)

D

=

€E

(12)

where E is the permittivity (F/m), U is the conductivity (mhos/&, and p is the permeability (H/m). Equations (6)-(9) represent a system of coupled firstorder differential equations. These four equations reduce to two second-order equations via the introduction of the potential functions A and V . Specifically, the vector potential A is defined by (13)

B = V x A .

This relation follows from (8) and the results of Helmholtz's theorem. Substituting (13) into (7) yields V

X

(E

+ jwA) = 0 ,

(14)

which implies that E

+ j u A = -VV,

(15)

which, in turn, defines the scalar potential V . Combining this with the constitutive relation (10) gives an expression for the total current density J: J = -juwA -

U

VV.

(16)

Substituting the expressions for B and J, (13) and (16), respectively, into (6) yields the second-order equation V x V x A = -jpuuA - puVV.

(17) This equation is simplified further by imposing the Coulomb gauge condition (which is needed to uniquely specify A ) : V.A=O. The resulting equation is V 2 A - jpuuA = puVV.

(18) (19)

Also, from (91, (121, (15), and (18), it follows that

v 2 v= 0.

(20) Equations (19) and (20) are equivalent to (6)-(9), and, in

principle, they can be solved simultaneously to obtain the field solution. However, it is also possible to obtain the field solution by solving (19) subject to a current constraint (without solving for Y explicitly). This is the approach taken here. However, before it is introduced, it is instructive to see how (19) applies region by region. For this application, there are three distinct regions to consider: the work coil, the work piece, and the surrounding air. Since the work coil is connected to an external source, the current density in it can be thought of as consisting of two components: an impressed part and an induced part. The impressed part, which is referred to as J,o,rc,,is due to an external source and is defined by the gradient of a scalar potential. The induced part is due to the time-varying B field in the coil itself. Therefore, in the work coil, V 2 A - jpuuA = -pJ,,,,,

where J,,,,,

(21)

= -uVV.

In the work piece, there is an induced current density denoted by J e d d y , but there is no source term. The appropriate equation for this region is V 2 A - jpuwA = 0

(22)

with

Jddy = -juuA. There are no conduction currents in air (19) simplifies to

(23) (U =

0), and so

V 2 A = 0.

(24)

For many applications, both A and J,,,,, are unknown. Therefore, since there are two unknowns, two independent equations are needed. An approach that is commonly used is to assume that the work coil terminals are connected to independent current source with, for example, I(?) = Io cos (ut). The idea is to solve (19) subject to this constraint, and then use the solution to compute the equivalent series impedance of the work coil-work piece. Once the impedance is known, it can be used in the circuit analysis described earlier to predict the performance of the system. The constant current constraint implies that

where J ( x , y, t) is the physically measurable current density given by J ( x , y, t) = Re [J(x, y) exp ( j w t ) ] . Accordingly, (25) can be rewritten as

I(?) =

s

workcoil

Re [ -juuA exp ( j w t ) ] dS

____

4347

EGAN AND FURLANI: COMPUTER SIMULATION OF INDUCTION HEATING SYSTEM

where the integration is over a cross-sectional area of the work coil, as indicated. Since there are two unknowns, A and Jsource, the two equations (19) and ( 2 6 ) must be solved simultaneously. These equations can be solved using finite element analysis. A key fact that is exploited in this approach is that the second integral in ( 2 6 ) is readily obtained for both 2D and axisymmetric problems. The details of this approach are beyond the scope of this paper, but can be found in detail in [7] and [8]. Briefly, the finite element technique amounts to dividing a given geometry (which may contain many different materials) into small elements (collectively known as the mesh) and assigning to each element a polynomial that will approximate the behavior of the vector potential A in that element. The coefficients of these polynomials are arranged in a matrix, and their values are obtained via the solution of a global matrix equation that reflects all of the information of the problem, including the location of each element relative to its neighbors, the underlying field equations, the boundary conditions, and the material properties. As an example, consider the induction heating system depicted in Fig. 3 . This geometry represents an actual work coil-work piece system that has been used extensively for noncontact heating/sealing. Note that the geometry of this system is fully three dimensional. However, it was modeled using an axisymmetric approximation. The geometry depicting this approximation is shown in Fig. 6 . Basically, the approximation amounts to ignoring the effects of the 90" bends in the comers of the wire. Since the actual circuit is square, an effective radius was used in the approximation such that the area of the axisymmetric circuit equals that of the square configuration. The FEA of this problem is straightforward. The MAXWELL software from ANSOFT Corporation was used for the FEA calculation [9]. In particular, the MAXWELL Axisymmetric Eddy Current program was used, and it implicitly solves (19) subject to the constraint (26). Since the geometry is cylindrically symmetric about a central vertical axis, the FEA can be demonstrated using only the right-hand side of the geometry (the left-hand side being a mirror image). A cross section of the geometry and the corresponding mesh for the right-hand side of the work coil-work piece system (Fig. 6 ) appear in Figs. 7 and 8, respectively. Note that the work coil consists of two turns of copper wire, and the work piece is a thin sheet of aluminum (0.00035 in thick). The appropriate boundary condition for this problem is the Dirichlet condition. Once the material properties and boundary conditions are set, a frequency is selected, and both the vector potential A and VI/'are computed. All of the other fields can be computed from these two functions. As an example, a solution was obtained using an operating frequency of 384 kHz. The magnetic flux pattern shown in Fig. 9 was computed for r = 0 (when the in-

Work Pjece (Foil not lo scale]

Work Coil

Source Current Applied To These Terminals

Fig. 6. Axisymmetric approximation for work coil-work piece system shown in Fig. 3 .

-

Cross-section of Rlght-Hand %de 01 Aluminum Foil (Work Piece]

1

Cross-section 01 Right-Hand Side Of COll

Fig. 7 . Right-hand side of axisymmetric representation of the work coilwork piece system.

Fig. 8 . Finite element mesh for axisymmetric analysis.

dependent source current is maximum). For purposes of comparison, the flux pattem for the dc case (0 Hz) is shown in Fig. 10. Note that at higher frequencies, the induced current in the aluminum develops a magnetic field of its own that tends to oppose that of the work coil. The equivalent impedance for the work coil-work piece, that is, the equivalent series resistance RL and inductance LL, can be computed from the field solution. Since RL is a series resistance, and since by assumption Z(t) is flowing through it, the energy dissipated per cycle

IEEE TRANSACTIONS ON MAGNETICS, VOL. 27. NO. 5, SEPTEMBER 1991

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tance, yielding

-

L -

1

10

2

sallspace

CL dv.

Equations (29) and (30) were applied to the work coilwork piece described above for a variety of frequencies and configurations. The computed and measured values of RL and LL are compared in the next section. F i g . 9. F l u x plot forf = 384

kHz

RESULTS Oscillator Simulation

\ I

I I

Fig. 10. F l u x plot forf

(e,)

=

0 Hz

is given by

1

PT

QL =

Z2(t)RLdt

0

where, by assumption, Z(t) = Io cos (wr) and T i s the period of one cycle. However, energy is dissipated in both the work coil and work piece. The energy dissipated per cycle in these two components is given by Q =

s,'ls

Jzdv

+

workcoil U

i

Jzdv

workpiece U

where

J

=

Re [J(x, y,

z ) exp (jut)]

and

s

d v implies integration over indicated spatial region [20].

By equating expressions (27) and (28) and using the fact that all of the time variation is time harmonic, one obtains an expression for the series resistance: Rt

=

1[i Io

workcoil

(JIzdv U

+

i

workpiece

-1 JdI 2 v ]

(29)

U

where the current densities have only spatial dependence as the time dependence has been integrated. A similar analysis can be applied to the series induc-

As stated above, the motivation for this study was to understand in a quantitative sense when efficient heating will occur for a given induction heating system with a given load. The system components have been described in earlier sections and the load geometry is shown in Fig. 3. Recall that the load consisted of a thin foil of aluminum (0.00035 in) situated above a two-turn rectangular work coil. Experiments were performed in which the spacing between the foil and work coil was varied and the degree of heating was noted. The following results were observed. 1) For spacing of less than 0.045 in, there was little or no heating of the foil. 2) The onset of heating occurred at a spacing of 0.045 in. 3) The foil heating rate rapidly increased for coil-tofoil spacings up to 0.085 in and gradually decreased for spacings beyond 0.085 in. In order to compare these observations to the theory, four cases were considered. For each case, the load voltage was computed and plotted as a function of time. The four cases are as follows: 1) foil-to-work coil spacing = 0.015 in (no heating should occur) 2) foil-to-work coil spacing = 0.045 in (onset of heating) 3) foil-to-work coil spacing = 0.085 in (efficient heating) 4) coil only, no foil present. The results of the analyses appear in Figs. 1 1 and 12. Specifically, in Fig. 1 1 , the load voltage decays rapidly when the spacing is 0.015 in. This implies that there would be little or no heating with this spacing. Similarly, the oscillation is still decaying (much less rapidly) when the spacing is 0.045 in. This indicates that some heating would occur, but that it is not an efficient heating condition. In Fig. 12, it is obvious there is no decay in the load voltage oscillations for a spacing of 0.085 in. One can therefore expect efficient heating for this condition. The same result is obtained when there is no foil present. This case is included as a reference. Note that there is excellent agreement between the predicted behavior of the load voltage and the experimental observations in terms of effective heating versus air gap spacing.

~

EGAN A N D FURLANI: COMPUTER SIMULATION OF INDUCTION HEATING SYSTEM T I M E ISPACING=O.O15

L O A D V O L T A G E VS. 100.

LOAD V O L T A G E V S . 250. I

INCH1 I

I

-

J

cn 50

60. 40.

0

2 W

a

' D 5

INCHI I

150. 100.

20.

3

50.

0.00

y4

0.00

4

50

T I M E ISPACING=0.085

200.

80.

cn c

4349

-20. -40. 0

5

-60.

-150.-

J

J

-80.

'

100.

0,000

.00002

.00006

.00004

T I ME LOAD V O L T A G E V S .

.00008

-200.-

I

-250. 0.000

.00002

'

I .00010

I

I I I

.00004

T I ME

SECONDS I

T I M E ISPACING=0.045

250.

LOAD VOLTAGE VS.

INCH1

.00006

.00008

.00010

I SECONDS I

T I M E ( C O I L ONLY

-

NO F O I L 1

600.

1 -

400.

1

A

in F

d>

200.

g

0.00

+ d

, ~200. 0 J 0

0 J

'

-250. 0.000

.00002

.00004

.00006

.00008

I .OOOlO

T I ME ISECONDS I

-400.

600.

I

0.000

I .00002

.00004

T I ME

.00006

.00008

.00010

SECONDS 1

Fig. 11. Plots of load voltages versus time for load conditions in which no heating occurred. Top: Coil-to-foil spacing was 0.015 in. Bottom: Coil-tofoil spacing was 0.045 in.

Fig. 12. Plots of load voltage versus time for load conditions in which heating occurred. Top: Coil-to-foil spacing was 0.085 in. Bottom: Coil-tofoil spacing was infinite (no foil).

Work Coil- Work Piece Impedance

quency of 416 kHz and a foil-to-work coil spacing of 0.045 in. The computed values are RL = 0.202 Q and LL = 0.222 p H , and the corresponding measured values are RL = 0.249 Q and LL = 0.269 pH. The third condition (efficient heating) was with a foilto-work coil spacing of 0.085 in and an operating frequency of 384 kHz. The computed values are RL = 0.155 Q and LL = 0.289 pH, while the measured values are RL = 0.185 Q and LL = 0.343 pH. The last condition was with the work coil only (no foil present) and a frequency of 377 kHz. The computed impedance is RL = 0.0204 Q and LL = 0.636 pH, with corresponding measured values of RL = 0.024 Q and LL = 0.729 pH. Note that there is a 13-20% difference between the computed and measured data. The main reason for this difference is that an axisymmetric approximation was used to model a rectangular work coil. The authors have computed the impedance for numerous axisymmetric work coils, and have found the error to be on the order of 15 % . Nevertheless, even with the crude approximation used here, the calculated impedance values are accurate enough to be used for a simulation of system performance.

The impedance of the load depicted in Fig. 3 was computed and measured at room temperature for several different spacings and frequencies. The work piece is a 0.00035-in-thick aluminum foil (a = 3.88 * lo7 mho/m) and the work coil, which is rectangular, consists of two turns of copper wire (a = 5.8 * lo7 mho/m). The wire has a 0.060 in diameter, and there is a 0.080 in spacing between the two turns. The work coil measures 3 in on a side (defined by the outermost wire). For the axisymmetnc approximation, the radii to the center of the inner and outer wire were 1.53 and 1.67 in, respectively. They were computed based on keeping the area subtended by the axisymmetric coil equal to that of the rectangular coil. Four different load conditions were analyzed corresponding to the four cases studied in the oscillator simulation described above. The first condition (no heat) was with a foil-to-work coil spacing of 0.015 in and an operating frequency of 450 kHz. The computed values of resistance and inductance are RL = 0.269 Q and LL = 0.153 pH, which compare to measured values of RL = 0.344 Q and LL = 0.189 pH. The second condition (onset of heating) was for a fre-

IEEE TRANSACTIONS ON MAGNETICS, VOL. 21, NO. 5, SEPTEMBER 1991

4350

CONCLUSIONS The behavior of an induction heating system can be accurately predicted from first principles using a combination of nonlinear circuit analysis and the impedance predicting capability of finite element analysis. The technique, as described in this paper, has been successfully applied to several induction heating configurations.

Once the initial models have been developed, the system can be studied parametrically and optimization can be achieved in a relatively short period of time. The advantage of this analysis is that it eliminates the costly and time-consuming trial-and-error approach that is typically employed in bringing a new or modified induction heating system on line.

I APPENDIX AMPEREXMODEL8867 TUBECHARACTERISTICS

ANODE VOLTAGE(V1

ANODE VOLTAGE ( V )

4351

EGAN AND FURLANI: COMPUTER SIMULATION OF INDUCTION HEATING SYSTEM

ANODE VOLTAGE ( V )

8867 CONSTANT CURRENT CHARACTERISTICS

APPENDIXI1 MACSYMA PROGRAMMING SESSIONTO SOLVEFOR 12, I , , I4 load('[egan.modeli]osmdint.mac"); (C3) values; 03)

(U) vl;

[Vl, V2, V3, V4, V5, V6, DI2, DI3, DI41

(D4)

V201 = - DI3DT MPS

((3) v2; 05)

V301 = DI4DT MSF

((33) v3;

+ DI4DT MPF + DIZDT LP

+ DIZDT MPS - DI3DT LS

V601 = DI3DT MSF - DIZDT MPF - DI4DT LF 06) (C10) solve ([vl,v2,~3l,[di2dt,di3dt,di4dtl);

IEEE TRANSACTIONS ON MAGNETICS, VOL. 21, NO. 5 , SEPTEMBER 1991

4352

(C12) di3:part@art(dlO,l),Z); (D12) DI3DT = - (LP (- MSF V601- LF V301)

+ (LF MF!S - MPFMSF)WOl)/(LP

+ MPF MPS V601+

('MSF2 -LFLS)

MPF' V301

- 2 MPF MPS MSF + LFMPS2

+ LSMPF') (C13) di4:part@art(dl0,1),3); (D13) DI4DT = - (LP (- LS V601- MSF V301) + mp52 V601 + MPF MPS V301 + W S MSF - Is MPF)V2Ol)/(LP (MSF' - LF LS)- MPF MPS 2 MSF+ LF MPS'

+ LsMPF2) ((214) fortran(di2); DI2DT = WS*(MSF*V601+ LF*V301) +MPP(-LS+V601-MSF*V301) + (MSF**2-LF 1 *LS)*K?Ol)/(LP*(MSF**2-LF*LS)-2*MPF*MPS*MSF + LF*MPS**2 + LS*MPF**2 2)

((215) fortran(di4); +(MPS*MSF DI4DT = -(LP*(-LS*V601-MSF*V30l)+MPS**2*V601+MPF*MPS*V301 -LS*MPF)*V2Ol)l(LP*(MSF**2-LF*IS)-2*MF'F*MPS*MSF+ LF*MPS**2 + L S M P 2 FO.2)

(CU) load('[egan.modelilosmdintt.mac"); (C17) a l ; (D17) DDDT = - (LP (- MSF V601- LF V301) + MPF MPS V601+ MPF' V301 + (LF MPS - MPF MSF) V2Ol)/(LP (MSF' - LF IS)- MPF MPS 2 MSF+ LF MPS'

+ IsMPF2) (C18) a2: (Dl8) V301 = I3 (RS + RL) + DI3DT LL (C20)solv~[al,a21,~di3dt,v3011~; @20) [[DI3DT = ((LP MSF - MPF MPS)V601

+ (MPF MSF - LF MPS) V201

+ LF LP (I3 RS + I3 RL) + MPF2 (- I3 RS - I3 RL)) /(LPMSF2-2MPFMPSMSF + L F M P S 2 + (LS + LL)MPF2-LFLPLS-LFLLLP), V301= ((LLLPMSF-LLMPFMPS)V601+ (LLMPFMSF-LFLLMPS)V201

+LPMSF2(I3RS+I3RL)+LFMPS2(I3RS+I3RL)+LSMPF2(I3RS+I3RL) + LF LPLS (- I3 RS- I3 RL) + MPFMPSMSF(- 2 I3 RS- 2 I3 RL)) /(LPMSF2-2MPFMPSMSF

+ LFMPS2+ (LS + LL)MPF'-LFLPLS-LFLLLPHI

(C21) di3:part@art(dZO,l),l); (D21) DI3DT = ((LP MSF - MPF MPS) V601 + (MPF MSF - LF MPS)V201

+ LF LP (I3 RS + I3 RL) + MPF' /(LPMSF2-2MPFMPSMSF

(-

I3 RS - I3 RL))

+ L F M P S 2 + (LS + LL)MPF'-LFLPLS-LFLLLP)

((22) v3vOl:part@art(d20,1),2); (D22) V301= ((LLLP MSF - LL MPF MPS) V601+ (LL MPF MSF - LF LL MPS) V201

+LPMSF2~3RS+I3RL)+LFMPS2(I3RS+I3RL)+ISMPF2(I3RS+I3RL) + LF LP IS (- I3 RS - I3 RL) + MPF MPS MSF (- 2 I3 RS- 2 I3 RL)) /(LPMSF2-2MPFMPSMSF (C23) fortran(di3);

+ LFMPS2 + (LS + LL)MPF'-LFLPLS-LFLLLP)

DDDT ((LP*MSF-MPF*MPS)*V601+ ~F*MSF-LF*MPS)*V201+LF*LP*(I3*RS + 1 I3*RL) + MPF**2*(-13*RSI3*RL))/(LP*W**2-2*MPF*MPS*MSF + LF*MPS** 2 2 + (LS+ LL)*MPF**2-LF*LP*SLF*LL*LP)

((24) fortran(v3vOl); V301 = ((LL*LP'MSF-LL*MPF*MPS)+VGOl+ (LL*MPF*MSF-LF*LL'MPS)*V201+ LP 1 *MSP*2*(I3*RS+ 13*RL)+WMPS**2*(IS*RS+ I3*RL) +LS*MPF**2*(I3*RS+ 2 I3*RL) +LF*LP*LS*(-13*RS13*RL) + MPPMPS*MSF*(-2*13*I*I3*RL))/ 3 (LP*MSF**2-2*MPFaMPS*MSF+ IPMPS**2 + (Ls+ LL)*MPF**2-LF*LP*SLF*L 4 L*LP)

1

EGAN AND FURLANI: COMPUTER SIMULATION OF INDUCTION HEATING SYSTEM

APPENDIX I11 ACSL PROGRAM program owillator simulation ( cle-Dyne A501 no foil constant rl=o.o41 ~ B p u i v a d resistance" L constant I1 =7.90.47 $"equivalent load inductance" constant cp =4.73e-O9 $"tank capacitor" constant rp=O.78 ("tank inductor resistance" constant 1p=47.7e-O6 $"tank inductor" constant m = 0.017 $"seeonday inductor resistance" constmt 1s=2.5147 Yseeondary inductor" constant rf= 0.25 $"fedback inductor resistance" constant If= 15.54e-06 $"feedback inductor" constant cfl = 1.$"feedback coupling capacitor" constant rml= l . l l e + 0 3 $"grid meter series inductor resistance" constant lm = 3.6e-03 $"grid meter series inductor" constant cml= 1.07e-10 $"grid meter aeries capacitor" constant cm2 = 1.oe-08$"grid meter shunt capacitor" constant rm2= 1.0 $"grid meter resistor" constant kps = 0.83 Vprimarylmndary coupling" constant kpf=0.58 $"primary/feedback coupling" constant kst=0.0 $"seeondary/feedback coupling" constant rg=4oooO.O $"series gate resistor" constant cg= 1.Oe-09 $"aeries gate capacitor" constant i2ic=O.O $"tank inductor current initial condition" constant i3ic=O.O $"secondary inductor current initial condition" constant i4ic=O.O $"feedback inductor current initial condition" constant i5ic=0.0 $"grid meter series inductor current initial cond" constant vcpic=O.O $"tank capacitor voItage initial condition" constant vc 'c 0 0 $"series gate capacitor voltage initial condition" constant vcRiclO.0 $"grid coupling capacitor voltage initial cond" constant vcmlic=0.0 $"meter series capacitor voltage initial cond" constant vcmZic=O.O $"meter shunt capacitor voltage initial cond" constant vO1 =O.O $"ground reference" constant vbs = -8.k+ 08 $"plate/cathode bias voltage ramp slope" table ipt,2,10,13/-800.0,0.0,100.0,200.0,~.0,400.0,500.0,~.0 ,...

700.0,800.0,5000.0,0.0,125.0,260.0,375.0,500.0,750.0,1000.0 ,... 15w.0,2OOO.0,3000.0,4000.0,5000.0,... 0.00,0.00.0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00 ,... 0.00,0.00,0.00.0.00,0.00,0.01,0.01,0.02,0.03,0.04 ,... 0.00,0.00,0.22,0.50,0.50,0.64,0.64,0.82,0.82,0.90,... 0 00 000022 0 50092 1 2 2 1 3 0 150 150 1 Eo,... 0 ~ ~ ~ 0 ~ ~ ~ 0 ~ 2 4 ~ 0 ~ ~ ~ 0...~ 9 8 ~ 1 ~ 4 8 ~ 1 ~ ~ ~ ~ ~ ~ ~ 2 8 ~ 2 ~ 6 4 ~ 0.00.0.00,0.24,0.56,1.00,1.sO,2.00,2.58,3.20,3.~ ,... 0.00,0.00,0.26,0.60,1.08,1.60,2.14,2.72,3.38,4.00,... 0.00,0.00,0.27,0.64,1.14,1.66,2.22,2.82,3.50,4.20,... 0.00,0.00,0.31,0.68,1.18,1.74f.36,3.00,3.76,4.44 .... 0.00.0.00,0.33,0.76,1.26,1.80,2.44.3.16,3.92,4.60 ,... 0.00,O.OO 0 39,O.M 138 2 00,2.62,3.36,4.10,4.82 ,... 0 00 0 00'048 0 95'1'50f:12 2 80,3.52 4 28 5 OO,... 0~00~0~00~0~sO~1~09~1~66,2.26~3~00,3.74~4~50~5~20/ tablei 269/-800000500.0600.0.700.0 ,... O..:. 800.0 h . 0 4% 0 600.0'750 0'1OOO b lsbd.0 2bOO.O,.._ 0.OOO:0.odo:0.01iob,0.0&,0.~,0.072~,... 0.OOO 0 OOO 0 00120 0 02900 0 OM00 0 06300.... 0.000'0'000'0~00130'0'oo2so'0~03OOO~0~0~00,... 0.OOO'O:OOO'O 00144'0'00264'0 00400 0 01700.... 0.000~0.OOO~0~00152~0~00272~0~oo44a~0~00,... 0.OOO 0 OOO 0 00160 0 00276 0 00500 0 00800,...

a.0

O.OOO'O'OOO'000148'0'00261'0~~'0'0016,...

0.000:0:OOO:0:~:0~001~~0~~92~0~00732,... 0.OOO.0.OOO.0.00016.0.00100,0.00200,0.00476/

constant tJ3top=1.044 nsteps nstp = loo00 cinterval cint=2.0e-07 minterval mint = LOe-20 algorithm ialg= 1 initial pi =4.0*atan(l.O) mps = kps*qrt(lp*ls) $"primarylsecondary mutual inductance" mpf =kpPaqrt(lp*lf) S"primary1feedback mutual inductance" msf= Wsqrt(ls*lf) $"secondary/feedback mutual inductance" ooId =0.0 b l d = 0.0 ipold=0.0 igold =0.0 pavint=O.O Igmavi=O.O ipavi = 0.0 end $"of initial" derivative vb = vbs*(ramdO.O)-ramp(5.0e-06)) V601=v6-v01 v301=v3-vOl V201=V2-v01 vbOl =vb-vOl

dvcfl.vcflic) l n !%2E vcml= integ(dVcm1,vcmlic) d v c d = i7/cm2 vcm2 = intag(dvcm2,vdic) dvcg=ilO/ veg= inw&,vcgic) il= ipi2 ... DI2=(MPS*(MSF*V601 +LF*V301)+MPF*(-LS*V6Ol-MSF*V3Ol)+(MSF**2-LF *Ls)lmOl)l(4P*(MSF**2-LF.LS)-2*hPF*MPS*MSF + LF*hPS**2 + LS*MPF**2) 12= integ(di2,dic)

4353

IEEE TRANSACTIONS ON MAGNETICS, VOL. 21. NO. 5 , SEPTEMBER I991

4354

D13- ((LP*MSF-MPF*MPS)*V601 +(MPFoMSF-LF*MPS)*V2O1 +LF*LP*(I3*RS+ ... I3*RL)+ MPFo*2*(-IS*~13*R)/(LP*MSF**Z-2*MPF*MPS*MSF +LF*MPS**... 2 + ILS+LL~*MPFo*2-LE"LP*ISLF*LL*LP) R**91

k =&g(di4,i4ic) di5= (v9-vlO)flm i5= inydi5,i5ic) i6=i4-1 16 i7 = i5 + i6-18 ilO=i i9 i8 = (vkvb)/rm2 i9 = (v8-vll)lrg vp=vl-vb Vl=dl+VCD v2=vl '2. * V3 =Vi; + (ieL*LP*MSF-LL*MPF*MPS)*V~l+ (LL*MPF*MSF-LF*LL*MPS)*VZOl+ LP ... *MSF**2*(I3*RS+ 13") +WMPS**2*(I3*RS+ I3*RL) +LS*MPF**Z*(I3*RS+ ... I3*FU.) + LF*LP*LS*(-I3*RS-I3*RL) + MPF*MPS*MSF*(-Z*I3*R2*13*~))/... (LP*MSP*2-2*MPF*MPS*MSF+ LF*MPS**2 + &3! + LL)*MPF**Z-LF*LP*LSLF*L... T.*T.P\

;42~i3*rs v5 =vdi3*rl v6=v7 + i4.d v7 =v8 +vcfl v8 =vb +vcml + vcm2 ~9 = v a i 5 * m i v10 =vb +vcm2 vii=v8-vq vg= v l 1-vb mv= inted(i3**2)*rl.0.0) &hedule &t :xp. -vb-vp ~ t @ ~ ~ p o r . i 2 . @1oooO.O) . d

i

i imt

s$v-pold)/(t-told) pold = p v I= (iav-igold)/(t-told) igold = i p v ipavi = (ipav-ipold)/(t-told) ipold = ipav told =t end $"of pavint" end $"of discrete" end

ACKNOWLEDGMENT The authors would like to thank J. Sproul, who provided careful and invaluable guidance in the preparation of this manuscript. The authors would also like to thank Dr. L. F. Pizziconi, whose comments led to significant improvements to the text.

REFERENCES [ l ] E. J . Davies and P. G. Simpson, Induction Heating Handbook. London: McGraw-Hill, 1979. [2] S. Zinn, S. L. Semiatin, I. L. Harry, and R. D. Jeffress, Elements of Induction Heating: Design, Control, and Applications. Palo Alto, CA: Electric Power Res. Inst., 1988. [3] G. M.Chute and R. D. Chute, Electronics in Industry, 4th ed. New York: McGraw-Hill, 1971. [4] "MACSYMA," developed by M.I.T. Lab. for Computer Science/ Mathlab Group, Cambridge, MA for NASA, Office of Naval Research, U.S.Department of Energy, and U.S. Air Force. [5] Advanced Continuous Simulation Language. Concord, MA: Mitchell and Gauthier Associates, 1986. [6] W. H. Hayt, Jr. and J. E. Kemmerly, Engineering Circuit Analysis, 2nd ed. New York: McGraw-Hill, 1971. [7] Z. J. Cendes and A. Konrad, Elecrric Machine Modeling and Power

System Constraints in Electric and Power Problems: The Mathematical Challenge, A. M. Erisman, K. W. Neves, and M. H. Dwarak-

I

anath, Ed. Philadelphia: SIAM, 1980, p. 460. 181 J. Weiss and Z. A. Cendes, "One-step finite element method for multiconductor skin effect problems," presented at the IEEE PES 1982 Winter Meeting, New York, NY, paper 82WM102-2, Jan. 1982. [91 "MAXWELL," developed by Ansoft Corp., Pittsburgh, PA, version 4, 1989.

[lo] D. H. Lowther and P. P. Silvester, Computer-Aided Design in Magnerics. Berlin: Springer-Verlag. 1986.

Note: Richardson Electronics, Ltd. recently purchased the Amperex power tube line, but they were allowed to retain use of the Amperex name. For more information, call (800) 348-5580. Lawrence R. Egan received both the B.S. and M.S. degrees in electrical engineering from Rochester Institute of Technology, Rochester, NY, in 1980 and 1987, respectively. His academic areas of concentration have been in electromagnetics and high-frequency, high-power devices. From 1981 to 1990 he was employed by the Manufacturing Research and Engineering Organization, Eastman Kodak Company, Rochester, NY. During that time, he developed induction, dielectric, and microwave heating systems for manufacturing and packaging. Mr. Egan is a Past President of the New York Pi chapter of Tau Beta Pi, and is currently a member of the International Microwave Power Institute. Edward P. Furlani received the B.S. degree in electrical engineering in 1977, the M.S. degree in physics in 1980, and the Ph.D. degree in physics in 1982, all from the State University of New York at Buffalo. His academic background is primarily in the area of mathematical physics, with an emphasis on quantum mechanics and quantum field theory. Since 1982 he has been employed at the Research Laboratories, Eastman Kodak Company, Rochester, NY. His research activities have included work in the areas of CCD scanning simulation, linear systems analysis, magnetic recording analysis, and applied partial differential equation theory. His current research interests are in the field of applied electromagnetic analysis. His experience in this area includes the analysis and design of electromechanical devices, the design of magnetizing fixtures, the analysis of high-energy permanent magnets, the analysis of nonuniformly magnetized materials, and the computation of device impedance using fieldbased techniques.