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IEEE TRANSACTIONS ON MAGNETICS. VOL 35. NO 5, SEPTEMBER 1999
A Dynamic Model for the Simulation of Induction Heating Devices J. Nerg, K. Tolsa, P. Silventoinen, J. Partanen and J. PyrhSnen Department of Electrical Engineering Lappeenranta University of Technology, P.O.Box 20,53851 Lappeenranta, Finland Abstract-A simulation procedure designed for the dynamic analysis of induction heating system is described, The procedure starts from the FEM based evaluation of the load impedance. This, as a function of the heating time is transferred to the d p uamic model describing the whole induction heating device, i.e. the power supply, its control system, the load and the impedance matching circuit. With the developed model the start-up of the heating process ns well as quick transients, e.g. fault situations can be examined. The applicabilityof the model was tested in the design of the induction heating installation developed for the annealing of aluminium plates. Temperature error less than 5 % has been achieved.
Index Terrns-Eddy currents, finite element method, induction
11. NUMERICAL CALCULATION OF THE LOAD PARAMETERS
In order to solve the dynamic behaviour of the induction heating load and the temperature profile of the workpiece, a FEM based analysis is used. A numerical analysis of coupled electromagnetic and thermal fields is carried out by solving the Maxwell and Fourier-Kirchhoff equations with two separate commercial finite element software packages. The electromagnetic field is assumed to he time-harmonic, which allows the employment of a complex vector potential A for the analysis. Hence, the task is reduced to solve the mutually coupled equations:
heating.
I. INTRODUCTION Induction heating is an efficient, easily controlled method for the heating of electrically conductive objects in processes such as metal hardening or annealing [l]. The analysis of an induction heating device is a complex process because induction heating is a combination of heat transfer and electromag,netics. They are'tightly inter-related since the heat sources are of electromagnetic origin. Furthermore both the electromagnetic and thermal material properties are highly temperature dependent. This makes the problem extremely severe, because the temperature dependence of the material properties causes that the induction heating load, i.e. the electrotechnical system formed by the induction coil and the workpiece, changes its properties during the heating cycle [2]., In order to design an optimal induction heating device for a new application a detailed analysis of the specifications to the process as well as the calculation of the dynamic behaviour of the load impedance must he performed [3,4]. The knowledge of the dynamics of the load can then he utilised in the design of a power supply, its control system and associated impedance matching circuit. In this paper, a dynamic model for the design of induction heating devices is presented. The model is based on the numerical evaluation of the load impedance during the heating cycle. The calculated load impedance as a function of the time is transferred to the model describing the whole induction heating device, i.e. the power supply, the control system, the load and the impedance matching circuit. Manuscriot received March 5. 1999: revised 29.4.1999.
AV2T+w= fmass
dT dt
cp-'
where v is the reluctivity, a is the conductivity, w is the angular velocity, J is the current density, h is the thermal conductivity, T is the temperature, w is the heat source density, pmn8is the mass density, cp is the specific heat and f is time. The electromagnetic field problem can also be handled in 3D using the edge elements [SI and the T-Q formulation 161. Equation (2) is solved on the following boundary conditions at the surface of the workpiece [7]:
-a- d T = asp-T,)+ cs(,4- T:) dn
,
(3)
Olllpllt
* ele~rncslpnrametet8 * Icmpuaturure distribution
t Input
Magnetic field CBICUIRUO~
Thermal 8eId calculnllon
Fig. 1. Flow chart of the iterative calculation procedure for coupled electmmaenetic and thermal oroblems. The innut consists of the inductor current.
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From the electromagnetic solution the induced power distribution within the workpiece is extracted. The power densities for each of the elements are then used together with the initial nodal temperatures as the input for the transient thermal field calculation, from where the new nodal temperatures are extracted. Tabulated material properties are used in the thermal field calculation in order to take the temperature dependence of the thermal material properties into account. Before performing a new magnetic field calculation the electromagnetic material properties are updated to correspond to the calculated temperature distribution. This iteration is continued until the heating cycle ends. To obtain the dynamic behaviour of the load impedance the resistance Rla and the inductive reactance X,, are then calculated at every time step of the electromagnetic solution using the equations:
(4) (BHdV
For the design of the control system a simplified lumped parameter representation of the induction heating installation is formulated (Pig. 3). The switching converter is replaced by a sinusoidal voltage supply with a constant frequency and amplitude. Fig. 3 contains, however, enough information about the heating installation so that the control loops can be tuned up by a simple transfer function analysis,
Fig. 3. A simplified representation of the load, the impedance matching circuit and the power supply. LI is the inductance of the matching inductor, RI is the resistance of the matching inductor and C is the capacitance of the rnsonant capacitor. The constant parameter values used in the sirnulations C=62.l9 WF, L1=22.8WHand RI=% mQ.
The inductor current I,, can be represented by the transfer function
J
Xld
= 2nf
-r,
(5)
IZ
IOU,
where is the volume containing the load and air areas, i.e. the total volume, p is the resistivity,fis the frequency and I is the effective value of the coil current. It must be noticed that the CUNent density J and resistivity P are valid for each merit value. In 2D solution the volume inteaal is simply the z-axis rotation Of the calculated RZ-plane (axisYmetric geometry) Or the multiplication Of the calcuhted XY-plane with the length Of the device geometry). 111. MODELFOR THE POWER SUPPLY, ITS CONTROLSYSTEM
(4 =
U i n (3)
Rid
+ L i d s + (RI + &s)(l+
RI, Cs+ Lld CS2)
. (6)
The control algorithm of the power supply is built up of two feedback loops, shown in Fig. 4. The first loop, i.e. the power control loop, compares the initial value of the voltage amplitude of the circuit with a predetermined amplitude reference waveform. The second loop prevents the dc-component which results from a small unbalance between different power transistors from building up at the output of the converter, The dc-component is measured from the output of the converter. This current signal is used to control the duty cycle of the transistor pairs.
AND THE IMPEDANCE MATCHING CIRCUIT
A schematic diagram of the switched-mode power supply used is shown in Fig. 2. The load is connected to a parallel resonant RLC-circuit, where the reactive power required by the inductor is produced in the capacitor. The matching inductor serves as a voltage divider and also as a current rate limiter for the power transistors.
Voltago amplilude of boresonua Cimil
Fig. 4. A block diagram of the control system.
0
IV. RESULTS
0==
._.
Fig. 2. A schematic diagram of the voltage-fed parallel resonant converter used. (a) is a three-phase rectifier, (b) is the smoothing reactor, (c) is the intermediate cizuit. (d) is the inverter. (e) is the matching inductor, (f) is the resonant capacitor and (g) is the load circuit consisting of the induction coil and the workpiece. The power supply and the impedance matching circuit are designed sa that the maximum inductor cumnt is 1500 A at a frequency of 15.3 kHz.
The developed dynamic model was tested during the designing o f an induction heating installation for the annealing of aluminium disks. The disks, 200 mm in diameter and 4 mm thick were heated with a series connection of two spiral inOf where the plate heated is placed between the inductors (Fig. 5 ) . The inductor is made of a fully annealed, high conductivity square (10 m X 10 mm, 0 6 mm) copper tubing, which is water cooled. The frequency was
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15.3 kHz and the inductor current was 570 A, supplied by the nature of the current is caused by the operation of the conpower supply shown in Fig. 2. Because of the structural sym- troller. metry a two-dimensional analysis was used. The number of imn I I I I I I I I elements was 10 181 in the electromagnetic analysis and 1 830 in the thermal analysis. The heating time was 30 s and the time-step size in thermal analysis was chosen to be 0.5 s. 3 '18 +
4
4
4
5
ti ti --h t (
I
3 I
n
I
I
'
UIPJ
I
I
z
1.5 Ti.D
1
'98
0.1
3
2,s
1.3
I
(mr)
Fig. 7. Simulated inductor current.
Fig, 5. The finite element mesh of the inductor-workpiece system in the R Z plane. Air areas are not shown. The distances a x in millimetres.
The measurement of the disk centre- and edge point temperatures was carried out with a Sensytherm IR-L6 infra-red temperature indicator, the documented uncertainty of which is 2OC. Measurements were repeated five times in order to minimise contingencies. The calculated and measured temperature values at the end of the heating cycle from the test case are shown in table 1. The error between the measured and calculated temperatures is less than 5 %. TABLE I CALCULATED AND MEASURED WORKPIECE TEMPERATURBS Calculated 198 "C 272 "C
The centre point (R = 0 mm) The edge point (R = 100 mm)
Measured 204 "C 267 "C
The input for the dynamic model describing the whole induction heating device, i.e. the calculated load impedance as a function of the heating time, is shown in Fig. 6.
n
I
I
-isno I os
I
1.3
z
2.3
3
3.3
cl
Timo (mi)
Fig. 8. Measured inductor current.
V. CONCLUSIONS The simulation procedure presented in this paper is proven to be a very efficient tool in the overall design of the induction heating devices. The calculated workpiece temperatures as well a i the simulated inductor current khowed to- be in good agreement with the measurements. The further publications will concentrate on the extending the model so that the load is actively compensated and that the frequency of the power supply is adapted to the load variation. REFERENCES 111 .. I. Davies and P. Simoson. Induction Heatine Handbook. London: McGraw-Hili. 1979. .121. T. P. Skoczkowski and M. P. Kalus. 'The mathematical model of induction heating of ferromagnetic pipes", IEEETrans. M a p ,vol. 25, no. 3., pp, 2745.2750, May 1989. [3] R. S. Ruffini, R. T. Ruffini and V. Nemkov, "Advanced design of induction heating coils", 1" lnt. Heat Treating Symposium, Indianapolis. 16-18 September 1997. (41 P. Siiventoinen, 1.Nerg, J.Pyrh0nen and J. PManen, "DC-link converters - effective power supplies for induction heating", in Proc. 1998 B E E Nordic Workshop on Power and Industrial Electronics, pp, 143-146, Espoo. Finland, 26-27 August 1998. [SI 1. P. Webb, Edge elements and what they can do for you", IEEE Truns. M a p , vol.. 29, no. 2., pp. 1460-1465, March 1993. [6] J. P. Webb and B. Forghani, T O method using hierarchal edge elements". IEE Proc.-Sci. Mear.Techno1. vol. 142, no. 2., pp. 133.141, March 1995. [7] F. Incropera and D.DeWitt, Fundamentals of mass and heat vansfer, 4" ed., New York Wiiey, 1996. ~
0
s
10
zn
IS Hcnllng ,,r"B
2s
in
(8)
Fig. 6. Calculated load impedance as a function of the heating time.
The simulation of the whole device was done using medium order Runge-Kutta method with a constant step of 2 p. The simulated and measured inductor current waveforms in steady state for 4 ms period are represented in Figs. 7 and 8. The peak value of the simulated inductor current is 1097 A and the corresponding measured peak current is 1010 A. The wave
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