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IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 2 , MARCH 1997
AStudy of ImpedanceAnalysis foran Induction Heating Device by Applying a New Interpolation Method TetsuyaImai and Kazuyuki Sakiyama Central ResearchLaboratory, MatsushitaElectricIndustrialCo., Ltd. 34 Hikaridai,Seika, Soraku, Kyoto61902,Japan IzuoHirota and Hideki Omori Home ApplianceLaboratory, MatsushitaElectricIndustrial Co., Ltd. 228Hinode,Toyonaka,Osaka, 561, Japan Abstract  We developeda practicalmethod to obtainimpedanceof induction heating structure of rice cookers. It is based on axisymmetricanalysisusing interpolationmethod, and we focused on the relationbetweenthe volume of the core structureand the coil impedance. We can obtain accurateresultsfor short computational timecomparedwith3D analysis. Since the discrepancy between the experimental resultsand the simulationresultsis less than 8%,we successfully applied this method to impedance analysis of the inductionheatingstructureofrice cookers.
ashortcomputationalt ime,andthelatter is for accurate calculation using the axisymmetriicanalysis. Then, we will discuss validity ofthephysicalassumplionsofthe proposedmethodafterclarifying the origins of approximation errors obtainedfromthe conventional axisymmetricanalysis Theaccuracy ofthismethodisinvestigated by comparingwith experimentalresults. Finally,we will illustrate the effectiveness of this method and determine the limitation by comparingwith the conventional 3D analysis. 11. I~MPEDANCE
ANALYSIS
I. INTRODUCIION A. Rice CookerHeatingStructureand theAnulysis Conditions Recently inductionheating has been widely applied to many kinds of cookware such asrice cookers,because inductionheating has highheatingefficiency,rapid temperaturecontrol,safety,and ease ofutensil cleaning. Induction heating cookware consists of two parts: a heating structure,which is composedof aheatedplate andaheatingcoil, andaninverterwhichsupplieshighfrequency powerto theheating coil. This design requires optimizations of both the heating characteristics and the impedance matching between a heating structureandaninverter[ 11.Weappliedthefiniteelementmethod to magnetic field analysis to controlthe highfrequency magnetic field around aheatingstructure,resultingin improving the heating characteristics. To match impedancebetween a heating structure andaninverter,we coulduse simulationswhichevaluateintegrated circuits. We, however, had to repeat a great number of trial experiments to determine the circuit parameters of a heating structurebecause the circuit parameters are influencedby various conditions, such as the shape of the components, component arrangements,materialproperties. Insteadofthe trial experiments, we could apply 3D analysis [2][4]to determine the parameters becausecookwarehascomplicated3Dstructure,butitneedsaIong computationaltime and alot of operations in preparing the finite element meshto obtainanaccurateresult. In thispaper, we will propose anewpracticalcalculationmethod usingthemagnetic field analysissimulationwith the finite element method to eliminatethe conventionaltrial experiments.To address 3D issues,we will introduce an axisymmetric analysis with an interpolationmethod. The former is because thestructures of rice cookers are arrangedmostly in misymmetrical order and it needs
Fig.1 shows an induction heating structure of a rice cooker. The structure consist!;of a doublelayer pot made of aluminum and stainlesssteel, aheating coil, and eight ferrite cores arranged in all directions as illustrated in Fig.2. Permeability and conductivity of components of the heating structureare assumed tobeconstantasshowninTable 3. Theimpedanceoftheheating structure is calculatedlwhen the heating coil is excited by a small current. Thus this anlalysis does not consider either effects of magnetic saturation or temperature on the permeability and conductivity of components of the heating structure.
\p /d\
,â€™ ..

(a) plane view
Fig.1 Induction heating structure. (vertical section)
(b) vertiial section Fig2 Core structure.
TABLE 1 MATERIAL FROPERTTES
Material
lielative Permeability
aluminum
1.o
3.57X10
stainless steel
100.
1.667 X lo6
ferrite cores
600.
0
00189464/97$10.00 0 1997 IEEE
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Conductivity (S/m)
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B. Method to Obtain Impedance from Magnetic FieldAnalysis Results
The formula leading to the impedance Z from the magnetic vector potentialAwhich is obtainedfrom themagneticanalysis is as follows: Z=R+ jwL
_ jwn
If exiting current I, is assumed to be constant, the inductance L proportional to the magnetic energy W, and the resistance R proportional to the eddy current loss P(Eqs.(2) and (3)) are overestimated.
@
(3)
Note that Sc and c represent the cross section of the coil and the pathalongthe flow directionof the excitingcurrent,respectively. j , w, nc, @, Io ,and A,, are the symbols which represent the imaginary unit, angular frequency, number of ampere turns, magneticfluxlinkage passingthroughthe coil, exciting current, and the normal component of A for Sc. The flux linkage @ in Eq.(l) is the average flux passing through Sc [5]. C. Preparation Investigation BeforeApplying the Interpolation Method
In this section, we will clarify the conditions in order to obtain the accurate impedance of the rice cookerheatingstructureusing the axisymmetric analysis. Let us assume that the core is cylindrically shaped in order to use the axisymmetric analysis. Fig3 shows the errors between the analysis and experimental results for various frequencies from 20kHz to 200kHz. The errors in both the resistance R and the inductance L are more than 20%, where R and L are derived from Eq.(l). Since the core is assumed to be cylindrical but the real core is composed of eight pieces of ferrite, the estimated volume of the core structure is more than the actual volume, Thereforethcse errors come from the following : (i) Analyzed flux densities around the core structure and the magnetic flux linkage passing through the doublelayer pot are overestimated by overestimation ofthe corevolume. (ii)As aresult of (i),boththe analyzedmagneticenergyW inthe wholeregionandthe eddy currentlossp induced by thefluxlinkage @ are overestimated.
To investigate the effect of the shape of the core structure on the impedance Z using the axisymmetric analysis results, we have considered two extreme core structures, cylindrical shape (Case A) and no core (Case B). Fig.4 shows resistance R and inductanceLas afunctionoffrequency for bothcase AandCase B. The difference of results between Case A and Case B is from 18 9% to 42% and is shown by hatching in Fig.4 impedance of the heating structure must lie in somewhere between the results of Case A and Case B.Thus, we can realize that the accurate R and L can not be acquired directly from Z, and Z, whichrepresent the impedances for Case A and Case El respectively.
0
50 100 150 200 frequency Wz) (a) resistance R
0
50 100 150 200 frequency (MIz) (b) inductance L
Fig.4 Effect of the shape of the core structure on the impedance of the heating structure.
D.Impedance Interpolation Method Sincethe actualimpedance ofthe heatingstructureliesbetween theresults ofCaseAandCaseBas shownabove, wecouldassume that the increment of the impedance is in proportion to the volume of the core. According to this assumption, the impedance Z can be represented by the linear interpolationformula with k as shown in W4). V Z(k)=kZ,+(lk)Z, k = vA (0s kll)] (4)
(
40 30
20 10 p=Qo
U 0
50
100
150
200
frequency (MIz) Fig.3 Error E based on the assumption that the core is cylindrically shaped.
Note that V ,V,, and k stand for the volume of the actual core structure, the volume of Case A, and the ratio of the volume of actual core structure to that of Case A, respectively. To validate this approximation, we compared the impedance obtained from this interpolation method with experimental results. F i g 5 shows the error of the impedance for various frequencies from 20kHz to 200kHz. The discrepancy between the results of this approximation and the experimental results is
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B. Effect of the Core Volume on the Impedance

E
4
W
2
0
50
100 150 200 frequency (IrHz)
Fig.5 Accuracy of the impedance using the interpolation method.
less than 8 9%. Therefore, we can conclude this interpolation method is practical and applicable from the view point of accuracy. 111. VERIFICATION
In order to figure out the effectiveness of this interpolation method, we have investigated the origin of the errors and the limitationof theassumptionusingthe3D analysiswhichtakesthe shape of the heating structure into account.
Here, we will discuss the effect of the core volume on the impedance quantitatively. Fig.7 shows the impedance as a functionof core volunne. The results were calculated by the 3D analysis which considlersthe exact shape of the analyzedmodel, or the actual values. Tlie hatched area in Fig.7 representsthe error due to the interpolation method. The increment of the impedance is saturatedwith increasing the core volume, as shown in Fig.7. Let us explain this phenomena by using 1he magnetic energy which is proportional to the inductance if we assume the heating coil is excited by a constant current I,. Most of the magnetic energy W is storedin the air and coil region as shown in Fig.8. When the core volume is increasing,wehaveto thinkoftwoopposite effects: oneisincrease of W by increasing the magneticflux linkage @ in the air, and the other is decreaseof Wby decreasingtheairregion.Sincethe former effect is1argerthanthc:lattereffectwhenthecorevolumeissmall. On the other hand, the former effect is getting small as k+l, resulting in thesaturationofthe W increment with increasingthe
A. Analyzed Model and the Analysis Conditions An analyzedmodelfor 3D analysisis composedof flat singlelayer pot made of stainlesssteel, flat heatingcoil, and four ferrite cores as illustrated inFig.6. It is simpler than the real rice cooker depictedinFigs.1 a n d 2 Thefrequencyforthismodelwas50Hz. The conductivityandpermeabilitygivento this model werethe same as the rice cooker model (Table 1).The volume of the four cores was changedby varying thewidthalongthe circumferenceofthe core. stainless steel 1
i
4
1W
0
0.2 0.4 0.6 0.8 1.0 k (a) resistance R
0.0
0.2 0.4 0.6 0.8 1.0 k (b) inductance L
Fig.7 Efftct of the core volume on the impedance using the 3D analysis.
4
c
S I
v _. n
ferrite cores
with interpolationmethod
28
I
4
t 64

I
oq
"
A, 0.00
3
,4
2
0.2 0.4 0.6 0.8 1.0
k Fig8 Magnetic energy distrubution of the analyzed model.
@) plane view of ;he core structure
in (c) vertical section of the core structure Fig.6 Analyzed model.
0.0
0.2 0.4 0.6 0.8 1.0 k
Fig.!) Estimation of the impedance error using the interpolationmethod.
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E
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core volume. Next, Fig.9 shows the error caused by the interpolationmethod. In our analyzedmodel, thelargest error ratio, 15%,is obtainedwhen kis about 0.2. Finally, let us discussthe inductance error betweenthe rice cooker model and this analyzed model. The tendency of the error due to this interpolationmethodwasinvestigatedbycomparingtheresults from the analyzedmodel and the rice cooker model. Figs.8 and 10show themagnetic energystored in eachcomponent of the analyzed model and the rice cooker model respectively. Figs.10 (a) and(b) correspond to CaseAandCaseB, respectively. Figs.8 and 10suggest that theratio ofthemagneticenergystoredin each component of the rice cooker model is different from that in the analyzed model. On the other hand, Fig. 11shows that the error of magnetic energy stored in the air in the analyzedmodel tends to distributesimilarly to that in the stainless steel. Eveniftheratio of the components is changed, the tendency of the error causedby this interpolationmethod is not variedvery much. Therefore, we can concludethat theerrortendency oftherice cookermodelagreesfairly to that of the analyzedmodel. Note that the heating structure of a rice cooker is an examplewithalargeerrorbecausekis equalto0.15.
50
0
100 150 200
0
50
100 150 200
frequency (kHz) (b) Case B
frequency (kâ‚ŹIz) (a) Case A
Fig.10 Magnetic energy distribution of the rice cooker heating structure,
IV. CONCLUSIONS In this paper, we have proposed a method to calculate the impedance of a rice cooker heating structure quickly and accurately. We combined the axisymmetric analysis with an interpolation method. The ' tric analysis has an olationmethodis used advantage of aquickcalculation to calculate the impedance ac om the results of the axisymmetric analysis, and we assumed that the increment of the impedance is in proportion to the volume of the core structure. This assumption is verified using the 3D analysis. We found that the errors between the results of the proposed method and the experiment is less than 8 9%. Therefore, we conclude that the axisymmetric analysis with an interpolation method is practical and applicable to calculation of the impedance of the heating structure. REFERENCES [l]H. Omori and M. Nakaoka, "New singleended resonant inverter circuit
and system for inductionheating cooking apparatus, " ICS,vo1.67, no.2, pp.277296,1989.
[2]Y.Kawase andT.Yamaguchi, "3D finiteanalysis for molten metalshapes in an electromagnetic melting system, " IEEE Trans. Magn., vol. 29, no. 2, pp. 1554.1557, March 1993. [SI A.Radovinsky, R.Pillsbury, Jr., and J.Schultz, "Eddy current heating in the cold structure in TPX, " IEEE Trans. Magn., vol. 30, no. 5, pp. 37013704, September 1994. [4]RJurgens, U. Ludtke and D. Schulze, "Threedimension the distribution of eddy currents and the heating effect on slit tubes when weldinglongitudinal seams, " IEEE Trans. Magn., vol. 3 3708, September 1994. [5] T.Nakata, N.Takahashi, K.Fujiwara and A.Ahagon, "3D finite element method for analyzing magnetic fields in electrical machines excited from voltagesources, " IEEE Trans. Magn., vol. 24, no. 6, pp. 25822584,1988.
I
0.0
I l l
I l l
0.2 0.4 0.6 0.8 1.0 k
0.0
0.2 0.4 0.6 0.8 1.0
k
(a) air and heating coil @) stainless steel Fig.11 Magnetic energy stored at each region in the analyzedmodel.
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