IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 2 , MARCH 1997 2143

AStudy of Impedance Analysis for an Induction Heating Device by Applying a New Interpolation Method

TetsuyaImai and Kazuyuki Sakiyama Central ResearchLaboratory, Matsushita Electric Industrial Co., Ltd.

3-4 Hikaridai, Seika, Soraku, Kyoto 619-02, Japan

IzuoHirota and Hideki Omori Home Appliance Laboratory, MatsushitaElectricIndustrial Co., Ltd.

2-2-8Hinode,Toyonaka, Osaka, 561, Japan

Abstract - We developed a practical method to obtainimpedance of induction heating structure of rice cookers. It is based on axisymmetric analysis using interpolation method, and we focused on the relation between the volume of the core structure and the coil impedance. We can obtain accurate results for short computational timecompared with3-D analysis. Since the discrepancy between the experimental resultsand the simulation resultsis less than 8%, we successfully applied this method to impedance analysis of the induction heating structure ofrice cookers.

ashortcomputational t ime,andthelatter is for accurate calculation using the axisymmetriicanalysis. Then, we will discuss validity ofthe physicalassumplions ofthe proposedmethodafterclarifying the origins of approximation errors obtainedfrom the conventional axisymmetricanalysis Theaccuracy ofthismethodisinvestigated by comparingwith experimental results. Finally, we will illustrate the effectiveness of this method and determine the limitation by comparingwith the conventional 3-D analysis.

11. I~MPEDANCE ANALYSIS I. INTRODUCIION

A. Rice CookerHeating Structureand theAnulysis Conditions Recently induction heating has been widely applied to many

kinds of cookware such as rice cookers, because induction heating has highheating-efficiency, rapid temperature control, safety, and ease ofutensil cleaning.

Induction heating cookware consists of two parts: a heating structure, which is composedof aheatedplate andaheatingcoil, andaninverterwhichsupplies high-frequency 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 control the high-frequency magnetic field around aheating structure,resulting in improving the heating characteristics. To match impedance between a heating structure andaninverter,we coulduse simulationswhichevaluate integrated 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,material properties. Insteadofthe trial experiments, we could apply 3-D analysis [2]-[4] to determine the parameters becausecookwarehas complicated3-Dstructure,butitneedsaIong computational time and alot of operations in preparing the finite element meshto obtainanaccurate result.

In this paper, we will propose anewpractical calculationmethod usingthe magnetic field analysis simulation with the finite element method to eliminate the conventional trial experiments. To address 3-D 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 double-layer pot made of aluminum and stainless steel, aheating coil, and eight ferrite cores arranged in all directions as illustrated in Fig.2. Permeability and conductivity of components of the heating structure are assumed tobeconstantasshowninTable 3. Theimpedanceoftheheating structure is calculatedl when 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 \ .. -

,’

(a) plane view

Fig.1 Induction heating structure. (b) vertiial section (vertical section) Fig2 Core structure.

TABLE 1 MATERIAL FROPERTTES

Material lielative Permeability Conductivity (S/m)

aluminum 1 .o 3.57X10

stainless steel 100. 1.667 X lo6

ferrite cores 600. 0

0018-9464/97$10.00 0 1997 IEEE

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B. Method to Obtain Impedance from Magnetic FieldAnalysis Results

The formula leading to the impedance Z from the magnetic vector potential Awhich is obtainedfrom themagneticanalysis is as follows: Z= R+ jwL - jwn @ _-

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, magnetic fluxlinkage 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 cookerheating structure using 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, Therefore thcse errors come from the following : (i) Analyzed flux densities around the core structure and the magnetic flux linkage passing through the double-layer pot are overestimated by overestimation ofthe corevolume. (ii)As aresult of (i), boththe analyzedmagneticenergy W inthe wholeregionandthe eddy current lossp induced by theflux linkage @ are overestimated.

40 30

20 10 p=Q---o U

0 50 100 150 200 frequency (MIz)

Fig.3 Error E based on the assumption that the core is cylindrically shaped.

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)

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 0 50 100 150 200 frequency Wz) frequency (MIz)

(a) resistance R (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 heatingstructurelies between 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 interpolation formula with k as shown in W 4 ) .

V Z(k)=kZ,+(l-k)Z, ( k = - vA ( 0 s k l l ) ] (4)

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 ig5 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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stainless steel 1W 4 1 c i

- E 4 W

2

4 S I

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.

n 28 v _. ferrite cores I

111. VERIFICATION

t I A ,

oq " 3

,4 4 2 64 -

In order to figure out the effectiveness of this interpolation method, we have investigated the origin of the errors and the limitationof theassumptionusingthe3-D analysiswhichtakesthe shape of the heating structure into account.

A. Analyzed Model and the Analysis Conditions

An analyzedmodel for 3-D analysis is composedof flat single- layer pot made of stainless steel, flat heatingcoil, and four ferrite cores as illustrated inFig.6. It is simpler than the real rice cooker depictedinFigs.1 and2 Thefrequencyforthismodelwas50Hz. The conductivity and permeability given to this model werethe same as the rice cooker model (Table 1). The volume of the four cores was changedby varying thewidthalongthe circumferenceofthe core.

@) plane view of ;he core structure

in (c) vertical section of the core structure

Fig.6 Analyzed model.

B. Effect of the Core Volume on the Impedance

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 3-D analysis which considlers the 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 saturated with increasing the core volume, as shown in Fig.7. Let us explain this phenomena by using 1 he 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 magnetic flux linkage @ in the air, and the other is decreaseof Wby decreasingtheairregion. Since the former effect is1argerthanthc:lattereffectwhen thecorevolumeis small. On the other hand, the former effect is getting small as k+l , resulting in thesaturationofthe W increment with increasingthe

with interpolation method

0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 k k

(a) resistance R (b) inductance L

Fig.7 Efftct of the core volume on the impedance using the 3-D analysis.

0.00 0.2 0.4 0.6 0.8 1.0 k

Fig8 Magnetic energy distrubution of the analyzed model.

0.0 0.2 0.4 0.6 0.8 1.0 k

Fig.!) Estimation of the impedance error E

using the interpolation method.

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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 discuss the inductance error between the rice cooker model and this analyzed model. The tendency of the error due to this interpolationmethodwasinvestigatedby comparing theresults from the analyzedmodel and the rice cooker model.

Figs.8 and 10 show themagnetic energy stored in eachcomponent of the analyzed model and the rice cooker model respectively. Figs.10 (a) and(b) correspond to Case AandCaseB, respectively. Figs.8 and 10 suggest that theratio ofthemagneticenergystoredin each component of the rice cooker model is different from that in the analyzed model. On the other hand, Fig. 11 shows 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 conclude that theerrortendency oftherice cookermodelagrees fairly to that of the analyzedmodel. Note that the heating structure of a rice cooker is an examplewithalargeerrorbecausekis equal to0.15.

0 50 100 150 200 0 50 100 150 200

frequency (k€Iz) frequency (kHz) (a) Case A (b) Case B

Fig.10 Magnetic energy distribution of the rice cooker heating structure,

I I l l I l l 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

k k (a) air and heating coil @) stainless steel

Fig.11 Magnetic energy stored at each region in the analyzedmodel.

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 advantage of aquickcalculation olationmethodis used 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 3-D 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 single-ended resonant inverter circuit and system for induction-heating cooking apparatus, "

ICS,vo1.67, no.2, pp.277-296,1989. [2] Y.Kawase andT.Yamaguchi, "3-D 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. 3701-3704, September 1994.

[4] RJurgens, U. Ludtke and D. Schulze, "Three-dimension 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, "3-D finite element

method for analyzing magnetic fields in electrical machines excited from voltagesources, " IEEE Trans. Magn., vol. 24, no. 6, pp. 2582-2584,1988.

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