Straightforward Modeling of Complex Permeability for ...

IEEJ Journal of Industry ApplicationsVol.7 No.6 pp.462–472 DOI: 10.1541/ieejjia.7.462

Paper

Straightforward Modeling of Complex Permeabilityfor Common Mode Chokes

Katsuya Nomura∗a)Member, Takashi Kojima∗ Member

Yoshiyuki Hattori∗ Non-member

(Manuscript received Oct. 16, 2017, revised March 19, 2018)

This paper presents a straightforward modeling method for the complex permeability of common mode chokes.The proposed model is obtained as an RL ladder network from a single RL parallel equivalent circuit by repeatedmanipulation called circuit dividing. Circuit dividing is the transformation of a single parallel RL circuit into twoseries-connected parallel RL circuits, and is used for expressing the complicated characteristics of inductive and resis-tive elements such as frequency-dependent complex permeability. The proposed model can be made to basically repeatthe parameter fitting up to two parameters, so that the fitting procedure is straightforward. This is a great advantagefor engineers because the model can be obtained without the need for special additional optimization programs, whichwere required in previous works. By using this model, the common mode impedance of a choke is fitted precisely,compared to the conventional single parallel RLC equivalent model. Moreover, not only the insertion loss of the EMIfilter but also a time waveform is simulated precisely by both frequency and transient analysis with the proposed model.

Keywords: common mode choke, electromagnetic interference, equivalent circuit, filter

1. Introduction

Electromagnetic noise due to power conversion circuits,such as inverters, may cause electromagnetic interference(EMI) with other electronic equipment. EMI filters are usedto reduce this noise, and common mode chokes, shown inFig. 1, are one of the main components of these filters. Thechoke consists of two windings and a magnetic toroidal core.The choke behaves as an inductor for the common mode cur-rent because the two fluxes created by the two line currentsenhance each other, but it does not behave as an inductor fordifferential mode current because the two fluxes cancel eachother out. Though there is a some leakage inductance due toleakage flux to the air, this leakage is outside the scope of thisstudy and not considered.

Circuit simulation is useful for efficient design of EMI fil-ters, but effective simulation requires precise circuit modelsof filter components. Models of common mode chokes mustdescribe common mode impedance, which is proportional tothe magnetic permeability, but this permeability is complex-valued and also frequency-dependent. As explained in thenext section, many models for common mode chokes havebeen developed to describe complex permeability. In addi-tion, some models using a resistance-inductance (RL) laddernetwork enable precise fitting of the permeability. However,modeling procedures of these models are obscure or requirean additional program for parameter extraction, so an easiermodeling method is required for wide use. This paper pro-poses a straightforward modeling method using an RL ladder

a) Correspondence to: Katsuya Nomura. E-mail: [email protected]∗ Toyota Central R&D Labs., Inc.

41-1, Yokomichi, Nagakute, Aichi 480-1192, Japan

Fig. 1. Common mode choke

to describe complex permeability for common mode chokes.The structure of this paper is as follows. Section 2 reviews

existing equivalent circuit models of complex permeabilityand clarifies the modeling challenges. In section 3, we ex-plain the main ideas behind the proposed modeling method.Section 4 illustrates the modeling procedure by describingand using the model for an actual choke. In section 5, exper-imental verification is carried out both for frequency analysisand time domain analysis. Finally, section 6 provides a con-clusion.

2. Review of Methods for Modeling ComplexPermeability

This paper focuses on precise modeling of frequency-dependent complex permeability like that shown in Fig. 2 (1),which shows the characteristics of FT-3KL cores from Hi-tachi Metals, Ltd. Both the absolute value and real partof permeability stay almost constant at low frequencies,whereas they decrease in the high-frequency region as fre-quency increases. Moreover, the imaginary part of the com-plex permeability exceeds the real part, so it is not negligiblewith respect to the resistive component of impedance, whichis proportional to the imaginary part of the permeability.

Conventional choke models typically use the parallel RLC

c© 2018 The Institute of Electrical Engineers of Japan. 462

Modeling of Complex Permeability for Chokes（Katsuya Nomura et al.）

Fig. 2. Frequency-dependent complex permeability

Fig. 3. Conventional equivalent circuit for a commonmode choke

Fig. 4. Impedance proportional to the complex permeability

Fig. 5. Impedance of the RL parallel circuit. R = 5 kΩ,L = 1.3 mH

circuit shown in Fig. 3 (2). In this circuit, C represents par-asitic capacitance and the parallel RL circuit describes thefrequency-dependent complex permeability. Figure 4 andFig. 5 show the impedance amplitude proportional to thecomplex permeability and the impedance of the RL paral-lel circuit, respectively. In both figures, frequency charac-teristics show that inductance is dominant at low frequenciesand resistance is dominant at high frequencies, but there isa large difference between the two, especially at high fre-quencies. In the impedance proportional to the complex per-meability shown in Fig. 4, resistance R and reactance ZL in-crease with frequency. On the other hand, in the impedanceof the RL parallel circuit shown in Fig. 5, the resistance staysconstant at higher frequencies; in this case, the resistancestays about 5 kΩ over 1 MHz. Moreover, the phase of bothimpedances, shown in Fig. 6, shows large differences evenat low frequencies. Hence, a single parallel RL circuit candescribe complex permeability characteristics qualitatively

Fig. 6. Phase of the impedances

Fig. 7. Equivalent circuit for a common mode choke in-cluding complex permeability

but not quantitatively.It is reported that the equivalent circuit including com-

plex permeability is described as Fig. 7 (3)–(6), which placesthe frequency-dependent inductance and resistance in series.Here, L( f ) and R( f ) represent impedances that are propor-tional to the real and imaginary parts of the permeability, re-spectively, and C represents parasitic capacitance. However,such frequency-dependent inductance and resistance cannotbe modeled in an ordinary way in circuit simulators. Sometechniques have described such frequency-dependent compo-nents as the arbitrary current source or voltage source with ans-function (7) (8), but these models are generally suitable onlyfor frequency analysis, and we need to obtain time wave-forms by use of a numerical Laplace transform inversion,which is relatively time-consuming and sometimes suffersfrom convergence issues (9).

Using a ladder RL network is the other approach for moreprecise fitting of complex permeability. Sullivan (10) proposeda Cauer-type ladder model but used Nelder-Mead optimiza-tion method written with MATLAB to fit circuit parame-ters, so it needs a special program for making the model.Labarre (11) also proposed a Cauer-type equivalent circuit, butthe fitting procedure is not clear. Stevanovic (12) proposed aladder network of RLC parallel circuits for expressing multi-ple resonances rather than modeling of frequency-dependentpermeability, but this model has high degree of freedom sothis model also can describe complex permeability. How-ever, this model also needs an additional optimization pro-gram using a genetic algorithm for parameter fitting. Tan andCuellar (13) (14) proposed a fitting technique based on the versa-tile method called vector fitting (15). In this technique, a mea-sured data is iteratively approximated as the rational functionby solving matrix equations; this function is expressed as theequivalent circuit by the circuit synthesis technique shownin (16). However, again, an additional program is needed toderive the equivalent circuit.

The drawbacks of the conventional models are summarizedas follows.

( 1 ) Single RLC models cannot describe frequency-dependent complex permeability precisely.

463 IEEJ Journal IA, Vol.7, No.6, 2018


( 2 ) Frequency-dependent models with R( f ) and L( f )are precise but undesirable for transient analysis dueto the use of numerical Laplace transform inversion.

( 3 ) Conventional RL or RLC ladder models are preciseand can be used in both frequency and transient anal-ysis, but their modeling process needs an additionalspecial program.

For these reasons, we propose a precise model with astraightforward modeling process that does not require addi-tional programs for parameter fitting and can be used in bothfrequency and transient analysis.

3. Overview of the Proposed Modeling Method ofComplex Permeability

In the proposed model, the equivalent circuit is expressedas an RL ladder circuit. The main difference between theproposed model and previous models is that our model canbe obtained from a single RL parallel circuit by repeatingthe transformation of separating a single RL parallel circuitinto two parallel RL circuits in series to improve the approx-imation quality, whereas the obtained equivalent circuit willbe essentially the same. We call this transformation circuitdividing, and in this section we explain how to use circuitdividing in the fitting method.3.1 Circuit Dividing The application of circuit di-

viding to a single parallel RL circuit is shown in Fig. 8. Thismanipulation facilitates expression of the more complicatedinductive and resistive characteristics shown in Fig. 4 andFig. 6, that cannot be expressed with a single RL parallel cir-cuit. Here, to adjust impedance only in the middle frequen-cies while keeping it constant in the low and high frequen-cies, we apply the constraint that the sum of the resistanceand inductance are equal before and after circuit dividing.

R = Ra + Rb · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (1)

L = La + Lb · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (2)

Without loss of generality, let La ≥ Lb.The influence of circuit dividing on the impedance charac-

teristic is shown in Fig. 9 and Fig. 10. The difference betweenFig. 9 and Fig. 10 is the relationship between Ra and Rb, i.e.,Ra < Rb in Fig. 9, but Ra > Rb in Fig. 10. The two lines forZRLa and ZRLb are the impedances of each single RL parallelcircuit, and the line ZRLab is the summation of them, so cir-cuit dividing has changed impedance from ZRL to ZRLab . InFig. 9(b), the region around 45 degrees is expanded by thecircuit dividing, and in this region, as shown in Fig. 9(a), theimpedance characteristics show a dip. On the other hand, asshown in Fig. 10, the impact of circuit dividing is relativelysmall compared to that of Fig. 9, so in order to enhance the ef-fect of circuit dividing, the following constraints from Fig. 9are always applied.

Ra ≤ Rb · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (3)

La ≥ Lb · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (4)

It is worth noting that, regarding the characteristics ofZRLab , ZRLa with higher inductance is dominant in the low-frequency region, whereas ZRLb with higher resistance isdominant in the high-frequency region. This feature is usedin the fitting process, as explained in the next subsection.

Fig. 8. Circuit dividing

(a) (b)

Fig. 9. Effect of circuit dividing on impedance whenRa < Rb and La > Lb, (a) amplitude and (b) phase

(a) (b)

Fig. 10. Effect of circuit dividing on impedance whenRa > Rb and La > Lb, (a) amplitude and (b) phase

(a) (b)

Fig. 11. Impedance after initial fitting, (a) amplitudeand (b) phase

(a) (b)

Fig. 12. Impedance after circuit dividing once, (a) am-plitude and (b) phase

3.2 Fitting Method Using Circuit Dividing Theimpedance characteristics proportional to the complex per-meability can be fitted by using circuit dividing as follows.First, we determine the frequency region of interest and con-duct the fitting with a single RL parallel circuit. Here, theinductance is adjusted to match the low-frequency charac-teristics, and the resistance is determined by the maximumamplitude of the impedance within the frequency region ofinterest. At this point, the fitting result, shown in Fig. 11, hasa large error both in amplitude and degree. Next, we conductcircuit dividing to improve the fitting precision, but it is dif-ficult to fit over the whole frequency region with one circuitdivision, so only the lower-frequency region is updated, asshown in Fig. 12. It is obvious that circuit dividing improvedthe fitting precision in the low-frequency region, but a largeerror still remains in the high-frequency region. Then, the



(a) (b)

Fig. 13. Impedance after circuit dividing twice, (a) am-plitude and (b) phase

Fig. 14. Transition of equivalent circuits by repeatedcircuit dividing

circuit dividing is repeated. Here, the line labeled “after di-viding 1” in Fig. 12(a) determines the low-frequency charac-teristics and the line labeled “after dividing 2” determines thehigh-frequency characteristics, that corresponds to the lineslabeled “ZRLa ” and “ZRLb” in Fig. 9(a). For this reason, weapply circuit dividing to the RL circuit of line 2 in Fig. 12(a),fixing the RL value of line 1. By this technique, the high-frequency fitting precision is improved while retaining the al-ready high precision in the low-frequency region. As shownin Fig. 13, the fitting precision in the high-frequency regionis improved by the second round of circuit dividing, and thewell-fitted region expands as a result. Thus, the proposedmethod enables us to develop a precise model by repeatedcircuit divisions, each of which expands the region of highaccuracy from the low frequencies to the high frequencies.

The fitting procedure is further explained using the equiv-alent circuit and circuit parameters shown in Fig. 14 as fol-lows. First, fitting is performed for the single RL parallelcircuit having resistance R0 and inductance L0. Next, the firstcircuit division is performed under the following constraints,which are based on the relationships in (1)–(4).

R0 = R1 + Rdiff1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (5)

L0 = L1 + Ldiff1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (6)

R1 ≤ Rdiff1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (7)

L1 ≥ Ldiff1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (8)

Here, the fitting parameters are R1 and L1 because R0 and L0

are given and Rdiff1 and Ldiff1 are determined by R1 and L1.Next, the second circuit division is performed on the paral-lel circuit consisting of Rdiff1 and Ldiff1 under the followingconstraints.

Rdiff1 = R2 + Rdiff2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (9)

Ldiff1 = L2 + Ldiff2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (10)

R2 ≤ Rdiff2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (11)

L2 ≥ Ldiff2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (12)

Here, the fitting parameters are R2 and L2. Thus, withR0, . . . ,RN−1, L0, . . . , LN−1 known, circuit dividing is per-formed on the parallel circuit with RdiffN−1 and LdiffN−1 underthe following constraints.

RdiffN−1 = RN + RdiffN · · · · · · · · · · · · · · · · · · · · · · · · · · (13)

LdiffN−1 = LN + LdiffN · · · · · · · · · · · · · · · · · · · · · · · · · · (14)

RN ≤ RdiffN · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (15)

LN ≥ LdiffN · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (16)

Here, the fitting parameters are RN and LN . This circuit divid-ing process is repeated until the fitting precision has improvedsufficiently for the frequency region of interest.

The proposed model has many circuit parameters, but it isnot necessary to optimize all of them at the same time basi-cally; the model requires the fitting of up to two parametersrepeatedly, which is simple and easy. This is the model’sgreat advantage for hand-fitting because it is difficult for en-gineer to fit many parameters simultaneously. The actual fit-ting procedure is explained in the next section.

4. Modeling Procedure for Common ModeChokes

The modeling procedure consists of following four steps.( 1 ) Measure the common mode impedance of the

choke.( 2 ) Fit the parasitic capacitance and remove it.( 3 ) Fit the derived impedance characteristics using cir-

cuit dividing.( 4 ) Make the equivalent circuit model using the ob-

tained circuit parameters.We performed modeling with an actual coil, which has

toroidal core F1AH0972 from Hitachi Metal, Ltd., made ofnanocrystalline magnetic material with two 9-turn windingsof 1.0-mm-diameter enameled copper wire.4.1 Measure the Common Mode Impedance of the

Choke Common mode impedance was measured overfrequencies of 150 kHz to 30 MHz using an impedance ana-lyzer (Agilent 4294A) as shown in Fig. 15. The measurementresult is shown in Fig. 16.4.2 Fitting of the Parasitic Capacitance Parasitic

capacitance is fitted from the results for the high-frequencyregion in Fig. 16(a). We consider the measurement circuit iscomposed as the circuit shown in Fig. 7, and our goal is tofit the frequency-dependent RL circuit as the RL ladder cir-cuit. To do this, the impedance of the capacitance is removedusing the following equation.

1Zmeas

= Ymeas = Y ′meas − jωC · · · · · · · · · · · · · · · · · · · (17)

where Y ′meas is the measured admittance, C is the fitted capac-itance, and Zmeas and Ymeas are the calculated impedance andadmittance, respectively. By this subtraction, Zmeas becomesthe impedance corresponding to R( f ) + jωL( f ) in Fig. 7. InFig. 16, an absolute value of impedance of over 5.1 MHz, the



Fig. 15. Measurement setup for common mode impedance

(a)

(b)

Fig. 16. Measurement result and parasitic capacitanceremoval, (a) amplitude and (b) phase

point at which the impedance is sufficiently capacitive be-cause the phase is less than −45◦, was used for fitting. Asshown in Fig. 16(a) and (b), parasitic capacitance removal de-creased the impedance drop-off in the high-frequency region,and changed the phase from negative to positive.4.3 Fitting Using the Circuit Dividing ProcedureImpedance fitting for Zmeas is conducted using the circuit

dividing procedure. Here, we demonstrate the hand-fitting toemphasize the advantage of the proposed modeling methodwhich does not require any additional programs; however,the fitting with an optimizer is also possible as shown in ap-pendix 1.

The fitting procedure is as follows.( 1 ) Determine R0 and L0; R0 is obtained from the max-

imum impedance amplitude over all frequencies, andL0 is from the impedance amplitude at low frequen-cies, where the inductive impedance is dominant.

( 2 ) Perform the first circuit dividing and optimize R1,Rdiff1, L1, and Ldiff1 to improve the fitting precision inthe low-frequency region.

( 3 ) Perform a second circuit dividing and optimize Ri,Rdiffi, Li, and Ldiffi to improve the fitting precision overa wider frequency range compared to the first division;repeat circuit dividing until the fitting precision con-verges compared to the previous result.

Resistance Ri, Rdiffi and inductance Li, Ldiffi are optimizedby adjusting the two parameters ki and li, which are definedas follows:

Ri =Rdiffi−1

2(1 − ki), Rdiffi =

Rdiffi−1

2(1 + ki),

0 ≤ ki ≤ 1 (i = 1, 2, . . . ,N) · · · · · · · (18)

Fig. 17. Absolute values of change rate of impedancesunder different k + l values

Li =Ldiffi−1

2(1 + li), Ldiffi =

Ldiffi−1

2(1 − li),

0 ≤ li ≤ 1 (i = 1, 2, . . . ,N) · · · · · · · (19)

For the unified discription, R0 and L0 are written as Rdiff0 andLdiff0, respectively. Here, the inequalities in (18) and (19) cor-respond to (15) and (16), respectively, and the initial valuesof ki and li are both set to 0.

The hand-fitting procedure for ki and li is as follows. Asshown in the appendix 2, the absolute value of impedance Zc

at the cutoff frequency—the frequency where the impedancesof resistance and inductance cross each other—is decreasedby the circuit dividing, and it almost depends on ki + li. Inother words, |Zc| changes only slightly for different valuesof ki under the same ki + li. Furthermore, as also shown inappendix 2, the phase of Zc always increases as ki increasesunder the same ki + li. By using these characteristics, we fitki+ li from |Zc| and then fit ki from the phase data at the cutofffrequency as described below.

First, at the cutoff frequency fc, the absolute value of theimpedance before circuit dividing is compared with the abso-lute value of the target impedance. In other words, we calcu-late the change rate A, which is defined as follows:

A =Zgoal

Zstart· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (20)

Zgoal = Zmeas −i−1∑m=1

Rm// jωLm · · · · · · · · · · · · · · · · · · (21)

Zstart = Rdiffi−1// jωLdiffi−1 · · · · · · · · · · · · · · · · · · · · · · (22)

Note that the second term of Zgoal is used to remove theimpedance of the RL circuit not subject to circuit dividing.Next, ki + li is determined using |A| by the transformationmap shown in Fig. 17. This curve is written with the follow-ing equation derived in appendix 2:

|A| = 4 − (ki + li)2

4 + (ki + li)2· · · · · · · · · · · · · · · · · · · · · · · · · · · · · (23)

Finally, the phase at the cutoff frequency is compared to ad-just ki.

Figure 18 shows the results of the first circuit dividing. Theabsolute value and phase of the impedance at the cutoff fre-quency are fitted by adjusting k1 + l1 and k1, respectively.Next, second circuit dividing is conducted. At first, k2 + l2 isadjusted using the absolute values of impedance data shownFig. 19, and then k2 is adjusted using the phase data shown inFig. 20(b).

In this case, circuit dividing was conducted two times, and



(a)

(b)

Fig. 18. Fitting results for impedance before and afterfirst circuit dividing, (a) amplitude and (b) phase. Ab-solute value of the impedances of R0 and L0 and corre-sponding cutoff frequency fc is plotted with dotted line

Fig. 19. Fitting results for subtracted impedance beforeand after second circuit dividing. Absolute value of theimpedances of Rdiff1 and Ldiff1 and corresponding cutofffrequency fc is plotted with dotted line

the precision of fitting was improved by the iterative circuitdividing as shown in Fig. 18 and 20. The fitting is poor forfrequencies over 10 MHz in Fig. 20, but this is trivial because,at these frequencies, the actual coil has the parasitic capaci-tive characteristics as shown in Fig. 16.4.4 Making the Equivalent Circuit Figure 21 and

Table 1 show the equivalent circuit and circuit parameters,respectively, of the common mode choke model used for ver-ification. Coupling coefficient k is set to 1. The obtainedequivalent circuit in this case is a three-stage Foster-type cir-cuit because the circuit is divided twice. In Fig. 21(b), induc-tors of the positive and negative lines are perfectively cou-pled, so this circuit affects only common mode current, notdifferential mode current.4.5 Verification of the Common Mode ImpedanceFigure 22 shows the measured and fitted impedance re-

sults, with the fitted results obtained from two models, a con-ventional single RLC parallel model and the proposed model.Parameters for the RLC model are also optimized; inductanceL and capacitance C were adjusted to match the impedance atlow and high frequencies, respectively, and resistance R wasoptimized to minimize the error using function (A1) after Land C were determined. As shown in Fig. 22, the proposed

(a)

(b)

Fig. 20. Fitting results for impedance before and aftersecond circuit dividing, (a) amplitude and (b) phase. Cut-off frequency fc is plotted with dotted line

(a) (b)

Fig. 21. Proposed equivalent circuits for (a) commonmode equivalent circuit and (b) full equivalent circuit

Table 1. Circuit parameters for equivalent circuit

model matched the measured impedance with better preci-sion compared to the conventional RLC model.

5. Experimental Verification of the ProposedModel

5.1 Insertion Loss of a Noise Filter with FrequencyAnalysis Experimental verification of the insertion lossdue to a noise filter with common mode chokes was carriedout using both the proposed and conventional models. Weused a T-type filter with two common mode chokes, as shownin Fig. 23, and the filter insertion loss was measured using thesetup shown in Fig. 24. The common mode chokes were thesame type of nanocrystalline coil used in the last section, and1-nF film capacitors were used for Y-type capacitors. Inser-tion loss of the filter was measured at two ports using a net-work analyzer (Agilent E5071C), with the average time setto the maximum of 999. The measured frequency region wasfrom 100 kHz, which is the analyzer’s lower limit, to 30 MHz.

The simulated insertion loss IL was calculated using the



(a)

(b)

Fig. 22. Fitting results for impedance of the commonmode choke, (a) amplitude and (b) phase

Fig. 23. Measurement circuit for insertion loss of T-typefilter

Fig. 24. Setup of system for measuring insertion loss ofT-type filter

following equation.

IL = 20 log10Vout1

Vout2· · · · · · · · · · · · · · · · · · · · · · · · · · · · · (24)

where Vout1 and Vout2 are the output voltage with and withoutthe filter, respectively (17). As a circuit simulator, LTSpice (18)

was used in this paper.Figure 25 shows the results for the measured and simulated

insertion losses. The maximum error was 8.4 dB for the con-ventional single RL parallel model, while the proposed modelshowed an error of 3.6 dB. This result showed that the pro-posed model can more precisely simulate the insertion loss ofthe noise filter. Although the phase data contained a substan-tial error between 1 MHz and 10 MHz, this is not a problemin practical applications because the major concern in a noisefilter is the amplitude of the insertion loss.5.2 Voltage Ripple Evaluation with Transient Analy-

sis Experimental verification of transient analysis witha common mode choke was also carried out using both the

(a)

(b)

Fig. 25. Measured and simulated filter insertion losses,(a) amplitude and (b) phase

Fig. 26. Measurement circuit for ripple voltage wave-form

proposed and conventional models. As shown in Fig. 26,a board mounted the common mode choke, which was thesame coil used previously, was connected to an oscilloscope(Tektronix DPO7104C) and a function generator (TektronixAFG3011C) by co-axial cables (Fujikura RG-58C/U), andthe voltage waveform was measured. The input waveformwas a trapezoidal wave with a frequency of 500 kHz and aduty cycle of 50%. It should be noted that parasitic capaci-tances between the signal line and the ground of co-axial ca-bles affected the voltage waveform, so the capacitances wereextracted from the impedance results shown in Fig. 27 and thevalues 55 pF and 114 pF were used in the circuit simulationas shown in Fig. 26. We measured the impedance betweenan inner signal pin and outer ground of the co-axial cablesusing an impedance analyzer (Agilent 4294A), and then theimpedance was fitted by hand. The lengths of these cableswere 525 mm and 1,032 mm, and according to the datasheet,their capacitance is 102 nF/km, so their capacitances wereestimated as 53.6 pF and 105.3 pF, respectively. The errorsbetween measurement results and these estimations are bothless than 10%.

Figure 28 shows the measured and simulated voltage wave-forms. The conventional model result shows a smaller ripplevoltage of 51 mV compared to the measured result of 75 mV,while the proposed model result of 73 mV almost matches themeasured value. The error percentages for the conventionaland proposed models were 31% and 2.6%, respectively. Theproposed model provided a more precise transient simulation



(a)

(b)

Fig. 27. Measured and fitted results of cable impedances,(a) amplitude and (b) phase. The solid curves are measuredresults and the dotted curves are fitted results

Fig. 28. Measured and simulated ripple voltage wave-forms

of the common mode choke.

6. Conclusion

In this paper, we presented a straightforward method ofmodeling complex permeability for common mode chokes.The proposed model was structured as an RL ladder networkderived from a single RL parallel equivalent circuit by repeat-ing a manipulation called circuit dividing. By this model,common mode impedance of a choke was fitted more pre-cisely compared to the conventional single RLC equivalentmodel. Moreover, the prosed model precisely simulated bothnoise filter insertion loss by frequency analysis and ripplewaveforms by transient analysis.

This proposed model requires a repeat of parameter fittingfor up to two parameters basically, so the fitting procedureis straightforward. This is very advantageous for engineersbecause the model can be employed without using the addi-tional special optimization programs such as a genetic algo-rithm required in previous works.

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Appendix

1. Parameter Extraction by an OptimizerIn this appendix, an optimizer is used to reduce the error of

fitting; however, adequate precision is also obtained by hand-fitting as shown in section 4.

We used the following function of the sum of the squarederrors.

Ferror =∑

f

{FRe + FIm

}· · · · · · · · · · · · · · · · · · · · · · · · (A1)

FRe =

(Re (Zmeas)|Zmeas| log10

Re (Zmeas)Re(Zfit)

)2

· · · · · · · · · · · (A2)

FIm =

(Im(Zmeas)|Zmeas| log10

Im(Zmeas)Im(Zfit)

)2

· · · · · · · · · · · · (A3)

Here, Zfit is the fitted impedance and the frequency regionis determined properly for each case. This function evalu-ates both real and imaginary values of the impedance, anddepending on the coefficient of the log function, FRe is largewhen the real part of Zmeas is dominant and FIm is large whenthe imaginary part of Zmeas is dominant. Additionally, be-cause the square of the log function is used, the value of thefunction is determined by the ratio of the real or imaginaryparts of Zmeas and Zfit; for example, the values of the func-tion are the same whether the ratio of Re (Zmeas) to Re (Zfit)



app. Table 1. Values of the function for fitting

(a)

(b)

app. Fig. 1. Fitting results for impedance after parasiticcapacitance removal, (a) amplitude and (b) phase. “1time” and “2 times” refer to the results after the first andsecond circuit dividing, respectively. The result of “2times” cannot be seen clearly because the results of “1time” and “2 times” are almost identical

is 10:1 or 1:10. app. Table 1 shows the values of the func-tion for some specific impedances. As we intended, FRe islarge when the real part of Zmeas is dominant, and FIm is largewhen the imaginary part of Zmeas is dominant. Also, the valueof the function is the same, regardless of whether the domi-nant part of Zfit is 10 times or 1/10 of Zmeas. Thus, the valueis determined by the ratio of the dominant parts of Zmeas andZfit.

The fitting procedure is the same as described in subsection4.3 except for the following additional last step.

( 4 ) After the results converge, optimize all parametersof R and L simultaneously.

Note that more than two parameters are optimized exception-ally in this step. Although optimization of many parame-ters sometimes causes convergence problems, these problemsare avoided here because appropriate initial parameter values,which are close to the desired solution, have already been ob-tained by the previous three steps. We used the generalizedreduced gradient (GRG) nonlinear optimizer, which is oneof the solvers that can be used in Microsoft Excel 2010, tominimize the value of function (A1).

app. Fig. 1 shows the transition of impedance characteris-tics under the fitting. In this case, circuit dividing was con-ducted two times and the maximum frequency of interest was100 kHz during the first division and 10 MHz during the sec-ond division. The first circuit division improved the fitting

app. Table 2. Circuit parameters for equivalent circuit

(a)

(b)

app. Fig. 2. Fitting results for impedance of the com-mon mode choke, (a) amplitude and (b) phase

app. Fig. 3. Relative errors of common mode impedance

precision compared to the initial fitting, while the second cir-cuit division produced a negligible improvement comparedto the first division, indicating that the results had converged.After that, all parameters were optimized for the frequenciesfrom 10 kHz to 10 MHz, which produced the “Last” lines inthe figure. The fitting is poor for frequencies over 10 MHz inFig. 1, but this is trivial because, at these frequencies, the ac-tual coil has the parasitic capacitive characteristics as shownin Fig. 16. The equivalent circuit is the same as that in theFig. 21 and the circuit parameters are shown in app. Table 2.

app. Fig. 2 shows the measured and fitted impedance re-sults, with the fitted results obtained from three models, theproposed models by hand-fitting and an optimizer, and a con-ventional single RLC parallel model. The both proposedmodels by hand-fitting and an optimizer matched the mea-sured impedance with better precision compared to the con-ventional RLC model.

app. Fig. 3 shows the relative errors of the two proposedmodels and a conventional model. The conventional modelresult showed approximately 50% error, while the proposed



(a)

(b)

app. Fig. 4. Impedances before and after circuit divid-ing for ki + li = 0.8 and ki = 0–0.8, (a) amplitude and(b) phase. The impedances before and after dividing areplotted with a thick black curve and thin color curves, re-spectively. Absolute value of the impedances of R andL and corresponding cutoff frequency fc is plotted withdotted line

models matched within 10% error.2. Background Information for Circuit DividingAs discussed in subsection 4.3, two parameters, ki and li,

need to be fitted by circuit dividing. Here, background in-formation for circuit dividing is presented and dependence ofki + li and ki on absolute values and phases of the impedanceat the cutoff frequency is shown.

app. Figs. 4 and 5 show the effect of circuit dividing onimpedance for different ki under a fixed ki + li. In app. Figs. 4and 5, values of ki + li are 0.8 and 1.2, respectively. In bothcases, the absolute values of impedance at the cutoff fre-quency fc are almost the same regardless of the values of ki.Furthermore, phases at fc increase as ki increases under thefixed ki + li.

Further analysis is performed using the change rate A′defined by the impedance before and after circuit dividing,Zbefore and Zafter, respectively, as follows.

A′ =Zafter

Zbefore· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A4)

Because Zbefore and Zafter are complex numbers, A′ is also acomplex number. app. Fig. 6 shows the change rate A′ plottedon the complex plane, where cases with the same ki + li areplotted with curves of the same color. app. Fig. 6 indicatesthat the absolute values of A′ decrease as ki + li increases.app. Fig. 7 shows the absolute value of the change rate A′ ver-sus ki + li. Although there are a few narrow ranges of valuesunder the same ki+li, |A′| tends to decrease as ki+li increases.

The change rate A′ is a function of ki and li, and we cangive one-to-one correspondence to the rate and ki + li, as in(23), by taking the condition ki = li. Derivation of (23) is asfollows, where we consider circuit dividing of R and L shownin Fig. 8. Impedance before circuit dividing is written as

(a)

(b)

app. Fig. 5. Impedances before and after circuit divid-ing for ki + li = 1.2 and ki = 0.2–1, (a) amplitude and(b) phase. The impedances before and after dividing areplotted with a thick black curve and thin color curves, re-spectively. Absolute value of the impedances of R andL and corresponding cutoff frequency fc is plotted withdotted line

app. Fig. 6. Change rate A′ under different ki + li and ki

values on complex plane. The points on the curves areplotted while ki changed by increments of 0.1

app. Fig. 7. Absolute values of A′ after circuit dividingunder different ki + li and ki values. The data points arethe same with those of Fig. 6

Zbefore =R × jωLR + jωL

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A5)

and from the condition R = ωL satisfied at the cutoff fre-quency, we obtain the following result.

|Zbefore| = R√2· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A6)

Impedance after circuit dividing is written as



Zafter =Ra × jωLa

Ra + jωLa+

Rb × jωLb

Rb + jωLb· · · · · · · · · · · · · · · (A7)

and from the conditions of

Ra =R2

(1 − ki) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A8)

Rb =R2

(1 + ki) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (A9)

ωLa = ωL2

(1 + li) =R2

(1 + ki) · · · · · · · · · · · · · · · (A10)

ωLb = ωL2

(1 − li) =R2

(1 − ki) · · · · · · · · · · · · · · · (A11)

we can obtain the following result.

|Zafter| = R√2

1 − ki2

(1 + ki)2· · · · · · · · · · · · · · · · · · · · · · · · · (A12)

We can derive (23) by substituting (A6) and (A12) into (A4)and assigning (ki + li)/2 into ki and A into A′.

Katsuya Nomura (Member) received the B.E. and M.E. degrees inelectrical engineering from Kyoto University, Kyoto,Japan, in 2008 and 2010, respectively. Since 2010,he has been working in Toyota Central R&D Labs.,Inc. He is currently pursuing the Ph.D. degree in me-chanical engineering at Osaka University. His currentresearch interest is electromagnetic compatibility inpower electronics. He is a member of the Institute ofElectrical Engineers of Japan (IEEJ), the Institute ofElectrical and Electronics Engineers (IEEE) and the

Institute of Electronics, Information and Communication Engineers (IEICE).

Takashi Kojima (Member) received the M.Eng. degree in electri-cal engineering from Tokyo University of Science,Noda, Japan, in 1995 and the Dr. degree in electri-cal engineering from Hiroshima University, Higashi-Hiroshima, Japan, in 2009. Since 1995, he hasbeen with Toyota Central R&D Labs., Inc., Nagakute,Japan. His research interests include power electron-ics simulation, power device modeling and electro-magnetic interference in hybrid vehicles.

Yoshiyuki Hattori (Non-member) received the M.E. and Ph.D. de-grees in electrical engineering from Nagoya Instituteof Technology, Nagoya, Japan, in 1991 and 2001, re-spectively. Since 1985, he has been with Toyota Cen-tral R&D Laboratories, Inc., Nagakute, Aichi, Japan,where he has been engaged in research and develop-ment of integrated circuits design, Si power semicon-ductor devices, and electromagnetic compatibility forautomobiles. He is now the Department Manager ofSystem and Electronics Engineering Department III.

Dr. Hattori is a member of the Institute of Electronics, Information and Com-munication Engineers, the Japan Society of Applied Physics, and Society ofAutomotive Engineers of Japan.


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