Equivalent circuit for high-voltage partial-core resonant transformers

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Published in IET Electric Power ApplicationsReceived on 28th October 2007Revised on 5th January 2008doi: 10.1049/iet-epa:20070430

ISSN 1751-8660

Equivalent circuit for high-voltage partial-coreresonant transformersS.C. Bell P.S. BodgerElectrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New ZealandE-mail: [email protected]

Abstract: An equivalent circuit model for the partial-core resonant transformer is presented. A model which omitscore losses is first introduced and is verified on a sample air-core resonant transformer. Partial-cores are thenintroduced into the transformer and the resonant characteristics are re-measured. The model is modified toaccount for the core losses and then verified against the measured data. Comparisons and transformationsbetween the model and the Steinmetz ‘exact’ transformer equivalent circuit are given. A procedure formeasuring the equivalent circuit inductances is described and then used to investigate the inductancevariation with core displacement characteristics of the sample transformer in both axial-offset and centre-gaparrangements. In a capacitive load test the sample transformer was linear to a significantly higher voltagewhen using the centre-gap arrangement.

1 IntroductionAn alternative design to a full-core transformer is a partial-core transformer, where the outer limbs and connectingyokes are missing. The magnetic circuit includes both thecore and the surrounding air. Consequently, partial-coretransformers have a lower value of magnetising reactancethan their equivalent full-core counterparts.

Partial-core transformers have been designed as step-uptransformers for energising capacitive loads, where they arereferred to as partial-core resonant transformers (PCRTXs).By matching the inductive reactance of the secondarywinding to the capacitive reactance of the load, the reactivepower drawn from the primary winding can be reduced toalmost zero. Applications include high voltage testing ofhydro generator stators [1, 2] and energising arc-signs [3,4]. In these examples the advantages over conventionalequipment–a full-core step-up transformer and a separatefull-core compensating inductor–are significant reductionsin weight and cost and increased portability. A PCRTX isdepicted in Fig. 1.

2 BackgroundIn insulation testing applications, the key requirement is toenergise an insulation capacitance C to a voltage V2 while

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ensuring the supply current Is is below the capacity of thelocal supply. A test circuit employing a PCRTX is shownin Fig. 2.

PCRTXs have been designed using the reverse method oftransformer design [5, 6]. Supply conditions and the physicalcharacteristics and dimensions of the core and windings areused to produce component values for the Steinmetz ‘exact’transformer equivalent circuit of Fig. 3.

Where, R1 ¼ primary resistance, R2 ¼ secondaryresistance, Rc ¼ core loss resistance, X1 ¼ primary leakagereactance, X2 ¼ secondary leakage reactance,Xm ¼ magnetising reactance and a ¼ transformer turns ratio.

Resonance is said to occur when the transformermagnetising reactance Xm is matched to the capacitivereactance of the (primary-referred) load capacitance [1].This description neglects the leakage reactances, which, inPCRTXs, can be significant when compared to themagnetising reactance value. Using the Steinmetztransformer equivalent circuit the resonant frequencies andloaded-circuit impedance can be calculated for a sampletransformer. However, it is difficult to obtain simpleanalytical expressions for the general case. This can make itdifficult to translate design specifications into required

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values or allowable ranges of values for the equivalent circuitcomponents. The Steinmetz transformer equivalent circuitrepresentation of the PCRTX can also be a source ofconfusion when it comes to measuring the reactance values.The traditional transformer open-circuit and short-circuittests may not give an accurate measure of the magnetisingand leakage reactance values [7].

3 Equivalent circuit without corelossesThe return path for the magnetic flux in a PCRTX is air.When compared to equivalent full-core transformers, thecoil coupling is reduced and the inductance values aresmaller. It is proposed that under normal operating fluxdensities the coil inductances are essentially linear.Consequently, the PCRTX can be more correctly definedas a coupled inductor than a transformer [8].

Figure 3 Steinmetz ‘exact’ transformer equivalent circuit,referred to primary winding

Figure 1 Isometric view of a PCRTX, showing core andwinding layout

Figure 2 PCRTX in a high voltage insulation testingapplication

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The PCRTX can be analysed in terms of self- and mutual-inductances. In the first analysis the core loss resistance Rc isomitted. The impedance of the insulation under test can berepresented as a resistance Rl in series with a capacitance Cl

[9]. The supply can be approximated as a Theveninequivalent circuit. The resulting circuit is shown in Fig. 4.

Where, L11 ¼ primary winding self-inductance,L22 ¼ secondary winding self-inductance andM12 ¼ primary-secondary mutual-inductance.

A general analysis of the circuit in Fig. 4 has beenpublished by Sherman in 1942 [10]. The PCRTX with acapacitive load is an example of an inductively coupledcircuit with a tuned secondary. The input impedance Zin isequal to the primary winding impedance Zp in series withthe coupled impedance Zc. The equivalent circuit is shownin Fig. 5.

The primary winding impedance is given as

Zp ¼ R1 þ jvL11 ð1Þ

The coupled impedance is given as [10]

Zc ¼v2M2

12

R02 þ jðvL22 � 1=vCl Þ

ð2Þ

where R02 ¼ R2þ Rl

After simplification, the input impedance is

Zin ¼ R1 þv2M2

12R02

R022 þ vL22 �

1vCl

� �2

þ j vL11 þv3M2

12Cl � v5M212C

2l L22

v2C2l R

022 þ ðv2L22Cl � 1Þ2

!ð3Þ

Figure 4 Proposed equivalent circuit for the PCRTX

Figure 5 Simplified series equivalent circuit for the PCRTX

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4 Measurement of equivalentcircuit inductancesThe resonant frequency of the coupled impedance is

v0;c ¼1ffiffiffiffiffiffiffiffiffiffiffiffi

L22Cl

p ð4Þ

After simplification, the input impedance at v ¼ v0,c is

Zin ¼ R1 þM2

12

L22ClR02

þ jL11ffiffiffiffiffiffiffiffiffiffiffiffiL22Cl

p ð5Þ

Parallel resonance in tuned circuits can either be defined bythe maximum impedance criteria or the unity power factorcriteria [11]. For the circuit of Fig. 5 these two definitionsof resonance do not coincide due to the primary windingself-inductance L11. There are two resonant frequencies forZin, namely v0,m and v0,u. The following inequality holds

v0;m � v0;c � v0;u ð6Þ

The inequality gives rise to a procedure for measuring theequivalent circuit inductances of the PCRTX of Fig. 4.The main advantage of this technique, termed the resonanttuning test, is that it does not require a supply rated for thevolt-amperes of the secondary circuit. The insulationcapacitance is replaced with a high voltage capacitor ofequivalent value and the circuit is energised from a variablefrequency sinewave generator. The resonant frequenciesv0,m and v0,u and the corresponding terminal conditions ofthe PCRTX are measured. The test results, along with themeasured values of winding resistance and load capacitance,are used to determine the secondary winding self-inductance and mutual-inductance values, given by

L22 ¼1

v2Cl

ð7Þ

M12 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið<fZing � R1ÞR

02

pv

ð8Þ

Ideally (7) and (8) would be evaluated at v ¼ v0,c, but thisfrequency is not directly measured in the resonant tuningtest. However, minimum and maximum values of L22 andM12 can be found by evaluating (7) and (8) at v ¼ v0,u

and v ¼ v0,m, respectively. The uncertainty in inductancevalues DL22 and DM12 is a function of the equivalentcircuit component values. The percentage uncertainty inthese values will determine the usefulness of the resonanttuning test as a method of measuring the equivalent circuitinductances. This can be determined by experimentingwith a sample PCRTX.

The resonant tuning test can also be used to measure theprimary winding self-inductance L11. A capacitive load isapplied to the low voltage winding and the high voltagewinding is energised using a sinewave generator andconventional full-core step-up transformer. The coil

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coupling can then be calculated as

k ¼M12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiL11L22

p ð9Þ

5 Equivalent circuit with corelossesCore losses in full-core power transformers can be calculatedusing magneto-static finite element analysis in combinationwith magnetisation and loss data, measured on singlesheets of material [12]. The magnetic field is typicallycomputed under open-circuit conditions [13]. The primaryleakage flux and mutual flux both have an influence on themagnetic field and hence core losses. To account for this, ithas been proposed that the core loss resistance Rc in thetransformer equivalent circuit of Fig. 3 should be moved tothe left side of the primary leakage reactance branch [13].

However, for the magneto-static case, the magnetic field inany transformer is a function of the instantaneous values ofboth the primary and secondary ampere-turns. For thePCRTX, the magnetic field is mainly due to the secondaryampere-turns, particularly for a circuit quality factor Q .

10. The core loss resistance Rc should therefore beaccounted for by a parallel resistance over the secondarywinding self-inductance, changing the load impedancefrom ZL to Z0

L, as shown in Fig. 6.

The new load impedance is given by

Z0L ¼ jZ0

Lj/fZ0L

ð10Þ

where

jZ0Lj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðRcR

02Þ

2þ ðRcXcÞ

2

ðRc þ R02Þ

2þ X 2

c

sð11Þ

fZ0L¼ tan�1

ðu1Þ � tan�1ðu2Þ ð12Þ

u1 ¼Xc

R02

ð13Þ

u2 ¼Xc

Rc þ R02

ð14Þ

Figure 6 Equivalent circuit of load: ZL–without core lossresistance, Z0L–with core loss resistance

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The effect of introducing core loss resistance on the resonantbehaviour of the transformer can be determined byexperimenting with a sample PCRTX.

6 T-equivalent circuit with corelossesThe series equivalent circuit of Fig. 5 can be used to calculatethe loaded-circuit input impedance but not the voltage ratiobecause the secondary terminals are absent. This can becalculated using a T-equivalent circuit. The circuit, shownin Fig. 7, is similar to the Steinmetz ‘exact’ transformerequivalent circuit but the reactance values change from fX1,Xm, X2g to fX0

1, X0m, X

02g and the core losses resistance is

moved to the right hand side of the secondary leakagereactance branch.

The reactance values and coupling factor are given by [14]

X 01 ¼ jvL11ð1� xkÞ ð15Þ

X 0m ¼ jvL11xk ð16Þ

h2X 02 ¼ jvL11xðx� kÞ ð17Þ

h ¼ x

ffiffiffiffiffiffiffiL11

L22

sð18Þ

The coupling factor h (not to be confused with the coilcoupling k) replaces the transformer turns ratio a which isnot measured in the above described tests. The choice of xis arbitrary, but one leakage reactance will become negativeif x is chosen outside the range k , x , 1/k. According tox ¼ k, 1 or 1/k, leakage reactances appear on the primaryside, split into two equal parts or on the secondary side.

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7 Experimental results7.1 Transformer specifications

A sample PCRTX was used to verify the proposed equivalentcircuit model. The transformer was designed for energisingarc-signs [4]. The specifications are shown in Table 1. Thecore was constructed in a single 604 mm length and thendivided into two 300 mm axial lengths using a water-jetcutter. Open-circuit tests, performed before and after thecore cutting, showed no difference in measured terminalconditions. The split core allowed for inductance variationvia centre-gap tuning as well as axial-offset tuning, used inprevious PCRTXs [1].

7.2 Resonant tuning test in air-coreconfiguration

The resonant tuning test was first performed on the PCRTXin the air-core configuration, at a reduced secondary voltageof 10 kV. The winding resistances were measured at dcusing a MPK 254 digital micro ohmmeter. The dcresistance measurements were assumed to be representativeof the ac resistances of the sample PCRTX at the supplyfrequency, where the calculated skin depths of the primaryand secondary winding materials were much greater thanthe conductor thicknesses. A calibrated Fluke-41Bharmonic analyser was used to take primary-sidemeasurements. The secondary voltage was measured using

Figure 7 T-equivalent circuit for the PCRTX, referred to theprimary winding

Table 1 Specifications of sample PCRTX

Ratings Core Inside (hv) winding Outside (lv) winding Aspectratio

primary voltage, V 230 length, mm 600 length, mm 700 length, mm 600 13

secondary voltage,kV

80 diameter, mm 72 number of layers 37 number of layers 2

operatingfrequency, Hz

50 laminationthickness, mm

0.50 leadouts onlayers

1,37 number of turns 160

number of turns 59,200 maximumcurrenta, A

126

maximumcurrenta, A

0.55

aAt a (short-time-rated) current density of 5 A/mm2


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a 10 000:1 capacitive voltage divider and Escort ’97multimeter. A Universal Technic ‘M1.UB02’ 5 A/5 V CTand additional Escort ’97 multimeter were used to measurethe secondary current. The load capacitance was formedfrom series or parallel combinations of capacitors from aninverted Marx impulse generator. The capacitance valuewas chosen so that resonance occurred at �50 Hz. Theload resistance was considered negligible and the loadcapacitance was estimated from secondary voltage, currentand frequency measurements as

Cmeas ¼I2

2pfV2

ð19Þ

The resonant tuning test was not repeated with the PCRTXin the reverse configuration due to the difficulty in acquiringlow voltage capacitors of sufficient rating. The required loadcapacitance could have been obtained by impedancematching high voltage capacitors with a conventional full-core step-up transformer, at the expense of circuitcomplexity. In this instance, the PCRTX was rated for theopen-circuit current at the reduced test voltage so theprimary winding self-inductance was obtained from open-circuit test results. The equivalent circuit impedance underopen circuit was assumed to be R1 þ jvL11. The primarywinding self-inductance was calculated as

L11 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVoc

Ioc

� �2

�R21

s

vð20Þ

The measured parameters are shown in Table 2. These wereused to model the resonant tuning test by calculating theloaded-circuit input impedance (via (3)) over the testedfrequencies. Themodel and test results are compared in Fig. 8.

The uncertainties in the inductance measurementhighlight the limitations of the resonant tuning test as amethod of measuring the equivalent circuit inductances inair-core resonant transformers.

The uncertainties can be reduced by obtaining a betterestimation of v0,c (and corresponding f0,c). Model resonantcharacteristics were plotted for each set of measured

Table 2 Measuredparameters of samplePCRTX in air-coreconfiguration

Parameter Value

R1, V 0.14

R2, kV 3.63

L11, mH 1.23

M12, H 0.21 + 7%

L22, H 58.1 + 9%

k 0.78 + 3%


terminal conditions for all the measured frequencies fromf0,m to f0,u. The best estimate of f0,c, denoted f �

0,c, wasdefined as the frequency whose corresponding modelresonant characteristic best matched the measured terminalconditions.

There is a high level of correlation between the test andmodel results when using f �0,u in the model. The shape ofthe resonant characteristic, and hence the circuit Q, hasbeen accurately predicted. This indicates that the proximitylosses and capacitor losses, which were not modelled, arenegligible when compared to the winding resistance losses.

7.3 Open-circuit test in partial-coreconfiguration

The sample PCRTX was configured to maximise thewinding inductance values by placing the two core sectionsinside the former and setting the centre-gap to zero. Thedevice saturation characteristics were measured in an open-circuit test. The results, shown in Fig. 9, indicate thatsaturation started to occur at a secondary voltage of62.6 kV. The core peak flux density, calculated using the‘transformer equation’ [15] at this voltage was 1.43 T,

Figure 8 Measured and predicted resonant characteristicsof sample PCRTX in air-core configuration at V2 ¼ 10 kVDf ¼ 2.41 Hz

a Magnitudeb Phase

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significantly lower than the typical saturation flux density forsilicon steel of 1.7 T. In this instance, the ‘transformerequation’ cannot be used to accurately predict the onset ofcore saturation. As the ‘transformer equation’ assumes auniform core flux density this may indicate that the samplePCRTX has a non-uniform core flux density and that partsof the core steel are in saturation at a secondary voltage of62.6 kV.

Harmonic voltages and currents were also measured. Theresults, shown in Table 3, indicate that the introduced coresteel has not caused significant third harmonic currents,confirming the assumption that the device can be modelledusing linear circuit elements provided that the operatingflux density is below the saturation level.

7.4 Resonant tuning test in partial-coreconfiguration

The procedures used to measure the inductance values for theair-core configuration were first repeated for the partial-coretransformer in several different core configurations. Themeasured values of secondary winding-self inductance were

Figure 9 Open-circuit test of sample PCRTX

Table 3 Primary-side total harmonicdistortion measurements for sample PCRTXwith 2 � 300 mm core pieces and centre-gap ¼ 0 mm under open-circuit conditions

Secondary voltage, kV Total harmonicdistortion

voltage current

10.1 3.6 0.8

20.4 3.6 0.8

30.1 3.9 0.8

40.2 4.4 1.0

50.3 4.0 0.9

60.4 3.8 2.3

62.6 3.7 3.2

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then used to calculate load capacitance values for eachconfiguration such that f0,c ¼ 50 Hz. The effect ofintroducing core losses was then investigated theoreticallyby plotting the ratios of the real and imaginary componentsof Z0

L/ZL against specific core losses using (11)–(14), asshown in Fig. 10.

The plots indicate that, at a typical value of specific coreloss of 1 W/kg, for the considered configurations, theeffect of the introduced core losses is to increase the realcomponent of ZL and leave the imaginary componentunchanged. Hence the core loss resistance can beincorporated into the circuit model by increasing the valueof R0

2 to R002 ¼ Re fZ0

Lg. With reference to (7) and (8), theresults also show that omitting the core loss resistance willhave a small impact on the measured value of secondarywinding self-inductance but a much larger impact on themeasured value of mutual-inductance.

The resonant tuning test was repeated on the samplePCRTX in two different partial-core configurations. Thefirst was an artificial case with a single core of length100 mm and the second was a more realistic case with twocores of 300 mm length and a centre-gap of 195 mm. Thecapacitor and proximity losses were not accounted for inthe model.

Figure 10 Effect of introducing core losses on the loadimpedance for the sample PCRTX for several coreconfigurations

a Real componentb Imaginary component


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As indicated in Fig. 10, the introduced core loss will have aminimal impact on performance for the first configurationand do not need to be modelled. The test and modelresults are compared in Fig. 11. The introduced core steelhas reduced the difference in resonant frequencies Df andthe winding inductances and coil coupling have increased.The circuit model accurately matches the test results.

For the second case the introduced core losses were largerand needed to be modelled. The core losses were estimated bysubtracting the i2R winding losses from the measured inputpower. A specific loss value of 0.4 W/kg was obtained atf ¼ f �0,c at V2 ¼ 10 kV. The resulting core loss resistancevalue of Rc ¼ 13.3 � 106 V was used to compute theeffective secondary winding resistance value ofR00

2 ¼ 4384 V. The model input impedance was thencalculated using (3) with R00

2 in place of R02. The test and

model results are compared in Fig. 12.

By including the core loss resistance, the equivalent circuitmodel is again verified over the limited frequency range. Themodel may not hold for other frequencies since the core lossresistance is a function of frequency. The difference inresonant frequencies Df in this configuration has reduced to

Figure 11 Measured and predicted resonant characteristicsof sample PCRTX with 100 mm core at V2 ¼ 10 kVDf ¼ 1.53 Hz

a Magnitudeb Phase

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an immeasurably low value and the coil coupling hasincreased to 0.92.

7.5 Inductance variation characteristics

The resonant tuning test was used to measure the secondarywinding self-inductance value of the sample PCRTX in bothcentre-gap and axial-offset arrangements for several differentcore displacements. The results are shown in Fig. 13. Centre-

Figure 12 Measured and predicted resonant characteristicsof sample PCRTX with 2 � 300 mm cores with a centre-gapof 195 mm at V2 ¼ 10 kV

a Magnitudeb Phase

Figure 13 Secondary winding self-inductance variation withcore displacement characteristics for the sample PCRTX with2 � 300 mm cores

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gap tuning provides more variation for the same offset. It istherefore preferred, and also has the advantage ofproducing no net axial magnetic forces between the coreand windings, which was an issue for previous designs.

7.6 Capacitive load test

A capacitive load test was performed on the sample PCRTXin two different configurations. In the first, the two 300 mmcore pieces were placed together and offset axially 355 mmfrom the winding centre. In the second configuration thetwo core pieces were centered with respect to the windingand a 195 mm gap was placed in the centre of the corepieces. Both configurations gave approximately the samevalue of secondary winding self-inductance, as shown inFig. 13. The load capacitance value was the same in bothcases and was chosen so that the power factor was leadingbelow the saturation voltage. The results are shown in Fig. 14.

The difference in primary currents between the twoconfigurations in the linear region of the graph is due tothe 8% difference in the value of secondary winding self-inductance, illustrating the sensitivity of the tuned circuit.The decrease in primary current at higher voltages is due tocore saturation. This reduces the inductance value, bringingthe loaded circuit closer to resonance. Saturation occured at

Figure 14 Capacitive load test for the sample PCRTX in twodifferent core configurations with the same value of loadcapacitance

a Magnitudeb Phase

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a significantly higher voltage in the centre-gapconfiguration, indicating a more uniform flux density. Inboth cases, measurable levels of harmonic distortion appearin the primary current waveform as the saturation voltage isreached.

8 ConclusionAn equivalent circuit model for the PCRTX has beendeveloped. The model is mathematically identical to theSteinmetz ‘exact’ transformer equivalent circuit but the coreloss resistance has been moved to the right hand side of thesecondary leakage reactance branch. The main advantage ofthe new model is that the input impedance can be moresimply expressed. This allows design specifications to bemore easily translated into required values or allowableranges of values of equivalent circuit components. Amethod has been presented for measuring the winding self-and mutual-inductances using a variable-frequency sinewavegenerator and a load capacitance. Secondary winding self-inductance variation with core displacement characteristicshave been measured for the sample transformer in thecentre-gap and axial-offset arrangements. Centre-gaptuning was found to be the most sensitive method ofinductance variation and allowed for operation at a highervoltage before the onset of saturation.

9 References

[1] BODGER P.S., ENRIGHT W.G.: ‘A resonant transformer forhigh voltage testing of generator stators’, Aust. J. Electr.Electron. Eng., 2004, 1, (3), pp. 179–185

[2] ENRIGHT W.G., BODGER P.S.: ‘Short time rated and protectedhigh voltage ac testing of generator stators using parallelresonant circuits’. Electricity Engineers Association Conf.(EEA), Christchurch, New Zealand, 18–19 June 2004, CD-ROM

[3] BELL S.C., ENRIGHT W.G., TUNSTALL K.R., BODGER P.S.: ‘Lightningarc drawings - dielectric barrier discharges for artwork’.15th Int. Symp. High Voltage Engineering, Ljubljana,Slovenia, 27–31 August 2007, CD-ROM

[4] LYNCH K., BODGER P.S., ENRIGHT W.G., BELL S.C.: ‘Partial coretransformer for energization of high voltage arc-signs’.15th Int. Symp. High Voltage Engineering, Ljubljana,Slovenia, 27–31 August 2007, CD-ROM

[5] BODGER P.S., LIEW M.C.: ‘Reverse as-built transformerdesign method’, Int. J. Electr. Eng. Educ., 2002, 39, (1),pp. 42–53

[6] LIEW M.C., BODGER P.S.: ‘Partial-core transformer designusing reverse modelling techniques’, IEE Proc., Electr.Power Appl., 2001, 148, (6), pp. 513–519


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[7] ENRIGHT W., ARRILLAGA J.: ‘A critique of Steinmetz model asa power transformer representation’, Int. J. Electr. Eng.Educ., 1998, 35, (4), pp. 370–375

[8] WITULSKI A.F.: ‘Introduction to modeling of transformersand coupled inductors’, IEEE Trans. Power Electron., 1995,10, (3), pp. 349–357

[9] EMERY F.T.: ‘Basics of power factor measurements onhigh voltage stator bars and stator windings’, IEEE Electr.Insul. Mag., 2004, 20, (31), pp. 40–45

[10] SHERMAN J.B.: ‘Some aspects of coupled and resonantcircuits’, Proc. IRE, 1942, 30, (11), pp. 505–510

[11] LEE R.: ‘A practical analysis of parallel resonance’, Proc.IRE, 1933, 21, (2), pp. 271–281


[12] MOSES A.J.: ‘Comparison of transformer loss predictionfrom computed and measured flux density distribution’,IEEE Trans. Magn., 1998, 34, (4), pp. 1186–1188

[13] ALONSO G., ANTONIO J.: ‘A new method for calculating ofleakage reactances and iron losses in transformers’. Proc.5th Int. Conf. Electrical Machines and Systems (ICEMS),Shenyang, China, August 2001, vol. 1, pp. 178–181

[14] MARGUERON X., KERADEC J.-P.: ‘Design of equivalent circuitsand characterization strategy for n-input coupled inductors’,IEEE Trans. Ind. Appl., 2007, 43, (1), pp. 14–22

[15] CONNELLY F.C.: ‘Transformers: their principles and designfor light electrical engineers’ (Sir Issac Putman & Sons Ltd,London, 1965, 1st edn.)

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