The Role of Magnetizing and Leakage Inductance in Transformer Coupling Circuitry

Petrus A. JANSE VAN RENSBURG† and Hendrik C. FERREIRA‡

†Department of Electrical Engineering ‡ Department of Electrical and Electronic Engineering

Border Technikon Rand Afrikaans University

P.Bag 1421, East London, 5200, South Africa P.O.Box 524, Auckland Park, 2006, South Africa

Phone: +27-82-200-6207, Fax +27-43-702-9226 Phone +27-11-489-2463, Fax +27-11-489-2357

E-mail: pvanren@ingulube.bortech.ac.za E-mail: hcf@ing.rau.ac.za

Abstract

As the internal impedances of a coupling transformer canhave a detrimental effect on coupling, these have to beproperly designed. This paper starts with the fundamentaldefinition of inductance, and continues to define leakageinductance and magnetizing inductance. The effect of bothleakage and magnetizing inductance on coupling isdiscussed. Design equations are derived and various methodsto increase and decrease these inductances are given1.

Keywords: modem design, coupling circuits, filters, leakageinductance, magnetizing inductance, transformers.

1. Introduction

In [1] and [2] it was shown that accurate modeling anddesign of leakage inductance is necessary in order to designan effective transformer-capacitor coupling circuit.Furthermore, transmission through this coupling circuit isruined if the transformer saturates, and for this reasonaccurate magnetizing inductance design is necessary.Although theoretical understanding is required for such acoupling transformer design, practical knowledge is oftennecessary to alter a design parameter. Therefore varioushands-on methods to increase or decrease these impedancevalues are discussed.

2. Summarized theory of inductance

2.1 Definition of inductance

Fundamentally, inductance is defined by the well-knownequation

1 This work was supported in part by the S.A. National ResearchFoundation under Grant No. 2053408.

dt

diLv L

L ⋅= (1)

and therefore inductance L is the constant of proportionalitybetween instantaneous voltage and the time derivative ofinstantaneous current for a certain inductor. Furthermore,Faraday’s Law states that

di

d

dt

dNv

λφ ≡⋅= (2)

where λ (number of flux-linkages) is defined as Nφ. Byequating (1) and (2), inductance can be expressed in terms ofλ as

di

d

di

dNL

λφ =⋅= , (3)

a non-linear equation which is material dependent. If λ isplotted against i for a certain inductor, the resultant curveexhibits a similar shape to the applicable B-H curve of themagnetic material used (see Fig. 1 below).

λλλλ ∝∝∝∝ B

λλλλSAT ∝∝∝∝ BSAT

iSAT i ∝∝∝∝ H

Fig. 1. Plot of λλλλ vs i to show non-linearity. Dotted line indicatesassumption that magnetic material is linear up to saturationlevel.

In the rest of this paper, a linear λ-i relationship will beassumed in order to make the modeling and mathematicsmanageable. It is therefore assumed that the inductor core

has a constant µR. (This assumption is accurate only for air-core inductors or cored inductors where excitation isminimal.) Based on this assumption, inductance can now bewritten as

iiNL

λφ =⋅= , (4)

the ratio between total flux-linkage and the current caused bysaid flux-linkage.

2.2 Inductance in transformers

In a transformer core, two main flux paths exist, and theassociated fluxes are called mutual (or magnetizing) flux andleakage flux. These two flux components are responsible fora transformer’s magnetizing and leakage inductancerespectively. For an inductor though, the magnetizing andleakage flux components add to make up the totalinductance. Refer to Fig. 2.

φφφφL1 φφφφM1 φφφφM2 φφφφL2

φφφφN1 φφφφN2

i1 i2

N1 N2

primary core secondary

MMF N1i1 ≈≈≈≈ N2i2

core window

Fig. 2. Simplified representation of a transformer’s flux pathsand MMF distribution across the core window.

In Fig. 2., φN1 represents the total flux induced by MMF N1i1.A certain component of φN1 ‘leaks’ back to the bottom leg ofthe core and is labeled φL1. This primary leakage flux φL1

therefore does not couple with the secondary winding N2.The remaining (mutual) primary flux φM1 however doescouple with N2 and induces MMF φN2 in winding N2.Similarly, only φM2, the magnetizing (mutual) portion of φN2

couples with N1, whereas the uncoupled flux φL2 leaks backthrough the core window.

Using (4), the magnetizing inductance LM of a transformercan now be written as

11 i

NL MM

φ⋅= (5)

where 21 MMM φφφ −= represents the resultant mutual or

magnetizing flux. Similarly, the resultant leakage inductanceLL can be written as

1

21

1

1121 i

Ni

NLLL LLLLL

φφ ⋅+⋅=+= (6)

where LL1 and LL2 are the primary and secondary leakageinductances respectively.

Assuming negligible losses and a linear λ-i relationship, thevoltages across the primary and secondary windings can beobtained through Faraday’s Law (2):

v Ldi

dtL

di

dt

N

N

di

dtL M1 11 1 2

1

2= + − ⋅

(7)

v Ldi

dt

N

NL

di

dt

N

N

di

dtL M2 22 2

1

1 2

1

2= − + − ⋅

(8)

These two equations, with resistors added in order to modellosses, give rise to the widely used transformer equivalentcircuit shown in Fig. 3 below. For an ideal transformer, theseries components need to be negligibly small, whereas theparallel components need to exhibit very large values.

RN1 LL1 LL2 RN2

RC LM Ideal

Fig. 3. General transformer equivalent circuit, assuming linearcore material, losses modeled with resistors.

The secondary leakage inductance can be referred to theprimary side, but this causes inaccuracies as the magnetizingcurrent does not flow through LL2. For a large magnetizinginductance though, the secondary leakage inductance may besafely referred to the primary side using

( ) 2

2

2

12 LL L

N

NreferredL ⋅

= (9)

3. Leakage inductance considerations

3.1 Influence on filtering

The leakage inductance of a coupling transformer forms aseries-resonant band-pass filter with the external (series)coupling capacitor. For frequencies close to resonance, theeffect of the magnetizing inductance is negligible. Seesection 4 as well as [1] or [2].

3.2 Calculation of leakage inductance

An alternative approach to calculate inductance, is to use twoenergy equations, namely the energy stored in an inductor(10) as well as the energy stored in a certain three-dimensional space through volume-integration of the B-Hproduct (11):

WLi=

2

2(10)

W BHdv H dvv

r

v

= =∫ ∫1

2 20 2µ µ (11)

By equating (10) and (11), and solving for L, inductance canbe expressed as

Li

H dvr

v

= ∫µ µ0

22 . (12)

As an example, the leakage inductance of the followingplanar transformer in Fig. 4 will be calculated. Thistransformer has separately wound primary and secondarywindings (no interleaving). Also see [3] for a differentwinding configuration.

secondary winding N2 w

X

b

c

a

N1i1≈≈≈≈N2i2 primary winding N1 core

Fig. 4. Cross-section through a planar transformer and MMFdistribution across window.

Assuming that µR = 1 for volumes v1 , v2 , and v3 (volumestaken up by windings N1, N2, and isolation between windingsrespectively) the leakage inductance can be calculated. Thetotal ‘leakage’ energy stored in volume v1, is given by

W dWv

1 1

1

= ∫ where dW H dv10 2

12= ⋅µ . (13)

As graphically depicted in Fig. 4 the MMF in volume v1 canbe written as

Hx

a

N i

w= ⋅ 1 1 for 0 < <x a . (14)

Take note that w, the width of a planar winding, representsthe effective magnetic path-length of the leakage flux. It isnow also assumed that the winding structure, viewed fromthe top, has a cylindrical shape with average turn-length T inorder that dv = A⋅dx = Tw⋅dx. It follows that

Wx

a

N i

wTw dx

a

10 1 1

2

0 2= ⋅ ⋅

⋅ ⋅∫µ (15)

⇒WN i

awTw x dx

a

10 1 1

22

02= ⋅

⋅ ⋅ ⋅∫µ (16)

⇒WN i

awTw a1

0 1 12

3

2

1

3= ⋅

⋅ ⋅ ⋅µ (17)

⇒ ( )WaT

wN i1

01 1

2

6= ⋅

µ (18)

Similarly, it can be shown that the leakage energy W2 storedin winding N2 is

( )WbT

wN i2

01 1

2

6= ⋅

µ (19)

as N1i1 ≈ N2i2. Furthermore, the energy stored in volume v3

can be calculated by

WN i

wTw dx

c

30 1 1

2

0 2= ⋅

⋅ ⋅∫µ as (20)

HN i

w= 1 1 for volume v3. Therefore (21)

( )WcT

wN i3

01 1

2

2= ⋅µ . (22)

The total ‘leakage’ energy stored in the transformer can beobtained by adding the individual energy components, andtherefore the total energy stored in the ‘leakage inductance’LL of the transformer is given by

2321 2

1iLWWWW LL =++= (23)

From (18), (19), (22) and (23) it follows that

LN T

wa b cL = ⋅ ⋅ + +µ 0 1

2

33( ) [H] (24)

where µ0 = 4πx10-7 is the permeability of free space and N1

the number of turns on the primary winding. T represents theaverage turn-length whereas w denotes core window width.As shown in Fig. 4, symbols a, b and c represent thethickness of the primary, secondary and insulation layersrespectively.

3.3 Enlarging leakage inductance

If a transformer’s leakage inductance is to be utilized as afilter inductor (instead of using an external, discreteinductor) its value often has to be increased. A fewtechniques will be discussed below:

• Increase the insulation thickness c between the primaryand secondary layers – refer to (24).

• Increase the number of turns N1 of the primary (andsecondary to keep winding ratio the same). See (24).

• Insert ferrite (or other material) with permeability µR

between primary and secondary windings to drasticallyboost the leakage flux. Equation (24) then becomes

)3(3

210 cbaw

TNL RL µµ ++⋅⋅= [H]. (25)

• Allow the external winding to protrude out of the core.See Fig. 5 below. If the original leakage inductance ofthe transformer without the protruding loop is negligible,the leakage inductance with protruding loop is given byan empirical equation [4]

HR

RNLL

250230

221

+= [µH] (26)

where R and H represent the radius and height of theprotruding loop respectively.

Fig. 5. Cross-section through transformer (top view) to indicatewinding protruding from core. External loop has a radius R.

3.4 Reducing leakage inductance

For high-frequency applications, the leakage inductance hasto resonate with the series coupling capacitor at the carrierfrequency. This implies that smaller reactive componentvalues are required. Although it would be the easiest andmost economical to reduce the coupling capacitance value,the leakage inductance may have to be reduced in certaincases. Some general guidelines are given below:

• Choose an appropriate core type that encloses thewindings to minimize external leakage.

• Minimize thickness of insulating layer betweenwindings.

• Interleave layers of primary and secondary windings inorder to lower the effective MMF.

Using the method described in section 3.2, it can be proventhat the theoretical leakage inductance of an interleavedtransformer (where the primary winding is divided into qlayers and secondary winding is divided into q+1 layers) isgiven by

( )Lq

N T

wa b cL = ⋅ ⋅ + +µ 0 1

2

66 [H] (27)

Comparing (25) with (24) it is evident that leakageinductance can be reduced by almost 75% by merelydividing the primary winding into q = 2 layers and thesecondary winding into three layers.

4. Magnetizing inductance considerations

4.1 Influence on saturation and filtering

As the total flux density in a coupling transformer core ismade up of the communication waveform’s flux densitysuperimposed on the filtered power waveform’s flux density,the influence of both have to be considered in order toprevent saturation. The influence of the filtered powerwaveform on core saturation can be gauged by its v/f factor,also called the volt-time product. Equation (28) refers to Fig.6(a) and summarizes the super-imposing influence of thefiltered power waveform (subscript P) and the referredcommunication / modem waveform (subscript M) on the totalflux density (subscript T), and is valid for sinusoidalwaveforms only.

+⋅=+=

M

M

P

PMPT

f

V

f

V

ANBBB

''

2

1

π(28)

The ratio of the filtered power waveform’s v/f value to thecommunication waveform’s v/f value will determine how thetwo waveforms share (or exceed) the total flux density, ascore cross-sectional area (A) and number of primary turns(N) are fixed for a certain design. Take note that in Fig. 6(a)

and equation (28) the voltage V’P refers to the peak filteredpower voltage, whereas V’M refers to the peak referredcommunication voltage. Furthermore, the minimum value offM has to be used in order to calculate the maximum fluxdensity.

As an example, the coupling circuit designed in [2] will bediscussed. This coupling circuit was designed to transmitand receive any 122dBµV or 1.26VPEAK CENELEC B, C, orD signal (95kHz-148.5kHz) to and from a 50Hz, 220VRMS,1Ω power line. Thus the received communication waveformwill have a maximum v/f factor of only 1.26V / 95kHz =13µV/Hz, yielding a flux density of BM ≈ 9mT for the chosencore set. For the filtered power waveform to have a v/f figure(and flux density contribution) of say twice this value, itneeds to be filtered to a voltage of only 1.3mVPEAK or by107.6dB. Alternatively, a power waveform attenuation levelof –100dB would represent 62.2µV/Hz (BP ≈ 43mT) whereas–90dB would represent 196.7µV/Hz (BP ≈ 136.2mT), all stillwithin acceptable limits as BSAT = 400mT for this ferrite core.

At a first glance, it seems as if this band-pass coupling circuitdesigned in [2], would only be able to filter the powerwaveform by a maximum of 60dB, as only three decades isavailable for filtering, and the roll-off slope is -20dB/decade.However, it must be emphasized that the model in [2] is onlyvalid for frequencies close to the resonant point. A morecomprehensive model of the coupling circuit that includesthe magnetizing inductance, is shown in Fig. 6(a). Thevoltage transfer function of this circuit (in the receivingdirection) is expressed by

RsLLLRCsLCLs

RCLs

V

V

MLMLM

M

P

P

++++=

)(

'23

2 (29)

where R represents the referred modem impedance. From(29) it is clear that the power-line impedance has noinfluence on the voltage transfer function in the receivingdirection. The amplitude response of (29) was simulated andis shown in Fig. 6(b). As can be seen in Fig. 6(b), thesimulated amplitude response of the coupling circuit underconsideration correlates well with the simple band-passmodel of [2] for frequencies above 1.6kHz, but a roll-offslope of –40dB/decade is evident below 1.6kHz.

This is due to the influence of the parallel magnetizinginductance at lower frequencies. At 1.6kHz, the reactance ofthis coupling transformer’s 100µH magnetizing impedancenearly equals the 1Ω (referred) load resistor, and atfrequencies below 1.6kHz, its reactance dominates theparallel branch (LM||R), attenuating these frequencies morethan for a pure resistor.

C LL

VP V’P + V’M

(superimposed)

ZP LM R’M

≡≡≡≡ R

(a)

Fig. 6. (a) More comprehensive model for the coupling circuitdesigned in [2] and (b) simulated amplitude response for thecoupling circuit designed in [2]. Here the reference level is 0dBas the 1:7 transformer winding ratio was not included in thesimulation.

It is not necessary to simulate above circuit for every design,as the magnitude of the filtered power waveform voltage fora certain frequency (voltage across the magnetizinginductance) can be calculated using the voltage divider rule:

PRLMLLC

RLMP V

XXXX

XXV ⋅

++=

)(' (30)

Subscripts LL and LM refer to leakage and magnetizinginductance respectively, whereas subscript R refers to themodem impedance referred to the power-line side. Once V’P

is known, its contribution to the flux density can becalculated with (28).

The simulated amplitude response (Fig 6(b)) of the couplingcircuit designed in [2] was verified practically and ameasurement is shown in Fig. 7. In Fig. 7 the general trendof Fig. 6(b) can be seen, although low-frequency values arenot as accurate because of a limited bandwidth whenscanning a wide frequency range. The coupling circuitdeviates from theoretical values for low frequencies. This ispartly due to the limited power output of the measuring

20

-10

-40

-70

-100 16 160 1.6k 16k 160k 1.6M 16M

Frequency (Hz) (b)

20lo

g1

0[V

’ P/V

P]

(d

B)

instrument - the transformer core cannot be driven deeplyinto saturation.

Fig. 7. Measured amplitude response of the coupling circuitdesigned in [2] The cursor was positioned at ≈≈≈≈ 50Hz andmeasured –83dB (+17dB reference level).

From the cursor measurement in Fig. 7 it is evident that forthe specific coupling circuit designed in [2], a totalattenuation of 100dB is attainable. Thus the power waveformis expected to occupy 43mT of flux density whilst thecommunication waveform is predicted to utilize 9mT. Thesum of these two figures, 52mT, will be the total flux densityin the coupling transformer’s core. This value will be thesame, regardless of whether the coupling circuit is used toreceive data or transmit data: although the voltages on themodem side is seven-fold those on the power line side, thenumber of turns is also seven-fold and thus cancels theinfluence of former.

4.2 Calculation of magnetizing inductance

It was shown in section 2 that

iNL

φ⋅= where ℜ

=ℜℑ= Niφ . (31)

Furthermore reluctance ℜ is defined by

e

e

A

l

µ=ℜ (32)

where µ represents permeability, le is the effective flux pathlength and Ae is the effective core cross-sectional area. From(31) and (32) follows that total inductance

e

eRLM

l

ANLLL

2

0 ⋅=+= µµ . (33)

Equation (33) gives reasonably accurate values formagnetizing inductance as leakage inductance LL is typically

negligible. Leakage inductance can be subtracted from L toobtain more accurate answers for LM.

4.3 Enlarging/Reducing magnetizing inductance

From equation (33) it is clear that magnetizing inductancecan be increased by either increasing µR, increasing A ordecreasing l (using a different transformer core), oralternatively increasing N (thinner conductors or biggercore). It must be mentioned that µR is also dependent on theair gap if a core set is used. In this case µR can be boosted bydecreasing the air-gap, or even clamping the core settogether.

Conversely, magnetizing inductance can be decreased byinserting an air-gap between certain legs of a core. Thistechnique of fine-tuning the air-gap for a certain magnetizinginductance is widely used. Take note that the air-gap doesn’talter BSAT but only lowers µe to skew the B-H curve [5].Although BSAT stays the same, a smaller magnetizinginductance would help to filter low frequencies moreeffectively, as the –40dB/decade slope starts at a higherfrequency. Refer to Fig. 6(b) and Fig. 7.

5. Conclusion

In order to design an effective coupling circuit, the internalimpedances have to be designed properly, as they can have adetrimental effect on coupling by either (i) filtering thecommunication signal or (ii) allowing the power waveformto saturate the coupling transformer. Therefore thefundamental definition of inductance, as well as leakageinductance and magnetizing inductance were discussed.Design equations were derived and various practical methodsto increase and decrease these inductances were discussed.

References

[1] P. A. Janse van Rensburg, H. C. Ferreira, “Coupling circuitry:understanding the functions of different components,” Proc.7th Int. Symp. Power-Line Comm., 2003, pp. 204-209.

[2] P. A. Janse van Rensburg, H. C. Ferreira, “Step-by-step designof a coupling circuit with bi-directional transmissioncapabilities,” Submitted for 8th Int. Symp. Power-Line Comm.,2004.

[3] A. S. Langsdorf, Theory of Alternating-Current Machinery,2nd ed., New York: McGraw-Hill, 1955, pp. 43-49.

[4] P. C. Theron and J. A. Ferreira, “The zero voltage switchingpartial series resonant converter,” IEEE Transactions onIndustry Applications, vol. 31, no. 4, pp. 879-886, July/Aug.1995.

[5] J. G. Kassakian, M. F. Schlecht, G. C. Verghese, Principles ofPower Electronics, Reading, Massachusetts: Addison-Wesley,1991, p. 572.

+15

-15

-45

-75

-105

-135 10 100 1k 10k 100k 1M 10M

Frequency (Hz)

20lo

g1

0[V

M/V

P]

(d

B)