Magnetizing Current Harmonics, Interharmonics and Subharmonics: An

Analytical Study Alexander E. Emanuel and Mayer Humi

�Abstract—This paper is a theoretical study of the magnetizing

current components. Three conditions were investigated: 1. Sinusoidal voltage supply. 2. Sinusoidal voltage with dc bias. 3. Sinusoidal voltage with sinusoidal bias. The magnetizing curve is approximated with the equation . The analysis leads to closed expressions that enable the quantitative evaluation of effects dependent on the core geometry and physical constants of the active materials.

)sinh(bBaH �

Index Terms—Transformer magnetizing current, Harmonics, GIC, Power Quality.

I. NOMENCLATURE

ba, core material physical constants (3)B magnetic induction, (T) H magnetic field, (A/m) h harmonic order

Ii, instantaneous and rms current, respectively, (A) )(I),(I),(I 10 zzz h Bessel functions of order 0, 1 and h,

respectively� mean-path length, (m)

nm, frequency multipliers (27) t time, (s)

Vv, instantaneous and rms voltage, respectively, (V) z parameter (11)

�� , core geometry and material dependent coefficients � Gamma function

��, flux and flux linkage (vs) saturation coefficient

,� angular frequencies (rad/s)Subscripts:

F falseh harmonic order

nm, function orderR rated conditions S sinusoidal bias

h,1 fundamental, h-order

A.E. Emanuel is with the Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, (e-mail: aemanuel@ieee.org).M. Humi is with the Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609. (e-mail: mhumi@wpi.edu).

II. BACKGROUND

HE magnetizing branch of a transformer or autotransformer can be modeled by an equivalent

inductor. The inductor’s core geometry is defined by an equivalent cross sectional area A and an equivalent mean-path length The core carries N turns.

)m( 2

).m(�

The inductor is designed to operate with a sinusoidal voltage rated (V rms) and frequency f (Hz). The rated maximum flux linkage is

RV

�� R

RmV2

� (Vs) f�� 2� (1)

with RmRmRm NABN �� �� (2)

where RmRm B and � are the rated peak values of the magnetic flux (Vs) and induction (T), respectively. The actual magnetization curve of the core’s material may be simulated (best fitted) with one of the many analytical expressions found in literature [1,2,3]. For this study the following well known [4,5,6] equation H=H(B) was favored

)sinh(bBaH � ; core material constants (3) ba,

The magnetizing current is obtained by substituting (3) in Ampere’s Law

)sinh( ����� HN

i � (4)

where Na /��� (A/m) and .1(Vs)/ � ANb� The peak magnetizing current at rated conditions

)sinh( RmRmI ���� (5)can be conveniently used to normalize the current i. From (4) and (5) results

)sinh()/sinh(

)sinh()sinh(

��

���� Rm

RmRmn I

ii ��� (6)

where the parameter

Rm�� � (7) is unit less and controls the nonlinearity degree of the i=i( � )characteristic. In Fig. 1 are presented a set of curves that describe the normalized magnetizing current versus the

T

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978-1-4244-7245-1/10/$26.00 ©2010 IEEE

normalized flux, )( nnn ii �� where

RmRmRmn B

B���

��

���

All the curves intersect at the point that corresponds to rated conditions, i.e. .1 and 1 �� nn i�

Fig. 1. Normalized magnetizing current versus normalized flux

III. ANALYSIS

A. Sinusoidal Excitation The input voltage is

dtdtVv �� � � )sin(2 (8)

with a corresponding flux linkage

)cos( tm ��� ��

�V

m2

� (9)

Substitution of (9) in (4) gives the magnetizing current )]cos(sinh[( tzi ��� (10)

where

Rm

mm AN

bVz��

�

�� ���2

(11)

One should not confuse the parameter with z Rm�� � . The magnetizing current (10) can be rewritten using the identity

(12) 2/)()sinh( xxx � ��thus

][2

)cos()cos( tztzi �� ��� � (13)

At this point we make use of a less known, but useful identity [7]

(14) ��

���

10

)cos( )cos()(I2)(Ih

htz thzz �� �

where

� ���

����

���

����

0

2

)1(!4/

2)(I

k

kh

h khkzzz (15)

is a modified Bessel function of order h [5] and has the following property: )(I)(Ih zz h� for and�,4,2,0�h

)(I)(Ih zz h � for �,5,3,1�h . Substitution of (14) in (13) leads directly to the magnetizing current expression that reveals the harmonics

In Fig.2 are shown the magnetizing current waveforms of a single-phase transformer rated 1.5 kVA , 480/115 V, 60 Hz, (core made out of Si-steel, M-19, gage 24). The experimental oscillograms are reasonably approximated by (6).

Fig. 2. Normalized magnetizing current of a single-phase transformer, (1.5 kVA, 480/115 Hz, 60 Hz. (a) Experimental: V=115, 126, 138 V, (1.0 A/div, 2 ms/div) (b) Theoretical: .2.1,1.1,0.1/,9 �� z

The normalized harmonic currents for ,/ 1IIh 5,3�h and and 7 10,5� and 40, are presented in Fig. 3.

In the range 2.1/1 �� RVV the harmonic currents vary linearly with the applied voltage. The effect of the normalized applied voltage

Fig. 3. Sinusoidal voltage: Normalized magnetizing current harmonics, vs. the normalized voltage 1/ IIh ./ RVV

▀▀

�� /// zVV RmmR �� on the rms value of the magnetizing current is shown in Fig.4. As expected a large will cause a more abrupt increase of versus V .rmsI

Fig. 4. Sinusoidal voltage: Normalized rms magnetizing current, vs. the normalized voltage Rrmsrms II / ./ RVV

B. Sinusoidal Voltage with dc Bias The applied voltage

)sin(2 tVVv dc � � (18) produces a current that has a dc component that in turn will produce a dc biased flux linkage

)cos( tmdc ���� � (19) Substitution of (19) in (4) gives

][2

)]cos(sinh[(

)cos()cos( tztz

mdc

dcdc

ti

������ �����������

�

�� (20)

Applying the identity (14) one obtains

(21))cos()(I)cosh(2

)cos()(I2

)(I)sinh(2

)(])cos()(I2[2

)(])cos()(I2)([I2

])cos()(I2)([I2

])cos()(I2)([I2

5,3,1

6,4,2

0

1

10

10

10

���

�

���

�

���

�

���

��

��

��

��

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

hhdc

hhdc

hh

hh

hh

hh

thz

thzz

thz

thzz

thzz

thzzi

dcdc

dcdc

dc

dc

����

����

����

����

���

���

����

����

��

��

The consequence of dc biasing is the generation (injection)

of even harmonics. When 0�dc� , 0)sinh( �dc�� ,1)cosh( �dc�� and equation (21) becomes identical with

(16). A key question to be answered next is what is the value of dc� ? One may be tempted to use (4) and consider the expression

��

���

�

��� dc

dcFI1sinh1

where RVI dcdc /� and R is the winding resistance. The correct approach, however, is to equate the mean value of the current i, given in (21), with i.e. ,dcI

dcdc Iz �)sinh()(I0 ��� or

��

���

�

)(Isinh1

0

1

zI dc

dc ��� (22)

Since for , the Bessel function , results that 1!z 1)(I0 !z

dc� < dcF� . The physical cause for this effect stems from the system’s nonlinearity, in which case superposition cannot be applied. Moreover, the following observations may help explain this result: When a nonlinear inductance with an asymmetrical magnetizing characteristic, )()( �� " ii , is supplied with a perfectly sinusoidal voltage (8), the flux is also perfectly sinusoidal, nevertheless the magnetizing current will have odd and even harmonics. When a nonlinear inductance with a symmetrical magnetizing characteristic

)()( �� � ii is supplied with a sinusoidal voltage (8) the magnetizing current contains only odd harmonics (16), however, when the inductance is supplied with a dc biased voltage (18), acdc vVv �� , the flux will have a dc and an ac component acdc ��� �� (19) and the excursion of the flux

ac� takes place along an asymmetrical curve, causing the generation of both odd and even harmonic currents (21). The direct current is controlled by Ohm’s Law, dcdc RIV � . The direct voltage does not obey Faraday’s Law

0/ �" dtdV dcdc � , hence has no correlation with dc� . The ac component of the flux ac� sustains the ac component of voltage dtdv acac /�� and controls the alternating current component . In steady state the magnetic circuit will self adjust itself to a particular value of

aci

dc� that insures that the biased waveform of ac� yields the current that fulfills the condition:

aci

0)(21 2

0

�# tdiac ��

� (23)

In Fig. 5 are shown the magnetizing current waveforms for the same 1.5 kVA transformer supplied with 115 V and different dc biasing values. Such conditions are typical for transformers with GIC (Geomagnetically Induced Currents.)

▀▀

Fig. 5. Magnetizing current waveforms. Sinusoidal voltage with dc bias (a) Experimental: V=115 V, 0.0, 0.5, 1.0 and 1.3 A, (0.5

A/div, 2 ms/div) (b) Theoretical:

�dcI0.1/,9 �� z ,

0.0, 1.8, 3.3, 4.5. �Rrmsdc II /

The rms magnetizing current for a dc biased nonlinear inductance (APPENDIX B.) has the expression

)sinh(12

1)2(I)cosh(2

1)2(I 00dcdcrms

zzI ������ �

�

�

2

0

0)(

12

1)2(I��

���

�

�

zIIz dc

�� (24)

The theoretical and experimental curves of the normalized harmonics versus the normalized direct current, at Rrmsdc II /

,9� are presented in Fig. 6. The observed discrepancies are due to the imperfect matching of the “sinh” curve with the actual transformer’s magnetization curve and, of course the fact that the hysteresis loop was not accounted for..

Fig. 6. Normalized magnetizing current harmonics vs. normalized dc bias.

C. Two Sinusoidal Voltages of Different Frequencies

Low frequency subharmonics, that bias the fundamental voltage, are encountered at busses where modulated loads such as cycloconverters, spot welders, or converters that in turn supply choppers are connected, [8,9]. The applied voltage

)sin(2)sin(2 tVtVv S � � (25) yields the flux

)cos()cos( tt Sm �� ���� (26)

where � /2 SS V� . Oscillograms of the simulated voltage v and current (for i

502�� � and 22�� rad/s, are presented in Fig. 7. For transformers using magnetic cores with sharp saturation a relatively small ratio may cause a significant rms value increase as well as injection of additional interharmonics.

VVS /

The current expression (APPENDIX C.) is

)]cos()cos(sinh[ tti Sm �� ������

$%

$&'

� ��

� �,3,2,10 )cos()(I)(I2

hhS thzz ��

��

�

��,3,2,1

0 )cos()(I)(Ih

Sh thzz

with and (27)

$(

$)*

�� � ��

nmSnm tnmtnmzz

,])cos())[cos((I)(I ��

%&'

��

�

�

,6,4,2,5,3,1

nm

%&'

��

�

�

,5,3,1,6,4,2

nm

where SSz ��� This current has three terms: First term contains the carrier fundamental frequency � and the odd harmonics �h ; the second term contains the odd harmonics and the third h

Fig.7 Sinusoidal voltage with sinusoidal bias: Voltage and current oscillograms. ; ;05.0/ �VVS 3.0/ �RVV 20� ; 502�� � and

22�� rad/s. term, the so called interaction term, consists of subharmonics and interharmonics + nm� . The third term leads to three distinct situations:

Case I: K�/� , �K integer.

The current wave has the period KT��� 22

�

� , and the

rms value (APPENDIX D.) is

▀▀

21)2()2(I 00

� Srms

zIzI � (28)

Fig. 8. Voltages, magnetizing current oscillograms and frequency spectra. 0.1/ and,3.0//,10 ���� RsRs VVzVV

(a) Case I: �� 5502�� . (b) Case II: �� )2/5(502�� . (c) Case I: Spectrum. (d) Case II: Spectrum. (In black

are the characteristic harmonics, �5,3,1�h

Case II: ,� �/ , �, rational number. Now the actual period of the current wave is

--�-�

��� 22 KKT

therefore

,��

�

---

KK

Case III: .� �/ , �. irrational number. The resulting current’s expression is still (27), nevertheless its waveform is not periodical, "--- // �KK .

IV. EXAMPLES

In Fig.8 are given the oscillograms and the frequency spectra for two examples: Case I, 502�� � rad/s,

102�� rad/s, 3.0/,10 ��� SzKz . The period is 100205 �/�T ms.

For Case II we have 202�� rad/s, the remaining parameters are the same. In this case 2/520/50/ ��--- KK and the period is 100/25 �/� ��T ms. The unusually large ratio 3.0// �� RSS VVz was used to evidence the dominant interharmonics.

In fig.9 are presented curves that describe the normalized dominant interaction components versus the ratio

RSS VVz // � for .10� The trend of these graphs confirms the fact that very low values of cause a significant jump on the amplitude of the components of frequencies

RS VV /

+ nm� . The larger the integer K or K - ,the stronger becomes this effect. A second observation is that these interaction components converge fast (note the logarithmic scale.)

Fig.9 Normalized magnetizing Current interaction components vs. the normalized bias voltage ./ RS VV

Theoretically Case III components can be reasonably evaluated using Case II properties by approximating the irrational number . with a rational number, i.e. ., 0 . For example if we study a modulated current (27) where for convenience only the major components are included

▀▀

)cos()(I)(I)cos()(I)(Ii 1010 tztz SS �� �])2cos()2)[cos((I)(I 12 ttzS �� � ��

Choosing � 10, 5,�Sz 502�� � and 2102�� , we have for /� an irrational ratio and we find

�535533906.3210

50��.

If we assume 55.30, then 55.3/�0 and the pseudo sampling duration for the implementation of the FFT is

14207120 �--�-� KKT ms In Tables I and II are summarized the errors due to this

approximation when the FFT was implemented by means of PSpice. Evidently the errors are caused by the fact that the actual period of is not 71 ms but 70.71067812… ms.

TABLE I. Frequency Errors Component Actual (Hz) Error %

14.14213562… -0.411� 50.0 0.0

�2 85.85786438… 0.067��2 114.14213562… -0.051

TABLE II. Amplitude Errors Component Actual 1/ IIh Error %

0.256 -1.324� 1.000 -0.21

�2 0.207 -1.163��2 0.207 -2.17

The actual measurement techniques used for monitoring subharmonics and interharmonics are beyond the scope of this paper. Evidently that a larger observation time T leads to improved accuracy.

V. CONCLUSIONS

This paper reviews the basic mathematical analysis that governs the equations of the magnetizing current waveforms and spectrum. Using an approximate expression for the current-flux characteristic it was possible to develop closed expressions for all the magnetizing current harmonics, subharmonics and interharmonics as well for the rms value.

It was shown that due to the nonlinearity of the curvethe dc bias flux is not proportional with the dc bias current. This observation may impact the design of relays that sense dc leakage flux.

HB /

Once the actual )(�ii � curve is best fitted by (4), it is possible to evaluate the magnetizing current spectrum and rms value using software packages, such as EMTP or SPICE, or run simple calculations based on the presented expressions.

VI. APPENDIX

A. (17)From (13) is obtained

1 224

)cos(2)cos(22

2 �� tztzi �� ���

���

�

�

���

�

� � �

�

� �,6,4,20

2)cos()2(I42)2(I2

4h

h thzz ��

and

1 2()*

%&' �� # 1)2(I

2)()(1

02

0

22 ztdtiIrms�

����

�

�

B. (24)

���

�� �� 2

4)cos(22)cos(22

22 tztz dcdci ������ �����

���

�

�

���

�

� � �

�

� �,6,4,20

2)cosh()cos()2(I1)2(I

2h

dch thzz ����

)cosh()cos()2(I1)2(I2

,6,4,20

2dc

hh thzz ����

$%

$&

'

���

�

�

���

�

� � �

�

� �

$(

$)*

� ��

�

)sinh()cos()2(I,5,3,1

dch

h thz ����

1 23 4)cosh(1)2(I2

)()(21

022

0

22dcrms ztdtiI �����

�

�

�� #C. (27)

][2

)]cos()cos(sinh[

)cos()cos()cos()cos( tztztztz

Sm

SS

tti

�

��

�����

������

��

])cos()(I2)(I[

])cos()(I2)(I[2

10

10

�

��

�

�

�

�

5$%

$&'

��

hShS

hh

thzz

thzz ��

5 � ��

�

])cos()(I2)(I[1

0h

h thzz �

$(

$)*

� ��

�10 )]cos()(2)(I[

hShS thzIz

3 )(I)(I)(I)(2 0000 SS zzzzI ��

��

�

�1

0 )cos()(I)(I2h

hS thzz �

��

�

1

0 )cos()(I)(I2h

hS thzz �

��

�

�1

0 )cos()(I)(I2h

ShS thzz

▀▀

��

�

1

0 )cos()(I)(I2h

Sh thzz

)cos()cos()(I)(I41,

tntmzz Snnm

m � ��

�

�

)}cos()cos()(I)(I41,

tntmzz Snnm

m ��

�

�

D. (28) The squared current is

2)cos()cos()cos()cos(2

24 ��

��� � tztztztz SSi ����� ��

���

�

�

���

�

�

$%

$&

'

���

�

�

���

�

�� ��

�

�

�

� 10

10

2)cos()2(2)2()cos()2(2)2(

4n

snSm

m tnzIzItmzIzI ��

$(

$)

*

���

�

�

���

�

�

���

�

�

���

�

�� ��

�

�

�

�

2)cos()2(2)2()cos()2(2)2(1

01

0n

snSm

m tnzIzItmzIzI �

For the Case I the squared current is integrated over the range Kt �� 20 ��and for Case II over the range Kt -�� �� 20 . Since 0)](2sin[ �+ nmK� and

0)](2sin[ �--+- KnKm� as well, the integrals of all harmonic terms have zero mean values and for both, Case I and II, we obtain

1 22)2()2(24 002

2 � Srms zIzII �

VII. REFERENCES

[1] A. Ivany, “Hysteresis Models in Electromagnetic Computation,”Akademiai Kiado, Budapest, 1997 (Chapter 2, Analytical Models).

[2] G.F.I. Widger, “Representation of magnetization curves over extensive range by rational-functions approximations,” Proc. IEEE, 1965, Vol.116,No.4, pp.156-60.

[3] R,J. Fischer, H. Moser, “The representation of magnetization curve by simple algebraic and transcendental functions,” Archiv fur Elektrotechnik, 1956 Vol.42, pp.286-99.

[4] A. Boyajian, “Mathermatical analysis of non-linear circuits,” General Electric Journal, Vol.34, No.9, Sept.1931, pp.531-37. [5] R. Ladzinski “Theory of Transductors,” Electryca, No.1, 1953 [6] J.C. de Oliveira, L.C.O. de Oliveira and M.S. Miskulin. “Physical

considerations and modeling of AC/DC double excitation in power transformers,” Proceedings of the Fourth International Conference on Harmonics in Power Systems, Oct. 1990, Budapest, Hungary, pp.101-6.

[7] M. Abramovitz, I. Stegun, “Handbook of Mathematical Functions,” Dover Publications, NY,1965, pag. 376 (9.6.34).

[8] D. Gallo, R. Langella, A. Testa, A. Emanuel, “On the effects of voltage subharmonics on power transformers : A preliminary study,” The 11th

Intnl. Conf. On Harmonics and Quality of Power, Lake Placid, NY, Oct. 2004.

[9] A. Testa, “Issues related to interharmonics,” IEEE PES Annual Meeting, July 2003, Toronto Canada, paper 0-7803-8465.

Alexander E. Emanuel (SM 1971, LF 2005), earned the B.Sc, M.Sc and D.Sc. all from the Thechnion, Israel Institute of Thechnology. From 1969 till 1974 he worked as a R&D Senior Engineer for High Voltage Engineering in Burlington, MA. In 1974 he joined Worcester Polytechnic Institute where he teaches Electrical Engineering topics and conducts Power Quality related research.

Mayer Humi, earned B.Sc and M.Sc from the Hebrew University,Jerusalem and Ph.D from the Weizmann Institute,Rehovot,Israel. Form 1969 to 1970 was a postdoc fellow at the University of Toronto. In 1971 he joined the faculty of Worcester Polytechnic Institute where he is teaching and conducting research in Applied mathematics. He also published two books on Differential Equations and Boundary Value Problems.

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