Abstract

Inrush current is an important issue for three-phase

transformers security and stability. It has close relationship

with flux variation of three-phase transformers. Harmonic

effects in power systems have been significantly increasing in

the last decade. It results in more complicated for

investigating inrush current. In this paper, SINm( t)

waveforms represent non-sinusoidal waveforms because

studies on SINm( t) waveforms appear to be simpler for

more predictable results. A flux analysis of three-phase

transformer for inrush current is proposed in this paper. The

results are very helpful to estimate harmonics effect for inrush

current.

Keyword: Transient analysis; Transformers; Harmonics

1. Introduction

Three-phase transformers are key components in power

systems and power plants. They are also widely applied to

uninterruptible power supply. Security and stability of

transformers are both important and necessary to system

operation. The large transient current of transformers due to

flux saturation in the core often causes the mal-function of the

protective relaying system, costing time and money as the

engineers have to examine closely the transformer and the

protective system, to check for faults. The large transient

current also causes serious electromagnetic stress impact and

shortens the life of transformers. The overvoltage [1]-[2]

resulting from the inrush current could happen and cause

serious damage to power apparatus. So it is very important to

solve the effect of inrush current.

In recent years, various protective systems for

transformers, based on the differential relaying system, were

developed. Various techniques based on complex circuits or

microcomputers [3]-[6] are proposed to distinguish inrush

current from fault current. However, the transformer still

must bear with large electromagnetic stress impact caused by

the inrush current.

Non-sinusoidal voltage and current have significantly

increased in power system due to broad application of solid-

state controlled devices and fluorescent lamps. Transformer is

the most sensitive component in response to power system

harmonics. As non-sinusoidal harmonics have been generated

from many sources, harmonics flow through many

transformers and cause a compound effect to the power system.

Focusing on the drawbacks of the above protection, an

active suppression method [7]-[8] is developed by our group.

The suppression method is very simple and effective. The flux

of transformers is controlled to non-saturation by developed

method. Non-sinusoidal excitation strongly influences the flux

variation of three-phase transformers. Therefore, it is

necessary to investigate the effect of non-sinusoidal excitation

to flux variation for inrush current. In this paper, a flux

analysis of three-phase transformers for inrush current under

non-sinusoidal excitation is proposed. The maximum flux

value of three-phase transformers is calculated and discussed

in detail. It is very helpful to estimate the effect of active

suppression method for suppressing inrush current.

2.The Proposed Method

2.1 Flux Analysis under Sinusoidal Excitation

Suppose that the connection circuit between three-phase

transformer ( Y) and power source is shown in Fig. 1. The

source line voltages in Fig. 1 are

VRS=V sin (1)

VST=V sin(

VTR=V sin(

R

S

T

A

B

C

tR = t

1

tS = t

1

tT = t

2

3-Phase

Transformer3-Phase

Source

Fig. 1. The connection circuit between three-phase transformer and power

source.

Estimation of Transformer Inrush Current under Harmonic Source

C. L. Cheng, S. C. Chern, Q. S. Chen, F. Y. Lin Department of Electrical Engineering

National Formosa University

Huwei, Yunlin, 632, Taiwan

clcheng@nfu.edu.tw

10811­4244­0178­X/06/$20.00 ©2006 IEEE PSCE 2006

Where the symbol V represents the peak value under

sinusoidal excitation. It is used as compared base value in

following discussion. The source has RST phase sequence.

The switches on lines R, S, T are closed at t1, t1 and t2

respectively. It is used as compared base value in following

discussion. The flux linkages of A, B, C windings of the three-

phase transformer, are computed as

A =t

tRSdtV

1=

Vcos t + cos t1 (4)

B =2

121 )(

t

tRS dtV +

t

tST dtV

2

=V

2

1( t2-cos t1)+cos( t2-2 cos(

V[

2

3

2

1cos cos( (5)

C =

2

121 )(

t

tRS dtV +

t

tTRdtV

2

=V

[2

1( t2 cos t1)+cos( t2- cos(

V[

2

3

2

1cos t1 cos(

Suppose that the switches on lines R, S, T are closed at the

same time, that is t2= t1, the flux linkages of A, B, C windings

of the three-phase transformer become

A =t

tRS dtV

1=

Vcos cos

B =V

[ cos( cos(

C=

V[

2

3

2

1cos cos(

V[ cos( cos(

Observing Eqs. (7)-(9), the reachable maximum value of flux

linkages is different depending on t1. Suppose that switches on

lines R, S and T are closed at t1= , simultaneously. The

reachable maximum absolute values of flux linkages on A, B,

C windings are computed as

A |V

B | 1.866V

C| 1.866

V

The reachable maximum value flux linkage on winding A is

V. It is the least value, the same as steady-state condition.

However, the reachable maximum value flux linkage on B and

C windings is 1.866V

, much more than normal value V

.

The relation between line voltage and flux linkage at

t1= t2= is shown in Fig. 2. The flux saturation results in

serious inrush current. Besides, the reachable maximum value

of flux linkage on A, B, C windings cannot increase or

decrease simultaneously. This means that the reachable

maximum value of flux linkage cannot be suppressed by

powering on switches A, B, C, simultaneously.

flu

x lin

ka

ge

/

line

vo

lta

ge

/ V

V

Fig. 2. The relation between line voltage and flux linkage at t1= t2= ,

m= 1. (line voltage :VRS /V, :VST /V, :VTR /V, flux linkage

:A

/V

|B

/V

| 1.866C

/V

| 1.866)

After analyzing flux linkage variation, selecting t1= t2=

the following results can be obtained from Eqs. (4)-(6).

A |V

B |V

C|

V

The reachable maximum absolute values of flux linkages on A,

B, C windings are the least value V

, the same as steady-state

condition. The relation between line voltage and flux linkage

at t1= t2= is shown in Fig. 3. This cannot cause flux

saturation. Hence the inrush current can be effectively

eliminated.

1082

flu

x lin

ka

ge

/

line

vo

lta

ge

/ V

V

Fig. 3. The relation between line voltage and flux linkage at t1= , t2= ,

m= 1. (line voltage :VRS /V, :VST /V, :VTR /V, flux linkage

:A

/V

|B

/V

| 1C

/V

| 1)

2.2 Flux Analysis under Non-sinusoidal Excitation

The distorted sinusoidal waveforms are described by m-

order of sine functions in a convenient and generalized manner

as:

VRS=Vmax SINm( t) (10)

VST=Vmax SINm( (11)

VTR=Vmax SINm( (12)

Where Vmax is the peak value of voltage, and m is the order of

sinusoidal waveform. Graphs of (10) are presented in Fig. 4.

This function covers a wide spectrum of distortions, which

exist in power system. The advantage of this approach

consists of the fact that only one parameter of the exponent m

can describe the distortion clearly. Strictly speaking, (10)-(12)

are not typical for all distorted waveforms. However, it is

more generalized and predictable for analyses and simulations.

In this paper, only the generalized functions of (10)-(12) are

used.

Fig. 4. The standard distorted excitation function SINm(wt) with same root-

mean-square value Vrms =110 V.

Observing Fig.4, distortion parameter m=1 represents pure

sinusoidal excitation. The discussion of sinusoidal excitation is

presented in above section. Under non-sinusoidal excitation,

distortion parameter m=10 and m=0.1 are discussed as

following respectively. Suppose that three excitations is

measured at same effective value Vrms =110 V. The peak value

Vmax of line voltage can be calculated with respect to distortion

parameter m=1,10,0.1 under same root-mean-square value

Vrms =110 V. The calculated results are shown in Table 1.

Table 1 shows the peak value Vmax = 155.7 volt at m=1, Vmax =

262.1 volt at m=10 and Vmax = 121.2 volt at m=0.1.

Table 1. The peak value Vmax of line voltage with respect to distortion

parameter m, under same effective voltage Vrms =110 V.

distortion

parameter m

Vrms (volt)

Vmax(volt)

2.2.1 Distortion parameter m=10

VRS=Vmaxsin10

(13)

VST=Vmaxsin10

( (14)

VTR=Vmaxsin10

( (15)

The source has RST phase sequence. The switches on lines R,

S, T are closed at t1, t1 and t2 respectively. The flux linkages of

A, B, C windings of the three-phase transformer, are computed

as

A =t

tRS dtV

1

max 1

10240

V (-2sin10 + 25sin8 -150sin6

+600sin4 -2100sin2 +2520 t) 1

|tt (16)

B=

2

121 )(

t

tRS dtV +

t

tST dtV

2

= - max 1

20480

V (-2sin10 + 25sin8 -150sin6 + 600

sin4 -2100sin2 +2520 t) 2

1|tt + max 1

10240

V [-2sin10(

+25sin8( -150sin6( +600sin4( -

2100sin2( +2520(2

|tt (17)

C =

2

121 )(

t

tRS dtV +

t

tTRdtV

2

= - max 1

20480

V (-2sin10 + 25sin8 -150sin6 +600 sin4 -

2100sin2 +2520 t) 2

1|tt + max 1

10240

V [-2sin10( +

1083

25sin8( -150sin6( +600sin4( -

2100sin2( +2520(2

|tt (18)

Observing Eqs. (16)-(18), the reachable maximum value of

flux linkages is different depending on t1. Suppose that

switches on lines R, S and T are closed at t1= t2= ,

simultaneously. The reachable maximum absolute values of

flux linkages on A, B, C windings are computed as

A|

V

B | 1.289V

C| 1.302

V

The reachable maximum value flux linkage on winding A is

V. It is a small value, less than steady-state condition.

This cannot cause flux saturation. However, the reachable

maximum value flux linkage on B and C windings is

1.289V

and 1.302V

more than normal value V

. The

relation between line voltage and flux linkage at t1= t2=

is shown in Fig. 5. The flux saturation results in

inrush current.

1.285

2.569

0

-

2.569

-1.285

1.285

2.57

0

-2.57

-

1.285

V

1

1

2

2

3

3

Fig. 5. The relation between line voltage and flux linkage at t1= t2= ,

m= 10. (line voltage :VRS /V, :VST /V, :VTR /V, flux linkage

:A

/V

|B

/V

| 1.289C

/V

| 1.302)

Selecting t1= t2= the following results can be

obtained from Eqs. (16)-(18).

A|

V

B| 0.911

V

C| 0.910

V

The reachable maximum absolute values of flux linkages on A,

B, C windings are less than V

. The relation between line

voltage and flux linkage at t1= t2= is shown in Fig. 6.

This cannot cause flux saturation. Hence the inrush current

can be effectively eliminated.

VFig. 6. The relation between line voltage and flux linkage at t1= , t2= ,

m= 10. (line voltage :VRS /V, :VST /V, :VTR /V, flux linkage

:A

/V

|B

/V

|C

/V

| 0.910)

2.2.2 Distortion parameter m =0.1

VRS=Vmaxsin0.1

(19)

VST=Vmaxsin0.1

( (20)

VTR=Vmaxsin0.1

( (21)

The source has RST phase sequence. The switches on lines R,

S, T are closed at t1, t1 and t2, respectively. The flux linkages

of A, B, C windings of the three-phase transformer, are

computed as

0.1

max1

sint

AtV tdt (22)

B=

2

1

1.0

max21 )sin(

t

tdttV

+t

tdttV

2

1.0

max ))3/2(sin( (23)

C =

2

1

1.0

max21 )sin(

t

tdttV

+ t

tdttV

2

1.0

max ))3/4(sin( (24)

Based on MATLAB computation from (22)-(24), the

reachable maximum value of flux linkage is different

depending on t1. Suppose that switches on lines R, S and T are

closed at t1= t2= , simultaneously. The reachable

1084

maximum absolute values of flux linkages on A, B, C

windings are computed as

A | 1.143V

B| 1.939

V

C | 1.939V

The reachable maximum value flux linkage on B, C winding is

1.939V

, much more than normal value V

The relation

between line voltage and flux linkage at t1= t2= is shown

in Fig. 7. The flux saturation results in serious inrush current.

V

Fig. 7. The relation between line voltage and flux linkage at t1= t2= ,

m= 0.1. (line voltage :VRS /V, :VST /V, :VTR /V, flux

linkage :A

/V

|B

/V

| 1.939C

/V

|

1.939)

Selecting t1= t2= the following results can be

obtained from Eqs. (22)-(24).

A | 1.143V

B | 1.309V

C | 1.309V

The reachable maximum value of flux linkage on B, C winding

is 1.309V

, more than normal value V

The relation between

line voltage and flux linkage at t1= 2 t2= is shown in Fig.

8. This causes minor flux saturation. The minor flux

saturation results in smaller inrush current.

V

Fig. 8. The relation between line voltage and flux linkage at t1= , t2= ,

m= 0.1. (line voltage :VRS /V, :VST /V, :VTR /V, flux

linkage :A

/V

|B

/V

|C

/V

|

1.309)

3. Results and Discussion

A 330VA, 110V/220V single-phase transformer and a

1KVA, 110V/220V three-phase transformer are used for on-

site measurement of inrush current in our laboratory.

3.1 Results under Sinusoidal Excitation

3.1.1 Improper Switching-on

For three-phase transformers, Fig. 9. shows the

experimental result of inrush current (Ip=16A), under t1

= t2 = , three-phase transformer banks. The peak value of

the inrush current of winding A, in Fig.9, is effectively

suppressed due to no flux saturation. However, the inrush

current of phases B and C cannot effectively be suppressed

simultaneously. It can be observed that the peak value of the

inrush current of winding C in Fig. 8, Ip= 16A, is more than 5

times the rated phase current.

Fig. 9. Experimental result of inrush current (Ip=16A), under t1 = t2= ,

three-phase transformer banks. (1: phase A, 2: phase B, 3: phase C)

3.1.2 Proper Switching-on

According to analysis results of flux linkage variation,

selecting t1 =4.2ms( t1 ), t2 =8.8ms( t2 ), the

experimental result of the inrush current (Ip=2A) is shown in

Fig. 10. The inrush current is effectively suppressed.

1085

Fig. 10. Experimental result of inrush current (Ip=2A), under t1 =

4.2ms, t2 =8.8ms, three-limb three-phase transformers.

(1: phase A, 2: phase B, 3:phase C)

3.2 Influence of Non-sinusoidal Excitation

In this paper, the influences of non-sinusoidal excitation

to flux linkage are investigated in detail. The results are

helpful for suppressing inrush current. Table 2 shows the

comparison of reachable maximum absolute values of flux

linkages on A, B, C windings under various conditions.

Observing Table 2, non-sinusoidal excitation strongly

influences flux saturation of core in three-phase transformers.

Under t1= 2 t2= 2 maximum flux linkage on A, B,

C windings is 1.866V

at m=1, 1.302V

at m=10, 1.939V

at

m=0.1. The flux saturation is more serious at m=0.1 and much

less serious at m=10. The similar results exist on other

switching-on conditions. In conclusion, the non-sinusoidal

excitation, more close to m=0.1, causes more serious flux

saturation. On the contrary, the non-sinusoidal excitation,

more lose to m=10, cause less flux saturation. Under

t1= 2 t2= maximum flux linkage on A, B, C

windings is 1V

at m=1, 0.911V

at m=10, 1.309V

at m=0.1.

Flux linkage 1V

is the smallest value among all sinusoidal

excitations m=1. Consequently, switching-on angle of t1= 2,

t2= , can be applied to suppress inrush current very well for

three-phase transformers. However, for non-sinusoidal

excitation m=0.1, the flux linkage 1.309V

at

t1= 2 t2= is not the least value. Comparing the flux

linkages under non-sinusoidal excitation m=0.1, flux linkage

1.309V

at t1= 2 t2= is more than 1.275V

at

t1=95 t2=180. The switching-on angle of t1= 2 t2=

is no more the best selecting for suppressing inrush current.

By changing switching-on angles, the maximum flux linkage

can be decreased from 1.309V V

t1=95, t2=180.

In conclusion, non-sinusoidal excitation strongly influences

flux saturation and flux saturation can be improved by proper

switching-on.

4. Conclusion

This paper proposed an effective flux analysis of three-

phase transformers in detail for estimating inrush current. The

effect of non-sinusoidal excitation to inrush current is

investigated and discussed in detail. In this paper, m-order

sine functions bring some representative discussions for this

topic. The results can be applied to improve the protective

relaying system of three-phase transformers [9]. Furthermore,

the experimental examination of non-sinusoidal excitation to

inrush current for three-phase transformers is being

investigated in our lab.

5. Acknowledgment

This work is supported by the National Science Council under

research project: NSC93-2213-E150-025.

References

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m=1 m=10 m=0.1

A B C A B C A B C

t1= t2 =90 1. 1.866 1.866 0.651 1.289 1.302 1.143 1.939 1.939

t1 =90

t2 =180

1. 1 1 0.651 0.911 0.910 1.143 1.309 1.309

t1 =90

t2=170

1 1.152 1.150 0.651 0.952 0.810 1.143 1.160 1.391

t1 =90

t2=190

1 1.151 1.151 0.651 0.800 0.965 1.143 1.391 1.162

t1=100

t2 =180

1.174 1.087 1.087 0.923 1.060 1.060 1.279 1.241 1.241

t1 =85

t2=180

1.087 1.044 1.044 0.796 0.850 0.850 1.210 1.342 1.342

t1 =95

t2 =180

1.088 1.044 1.044 0.797 0.996 0.996 1.211 1.275 1.275

Table 2. The comparison of reachable maximum absolute values of flux linkages

( V / ) on A, B, C windings under various conditions.

1086

SYSTEMS, April 18-20, 2005, Krabi, Thailand, pp.14-18.

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Suppressing Method of Three-Phase Transformers Inrush Current”,

IEEE Region 10 Technical Conference on Computers,

Communications, Control and Power Engineering Proceedings,

2002, Beijing, China, pp. 2030-2033.

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for Three-Phase Transformers”, IEEE/PES Transmission and

Distribution Conference and Exhibition 2002: Asia Pacific,

Conference Proceedings, 2002, Yokohama, Japan, pp. 1808-1813.

1087