CHAPTER2
2.1 INTRODUCTION
Inrush currents under consideration are high magnitude transient currents that flow in the primary circuit of a transformer when it is energized under no-load or lightly loaded condition. The peak of the inrush current is found to be as high as 10 times that of the rated current of the transformer lasting for a few cycles. Inrush currents are unsymmetrical, contains harmonics and de component. The decay rate of inrush current is determined by the ratio of resistance to inductance of the primary winding and source.
The knowledge of peak inrush currents, decaying time, its harmonic order and magnitude are required to avoid mal-operation of protective relays and temporary over voltages during commissioning of cable connected transformers. In this chapter a numerical method to estimate the worst case inrush current is described which presents satisfactory results of inrush current calculated over a longer period of time by considering saturation and residual flux. With this, remedial measures can be adopted to avoid the problem of temporary over voltages during transformer energization. The equation for inrush current is developed based on the characteristics of the inrush current present in a laboratory type transformer both by simulation and experiment.
2.2 FACTORS INFLUENCING INRUSH CURRENT
The magnitude and duration of transient inrush current depend on 1) the point-on­ voltage wave at the instant of switching 2) the magnitude and polarity of the residual flux in the transformer core at the instant of energization 3) the inherent primary winding air­ core inductance which depends on volts I turn , that is energization on low voltage side causes more inrush current compared to high voltage side energization 4) the primary winding resistance and the impedance of the circuit supplying the transformer 5) the
10

maximum flux-carrying capability of the core material. Inrush currents are originated by the high saturation of iron core during switching- in of transformers [I].
Switching- on angle
Inrush current decreases when switching-on angle on the voltage wave increases.
The current response of RL circuit when switched at an angle (a.) is given by i(t)=l m where i(t) : instantaneous current in ampere a switching-on angle in degree
I m the maximum steady-state current in ampere e power factor angle in degree
R resistance of the winding in ohm
X reactance of the winding in ohm t : time in seconds rot: in degree
(2.1)
The transient vanishes when a = e and therefore doubling effect will not take place for highly reactive RL circuits when switched at a = 90 ° (8=90 ° ). Moreover for any
RL circuit the peak switching current Ip, also confines to steady-state maximum level I m when a = ± 90 ° . Thus the inrush current of a single-phase transformer can easily be eliminated when switched at a± 90 ° , using an instant-controlled switching circuit.
Residual flux
This parameter is the most difficult to determine exactly. Inrush current is significantly aggravated by the magnitude and polarity of residual flux density. This in turn depends on core material characteristics, power factor of the load at interruption and the angle at which the transformer was switched off. The total current i
0 is made up of the magnetizing current component i m and the hysteresis loss component ih. The current interruption generally occurs at or near zero of the total current waveform . The magnetizing current passes through its maximum value before the instant at which the total current is switched off for no load, lagging load and unity power factor load
11

conditions, resulting in maximum value of residual flux as per the flux- time curve of
Fig.2.1. For leading loads, if the leading component is less than the magnetizing
Fig.2.1 Flux-time curve component, at zero of the resultant current the magnetizing component will have reached the maximum value resulting in the maximum residual flux. On the contrary, if the leading current component is more than the magnetizing component, the angle between maximum of the magnetizing current and zero of the resultant current will be more than
90 ° . Hence, at the interruption of the resultant current, the magnetizing component will not have reached its maximum resulting in a lower value of residual flux density. The point on voltage waveform at which a transformer is switched-off influences the amount of flux that remains in the core. Typically, the amount of flux that remains in the core is anywhere from 30% to 80% of the maximum core flux and can be positive or negative.
Its maximum value is usually taken as about 80% and 60% of the saturation value for cold rolled and hot rolled materials respectively. It is also a function of joint characteristics. In practice the transformer supplying de load through rectifier circuits appear to have a remnant flux close to the minimum value, when switched on no-load.
The value will depend on the transformer magnetizing inductance, the capacitance and the resistance of the bridge.
Saturation flux
Saturation flux density is how much magnetic flux the magnetic core can handle before becoming saturation and not able to hold any more. This depends on several factors including core material temperature, electrical and magnetic condition on the
12

transformers. When the core saturates, the transformer no longer acts like an inductor with a linear increase in current over time. The magnetic field cannot increase further and current is limited by the source impedance of the power supply and the resistance of the transformer wire. This leads to a very large current. It is usually essential to avoid reaching saturation since it is accompanied by drop in inductance. The rate at which current in the coil increases is inversely proportional to the inductance (di/dt = v/L). Any drop in inductance therefore causes the current to rise faster, increasing the field strength and so the core is driven even further into saturation causing fall in inductance.
2.3 PHYSICAL PHENOMINA
When a transformer is re-energized by a voltage source, the flux linkage must match the voltage change according to Faraday's law
Y m sin (rot+a) = i
0
R+ N d:tm where , V m
<D m
N
1 0 maximum value of steady state voltage in volt instantaneous value of flux in weber primary winding turns no load transformer current in ampere
The solution of the equation assuming linear magnetic characteristics [1] is
(2.2)
<D m
= (<D mp
COS
R
<D r
) e-(x:)t _
(<D mp
COS (rot+a)) where <D m p: peak flux
(2.3)
<D r
: residual flux
The equation (2.3) shows that the first component is a flux wave of transient de component, which decays at a rate determined by the ratio of resistance to inductance of primary winding and a steady state ac component. The de component drives the core strongly into saturation, which requires a high exciting current. Inrush current waveform is completely offset in first few cycles with wiping out of alternate half cycles because the flux density is below saturation value for these half cycles. Hence, the inrush current is highly asymmetrical and has a predominant second harmonic component.
13

Time constant (L/R) of the circuit is not constant; the value of L changes depending on the extent of core saturation. During the first few cycles, saturation is high and L is low. Hence, initial rate of decay of inrush current is high. As the losses damp the circuit, saturation drops, L increases slowing down the decay. Therefore the decay of inrush current starts with a high initial rate and progressively reduces; the total phenomenon lasts for a few seconds. Inrush currents are generated when transformer is switched on at an instant of voltage wave, which does not correspond to the actual flux density in the core at that instant. When a transformer is switched off, the excitation
· current follows the hysteresis curve to zero. The flux density value will change to a non­ zero value or zero corresponding on the material of the core. Suppose the transformer is switched on again at the zero crossing instant of voltage, then flux should be at its negative peak as flux lags behind applied voltage by 90 ° . In the subsequent half cycle, flux varies from -<l> m to +<l> m
, But as per the constant flux linkage theorem, magnetic flux in an inductive cannot change instantaneously. Hence, the flux instead of starting from negative maximum value starts from 0. Then it will rise to 2 <I> m in the subsequent half cycle.
The flux will be unsymmetrical at the time of turn on. It will rise to 2 <I> m initially and then depending upon the circuit parameters will reduce to <l> m
, The flux value of 2 <I> m will drive the core into saturation thereby causing currents of very high magnitude to flow in the primary circuit. The inrush currents are highly unsymmetrical because in the positive half cycle the flux has a very high value, thus the induced currents will be very high. But in the negative half cycle, the flux has a very low value, thus the currents are also low. As the flux falls to normal steady state value, the magnitude of inrush current also decreases. The explanation given in the section can be justified mathematically using the following equations:
Let the source voltage be given by v(t) = V m sin ( rot+a)
Then the instantaneous flux in the circuit is given by
<I> m
1 rt
= N Jo v(t)dt
Substituting (2.4) in (2.5)
(2.4)
(2.5)
14

<I> m
= cos(wt + a)] (2.6)
Maximum flux that can be build up in the core when a voltage is applied at its zero crossing instant is,
<I> m p = 2 <I> m
(2.7)
If there is some amount of residual flux, say <I> r
, in the transformer core at thetime of switch on, then the flux will rise to a value of 2<I> m
+ <l> r
. This will drive the core into deep saturation, thereby causing the currents even higher than that of earlier case as shown in Fig.2.2. The figure shows inrush currents produced in the transformer core with and without residual flux. In the presence of remenance of flux
(2.8) l(l)
Inrush current
(with remanance)
Fig.2.2 Graphical description of inrush current phenomena
2.4 EXPERIMENTAL DETERMINATION
In order to study the different behaviour of transient magnetising current of a transformer during energization on no load first it is determined experimentally. For this a laboratory type single phase 230V /230V, l kV A 50 Hz transformer with a tertiary winding of 115V is considered. The parameters of the transformer are determined from open and short circuit tests. The values are calculated for a base current of I base
=4.348A and Z base
=52.89 and are shown in Table I of Appendix. The experimental set up for measuring the primary current I a and energization voltage E a of the transformer is shown in Fig.2.3. As the rated current of the transformer is 4A inrush current is assumed to be
50A peak (1.4142*4*8) a hall sensor HE055T and a potential transformer 240V/6V
15

are used for converting the high level current I a and voltage E a of the transformer in the range of± 15V for compatibility with the data acquisition card. As the output current of the hall sensor is 50mA for an input current of 50A a resistor R
1
(6 Ohm ) is connecte� to the output of the hall sensor so that the maximum voltage drop across R1 is 300mA. In order to limit the operational amplifier output voltage within the range of +/-15 V which is compatible for data card, the operational amplifier gain is taken as 45. As the potential transformer has a 240V /6V turns ratio voltage gain multiplier is taken as 40 ( 40*6 = 240).
The circuit diagram of the set up for plotting the inrush current and voltage signals using
N
Ia
�---ol---/
Ill
TEST
TRANSFORMER p
TRANSISTOR
PCl1711
Fig.2.3 Experimental diagram for measuring voltage and current
Analog
Input
Analog lnpJt1
Mv.anteoh
PCl-1711 (a,uto) i--------1� ln10ut1
:.::od
Dlglta
Output
Digital o,tput
.Adv ante ch
Cl-1711 [auto)
S::,opei!5
Analog
Input
Analog Input
.Qdvanteoh
PCl-1711 lautol S::ope3 S::,opeO
Fig 2.4a) Block diagram for plotting voltage and current signals using the MATLAB-Simulink real time workshop
16

ln1
Out1
Logical
Operator
Constant3
Fig. 2.4(b) Switching circuit for energizing the transformer at different inception angles
MA TLAB-Simulink real time workshop is shown in Fig.2.4. A PCI 1711 data card is used to interface the voltage and current signals with the host computer. In order to plot the inrush current at different inception angle, a zero crossing detector (zed) is used. The expanded circuit of zero crossing detectors is shown in Fig.2.4b. It uses two switches to convert the sinusoidal voltage signal and the delayed voltage signal using the transport delay into a square waveform. This in phase and delayed waveforms are connected to a logical XOR gate and again passed to an AND gate with that of in phase square waveform. The required delay is obtained in the transport delay block in Fig.2.4b and a mono pulse is generated which is modified by the discrete monostable. The output of this block generates the required digital output for switching the transformer through a relay.
As this signal is low to drive the relay, a NPN transistor TIP12 I is used to amplify the signal as shown in Fig.2.3. The inrush currents for different energization angles are determined and are shown in Table 2.1. The peak inrush current of 45A occurs at a switching angle near to o 0 and 180 ° . Fig.2.5 shows the waveform for a switching angle of
0 ° . First plot shows the switching signal to close the relay, second plot the energization
(A)
DC peak (A)
Settling time (s)
Table 2.1 Measured values of inrush currents
Switching Angle
Peak current First cycle
Second cycle ou
45
18.5
9.6
0.5
90 u
-2.5
-1.26
-0.2
0.45
180
-25
-7.3
-3.5
°
0.47
17
270
5.2
1.4
0.5
°
0.43

voltage and the third plot shows the inrush current. There is a delay between the instant of switching signal and the actual time of relay operation. Inrush current for a switching angle of 90 ° is shown in Fig.2.6 and peak of inrush current in this case is 2.5A. The parameters of the PCI 1711 card, hall effect current sensor HE055T, TIP121 and relay
Fig.2.5 Transformer energization at 0 °
0.1
(c)
0.2 0.3
Time (s) a) Switching signal b) Voltage c) Inrush current
,. ..
...................................... T ................................................ T
.......................
..,
....... .
,A
0
Q 1
--·-,---·-·-····-··-r····--·-·-·--,-------r-------..---
V
005 f
0
-005
.(I t
0 01
(b) t) H, 02
Fig.2.6 Transformer energization at 90 ° a) Voltage and switching signal b) Inrush current
18

Harmonic analysis
The inrush current I a during the peak value is subjected to harmonic analysis by passing the current signal to the discrete Fourier block in MA TLAB-Simulink. The harmonic spectrum for the inrush current is shown in Fig.2. 7. The spectrum shows the de component is larger than the fundamental component of current. Currents in the
Fundamental (50Hz) = 0.003704 , THO= 100.63%
100
140
80
60
�
::a:
40
20
0
0 200 400 600
Frequency (Hz)
800
Fig.2. 7 Harmonic spectrum of inrush current
1000 frequency range of 100,300,400,500 and 800 are stronger. Fig.2.8 shows the plot of de component. The peak value reaches to 11.5A instantaneously and exit for a short duration of approximately 0.05s, then decreases exponentially.
0.25�-�--�-�--�-�---.-----,
0.2
0.15 o 0.1
0.05
0 o 0.1 0.2 0.3
Time(s)
0.4 0.5
Fig.2.8 de component of inrush current
0.6 0.7
19

Effect of power factor and angle of current interruption on inrush current
In order to find the effect of power factor of the load, first the test transformer is demagnetized to bring the residual flux equal to zero. This is achieved by applying a 50
Hz low voltage to the secondary winding through an autotransformer and then decreasing gradually to zero. Then the transformer is energized at different switching angles each time connecting to a resistive load and demagnetizing the core before next energization.
In each case the peak inrush current is noted. Next an inductive load is connected instead of resistive load and the experiment is repeated. To study the effect of current interruption angle the residual flux is made zero. Then the transformer is energized and connected to a resistive load and interrupted at a definite angle. Again the transformer is energized and peak inrush current is noted. This is repeated for different instants of current interruption with resistive and inductive loads. The results reveal that the nature of load current and interruption angle has effect on inrush current. Since in real situations it is difficult to control the switching off the load current, further investigations are not carried out in this direction. The results are given in Table 2.2
Table 2.2 Variation of inrush current due different load interruption angle le 0
42
90
-5
0
180
-30
270
6
2.5. SIMULATION RESULTS
In this section the investigation related to the determination of the inrush current characteristics is carried out with time-domain simulations in PSCAD environment. The transformer and the energization circuit modeling details are discussed in this section.
Transformer model
The transformer representation used for the investigations is the model available in PSCAD. The value of parameters for simulation is those of a transformer used for experimental determination which is given in Appendix. Magnetic core residual flux is represented by a de current source in parallel with the low voltage winding; the current is chosen to establish the required level of residual flux linkage [15]. The polarity of the
20

residual flux is changed by reversing the de source. The residual flux and air core reactance in the model transformer is achieved by setting the de current source such that both the simulation and experimental peak inrush currents are the same. In this thesis, this transformer model is taken throughout for simulation studies and the results are validated.
Circuit model
The circuit diagram considered for inrush current investigations is shown in
Fig.2.9. The test transformer is connected to the line to ground voltage of the secondary side of an 11 kV /415V delta-star solidly grounded distribution transformer. This is taken as the voltage source and this transformer impedance is taken as the source impedance.
P = 5.256e-008
Q = 8.243e-01 o
V= 0.2268
BRK1
-=-
()
.i,.
'*le
[;> la
···· l aR·�� f
C .ClCl 1 ·· .... · I a R
I::
�1
-=-
FFT
F = 50.0 [Hz] s
Ma 7
(7)
Ph
(7) la
HarmonicTo tal ..
l
7
,._JQ
�aryvoltage
A',n;:::i l�ary current
Fig.2.9. Circuit for simulating transformer inrush current
The voltage, current and power signals are obtained from the power system meters. The primary voltage is measured by the voltmeter E a
, current by ammeter I a and the power by the multimeter A/V. The medium voltage side line resistance and inductance are taken from data book [69] which is I ohm and 0.0031 H respectively. In the figure, the primary
21

side switching angle is controlled by a timer switch such that voltage is applied at the instant of its zero-crossing or peak of voltage in both positive and negative cycles. The inrush current is analysed by Fast Fourier Transform (FFT) method.
Simulation of inrush current
Different cases of inrush current are simulated by varying those major parameters that influence the characteristics of inrush current. These parameters are angle of switching, magnitude and polarity of residual flux. First the primary side of the test transformer is energized by a voltage of 230 V at a switching angle of o 0 without any residual flux. Fig.2.10 shows the variation of the applied voltage together with the inrush current and core flux which is the integral of voltage. It is observed from the waveform that at the time of switching on the primary voltage at zero cross over instant, without residual flux, the flux rises from zero value to a maximum value of l .8pu and is highly unsymmetrical. The reason for this is that the flux present in the transformer core before the application of the primary voltage was zero. By virtue of constant flux linkage theorem, the flux remains at the same value when the primary voltage is applied. Since the flux is expected to lag the applied voltage, the flux is to be at a value of -lp.u,
-400
30
:w
10
0
11 w g
C:
�
2.0
•s;sJ
1.0
0.0 n,ne<s)
.1.0
0.100
•:
0.150 0.200 O.Z50 0.300
Fig.2.10 Transformer energization without residual flux at switching angle 0 ° b) Inrush current c) Core flux a) Voltage
22

which is to rise to I p.u. when the voltage is applied. However since the initial flux has a value of zero, it will rise to a maximum value of 1.8 p.u. in the first half cycle and is highly unsymmetrical. This causes inrush current of peak magnitude equal to 27 A and
14A to flow in primary circuit during first and second cycles respectively. Then due to damping in the circuit the inrush current is found to reduce to their steady state value of
1. IA. The peak value of active power and reactive power demand read by the multimeter is 350 W and 2075 VAr respectively. The worst case scenario when the transformer is switched-in at a switching angle of 0 ° with positive residual flux of 0.8pu is shown in
Fig.2.11. In this case, at the zero voltage cross over the core flux starts from 0.8 pu and reaches to a peak value of 2.2 pu. The peak value of inrush current for the two successive cycles is 45A and 24A and settles at 0.3s. The transformer is also energized for a negative residual flux at a switching angle of 0 ° and Fig.2.12 shows the plot of flux and inrush current. The variation of active power and reactive power demand during energization at o 0 switching angle with positive residual flux is shown in Fig.2.13. The figure shows that there is a peak demand of 850W active power and 3000 Y Ar reactive power from the source which gradually decreases.
50� -"=-
---····--···-��--�-------����������--'
40
30
20
10
-10
0 rune(s)
-1.50
0.1 oo 0.150 0.200 0.250 0.300
Fig.2.11 Transformer energization with positive residual flux at switching angle
0 °
(a) Core flux
(b) Inrush current
23

g:
-17.5
��--
!
-0.50
)(
1.00
0.50
0.00
-1.00
-1.50
-2.00
Time(s)
0.100 0.150 0.200 0.250
--�--�-..-
.....
Fig.2.12 Transformer energization negative residual flux at switching angle 0 ° b) Core flux a) Inrush current
.................................................. ____ ... __ ........ _ _J
2.0k
1.0k
0.0
-1.0k �
-®.
-2.0k
-3.0k
-4.0k
....
... ___ ..................... _ ... "'-''' ·=·· -====-===="-;===�==·=···a,· .. ·=· .. ·-=···=-..·
=
··"",'"'' "·
=
...... =a,==,----;
0.100 0.150 0.200 0.250 0.300 0.350 0.400
Fig.2.13 Transformer energization with positive residual flux at switching angle 0 ° power b) Reactive power a) Active
Harmonic analysis
The harmonic content of the transformer inrush current is calculated using Fast
Fourier Transform (FFT) method Fig.2.14 shows the waveform of the lower three frequency components present in the inrush current. The results presented are for the worst case switching of o 0 with positive residual flux to get the maximum inrush current.
It shows that the peak value of any individual harmonic component during one cycle is generally different from its peak during another cycle.
24

25.0
20.0
15.0
10.0
5.0
Time(s)
0.100 0.120
....--
0.140
-�T-'-���-----�"-'-=--=--·=--·=, -·- -;= -
0.160 0.180 0.200 0.220 0.240 0.260
..
Fig.2.14. Harmonic components of inrush current.
Due to the non-symmetrical wave shape, the transformer inrush current contains all harmonic components i.e. fundamental, 2 n° , 3 rc1 , 4 th , 5 th etc as well as a de (zero frequency) component. The magnitude of the peak significant (fundamental) component
(blue) is 22 A, second (green), and third (red) order frequency components are 12A, and
9.2 respectively. The fourth, fifth, sixth and seventh order harmonics are 1.3A, 0.94A,
0.1 and 0.36A respectively. The spectrum also contains a de component of peak value
16A and this is shown in Fig.2.15. The second harmonic current creates negative sequence current and the DC component causes the sympathy current in the power system. The THO measured is 71 %. The frequency spectrum is shown in Fig. 2.16.
-···-· - -·-- -·------·----··---· ----·-------·-----·-------··---·--------------------_:J
16.0
I
- -·-··-·····--- ·- ·----··-·-·-··--····--------·-----------------·____j
12.0
� 8.0
4.0
Q
-
-
Tme(s)
0.0
0.100 0.150 0.200 0.250 0300 0.350
Fig, 2.15 de component of inrush current
25

1:50.0
0.0
Harmonic order (7) 2.76879
Fig.2.16 Frequency spectrum of inrush current
Comparative evaluation
In order to study the performance at various switching conditions, the transformer is energized at different inception angles. Parameters like peak inrush currents, peak demand of active power, peak demand of reactive power; peak de values, second harmonic component, Total Harmonic Distortion (THO) and settling time are determined. Since inrush current energy parameter J 1 2 t is a measure of the temperature rise it is also calculated for one cycle. Table 2.3 summarises the parameter values for different switching instants without any residual flux. The effect of adding positive residual flux and negative residual flux in the transformer core on inrush current is represented in Table 2.4 and Table 2.5 respectively.
Table 2.3 Switching parameters for zero residual flux
Switching Angle
Peak current First cycle
(A)
Harmonics
%
Second cycle
Second
THO
DC peak (A)
Peak reactive power (V Ar )
Peak power (W)
JI 2 t
Peak flux pu
First
Second
Settling time (s) ou
27
14
64
85
5.6
2075
350
0.24
1.8
1.7
0.9
90 °
-12
-6
50
65
-2
774
157
0.04
-1.6
-1.53
0.9
180 °
-38
15
66
84
-8.5
3082
591
0.4
7.9
-1.7
0.9
270 °
12
7
47
65
2.1
776
157
0.04
1.6
1.5
0.9
26

(A)
Harmonics
%
Peak Flux pu
JI 2 t
Table 2.4 Parameters for 0.8pu positive residual flux
Switching Angle
Peak current First cycle
Second cycle
Second
THO
Peak DC(A)
Peak reactive power (VAr)
Peak power (W)
First
Second
Settling time (s) ou
45
25
43
80
10.7
3000
850
0.78
2.03
1.76
0.6
90 °
-2.7
-1.2
18.7
25
0.137
237
59
0.004
1.3
1.29
0.13
180 °
-20
-11
59
83
-4
588
153
0.12
-1.7
-1.6
0.9
270 °
30
16
68
82
6
822
213
0.34
1.8
1.6
1
Jh
Table 2.5 Parameters for 0.8pu negative residual flux
Switching Angle
Peak current First cycle
(A)
Harmonics
Second cycle
Second
% THO
Peak DC (A)
Peak reactive power (V Ar)
Peak power (W)
Peak Flux pu
First
Second
Settling time(s) au
9.5
6.2
36
45
2
296
68
0.04
1.58
1.29
0.4
90 °
30
-4
42
54
-4.7
440
100
0.31
-1.9
-1.5
I
180 v
-45
-17
62
69
-11
795
178
0.84
-2
-1.7
I
270 v
2.7
2.5
17
22
0.75
254
57
0.004
1.34
1.3
0.3
The Tables indicate that inrush currents are large during energization, depends on polarity of residual flux and angle of switching. The large value of inrush currents and reactive power demand in the system causes voltage sag problems. Various parameters during transformer energization at different inception angle are compared from the tables and a bar chart is drawn which is shown in Fig.2.17. In this figure x-axis is the switching angle and y-axis is the magnitude of current. The bar chart shows that the most favourable condition for switching a transformer is at 90 ° or 270 ° .
27

~
'...;/
~
Q)
0 t::
;:J
10
0
·10
·20
-30
... 0
SO
0
3t1
20
·50
• For ;:ero residu"llIux
.Forpositiive
re~du~fl o
90 130 2701
Switching angle (degrees)
Fig.2.17. Bar chart inrush currents for various switching instants.
Effect of harmonics
The effect of harmonics in the grid voltage during energization of transformer is also studied. The point of coupling of the transformer is polluted by connecting a non linear load to achieve a THD of 5%. The harmonically polluted voltage waveform is shown in Fig.2.18. The values of inrush current for various switching angles and residual flux are also simulated. Observation of the values shows that the value of inrush current is slightly feduced and there is change in the harmonics content of inrush current.
~
Q)
~
0>
'"
400
•
300
200
100
0
-100
-200
-300
Time(s) 0.100
0.110
0.120
0.130
0.140
0.150
0.160
0.170
0.180
0.190
0.200
Fig.2.18.Harmonically polluted grid voltage
28

2.6 MATHEMATICAL FORMULATION
The inrush current and the de component waveform obtained both by simulation and experiment for an energization angle of o 0 is shown in Fig.2.19 and Fig.2.20 respectively. It is assumed that the residual flux is in the same direction as that of the
................................. I ·�
.................................................. .1 :
.
Time(s)
-2.0
0.100 0.150 0.200 0.250 0.300 0.350
Fig2.19 Simulated waveform (a) Inrush current (b) de component g
1
0.1 0.2 0.3
Time (s)
(a)
0.25 ,----.--�--�---,---�---.-----.
0.2
0.05
I
Time(s)
(b)
Fig.2.20 Experimental waveforms a) Inrush current b) de component
29

initial flux change, thus giving maximum possible value of inrush current. The comparison of inrush current and de component waveform obtained by both experiment and simulation are similar to that of an exponentially decaying half wave rectified waveform. The inrush current is completely offset in first few cycles with wiping out of alternate half cycles. Due to the non-symmetrical wave shape, this wave form contains, besides the fundamental, harmonics of the order kn± 1 (k = l , n = l, 2, 3 ... ) where k is the no. of pulses per cycle. The
(1 n-1 t are of the negative-sequence type, whereas ( 1 n+ 1 )
1h harmonics are of the positive sequence type. Hence the inrush current contains all harmonic components; even harmonics appear to be the dominant and a de (zero frequency) component. The exponential decay of inrush current is taken same as that for the transient flux from (2) which is proportional to e - ""I t . The waveform of the de current shows that the time constant (L/R) of the circuit is not constant; the value of L changes depending on the extent of core saturation. During the first few cycles, saturation is high and L is low. Hence the initial peak current is large and the rate of decay of inrush current is quite high. As the losses damp the circuit saturation drops, L increases slowing down the decay. Therefore the decay of inrush current starts with a high initial rate and progressively reduces.
If a periodic wave form is represented by an analytical expression f (t), then a Fourier series can be applied to obtain a descriptive equation for f(t). In order to interpret this, the following mathematical model is considered. ao = f(t). dt (2.9) an = T f(t). cos nwt. dt for n = 1,2,3 etc.
(2.10) bn
= � f
0
T f(t). sin nwt. dt for n
=
1,2,3 etc.
where ao : de component
(2.11) a n
: Fourier coefficient of cosine terms b n
: Fourier coefficient of sine terms
Then the periodic function is represented by the Fourier series [70] f(t) = ao
+ i:�=
1
Ca n cos nwt + b n sin nwt )
If the inrush current waveform is considered as a half wave rectified waveform which is exponentially decaying, the equation satisfying Fourier series comprises of a de
30

term, cosine terms, b n coefficient for n = I only (for n greater than I, b n =
O) and an exponential function to represent the decay. From these terms an approximate mathematical model to realize the inrush current waveform can be deduced as, i(t) = (� + � cos(wt - 90) + L�=
2
210 n(l-n ) cos
e -It
(2.12)
The first term of the equation inside the parenthesis (
10
Tr
) is the de component which is the average value of the function, second term a sinusoid at the frequency of the waveform representing the fundamental component and the third term corresponds to the
R harmonic components. The multiplying exponential term e -I t is the time for which inrush current flows. Considering the initial steady de value and the non-linear inductance L 1
, i(t) = more approximate equation is
(1;
((u(t) - u(t - t
0
) + e-tiCt-to). u(t - t
0
)) +
(�. cos(wt - 90) + L�=
2 C
210
2
)
R cos nwt) e
-L 1 t (2.13) where u(t) is the step input, to = duration of the steady de value. This is obtained from the various simulation graphs of de current waveforms, the average value of which is
The non-linear inductor L I is represented as follows [9]
L 1 =
=
L s
O<t<to
L to<t<oo where L s and L n are saturation and nominal inductance. Normally nominal inductance
L = V/2*3.14£ n
, Ls = V/2*3.14£
5 where
£ n
£
5
Normal flux
Saturation flux
L 1 non linear inductance of transformer
Io is the peak inrush current of the first cycle given by [1]
Io = (K
1
(1 - cos �))/Xs where
K
1
A w
X s air core reactance
= (1-1-0N 2 A w fh w
)2rrf correction factor for the peak value = 1.1 area inside the mean turn of excited winding
31

� h w f absolute permeability height of energized winding frequency
� instant at which the core saturates, is where, 8
�
5
= K2cos1 {(8
5
-B mp
-B r
)/B mp
} saturation flux density= 2.03 Tesla
B mp
B r peak value of steady state flux density in the core (1.7 T) residual flux density (0.8 B m p = l.36T)
For cold rolled material, maximum residual flux density is usually taken as 80% of the rated peak flux density.
K2 = correction factor for saturation angle = 0.9
The validity of the formulated inrush current equation (2.13) is established by plotting in MATLAB and this is shown in Fig.2.21. The first plot shows the inrush current and the second plot shows de component of the inrush current. The simulated inrush current waveform (Fig.2.19), waveform obtained by experiment (Fig.2.20) and the computed waveform (Fig.2.21) are compared. The inrush current waveform obtained
60 - - -- r · --------.----- --- -----r- - --�-.----�
- -
-,
I I
�
1
I
I
- - - - - -� - - - - -r --- - - -� - - - - - -� - - - - - -r --- - - -� - - - - - -r - -- - -
1
I
I I
I
I
I
0 ---
0 0.02
�20 __ __
---- ·1
I
I
0.06
- -· 1·
I
0.08
(a)
0.1 0.12 0.14
·- --------r- -· ------,--··-�----,-------,-
0.16
8.
10 r - - - - - -r - - ----r - -----r -----
O _____ ____[__ ______
0 0.02
, I
_J ______ _J___ ________
0.04 0.06
:
0.08
(b)
:: _::::::::::::::::::�==�=�_j
0.1 0.12 0.14 0.16
Time (s)
Fig.2.21 a) Formulated equation of current waveform a) Inrush current b) de component
32

mathematically is practically matching with that of the simulated and experimental waveforms. In all the cases duration of settling time of the inrush current is 0.15s, time taken by the de component to settle to zero or the inrush current to reach the steady state no-load current . Hence the validity of the equation for the calculation of inrush current in single phase transformer is verified. Advantage of this equation is, for a given transformer the peak inrush current, de component, their settling time and magnitudes of different harmonic component can be estimated in terms of the energization voltage and physical parameters of the transformer. Knowing the harmonic components total harmonic distortion (THO) at the time of energization can be estimated. Various parameters calculated during energization by the equation for the sample transformer is given in Table 2.6. Similarly the parameters obtained by simulation are given in Table
2.7. Comparison of the values given in the Tables shows the validity of the formulated equation. Photograph of the experimental set up is shown in Fig.2.22
Table 2.6. Parameters obtained by calculation due to energization of transformer
Inrush de
Peak current component value
(A)
45 14
Funda mental
2nd 3rd
Harmonics
4th 5th THD
9.5 4.75 1.81 1.2 0.94 ettling time(s)
0.9 0.6
-
24
0.9 0.82
I
0.62
0.42 0.22
Table 2.7. Parameters obtained by simulation due to energization of transformer
Peak value
(A)
Inrush current component
45 de
16
Funda mental
2nd '
12
3 rd
9.2
Harmonics
4th
1.3
5th
0.94
THD
0.8 settling time(s)
0.16 0.25
-
22
O.ll6
0.16 0.16 0.16 0.16
I
33

Fig.2.22 Photograph of the experimental setup
2.7. CONCLUSION
In thi chapter the general characteristics of transformer energization current on noload both by experiment and simulation has been determined. A circuit for switching-in the transformer at various inception angles using real time MATLAB-Simulink Lab work shop has been developed. Maximum possible inrush current and its dependency both by simulation and experiment are found out. The inrush current depends on the angle of switching, load power factor, angle of load interruption and residual flux. The load current interrupting time has small effect on inrush current.
The harmonic analysis of the inru h current shows the presence of all lower order harmonics and dc component.
During the period of inrush current, the demand of peak of reactive power is large and it give an indication of transformer saturation. A mathematical equation, in terms of physical parameters of transformer, which takes account of the presence of dc component and harmonic components of the inrush current of a switch-in power transformer, has been developed. The equation makes it possible to estimate the peaks of decaying inrush currents, its duration, different harmonic components and the dc component. Networks can be checked for danger of inrush current over voltages due to resonant conditions. It is possible to foresee the eventual needed remedial measures when energizing or commissioning off-shore transformers with limited generation and long sea interconnecting cables.
34