ISSN:2249-5789
Swati Devabhaktuni et al, International Journal of Computer Science & Communication Networks,Vol 1(3), 264-269
Design of Excitation Capacitance for Self-Excited Induction Generator
Swati Devabhaktuni1 , S.V.Jayaram kumar2
1
Associate professor,Gokaraju Rangaraju Insitute Of Engineering And technology,Hyderabad,A.P.,India
2
Professor, J.N.T.Ucollege of engineering,Hyderabad,A.P.,India
E-mail: swatikjm@gmail.com1, svjkumar101@rediffmail.com2
Abstract
This paper presents simple and accurate approach to
compute the minimum value of capacitance required
for initiating the voltage build-up in a three-phase selfexcited induction generator. Based on the steady-state
equivalent circuit model different numerical methods
for solving frequency are known from previous
literature, which are of 6th order polynomial. In this
paper the order of the polynomial is reduced to the 4th
order frequency with a new, simple and direct method
is developed to find the capacitance requirement.
Critical values of the impedance and speed, below
which the machine fails to self excite irrespective of the
capacitance used, are found to exist. Closed form
solutions for capacitance are derived for no-load and
RL loads. Experimental results obtained on a 3.5kW
induction machine confirm the feasibility and accuracy
of the proposed method.
Keywords— capacitance requirements, self-excitation, induction
generator, steady state analysis, saturation
1. Introduction
In recent years, the strong drive to conserve the
global energy resources has initiated rigorous research
on electricity generation using wind and mini hydro
power. Much emphasis has been placed on the squirrel
cage induction machine as the electromechanical
energy converter in such generation schemes[1].
Notable advantages of the induction generator over the
synchronous generator are low cost, robustness,
absence of moving contacts and the need for d.c.
excitation[4]. Owing to its many advantages, the self
excited induction generator has emerged from among
the known generators as suitable candidate to be driven
by wind power.
Beside its application as a generator, the
principle of self-excitation can also be used in dynamic
braking of a three phase induction motor. Therefore
methods to analyze the performance of such machines
are of considerable practical interest. The terminal
capacitance on such machines must have a minimum
value so that self-excitation is possible [5].
In is paper, a simple method to compute the
capacitance requirements of a self-excited induction
generator are introduced. The proposed method differs
from previously published methods in the following
aspects [4]:
(a).It is based on the nodal admittance method for
steady-state analysis of the SEIG without considering
the saturation.
(b).The load and excitation capacitance branches in the
equivalent circuit are decoupled to facilitate the
solution of the self-excited frequency.
(c).No trial-and-error procedure is involved; hence it
may be regarded as a direct method.
(d).Reduced computational effort as only a 4thdegree
polynomial need to be solved to yield the value of
capacitance.
When the nodal admittance concept is used in
the analysis of the equivalent circuit, the process of self
excitation is satisfied by equating the sum of the nodal
admittances to zero. Using K.C.L at node 1.we get a
complex equation from which two simultaneous
equations for C and f are obtained [3]
When the SEIG has successfully built up its
voltage, the next question of interest is to maintain the
terminal voltage at a preset value as the load increases.
Using the same proposed method for computing
capacitance, an iterative procedure is also developed
for calculating the capacitance requirements of the
SEIG for maintaining a given terminal voltage under
load [2]. Experimented results obtained on a laboratory
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Swati Devabhaktuni et al, International Journal of Computer Science & Communication Networks,Vol 1(3), 264-269
machine are presented to verify the accuracy and
validity of the present approach. This paper introduces
a new and simple and direct method of finding
minimum capacitance required for self excitation.
2. Three Phase Self-excited Induction
Generator Model
For the modelling of the self-excited induction
generators, the main flux path saturation is accounted
for while the saturation in the leakage flux path , the
iron and rotational losses are neglected. Therefore in
the following analysis the parameters of the induction
machine are assumed constant except the magnetizing
inductance which varies with saturation [5].
2.1. Steady-state circuit model:
The steady state circuit of a self-excited induction
generator under RL load is shown in Fig.1
condition yields the minimum value of excitation
capacitance below which the SEIG fails to self-excite
For the circuit shown in Fig.1.,by Kirchhoff’s law,
the sum of currents at node(1) should be equal to zero,
hence
VY=0
(1)
Where Y is the net admittance given by
Y=YL+YC+Y2
(2)
The terminal voltage cannot be equal to zero hence
Y=0
(3)
By equating the real and imaginary terms in
equation(3) respectively to zero.
Real(YL+YC+Y2)=0
Imag(YL+YC+Y)=0
2.3. Proposed method to find general solution for
capacitance
The real part yields
A4f4+A3f3+A2 f2+A1 f+A0=0
(4)
And the imaginary part yields
C=
𝑎𝑓 +𝑏
𝑐𝑓 3 +𝑑𝑓 2 +𝑒𝑓
(𝑌𝑅 +𝑌𝑀 )𝑌𝑠
Y2=
𝑌𝑅 +𝑌𝑀 +𝑌𝑠
𝑘𝑓
Slip,𝑠 = 𝑘𝑓 −𝑁
;
(5)
(6)
(7)
Here the machine core losses are having been
ignored. Considering these losses increases the
mathematical work involved in obtaining the results,
without increasing the accuracy of the analysis
substantially. All the circuit parameters are assumed to
be constant and unaffected by saturation. Machine
parameters except capacitance and frequency all are
known values [6].
Where k=30;
The derivation for these constant coefficients A4 to
A0 is given in Appendix-A.Equation(4) can be solved
numerically to yield all the real and complex
roots.Only the real roots have physical significance and
the largest positive real root yields the frequency. The
corresponding capacitance can be calculated.
An investigation on the solutions for various load
impedances and speed conditions reveals that for RL
loads,there are in general two real roots and a pair of
complex roots.
The computed results reveal that there exist
critical values of load impedance or speed below which
the induction generator fails to excite irrespective of the
value of capacitance used.
2.2Mathematical model
3. Computed Results and Discussions
Fig1.shows the per phase equivalent circuit
commonly used for the steady state analysis of the
SEIG.For the machine to self excite on no load, the
excitation capacitance must be larger then some
minimum value, this minimum value decreasing as
speed decreases[4].For on load self-excitation, the
impedance line corresponding to the parallel
combination of the load impedance and excitation
capacitance should intersect the magnetisation
characteristic well into the saturation region[7].The
In this paper, the computed results are obtained by the
procedures and calculations outlined above, number of
experiments are conducted using three phase induction
machine coupled with a wind turbine. The induction
machine was three, phase3.5kW, 415V, 7.5A,
1500r.p.m, star connected stator winding.
A 3-Φ variable capacitor bank or a single capacitor was
connected to the machine terminals to obtain selfexcited induction generator action.
The measured machine parameters were:
Fig.1.Equivalent circuit of SEIG
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Swati Devabhaktuni et al, International Journal of Computer Science & Communication Networks,Vol 1(3), 264-269
r1=11.78Ω; r2=3.78Ω; L1=L2=10.88H. Lm=227.39H
Consider the case when the machine is driven at rated
speed with a connected load impedance of 200Ω.Solve
the frequency polynomial using MATLAB software.
The solution yielded the following complex and real
roots.
f1=50.06Hz;
f2=17.33Hz;
f3=1.275+j0.3567Hz;
f4=1.275-j0.3567Hz;
As only the real roots have physical significance and
the largest real root yields the maximum frequency that
corresponds to the minimum frequency.
Since all these values and capacitance and sufficient
to guarantee self-excitation of induction generator, it
follows that the minimum capacitor value required. It is
seen that only the larger positive real root gives the
feasible value of the capacitance. The smaller real root
on the other hand gives the value of the excitation
capacitance above which the machine fails to excite.
However such condition is unpractical as the
corresponding excitation current would far exceed the
rated current of the machine.
If the polynomial is having no real roots, then no
excitation is possible. Also, there is a minimum speed
value, below which equation (4) have no real roots.
Correspondingly no excitation is possible.
It is noted that for R-L loads, there are in general
two real roots and one pair of complex conjugate roots.
This restricts the set of two capacitors. It is also noted
𝑘𝑓
that Ns<N,the Slip,𝑠 = 𝑘𝑓 −𝑁 is always negative as it
Fig.4.Variation frequency with speed
should be for generator action.
Fig.5.Variation of capacitance with speed
Fig.2.Variation of frequency with load impedance
Fig.3.Variation of capacitance with load impedance.
The computed values reveals that there exist critical
values of load impedance or speed below which the
induction generator fails to excite irrespective of the
value of capacitance used.Fig.2.shows the computed
variation of the self excited frequencies f and f2 with
load impedance at 80Ω.It is noticed that the roots vary
only slightly with the load impedance. One is slightly
decrease with load impedance and another is increasing
with the same. When load impedance is less than80Ω,
however the two roots will approach rapidly. At load
impedance 80Ω, the two roots are equal, while all value
below 80Ω yield imaginary roots. The value of load
impedance that results in repeated real roots of the
polynomial thus defines a region of no-generation and
it may be termed the critical load impedance for a given
speed and power factor.
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Swati Devabhaktuni et al, International Journal of Computer Science & Communication Networks,Vol 1(3), 264-269
Fig.3.shows the computed variations of capacitance
with the load impedance. It is noticed that, in general,
capacitance increases with decrease in load impedance.
The increase in more gradual at large values of
impedance but becomes more abrupt as the critical
value 80Ω is approached.
Fig.4 shows the computed variation of self-excited
frequencies with speed for load impedance 200Ω.Again
a region of no-generation is identified and the critical
speed yields the repeated roots of the polynomial may
be termed the critical speed for a given load impedance.
Fig.5.shows the computed variations of minimum
capacitance with speed at different load impedances. It
is seen that the capacitance increases rapidly with the
decrease in speed. At speeds nearly to the critical value,
minimum capacitance is very large, typically hundreds
of microfarads. In practice, however it is unlikely that
the SEIG will be operated at such low speeds.
As for the no-load case close solutions exist for the
self excitation frequency which is maximum and
capacitance which is minimum. The self excitation
frequency and the critical speed for the inductive load
were same as for the no-load case.
Fig.6.Variation of capacitance with highest real root
of frequency
C must be slightly greater than the minimum
capacitance. Exact expressions for capacitor values
under no-load, resistive loads and corresponding output
frequencies are derived.
Fig.8.Variation of magnetizing reactance with load
impedance
Fig.8.shows the variation of magnetizing reactance
with the value of the load impedance. As the load is
increasing the magnetizing reactance is also increasing.
Below the value of load impedance 80Ω there will not
be any excitation.
Fig.9.Variation of magnetizing reactance with
Capacitance
Fig.9..shows the variation magnetization reactance
of generator with various capacitances .If the
capacitance value is below the minimum value of the
capacitance, the magnetization reactance is greater the
unsaturated reactance, in which case the machine is
failed to excite and the voltage will be zero.
Fig.7.Variation of capacitance with lowest real root of
frequency
Fig.6. and fig.7 the minimum capacitance required for
the self-excited induction generator. These values can
be used to predict the theoretically the minimum values
of the terminal capacitance required for selfexcitation.Ofcourse,for stable operation of the machine
Fig.10.Variation of Capacitance with slip
Fig.10.shows the variation of the capacitance with
slip. Capacitance is maximum having slip s=1.and then
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decreases as the slip decreases. As the slip decreases
the values of the frequency also decreases.
Since only one single root is required, the NewtonRaphson method was used in solving the polynomial
equation. Depending on the circuit conditions, the final
value of the frequency could be obtained in 4to10
iterations. It is advised to use MATLAB software to
solve the polynomial.
5. Computer Algorithm
In order, to develop a computer algorithm to
determine capacitance for self-excitation of SEIG using
the techniques described in section it is desirable to
have a program or subroutine to calculate the roots of a
polynomial with complex coefficients .The flow chart
of the computer program is given by
4. Experimental results And Discussions
Experiments were performed on the above
mentioned induction machine to verify the validity of
the computed results. It is found that if a sufficiently
large residual flux existed in the rotor core, the machine
would always self-excite whenever the capacitance was
slightly higher than the computed value.
The value of the capacitance required for the
machine to self excite from the computation result is
obtained as C=14.35μF.Similrly from the experiment,
using the magnetization curve, computed resulted is of
C=15μF.Hence proved that the experimental
capacitance value must be greater than the computed
capacitance value. The magnetization curve drawn
from the experimental result is as shown in Fig.11.
Fig.11.Magnetization Characteristic
The calculation of air-gap voltage is given in
APPENDIX-A.For different values of capacitances the
experiment were conducted and it was found that the
value of the frequencies calculated from the polynomial
and experimental verification are nearly equal.
Very good correlation between the computed
and experimental results is observed as shown in fig.12.
This verifies the accuracy of the proposed method for
computing minimum value of the capacitance for
SEIG.
Experimental Frequency
Calculated Frequency
29.98
37.01
41.5
47.11
49.88
30.45
36.67
41.2
47.01
50.09
Fig.12.Comparision between Experimental and
calculated frequency
Fig13.Flow chart to determine the frequency and
capacitance
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Swati Devabhaktuni et al, International Journal of Computer Science & Communication Networks,Vol 1(3), 264-269
6. CONCLUSIONS
A method for computing the minimum value of
capacitance to initiate self-excitation in the SEIG has
been described. The method is based on the steady state
equivalent circuit, but features the separate
consideration of the load and excitation capacitance
branches, which enables the frequency to be
determined by solving a single 4th order polynomial
.Computation studies on the experimental machine
reveals that there exist critical values of load
impedance and speed below which self-excitation is
impossible irrespective of the capacitance used. Using
the same analysis technique, an iterative procedure has
also been developed for estimating the capacitance
requirements for maintaining the terminal voltage
constant when the SEIG is on load. The validity of the
proposed methods are confirmed by experimental
results obtained on a 3.5kW laboratory induction
machine.
APPENDIX-A
References
[1].
A.K. At Jabri and A.I. Alodah, “Capacitance
requirements for isolated self-excited induction
generator,” Proc. IEE, Vol. 137,Part B, No. 3,
pp.154-159, May 1990.
[2]
AI-Bahrani, A.H., and Malik, N.H, “Selection of the
Excitation
Capacitor for Dynamic Braking
ofInduction Machines”, Proc. IEE, Vol. 140, Part.
B,No. 1, pp. 1-6, 1993
[3]
AI-Bahrani, A.H., and Malik, N.H, “Steady state
analysis and Performance Characteristics of a three
phase Induction Generator Self-Excited with a single
capacitor”, IEEE Trans. on Energy Conversion, Vol4,
No. 4, pp. 725-732, 1990
[4]
T.F. Chan, “Capacitance requirements of SelfExcited Induction Generators”, IEEE Trans. on
Energy Conversion, Vol. 8, No. 2, pp. 304-310, June
1992.
[5].
A.K. Tandon, S.S. Murthy, and G.J. Berg, “Steady
State Analysis of Capacitor Self Excited
InductionGenerators,” IEEE Trans. on Power App.
and Sys.Vol. PAS-103, No. 3, pp. 612- 618, 1984.
[6].
Rahim, Y.H.A., “Excitation of Isolated Three-phase
Induction Generator by a single capacitor”, IEE
Proc.,Pt. B., Vol.140, No. 1, pp. 44-50, 1993.
[7].
Malik, N.H; and Mazi, A.A. “Capacitance
requirements for isolated Self excited Induction
Generators”, IEEE Trans. on Energy Conversion,Vol.
EC-2, No. 1, pp. 62-69, 1987.
To compute the coefficients A4 to A0 of
equation(4),the following equations are first defined:
a=2πk(LMr1+L1r1+L2r1+LMr2+Lr2+rLLM+rLL2);
b=-2 πN*rL(LM+L2)
c=-8 π3k(LLMr1+LL2r1+LLMr2-rLL1LM-rLL2LM)
d=-8 π3N(rLL1LM+ rLL2L1+ rLL2LM+LL2LM)
e=-2πkrLr1r2
g=-4π2k(L1LM+L1L2+L2LM+LLM+LL2)
h=4π2N(L1LM+L1L2+L2LM+LLM+LL2)
i=r1r2+rLr2
j=-16π4k(LL1LM+LL2LM+LL2L1)
l=16π4N(LL1LM+LL1L2+LL2LM)
m=4π2k(Lr1r2+rLLMr1+rLL1r2+rLL1r2+rLL2r1+rLLmr2)
p=-4π2NrLLMr1;
A4=cg-aj
A3=dg+hc+-al-bj;
A2=eg+hd+ic-ma-bl;
A1=he+id-pa-bm
A0=ie-bp;

Air gap voltage:
The piecewise linearization of magnetization
characteristic of machine is given by
E1=0
Xm≥260
E1=1632.58-6.2Xm
233.2≤Xm ≤260
E1=1314.98-4.8Xm
214.6≤Xm ≤233.2
E1=1183.11-4.22Xm
206≤Xm ≤214.6
E1=1120.4-3.9.2Xm
203.5≤Xm ≤206
E1=557.65-1.144Xm
197.3≤Xm ≤203.5
E1=320.56-0.578Xm
Xm ≤197.3
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