Analysis of Subharmonic Oscillations in a Ferroresonant
Circuit
Rajesh G. Kavasseri
Department of Electrical and Computer Engineering
North Dakota State University, Fargo, ND 58105 - 5285, USA
(email: [email protected])
Abstract
Ferroresonance is a nonlinear oscillatory phenomenon that occurs in capacitively coupled trans-
formers or reactors under certain conditions. In this paper, an averaging method is utilized
to compute the domain in 2-D parameter space where subharmonic (period -3) ferroresonant
oscillations could persist. The accuracy of the analytical results is verified using numerical sim-
ulations and the power spectral density. It is shown that the proposed method yields a quick
means to determine (i) the proximity to initiation of subharmonic resonance and (ii) the effect
of core loss on the domains of subharmonic oscillations.
Index terms : ferroresonance, subharmonics, averaging.
Nomenclature
• Vs : peak value of the sinusoidal supply voltage
• ωf : angular frequency of the supply voltage
• C : coupling capacitance
• R : transformer shunt resistance representing core loss
• λ : flux linkage in the nonlinear inductor
• iM : magnetizing current
• a : coefficient of the linear part of magnetization characteristics
• b : coefficient of the nonlinear part of magnetization characteristics
• n : exponent corresponding to the nonlinear magnetization characteristics
1
• ω0 : natural frequency of the ferroresonant circuit
1 Introduction
Ferroresonance is a nonlinear oscillatory phenomenon leading to dangerous overvoltages in power
equipment such as transformers and saturable reactors. Several site experiences of ferroresonance [1],
[2] including some leading to catastrophic consequences [3] have been documented in the literature.
In principle, ferroresonance is thought to occur when a nonlinear saturable inductor is fed from
a sinusoidal source through a capacitive line as shown in Fig.1. In Fig.1, the nonlinear inductor
represents the core of an instrument voltage transformer, or a power transformer under light or no
load conditions, the sinusoidal source represents the supply voltage, and the capacitance represents
the coupling between the supply and the core of the transformer. Then, the equivalent circuit shown
in Fig.1 can represent several scenarios that lead to the onset of ferroresonant oscillations such as
single pole autoreclosure operation [4], stuck breaker condition in a voltage transformer [5], or a
disconnected transmission line in the same right of way as an energized line [6], [7]. Consequently,
the circuit shown in Fig.1 has been a useful representation in the literature for several studies
concerning ferroresonance. Some of the early analytical attempts to study ferroresonance were
reported in [8] and [9]. Germond, [9] using the Galerkin’s method showed that while fundamental
frequency oscillations could predominate, subharmonic oscillations of period 3 or 5 could also be
expected in response to variation in switching instants. In [4], the harmonic balance method was
used to establish conditions leading to fundamental frequency ferroresonance. A bifurcation theoretic
framework was developed by Kieny in [10] to study the global behavior and safety margins for a
ferroresonant circuit using the applied source voltage as a bifurcation parameter. Considering the
wide variations in operating conditions that power systems are subject to, both, the applied source
voltage and the coupling capacitance need to be treated as variables in the analysis of ferroresonant
phenomena. Various ferroresonance modes that arise in the system can then be identified by time
domain simulations. However, an extensively large number of them need to be conducted to obtain a
feel for the effect of parameter variations, as rightly pointed out by the authors in [11]. Therefore in
that paper, the authors compute the domain of fundamental frequency ferroresonance using the SVA
method, separatrix calculations and compare it with domains obtained by brute force time domain
simulations. Treating the source voltage and the coupling capacitance as parameters, the authors
compute the region in 2-D parameter space where period-1 ferroresonant oscillations can persist.
In addition to fundamental frequency (or period-1) ferroresonant oscillations, it is well known that
subharmonic oscillations [12] and even chaotic oscillations can arise in a ferroresonant circuit [13] and
[14], [15]. A good introductory survey and the importance of transformer modeling on ferroresonance
is summarized in [16]. Subharmonics leading to period doubling bifurcations and consequently, chaos
2
was demonstrated in [17]. Muchnik [18] established the occurrence of subharmonic oscillations in
a ferroresonant circuit using numerical simulations. Chakravarthy and Nayar [19], [20] applied the
slowly varying amplitude (SVA) method to a capacitor voltage transformer circuit and characterized
the modes of ferroresonant oscillations for various values of the applied source voltage while the
coupling capacitance was held fixed. In this paper, the source voltage and coupling capacitance
are treated as parameters and the region in 2-D parameter space where subharmonic (period -3)
oscillations can persist is explicitly identified. The analysis is based on the well known technique
of averaging for subharmonic resonance [21], [22]. Numerical simulations and the power spectral
density analysis are conducted to verify the accuracy of the analytical results. It will be seen that the
domain in parameter space obtained by the analytical method provides a quick means to demarcate
the region in parameter space where subharmonic oscillations can be initiated. Moreover, it can be
used to assess the effect of core loss on ferroresonant oscillations as will be shown later on in the
paper. The rest of the paper is organized as follows. In section 2, a brief outline of the application
of the averaging technique is presented. In section 3, the analytical results are presented along with
numerical simulations. In section 4, a discussion of the results is provided and the conclusions are
summarized in section 5.
2 System equations and Averaging
In this section, the system equations along with a brief outline of the averaging method are presented.
In the circuit shown in Fig.1, the nonlinear inductor and the parallel resistor (R) represent the
transformer. It is to be noted that the magnitude of the iron core losses which consists of (a)
hysteresis losses and (b) eddy current losses, depend both on the flux density as well as the supply
frequency. While modeling this dependence may be useful in creating a more accurate core loss
model, here, the losses are aggregated and represented by a resistor R for simplicity of analysis,
following [11]. The nonlinearity of the core is modeled by the following equation
iM = aλ+ bλn (1)
The capacitor C represents the coupling effect between the source voltage at a supply frequency of
ωf . The supply voltage e(t) is given by e(t) = Vs cos(ωf t). Then the dynamics of the circuit in Fig.1
can be described by the following equation, [11].
λ+ kλ+ a1λ+ b1λn = G cos(ωf t) (2)
where, k = 1/RC, a1 = a/C, b1 = b/C, and G = Vsωf . When the index n of the magnetization
curve is three, Eqn.(2) is functionally similar to the periodically forced Duffing oscillator.
Remark : In the context of ferroresonance, the index n is usually obtained by approximating
3
the magnetizing characteristics by a two term polynomial of suitable order. In [11], indices corre-
sponding to n = 3 and n = 13 were used in the calculations. In [15], the effect of the index n on
ferroresonant oscillations was examined and lower exponents were found satisfactory in representing
small capacity transformers. In this paper, the index n is set to three for ease and tractability of the
analysis. Thus, Eqn.(2) with n = 3 is used to determine analytically the regions in 2-D parameter
space (C, Vs), where subharmonic resonance can persist.
Detailed analysis has been done on the Duffing oscillator since the classical work of Duffing. In
particular, primary resonance, third order subharmonic and superharmonic resonances have been
exhaustively studied in the classical texts of [21] and [22] by using perturbation techniques such as
the methods of averaging and multiple scales. A brief outline of the averaging method to analyze
third order subharmonic resonance is provided here. The classic Duffing equation is described by
d2u
dt2+ k1u+ k3u
3 = F cos(ωf t) (3)
After a suitable amplitude and time scaling (see [22]), the dimensionless form of the Duffing equation
including the effect of a small viscous damping can be expressed as
v + ω20v + 2εµv + αεv3 = F cos(ωt) (4)
The averaging method starts by noting that when ε = 0, the general solution to Eqn.(4) is given by
v = A cos(ω0t+ β) + 2Γ cos(ωf t) (5)
where Γ = F (ω2o − ω2)−1/2. When ε 6= 0, the solution is still represented in the same form as in
Eqn.(5), except that now the variables A and β are treated as time varying instead of constant.
Thus, replacing A by A(t) and β by β(t) in Eqn.(5), the derivatives v and v are computed and
substituted in Eqn.(4) to solve for ˙A(t) and ˙β(t). It can be shown [21] that resonant conditions arise
in the system when
• ω = ω0 : termed as fundamental frequency, or period-1 ferroresonance (which has been ana-
lyzed in [11])
• 3ω = ω0 : termed as superharmonic resonance.
• ω = 3ω0 : termed as subharmonic resonance (which is the subject of this paper)
• ω = 0: which is not of relevance to ferroresonance
For the case when ω = ω0/3, following the development in [21], the averaged equations for the
system in Eq.(2) can be written as
A = −kA/2− 3b1ΓA2 sin(β)/4ω0 (6)
β = (ωf − 3ω0)− 9b1ω0
[Γ2 −A2/8− γA cos(β)/4] (7)
4
where ω20 = a1 = a/C, Γ = G/2(ω2
0 − ω2f ). Setting the right hand side of Eqns.(6) and (7) to zero
yields the steady state amplitude and phase of the subharmonic oscillations described by
−kA/2 = 3b1ΓA2 sin(β)/4ω0 (8)
(ωf − 3ω0) =9b1ω0
[Γ2 −A2/8− γA cos(β)/4] (9)
For a nontrivial subharmonic oscillation to exist, it is evident that A > 0. Eliminating β from
Eqns.(8) and (9), one can arrive at a quadratic in A2 and derive a condition [21] for the existence
of nontrivial subharmonic oscillations, the result of which is described by
63b1G2
8ω0k(ω20 − ω2
f )=
2(ω − 3ω0)
k±
√[4(ω − 3ω0)2
k2− 63] (10)
The condition described in Eqn.(10) defines the boundary of the region where subharmonic oscilla-
tions can exist. In this paper, the parameters of interest in Eqn.(10) are (i) the coupling capacitance
C which decides the natural frequency of oscillation ω0, (ii) the amplitude of the source voltage Vs
and (iii) the core loss resistor R. In the following section, we shall study the dependence of the
subharmonic domains on the parameters of interest and consequently, its implications to ferroreso-
nance.
3 Main results : Regions of subharmonic resonance
In this section, we shall utilize Eqn.(10) to study graphically the regions in parameter space where
subharmonic oscillations exist. Out of the three parameters of interest mentioned in Sec.(2), we
shall treat (i) Vs: the magnitude of the source voltage and (ii) C : the coupling capacitance as
variables and obtain the region in 2-D parameter space (C, Vs) where subharmonic oscillations can
exist. While [11] indicates the possibility of period-3 resonant regions obtained through brute-force
simulation, the present paper attempts to obtain analytically the region in parameter space where
such oscillations can possibly persist. In several studies [23], [24], [15], [4] and [11], the core loss
resistor has been shown to exert a significant influence on ferroresonance. Therefore in this paper,
the domains of subharmonic resonance are computed where the range of the core loss resistor is
varied from one fourth, to four times its nominal value. For each value of the core loss resistance,
the boundary of the region where subharmonics exist is plotted by using Eqn.(10). The results of
the computation are shown in Fig.2.
The region in parameter space where (1/3) subharmonic oscillations exist for a given value of the
core loss resistor is simply the interior of the “V” shaped curves (marked in the diagram for R = 2.5
K Ω) in Fig. 2. From Fig. 2, we note that for a given value of the core loss resistance, there is
a corresponding minimum value of capacitance required to initiate sub-harmonic oscillations. This
can be computed easily from Eqn. (10) by requiring that the second term on the RHS of Eqn. (10)
be positive. This is summarized in the following observation.
5
Observation 1 : For a given value of the core loss resistor R, the minimum value of capacitance
(Cmin) required to initiate sub-harmonic ferroresonance is given by Cmin = q2(R) where q(R) is
given by
q(Rm) = [3R√a+
√9R2a+ 2ωRk1]/2ωRm (11)
where k1 =√
63. Eqn. (11) can be obtained by setting
4(ω − 3ω0)2 > 63k2 and simplifying the resulting quadratic. The values of Cmin obtained by this
computation are summarized in Table I. We note from Eqn. (11) that q(R) does not depend on
the magnitude of the source voltage Vs. Therefore, the values of Cmin = q2(R) in Table I simply
correspond to the minima of each of the “V” shaped curves in Fig.2.
In the following section, we shall use numerical simulations and the power spectral density to verify
the existence of subharmonic oscillations in the domains (see Fig. 2) obtained by the averaging
method.
3.1 Numerical simulations
In this section, we shall numerically integrate Eqn.(2) for different values of the source voltage and
coupling capacitance. We shall consider three cases namely, case (I): Vs = 1 p.u, case (II): Vs = 2
p.u. and case (III): Vs = 3 p.u. For each case, we shall study how the value of the coupling capaci-
tance C affects the ferroresonant oscillations. For each simulation, the flux (λ) and voltage (λ) are
shown in one plot while the power spectral density of the voltage waveform and a phase plane plot
of λ, λ are shown in the second plot. The numerical values of all parameters used in the simulation
studies are based on [19].
Case I: Vs = 1p.u : In this case, the source voltage Vs and the core loss resistor are held at their
nominal values of 1 p.u. and 10 K Ω respectively. From Table I, we know that when Rm = 10KΩ, we
need at least C = 3.706 µ F to initiate subharmonic resonance. Therefore, the value of C is varied
slowly from C = 3.7µ F until the oscillations become noticeable in the simulations. In this case, the
onset of subharmonic oscillations is noticed at C = 4.1µ F as shown in Fig.3. The corresponding for
power spectral density and phase plane plots are seen in Fig.4. The power spectral density (PSD)
plot in Fig.4 clearly evidences the presence of the 20 Hz (i.e. one third) subharmonic oscillation.
Note that the PSD plot has been normalized. This has been done mainly to compare the relative
contents of the fundamental and the subharmonic oscillations. Next, the value of the capacitance is
increased and similarly, the graphs corresponding to C = 7.2µ F are shown in Fig.5 and Fig.6. It
is of interest to determine the fate of subharmonic oscillations as the capacitance is varied further.
Therefore, the simulations are continued for increasing values of capacitance until such a value when
6
the subharmonic content is almost annihilated. Such a simulation is shown in Fig.7 and Fig.8 when
the capacitance is equal to 7.2 µ F.
Case II: Vs = 2p.u : A similar procedure as described in Case I is conducted when Vs = 2 p.u. In
this case, the subharmonic is seen to be initiated when C = 6.7µ F as seen in Figs.9 and 10. Similar
plots when C = 12.7µ F and C = 22µ F are shown in the Figs .11, 12, 13 and 14.
Case III: Vs = 3p.u : In this case, the subharmonic is initiated at C = 8µ F and the plots when
C = 18µ F are shown in Figs.15 and 16.
4 Discussion
In Fig.2, we observe that the regions of subharmonic resonance are “nested” as the core loss resistor
increases from 2.5 K Ω to 40 K Ω. Understandably, this means that the domains of subharmonic
ferroresonant regions rapidly shrink as the core losses increase which is consistent with observations
noted in the literature, [23], [4] and [24]. When Vs = 1 p.u. and R = 10k Ω, we observe from Fig.2
that the capacitance necessary to initiate subharmonic ferroresonance is about 4 µ F. This fact is
verified in the simulations as shown in Figs.3 and 4 where the subharmonic oscillation is seen to be
initiated at C = 4.1µ F. From the PSD plot in Fig.4, we observe that the content of the 20 Hz (which
corresponds to the third subharmonic) is about 40 % of that of the fundamental frequency (i.e 60
Hz) component. As the capacitance is increased further, the content of the 20 Hz subharmonic
increases relative to the 60 Hz fundamental oscillation. When C = 5.7µ F, the content of the 20 Hz
subharmonic equals that of the fundamental beyond which, the 20 Hz subharmonic dominates the
60 Hz fundamental. The simulations shown in Fig.5 and 6 which corresponds to C = 7.2µ F clearly
illustrate this. In Fig.6, it is seen that the content of the 60 Hz fundamental is actually about 60 %
of that of the 20 Hz subharmonic. Interestingly, as the capacitance is increased further, the content
of the subharmonic diminishes. As we can note from Fig.7 and Fig.8, when C = 13.5µ F, the 20 Hz
subharmonic is nonexistent. A similar feature is noticed for the cases when Vs = 2 p.u and Vs = 3p.u.
When Vs = 2p.u, the subharmonic is seen to be initiated at C = 6.7µ F. To compare this with the
value predicted by the averaging method, we note from Fig.2 that at Vs = 2p.u, the capacitance
necessary to initiate subharmonic resonance is about 4.5 µF. This difference may be attributed to
the fact that for values of capacitance between 4.5µ F and 6.7µ F, the subharmonics obtained by
numerical simulation are too faint to be detected by the PSD method. The subharmonic grows
in content beyond C = 12.7µ F for which the simulations are shown in Fig.13 and Fig.14. When
C = 22µ F, the simulations also reveal the presence of a small 30 Hz subharmonic as evidenced
by Fig.14 in the PSD and the phase plane plot. As the capacitance is increased further, the 20 Hz
subharmonic is annihilated at about C = 25µ F (plot not shown here owing to space limitations).
The case when Vs = 3 p.u is similar to the other cases and hence, detailed simulations are not
7
presented. A sample simulation corresponding to C = 18 p.u. in shown in Figs.15 and 16. From
the simulations, we observe that in all cases, the 20 Hz subharmonic follows a pattern of initiation,
growth and annihilation as the coupling capacitance is varied. The averaging method on the other
hand yields a region in parameter space where subharmonics could exist. It cannot readily predict
either of the following attributes of the subharmonic oscillation namely (i) the stability and (ii) set
of initial conditions that can lead to subharmonic oscillations. The absence of subharmonics in the
numerical simulations for certain values of capacitance is precisely due to the combination of one or
more of the following namely (i) instability of the subharmonics or (ii) choice of initial condition in
the simulation (in this case, [1, 0], which corresponds to an unenergized transformer with residual
flux).
5 Conclusions
The method of averaging has been applied to a ferroresonant circuit to obtain domains of 1/3
subharmonic resonance in terms of the system parameters namely the source voltage and coupling
capacitance. Numerical simulations are performed and the power spectral density technique is
applied to assess the content of the subharmonic oscillations relative to the fundamental frequency
oscillation. It is seen from the numerical simulations that the subharmonics do exist in the regions
predicted by the averaging method. The averaging method is not seen to be effective in predicting
the set of initial conditions that lead to subharmonic oscillations or their stability. Nevertheless,
the technique is seen to be effective in quickly identifying the regions where subharmonics could
exist, if ever present. Therefore, the averaging method could be a employed as a preliminary tool to
narrow the range of system parameters before conducting elaborate numerical simulations. Studying
the ferroresonant modes and their corresponding domains using an improved core loss model that
incorporates frequency and core-flux dependence would would be an interesting subject for further
research.
Acknowledgement
The author is thankful for the constructive comments suggested by the referee. The financial support
from ND EPSCOR through NSF grant EPS 0132289 is gratefully acknowledged.
Appendix A : System data
The data for the transformer is adapted from [19].
Burden : 200 VA, Voltage ratio : 3.3 kV/110 V.
Magnetizing characteristics : n = 3, a = 0.03, b = 0.06
8
Nominal core loss resistance : 10 kΩ
Nominal coupling capacitance : 1.7 µ F.
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10
Figure 1: Ferroresonant circuit model
Figure 2: Subharmonic ferroresonant regions
11
Figure 3: flux and voltage oscillations when Vs = 1p.u,R = 10K,C = 4.1µ F
Figure 4: power spectral density and phase plane plot when Vs = 1p.u,R = 10K, C = 4.1µ F
12
Figure 5: flux and voltage oscillations when Vs = 1p.u,R = 10K, C = 7.2µ F
Figure 6: power spectral density and phase plane plot when Vs = 1p.u,R = 10K, C = 7.2µ F
13
Figure 7: flux and voltage oscillations when Vs = 1p.u,R = 10K, C = 13.5µ F
Figure 8: power spectral density and phase plane plot when Vs = 1p.u,R = 10K, C = 13.5µ F
14
Figure 9: flux and voltage oscillations when Vs = 2p.u,R = 10K, C = 6.7µ F
Figure 10: power spectral density and phase plane plot when Vs = 2p.u,R = 10K, C = 6.7µ F
15
Figure 11: flux and voltage oscillations when Vs = 2p.u,R = 10K, C = 12.7µ F
Figure 12: power spectral density and phase plane plot when Vs = 2p.u,R = 10K, C = 12.7µ F
16
Figure 13: flux and voltage oscillations when Vs = 2p.u,R = 10K, C = 22µ F
Figure 14: power spectral density and phase plane plot when Vs = 2p.u,R = 10K, C = 22µ F
17
Figure 15: flux and voltage oscillations when Vs = 3p.u,R = 10K, C = 18µ F
Figure 16: power spectral density and phase plane plot when Vs = 3p.u,R = 10K, C = 18µ F
18
Table 1: Minimum capacitance required to initiate subharmonic ferroresonance
R (K Ω) Cmin (µ F)
2.5 8.1441
5 5.2690
10 3.706
20 2.8553
40 2.3971
19
List of Figures
1 Ferroresonant circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Subharmonic ferroresonant regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 flux and voltage oscillations when Vs = 1p.u,R = 10K,C = 4.1µ F . . . . . . . . . . 12
4 power spectral density and phase plane plot when Vs = 1p.u,R = 10K, C = 4.1µ F 12
5 flux and voltage oscillations when Vs = 1p.u,R = 10K, C = 7.2µ F . . . . . . . . . 13
6 power spectral density and phase plane plot when Vs = 1p.u,R = 10K, C = 7.2µ F 13
7 flux and voltage oscillations when Vs = 1p.u,R = 10K, C = 13.5µ F . . . . . . . . . 14
8 power spectral density and phase plane plot when Vs = 1p.u,R = 10K, C = 13.5µ F 14
9 flux and voltage oscillations when Vs = 2p.u,R = 10K, C = 6.7µ F . . . . . . . . . 15
10 power spectral density and phase plane plot when Vs = 2p.u,R = 10K, C = 6.7µ F 15
11 flux and voltage oscillations when Vs = 2p.u,R = 10K, C = 12.7µ F . . . . . . . . . 16
12 power spectral density and phase plane plot when Vs = 2p.u,R = 10K, C = 12.7µ F 16
13 flux and voltage oscillations when Vs = 2p.u,R = 10K, C = 22µ F . . . . . . . . . . 17
14 power spectral density and phase plane plot when Vs = 2p.u,R = 10K, C = 22µ F . 17
15 flux and voltage oscillations when Vs = 3p.u,R = 10K, C = 18µ F . . . . . . . . . . 18
16 power spectral density and phase plane plot when Vs = 3p.u,R = 10K, C = 18µ F . 18
20
List of Tables
1 Minimum capacitance required to initiate subharmonic ferroresonance . . . . . . . . 19
21