ferroresonance

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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in a nonlinear resonant circuit, IEEE Trans. on Power Delivery 2002; 17(3): 865-871.

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9

[13] Mork BA, Stuehm DL. Application of nonlinear dynamics and chaos to ferroresonance in dis-

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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


13



14



15



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








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

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