ON THE CALCULATION OF A FERRORESONANT FILTER-STABILIZER

V. I. Lukovnikov

(Presented by the scientific and technical seminar of the Department of Electrical Equipment)

Figure 1a shows the schematic diagram of a ferroresonant filter-stabilizer (FRFST), which makes it possible simultaneously to protect the inverter from current overloads and to convert its rectangular output voltage into a stabilized sinusoidal voltage. The proposed device belongs to the group of ferroresonant

Fig. 1. Ferroresonant filter-stabilizer: a) schematic diagram, b) equivalent circuit

stabilizers with a magnetic shunt, which are described in sufficient detail in [1, 2, 3]. A distinctive feature of the device is that, for filtering the output voltage, a circuit consisting of the operating capacitance \(C_p\) and the inductance \(L_p\) of winding \(W_p\), placed on the magnetic shunt, is used. Through the circuit \(C_p L_p\), tuned to resonance at the third harmonic, the third harmonic is “sucked out” of the voltage curve \(U_n\), while all the others are attenuated by the filtering voltage \(U_p\) induced in winding \(W_p\).

The purpose of the present work is to derive the filtering conditions and to obtain corrections to A. G. Lurie’s method [2] for calculating ferroresonant stabilizers in order to determine the basic parameters of the FRFST.

In deriving the characteristics of the FRFST, assumptions are made that there are no losses in the ferromagnetic core, and that the curves of currents, voltages, and magnetic fluxes are sinusoidal. The equivalent circuit (the parameters are referred to winding \(W_2\)) is shown in Fig. 1, b, where the notation is: \(X_{s1}\) is the leakage reactance of the primary winding \(W_1\), \(L_0\) is the inductance of the saturated reactor, \(C_p\) is the working capacitance, \(R_n\) is the load resistance, \(U_k\) is the compensating voltage, and \(U_p\) is the filtration voltage.

The magnetization curve is approximated by a function of the form \(H = a \times B^m\), where \(a\) and \(m\) are constants. Solving the system of equations, where the designations of the currents and voltages correspond to Fig. 1b,

\[ \dot I_1 = \dot I_0 + \dot I_c + \dot I_n, \]

\[ \dot U_1 = \dot U_0 + jX_{s1}\dot I_1, \]

\[ \dot U_0 = j\omega L_0 \dot I_0, \]

\[ \dot U_0 = \dot U_p + \dot I_0\left(j\omega L_p + \frac{1}{j\omega C_p}\right), \]

\[ \dot U_0 = \dot U_k + \dot I_n R_n, \tag{1} \]

with respect to \(\dot U_1\), for the effective values of the variable quantities we obtain the relation:

\[ U_1 = \kappa_0 B_0 \left\{ [1-\rho(1+\kappa_1)+AB_0^m]^2+\gamma^2 \right\}^{0.5} \left[ (1-\kappa_1\rho)^2+\kappa^2\gamma^2 \right]^{-0.5}, \tag{2} \]

where

\[ \kappa = U_k U_1;\quad \kappa_1 = U_p X_{s1} I_1;\quad \gamma = X_{s1} R_n;\quad \rho = X_{s1}(\omega^{-1}C_p^{-1}-\omega L_p) \]

are the coefficients of compensation, filtration, load, and resonance; \(B_0\) is the effective value of the magnetic induction in the saturated core; \(\kappa_0 = \omega S_2 W_2\), and \(A\) are proportionality coefficients.

Since \(\dot U_n = \kappa_0 \dot B_0 - \kappa \dot U_1\), the effective value of the output voltage is

\[ U_n = \kappa_0 B_0 \left\{ (1-\kappa_1\rho)- \kappa[1-\rho(1+\kappa_1)AB_0^m] \right\} \left[ (1-\kappa_1\rho)^2+\kappa_0^2\gamma^2 \right]^{-0.5}, \tag{3} \]

the load current is

\[ I_n = \eta U_n X_{s1}^{-1}, \tag{4} \]

and the primary current is

\[ I_1 = \kappa_0 B_0 X_{s1}^{-1} \left[ (AB_0^m-\rho)^2+(1-\kappa)^2\gamma^2 \right]^{0.5} \left[ (1-\kappa_1\rho)^2+\kappa^2\eta^2 \right]^{-0.5}. \tag{5} \]

The quality of the FRFST is evaluated from the regulating \(U_n=f_1(U_1)\), external \(U_n=f_2(I_n)\), and load \(I_1=f_3(\eta)\) characteristics, whose calculation is carried out according to relations (2)—(5). Figure 2 presents the FRFST characteristics calculated for three variants of the stabilizer using parameters close to the experimental ones:

\[ A \ne 1\; m\,l^{-3};\quad \kappa_0 = 30\; m^2\,\mathrm{s}^{-1};\quad m=3;\quad \kappa=\kappa_1=0.2;\quad U_1=30\;\mathrm{V};\quad X_{s1}=300\;\Omega,\quad \eta=1. \]

The characteristics designated \(1a, 1b, 1v\) correspond to an FRFST without compensation and filtration windings with resonance coefficient \(\rho=1\) and \(\eta=0\). The characteristics \(2a \div 2v\) and \(3a \div 3v\) describe loaded FRFSTs with windings \(W_k\) and \(W_p\), and resonance coefficients of 2 and 1, respectively.

Analyzing the curves, it can be seen that introducing the windings \(W_k\) and \(W_p\) increases the stabilization coefficient, the stiffness of the external characteristic, and decreases the magnitude of the primary current \(I_1\) during a short circuit in the load \((\eta=\infty)\), while at \(\rho=1\) the multivalued region disappears in the characteristics.

To clarify the filtering conditions it is necessary to obtain an analytical expression for \(u_{\mathrm{n}}(t)\). The instantaneous values of the currents, voltages, and magnetic fluxes are related by the system of differential equations:

Fig. 2. Basic calculated characteristics of the FRFST

\[ \begin{aligned} i_1 &= i_0+i_c+i_{\mathrm{n}},\\ i_0 &= D\psi^\gamma,\\ u_1 &= \frac{d\psi}{dt}+L_{S1}\frac{di_1}{dt},\\ u_1 &= L_{S1}\frac{di_1}{dt}+L_p\frac{di_c}{dt} +\frac{1}{C_p}\int i_c\,dt+\kappa_1\left(u_1-\frac{d\psi}{dt}\right),\\ u_1 &= i_{\mathrm{n}}R_{\mathrm{n}}+\kappa u_1+L_{S1}\cdot\frac{di_1}{dt}, \end{aligned} \tag{6} \]

where \(i_0=D\psi^\gamma\) is a function approximating the magnetization curve for instantaneous values of induction and field strength; here \(\psi\) is the instantaneous value of flux linkage in the saturated core, \(D\) and \(\gamma\) are constants, \(i_0\) is the instantaneous value of the current in winding \(W_2\), and \(L_{S1}\) is the leakage inductance.

From system (6) the general differential equation is derived

\[ \frac{d^3\psi}{dt^3} +\left(a+b\psi^{\gamma-2}\right)\frac{d^2\psi}{dt^2} +e\frac{d\psi}{dt} +f\psi^\gamma+g\psi = \kappa\frac{d^2u_1}{dt^2} +h\frac{du_1}{dt} +\kappa e u_1 +f\int u_1\,dt, \tag{7} \]

in which

\[ a=R_{\mathrm{n}}L_{S1}^{-1} +R_{\mathrm{n}}L_p^{-1}(1+\kappa_1); \qquad b=R_{\mathrm{n}}D\gamma(\gamma-1); \qquad e=L_p^{-1}C_p^{-1}, \]

\[ f=DR_{\mathrm{n}}L_p^{-1}C_p^{-1}; \qquad g=R_{\mathrm{n}}C_p^{-1}L_p^{-1}L_{S1}^{-1}; \qquad h=\kappa_1R_{\mathrm{n}}L_p^{-1}+R_{\mathrm{n}}L_{S1}^{-1}. \]

Representing the rectangular input voltage \(u_1(t)\) in the form of a Fourier series, we transform expression (7):

\[ \frac{d^3\psi}{dt^3} +\left(a+b\psi^{\gamma-2}\right)\frac{d^2\psi}{dt^2} +e\frac{d\psi}{dt} +f\psi^\gamma+g\psi = 4U_{1m}\pi^{-1}\{\kappa(\omega^2- \]

\[ -e)\!\left\{2\sin\omega t+10\cdot 3^{-1}\sin 3\omega t+\ldots+\left(2p+1+\frac{1}{2p+1}\right)\sin(2p+\right. \]
\[ \left.+1)\omega t+\ldots\right] +(h\omega-g\omega^{-1})\!\left[2\cos\omega t+10\cdot 9^{-1}\cos3\omega t+\right. \]
\[ \left.\left.+\ldots+\left(1+\frac{1}{4p^{2}+4p+1}\right)\cos(2p+1)\omega t+\ldots\right]\right\}. \tag{8} \]

Here \(U_{1m}\) is the amplitude of the rectangular voltage \(u_1(t)\); \(p=1,2\ldots\)

The solution of equation (8) has the form:

\[ \psi=L_1\sin\omega t+L_3\sin3\omega t+\ldots+L_{2p+1}\sin(2p+1)\omega t+ \]
\[ +\ldots+N_1\cos\omega t+N_3\cos(2p+1)\omega t+\ldots, \tag{9} \]

where the coefficients \(L_1, L_3,\ldots N_1, N_3,\ldots\) are determined from the system of algebraic equations obtained after integrating a system of integrals of the form [1]:

\[ \int_{0}^{2\pi} F(t)\sin\omega t\,dt=0, \]

\[ \int_{0}^{2\pi} F(t)\cos\omega t\,dt=0, \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

\[ \int_{0}^{2\pi} F(t)\sin(2p+1)\omega t\,dt=0, \]

\[ \int_{0}^{2\pi} F(t)\cos(2p+1)\omega t\,dt=0, \tag{10} \]

\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

in which \(F(t)\) is the expression obtained from (8) after substituting the dependence (9) into it.

Since \(u_{\mathrm{n}}(t)=\dfrac{d\psi}{dt}-\kappa u_1(t)\), taking (9) into account we obtain that

\[ u_{\mathrm{n}}(t)=-(4\kappa\pi^{-1}U_{1m}+N_1\omega)\sin\omega t-[4\kappa U_{1m}\pi^{-1}(2p+1)^{-1}+ \]
\[ +(2p+1)N_{2p+1}\omega]\sin(2p+1)\omega t\ldots+L_1\omega\cos\omega t+ \]
\[ +L_{2p+1}\omega(2p+1)\cos(2p+1)\omega t+\ldots, \tag{11} \]

whence the filtering conditions are

\[ L_3=L_5=\ldots=L_{2p+1}=\ldots=0, \]
\[ 4\kappa U_{1m}+9\omega N_3=0, \]
\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]
\[ 4\kappa U_{1m}+(2p+1)^2\omega N_{2p+1}=0, \tag{12} \]
\[ \cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot\quad\cdot \]

In a number of cases one may be satisfied with approximate filtering conditions, which are obtained in analyzing expression (3). \(U_{\mathrm{n}}\) becomes zero when the relations are simultaneously satisfied

\[ \rho=\infty,\quad \kappa=\kappa_1(1+\kappa_1)^{-1}. \tag{13} \]

The quantity \(\rho\) tends to infinity at resonance in the circuit \(L_{\mathrm{p}}C_{\mathrm{p}}\). By calculating \(W_{\mathrm{p}}\) in a certain way, resonance is achieved

on the 3rd harmonic, then it will be absent from the output-voltage curve; the remaining harmonics are thereby attenuated, since the second relation (13) does not depend on frequency.

To verify the results obtained, an FRFST with an output power of 30 W was manufactured. The magnetization curve for the rms values of induction and field strength was obtained experimentally with the excitation winding supplied by a rectangular voltage; it was approximated by a function of the form \(H = 4000 B^4\). The FRFST was made on a type Sh-30×30 core of E332 steel; the air gap on one of the side limbs was \(2.5\) mm. In this case \(W_1 = 150 \times 2\), \(W_2 = 260\), \(W_3 = 200\), \(W_{\mathrm{k}} = 20\), \(W_{\mathrm{p}} = 100\), \(C_{\mathrm{p}} = 60\) μF, 250 V.

Fig. 3. Experimental characteristics of the FRFST

The characteristics of the manufactured stabilizer are presented in Fig. 3. The discrepancy between the calculation and the experiment is about 10%. Consequently, the parameters of the FRFST can be calculated from the relations obtained, but in manufacture taps from windings \(W_2\), \(W_{\mathrm{k}}\), and \(W_{\mathrm{p}}\) should be provided in order to make adjustment possible.

REFERENCES

1. L. A. Bessonov. Electric Circuits with Steel. Moscow–Leningrad, GEI, 344 pp., 1948.

2. A. G. Lur’e. Theory of Ferroresonant Voltage Stabilizers. Moscow–Leningrad, GEI, 130 pp., 1958.

3. D. I. Bogdanov, G. K. Evdokimov. Ferroresonant Stabilizers. Moscow–Leningrad, GEI, 180 pp., 1958.