STUDY OF STATIC OPERATING MODES AND CHARACTERISTICS OF TRANSFORMERS CONTROLLED BY MAGNETIC BIASING OF SHUNTS

V. P. OBRUSNIK

(Presented by the scientific seminar of the Electromechanical Faculty)

Although transformers controlled by magnetic biasing of shunts (TRPSh) are widely and successfully used in a number of industrial installations, the analysis of the operating modes and static characteristics of these devices has not yet been described in the literature with sufficient clarity. The known articles [1, 2] do not give a correct conception even of the operating principle of TRPSh, since they do not reveal the physical essence of the actual processes occurring in transformers with magnetically biased shunts. The author has attempted to present these questions in such a way that specialists engaged in the design and operation of automatic systems with magnetic regulators could obtain a fuller and more correct understanding of the regulating capabilities of TRPSh.

Studies show that, although the operation of TRPSh is characterized by a redistribution of fluxes (between the main magnetic core and the shunt), and for combinations of a transformer and a magnetically biased saturating reactor by a redistribution of voltages (between the reactor and the transformer), both regulators have identical laws of variation of their output quantities—the load current and voltage.

We shall first show this for operating modes and static characteristics of TRPSh having magnetic cores with a hysteresis-free rectangular magnetization curve, assuming that the supply-network voltage is sinusoidal, the load is active, the induction in the main magnetic core at complete saturation of the shunts does not go beyond the knee of saturation, and the resistance of the magnetic-biasing windings is negligibly small for currents of even harmonics. Under these assumptions, the equivalent circuits for three varieties of TRPSh [3] may be represented by Figs. 1a, b, c, where before saturation of the shunts \(X_{12}, X_{1s}, X'_{2s} = \infty\), and after saturation \(X_{1s} = X'_{2s} = 0,\ X_{12} = \infty\).

We shall conduct the reasoning by dividing the operating modes of the transformer into several states.

State 1: the load is disconnected; there is no magnetic biasing of the shunts. Here the total flux of the TRPSh is distributed in proportion to the cross sections of the magnetic cores located inside the primary winding.

Fig. 1.

The part of it that closes through the main magnetic core links with the turns of the secondary winding and induces in it a sinusoidal e.m.f.

for Fig. 1 a, c

\[ E_{2xx}=U_1\cdot\frac{W_2}{W_1}\cdot\frac{S_0}{S_{\mathrm{sh}}+S_0}, \tag{1} \]

for Fig. 1 b:

\[ E_{2xx}=U_1\cdot\frac{W_2}{W_1}, \tag{2} \]

where \(S_0, S_{\mathrm{sh}}\) are, respectively, the cross section of the main magnetic core and of the shunt.

The no-load e.m.f. will be nonsinusoidal only in those cases where the saturation induction of the shunt or of the main magnetic core is less than the no-load induction, that is,

\[ B_{s0}\ \text{or}\ B_{s.\mathrm{sh}}<B_{xx}= \frac{U_1}{4.44 f_1\cdot W_1\cdot\left(S_{\mathrm{sh}}+S_0\right)}. \tag{3} \]

For TRPSh this is excluded, since the main magnetic core is calculated from the condition:

\[ U_1 \leq B_{s0}\cdot S_0 \cdot W_1 \cdot 4.44 f_1, \tag{4} \]

therefore, always

\[ B_{s0} \geq B_{xx}\cdot \frac{S_{\mathrm{sh}}+S_0}{S_0}. \tag{5} \]

State 2: the load is connected, there is no biasing. With the secondary circuit closed, the transformer tends to shed the load. Therefore, if

\[ S_{\mathrm{sh}} \geq \frac{U_1}{4.44 f_1 \cdot W_1 \cdot B_{s\mathrm{sh}}}, \]

then in a TRPSh with the circuits in Fig. 1a the entire flux will close through the shunt, while with the circuit in Fig. 1b subtraction of the main flux and the shunt flux occurs inside the secondary winding. In both cases the transformation ratio is zero and \(E_2=0\).

If

\[ S_{\mathrm{sh}} \leq \frac{U_1}{4.44 f_1 \cdot W_1 \cdot B_{s\mathrm{sh}}}, \]

then what was said above is valid only up to the moment of shunt saturation, after which a change in flux will be observed only in the main magnetic core and the transformation ratio rises abruptly to the value of the turns ratio \(K_t=W_1/W_2\). The waveform of the secondary e.m.f. under active load is shown by the dashed line in Fig. 5. Here the saturation angle \(\alpha_{s0}\) and the effective value of the secondary e.m.f. can be found from the expressions

\[ \alpha_{s0}=\arccos\left(1-2\frac{U_{s\mathrm{sh}}}{U_{xs}}\right), \tag{4} \]

\[ E_{2\min}=\frac{U_1}{K_t}\cdot \sqrt{\frac{1}{\pi}\left(\pi-\alpha_{s0}+\frac{1}{2}\sin 2\alpha_{s0}\right)}, \tag{5} \]

where \(U_{s\mathrm{sh}}\) is the voltage at which saturation of the shunt should occur,

\[ U_{s\mathrm{sh}}=4.44\cdot f_1\cdot W_1\cdot B_{s\mathrm{sh}}\cdot S_{\mathrm{sh}}; \tag{6} \]

\(U_{xs}\) is the actual voltage determining the induction of the shunt cores,

\[ U_{xs}=U_1\cdot \frac{X_s}{Z_{\mathrm{in}}}=U_1\cdot K_z; \tag{7} \]

\(Z_{\mathrm{in}}\) is the resistance at the input of the equivalent circuit;

\(X_s\) is the magnetic leakage reactance, equal to \(X_{1s}\) for the circuits in Fig. 1a, b and \(X'_{2s}\) for the circuit in Fig. 1b.

In the case under consideration, for the circuits in Fig. 1a, b \(K_z=1\), and for the circuit in Fig. 1c

\[ K_z=\frac{1+\beta}{2+\beta}, \]

where \(\beta=S_{\mathrm{sh}}/S_0\).

The slope of the external characteristic under a change in load will be determined only by the losses due to air leakage and by the active resistance of the working windings. For example, the expression for the voltage across the active resistance of the load has the form:

\[ U_2 = E_{2\min}\cdot \frac{R'_{\text{н}}}{\sqrt{X_b^2 + (r_a + R'_{\text{н}})^2}}, \tag{8} \]

where

\[ X_b = X_{1b} + X'_{2b}; \qquad r_a = r_1 + r'_2. \]

Figure 2 shows that when the load is switched on, the operating point will jump from \(E_{2xx}\) to \(E_{2\min}\) and then, along the external characteristic of the unmagnetized transformer, will pass to the point of its intersection with the volt-ampere characteristic of the load (trajectory \(1—3—3\)).

Fig. 2.

State 3: the load is disconnected; the shunts or the main magnetic core are bias-magnetized.

First of all, if one of the two magnetic cores (the main core or the shunt) is bias-magnetized by direct current, then the non-bias-magnetized magnetic core must have a cross section that allows the entire alternating flux to pass at an amplitude value of the induction lower than the saturation induction. Otherwise, a large no-load current is inevitable, which will have to be limited by a ballast resistance in the primary circuit.

If the indicated condition is fulfilled for the TRPSh in Fig. 1a, with sufficiently small biasing of the shunt, the entire flux can be forced into the main magnetic core or, conversely—by biasing the latter—the entire flux can be forced into the shunt. In the first case the secondary

the emf will increase stepwise from \(E_{2xx}\) to \(E_{2\max}\), and in the second it will decrease from \(E_{2xx}\) to 0. Smooth regulation of the output voltage at no load with a hysteresis-free rectangular magnetization curve is impossible.

For the TRPSh in Fig. 1 b, biasing with the secondary winding open has no meaning.

State 4: the load is connected; the shunts are biased. The pattern of variation of the principal quantities for this state is shown in Fig. 3.

Fig. 3.

Here the periodic stepwise change of the coupling coefficient between the working windings is characteristic. While the shunts are not saturated, the transformation coefficient is equal to zero; when the shunts are saturated, it rises stepwise to the turns ratio \(k_t = W_1 / W_2\). At the moment,

when the amplitude value of the primary voltage is equal to zero, both cores of the shunt are unsaturated, and their induction is determined by the constant component \(B_d\), which corresponds to a certain position of point 1 on the idealized magnetization curves A and B. As the amplitude of the primary voltage increases, the flux in the main magnetic core either does not change at all, closing through the shunt cores, which will be true for a TRPSh with magnetic leakage of the primary or both working windings (Fig. 1a, c), or its change is compensated within the secondary winding by a change in the shunt flux, which is observed in a TRPSh with magnetic leakage of the secondary winding (Fig. 1b). As a result, the working flux \(\Phi_0\), transforming voltage into the secondary circuit, does not change with time, and the e.m.f. at the output is equal to zero. The induction in the shunt cores on curves A, B changes in the direction from point 1 to point 2; both of them are unsaturated.

At point 2, since

\[ B_{s\mathrm{sh}} - B_d < \frac{U_1}{4.44 f_1 \cdot W_1 \cdot S_{\mathrm{sh}}}, \]

core A becomes saturated. Further change of the flux in the shunt cannot occur, despite the fact that core B is not saturated, since in the bias winding located on it the transformed e.m.f. is no longer compensated by the e.m.f. in the bias winding of core A, and an equalizing current appears, preventing further change of the flux in core B. From instant 2, in a TRPSh with magnetic leakage of the primary or both working windings, the flux will change only in the main magnetic core, while in a TRPSh with magnetic leakage of the secondary winding, compensation of the main flux will no longer occur within the latter. As a result, the winding coupling coefficient increases abruptly from zero to unity, the secondary e.m.f. increases abruptly and, with an active load, so does the secondary current. On the magnetization characteristic of shunt A, the operating point passes from position 2 to position 3. Further change of the secondary e.m.f. proceeds according to a sinusoidal law until the end of the half-period; the operating point on characteristics A and B moves from position 3 to position 4. At the beginning of the next half-period both shunt cores are again unsaturated, and the pattern of variation of the quantities is repeated, but already in the region of negative values.

If the load resistance is constant, then the saturation angle of the shunts under dc biasing can be found from the expression

\[ \alpha_{sn} = \arc \Cos \left( 1 - 2 \frac{B_{s\mathrm{sh}} - B_d}{U_{xs}} \cdot 4.44 f_1 W_1 S_{\mathrm{sh}} \right). \tag{9} \]

For idealized TRPSh’s having no feedbacks, on the basis of the law of equality of m.m.f.’s, with an active load it will be true that

\[ IW_d = IW_{2\mathrm{cp}} = IW_{1\mathrm{cp}} = \frac{2\sqrt{2}\cdot U_{xs}}{\pi \cdot R} (1 + \Cos \alpha_{sn}) W_1, \tag{10} \]

where \(R = R'_n + r_a\), and \(X_b\) is taken equal to 0,

From expression (10) one may obtain

\[ \alpha_{sn} = \arc \Cos \left( 2 \frac{IW_d R}{0.9 \cdot K_z \cdot U_1 W_1} - 1 \right). \tag{11} \]

Analysis of equality (11) shows that the change in the saturation angle of the shunt of an idealized TRPSh is limited by definite values of the resistance of the operating circuit and of the biasing current.

If the total resistance of the operating circuit of the TRPSh varies from \(\infty\) to

\[ R_{\mathrm{kp}1}=\frac{K_z\cdot 0.9\,U_1 W_1}{I W_d} \quad \text{for } I W_d=\mathrm{const}, \]

or the biasing current \(I_d\) varies from

\[ I_{\mathrm{dkp}2}=\frac{0.9K_z U_1\cdot W_1}{W_d\cdot R} \quad \text{and higher for } R=\mathrm{const}, \]

then the saturation angle is equal to 0, the load current is sinusoidal and is determined by the expression

\[ I_2=\frac{U_1}{\sqrt{R^2+X_b^2}} . \tag{12} \]

When \(R\) changes from

\[ R_{\mathrm{kp}2}=\frac{(\cos\alpha_{s0}+1)\cdot 0.9K_z U_1 W_1}{2I W_d} \quad \text{and lower, when } I W_d=\mathrm{const}, \]

or when \(I_d\) changes from

\[ I_{\mathrm{dkp}1}=\frac{(\cos\alpha_{s0}+1)\cdot 0.9U_1K_zW_1}{2W_d\cdot R} \quad \text{and lower, when } R=\mathrm{const}, \]

the saturation angle remains unchanged and equal to \(\alpha_{s0}\), while the load current is determined by the expression

\[ I_2=\frac{U_1}{\sqrt{R^2+X_b^2}}\cdot \sqrt{\frac{1}{\pi}\left(\pi-\alpha_{s0}+\frac{1}{2}\sin 2\alpha_{s0}\right)} . \tag{13} \]

The external characteristic of an idealized TRPSh without feedback at \(I W_d=\mathrm{const}\) has the form of the stepped curve shown in Fig. 3 by the numbers 2, 4, 5, 6. Here, for \(R_{\mathrm{kp}1}>R>R_{\mathrm{kp}2}\), the saturation angle of the shunt changes automatically according to (11), ensuring, by the law of equality of magnetomotive forces, constancy of the mean value of the load current, whose effective value is then equal to

\[ I_{2\mathrm{ef}}=I_{2\mathrm{av}}\cdot K_f =I_d\cdot\frac{W_d}{W_2}\cdot K_f, \tag{14} \]

where

\[ K_f= \frac{ \sqrt{\dfrac{\pi}{2}\left(\pi-\alpha_{sn}+\dfrac{1}{2}\sin 2\alpha_{sn}\right)} }{ 1+\cos\alpha_{sn} }. \tag{15} \]

With the introduction of positive current feedback, the influence of the load resistance on the shunt saturation angle is appreciably reduced and disappears completely when the feedback coefficient is equal to unity, as is seen from the expression

\[ \alpha_{sn}=\arccos\left( \frac{2I W_d}{K_z U_1\,0.9W_1(1-k_{\mathrm{oc}})}-1 \right). \tag{16} \]

With positive current feedback with a coupling coefficient equal to unity, the saturation angle depends only on the control signal, which changes the dc component of the saturation induction; therefore, for each value of the biasing current one can obtain an external characteristic of the TRPSh, whose stiffness will be determined by leakage losses through the air and in the active resistance of the operating windings (see I, II, III, IV in Fig. 2). These characteristics are analogous to the external characteristics of an ordinary transformer regulated by the turns of the secondary winding, and differ from the latter only in the shape of the output voltage.

As can be seen, in the TRPSh it is possible to obtain characteristics of two types: the characteristics of a current regulator, when the transformer has no feedbacks, and the characteristics of a voltage regulator, if it is made with deep current feedback. At the points of intersection these characteristics have identical values of the saturation angle. A decrease in the output voltage of the TRPSh is always associated with an increase in the saturation angle, and thereby with a decrease in the power factor of the apparatus, which under deep regulation may reach very small values.

Fig. 4.

The characteristics of the TRPSh depend substantially on the type of load. For example, with a purely inductive or active-inductive load, when \(Z_{\mathrm{н}} \ll X_{1s}, X'_{2s}\), the pattern of variation of the output quantities will have the form shown in Fig. 4. At the instant of saturation of the shunts, the instantaneous value of the secondary emf increases abruptly from zero to \(U_1 \sqrt{2}\,\dfrac{W_2}{W_1}\cdot \sin \alpha_{sn}\), but the current rises gradually, and by the end of the half-period it is not equal to zero. In the second half-period, owing to the decaying current of the load inductance,

the load flux in the main magnetic core decreases for some time, and an e.m.f. is induced in the secondary winding until the load current falls to zero, after which the coupling coefficient between the windings becomes equal to zero. Since the curves of the output quantities change, their rms values change, and the form of the external and load characteristics changes. But it should be emphasized that the very principle of regulation of the TRPSh—the principle of controlling the instant of saturation of the shunts—is preserved for any load character.

The curves in Figs. 3, 4, and 5 are well confirmed by oscillograms when the magnetic cores of TRPSh are made of permalloy and even of textured steels.

Leaving in force the assumptions made above, let us now analyze the operation of the TRPSh with an active load, approximating the magnetization curve by a piecewise-linear function for which, at \(B < B_s\), \(B/H \ne \infty\), and at \(B > B_s\), \(B/H = 0\). We shall also assume that the quality of all magnetic cores is the same. Then, before saturation of the shunts,

\[ X_{1s}=X'_{2s}=\varepsilon X_{12}=2\pi f_1\cdot w_1^2\cdot \frac{S_{\text{sh}}}{l_{\text{sh}}}\cdot \frac{B_{ms}}{H_{ms}}, \tag{17} \]

where

\[ \varepsilon=\frac{S_{\text{m}}\cdot l_0}{S_0\cdot l_{\text{sh}}} \]

is a dimensionless coefficient,

\[ l_0,\ l_{\text{sh}} \]

are the lengths of the lines of force of the main magnetic core and of the shunt.

At no load, in the absence of biasing, the secondary e.m.f. of the transformer will be determined by the expression

\[ E_{2xx}=\frac{W_2}{W_1}\,U_1\,\frac{X_{12}}{X_{1s}+X_{12}} =\frac{U_1}{K_0(1+\varepsilon)}. \tag{18} \]

When the load is connected and biasing is absent, the secondary e.m.f. cannot at once assume the value determined by expression (5), since \(X_{1s}\), \(X'_{2s}\), and \(X_{12}\) are not equal to infinity.

Fig. 5.

Up to a certain load resistance, denoted by \(R_{ns}\), the shunt is not saturated: at the output there will be a sinusoidal voltage whose amplitude value decreases as \(R_n\) decreases. At \(R_n \le R_{ns}\), the shunt begins to saturate, and the form of the output voltage assumes the shape of the curve enclosing in Fig. 5 the shaded area. When \(R_n \ll R_{ns}\), it approaches the curve shown by the dashed line and determined by expressions (4,5).

The value \(R_{ns}\) can be found from the equivalent circuits, taking

\[ (U_{2xs};\ U_{1xs})=4.44 f_1\cdot W_1\cdot B_{\text{sh}}\cdot S_{\text{sh}} \]

and assuming that \(X_{1s}\), \(X'_{2s}\), and \(X_{12}\) are determined from expression (17).

For example, for the circuit in Fig. 1b one may obtain

\[ R'_{\mathrm{ns}} = X_{12}\cdot \sqrt{\left(\frac{U_1}{U_{\mathrm{xs}}}\right)^2 - 1}. \tag{19} \]

The external characteristics here consist of two sections. For values of the resistance \(R_{\mathrm{n}} > R'_{\mathrm{ns}}\), the output voltage is calculated from the equivalent circuits in the usual way, while for \(R_{\mathrm{n}} < R'_{\mathrm{ns}}\) it must already be determined as the root-mean-square area of a curve consisting of two truncated sinusoids (see the hatched area in Fig. 5), and then

\[ U'_2 = \frac{R'_{\mathrm{n}}}{\sqrt{X_b^2 + R^2}}\cdot \sqrt{ E_{2\mathrm{dn}}^2\cdot \frac{1}{\pi} \left(\alpha_{s0} - \frac{1}{2}\sin 2\alpha_{s0}\right) + E_{2\mathrm{in}}^2\cdot \frac{1}{\pi} } \;\times \]

\[ {}\times \left(\pi - \alpha_{s0} + \frac{1}{2}\sin 2\alpha_{s0}\right), \tag{20} \]

where \(E_{2\mathrm{dn}}, E_{2\mathrm{in}}\) are, respectively, the secondary-winding electromotive forces found from the equivalent circuits for unsaturated and saturated shunts.

The value \(\alpha_{s0}\) in expression (20) is determined from (4). If the characteristics are plotted, then in Fig. 2 they will have the form of curves a, b, designated in accordance with the equivalent circuits.

In a TRPSh with real magnetic cores, smooth regulation of the no-load voltage is possible, since the losses in the steel here play the role of a ballast load. To displace the alternating flux from the biased magnetic core, a current satisfying the equality will be required

\[ I_d = \frac{I_{10}}{k_f}\cdot \frac{W_1}{W_d}\,(1.1 \div 1.2), \tag{21} \]

where \(I_{10}\) is the effective value of the primary no-load current of the TRPSh at full saturation of the biased magnetic core.

For load resistances considerably smaller than the magnetic leakage resistance of the TRPSh, approximately at \(Z_{\mathrm{n}} < (0.2 \div 0.25)\cdot (X_{1s}, X'_{2s})\), the physics of the processes for the second method of approximating the magnetization curve will in practice differ in no way from that described above in states 3, 4.

The static characteristics of a TRPSh with real magnetic cores can be divided into two regions of their existence: large loads and small loads. In the region of small loads they are determined entirely by the magnitude of the losses in the steel and do not lend themselves to sufficiently accurate calculations. Distortion of the external characteristics of a TRPSh in the small-load mode is easily eliminated by shunting one of the secondary windings with a nonlinear ballast resistance having the current-voltage characteristic of a barretter, shown in Fig. 2 under the symbol \(Z_b\).

Such shunting also makes it possible, in a TRPSh of any design version, to regulate the secondary no-load electromotive force within the range from 0 to \(E_{2\max}\) without biasing the main magnetic core.

The analysis presented for the static characteristics of the TRPSh is well confirmed by experiments.

Figure 6 presents the calculated and experimental (marked by points) characteristics of a TSMSh with magnetic leakage of the primary winding, having the following data:

\[ W_1 = W_2 = W_{oc} = W_d = 600 \text{ turns}, \quad r_1 = 6.4 \ \Omega, \quad r_2 = 3.6 \ \Omega, \quad r_d = r_{oc} = 3.4 \ \Omega \]

\[ S_o = 14 \ \text{cm}^2, \qquad S_{\text{sh}} = 0.6 S_o; \]
the magnetic cores are made of identical Π-shaped plates of E-320 steel.

Fig. 6.

In the calculations it was assumed that

\[ B_{\text{sh}} = B_{so} = 19500 \ \text{Gs}, \quad X_{1s} = X'_{2s} = \varepsilon X_{12} = 1700 \ \Omega, \quad \varepsilon = 0.6, \quad U_1 = 220 \ \text{V}. \]

The characteristics in mean values in Fig. 6 are shown by dashed lines. In plotting, the instantaneous values of the quantities were referred to their nominal values, while the quantities in mean values were multiplied by a coefficient of 1.11.

As can be seen, the character of the experimental curves fully corresponds to the calculation. Only certain quantitative deviations are observed; these will be greatest for ordinary transformer steels and smallest for special steels of the permalloy type, etc.

On the basis of the analysis carried out of the operation of the TSMSh, the following conclusions may be drawn:

1. Characteristic of a TRPSh is a periodic stepwise change of the transformation ratio from minimum to maximum, which occurs at the moment the shunts become saturated. Regulation of the effective and average value of the output voltage occurs through changes in its waveform and volt-second area, which in principle distinguishes the TRPSh from regulated transformers with movable windings and movable sections of magnetic cores. The regulating properties of the apparatus are manifested most clearly when the magnetization curves of its shunts have a rectangular form.

2. The TRPSh can operate as a current regulator if it has no feedback, and as a voltage regulator if it is implemented with deep feedback with respect to the load current. In the latter case, its output voltage is determined only by the control signal.

3. Despite the different nature of the physical processes, the static characteristics of the TRPSh are, in their character, no different from the characteristics of biased saturation chokes operating in combination with ordinary transformers. This makes it possible to use many problems of the theory of magnetic amplifiers [4, 5] in the theory of the TRPSh.

REFERENCES

1. A. M. Bamdas, V. A. Somov, A. O. Shmidt. Transformers and stabilizers controlled by magnetic biasing of shunts. Gosenergoizdat, 1959.

2. S. B. Shapiro. Steady-state and transient processes in a transformer with a biased shunt. Elektromekhanika, No. 4, 1962.

3. A. I. Zaitsev, V. P. Obrusnik. Problems in the theory and application of transformers controlled by magnetic biasing of shunts. Izvestiya TPI, vol. 117, 1963.

4. M. L. Rozenblat. Magnetic amplifiers, Parts I, II, Sovetskoe Radio, 1960.

5. G. F. Strom. Magnetic amplifiers. Foreign Literature Publishing House, 1957.