PROCEEDINGS

of the Tomsk Order of the Red Banner of Labor
Polytechnic Institute named after S. M. Kirov

Vol. 117
1963

QUESTIONS OF THE THEORY AND APPLICATION OF TRANSFORMERS REGULATED BY SHUNT BIASING

A. I. ZAYTSEV, V. P. OBRUSNIK

(Presented by the scientific seminar of the Electromechanical Faculty)

An urgent problem in electrical engineering is the creation of new types of static apparatus and devices for smooth regulation of alternating-current voltage in high-power installations. They must be simple in design, reliable in operation, economical, and sufficiently fast-acting. Among electromagnetic devices, saturation chokes and biased transformers alone meet such requirements to the greatest degree. The latter are less well known in the literature and in practice, but invariably attract attention, since regulated transformers are needed for the most diverse installations.

Two groups of biased transformers are distinguished: transformers regulated by redistributing the primary voltage between two separate units [4, 5], and transformers regulated by branching off part of the magnetic flux into a shunt with variable magnetic reluctance [1, 2, 3]. If, in transformers of the second group, the magnetic reluctance of the shunt is changed by biasing with direct current, then they are called “transformers regulated by shunt biasing,” abbreviated “TRShB.”

TRShB differ favorably from analogous transformers of the first group in that they constitute a single structural unit, and a number of their designs require appreciably lower expenditures of active materials for manufacture while having higher efficiencies and power factors.

The first TRShB designs have been known for more than 30 years; however, all of them had a common drawback consisting in excessively large magnetic leakage, caused by placing their primary and secondary windings on different limbs of the main magnetic core. Poor electromagnetic coupling between the windings proved to be one of the reasons hindering the broad introduction of such transformers, since large leakage is associated with a considerable increase in the consumption of active materials and a low power factor.

Comparatively recently, at the Gorky Polytechnic Institute, under the direction of Prof. A. M. Bamdas, a number of original designs of TRPSh were created, in which the magnetic shunts, together with the biasing windings, are located in the developed window of the primary or secondary winding [1]. These transformers have considerable advantages over earlier designs; their efficiency and power factor differ little from the same parameters of ordinary transformers. At present, TRPSh of the new design are being successfully tested in industrial control systems.

The basic theoretical principles concerning transformers regulated by shunt biasing were developed at GPI and set forth in [1, 2, 3].

Also of interest is the work [7], where, in order to explain the basic regularities characterizing the operation of TRPSh, the author proposes a theory of magnetic amplifiers. However, a number of questions in the existing theory of TRPSh require further development.

Three Varieties of TRPSh

A large number of different design variants of both single-phase and three-phase transformers regulated by shunt biasing—TRPSh—are already known in the literature. However, the use of TRPSh as regulators in automatic control systems is made difficult by the fact that to this day there is no sufficiently complete characterization of these devices. It is impossible to approach all TRPSh with a single measure. This is evident already from the fact that not all of them possess identical properties. For example, for the TRPSh in Figs. 1, в and 1, г three conditions are fulfilled:

1) The total alternating magnetic flux in the steel, as the load changes within nominal limits, remains practically constant, regardless of whether we bias the shunt or the main magnetic core. Depending on the degree of biasing, this flux is merely redistributed between the main magnetic core and the shunt magnetic cores.

2) The secondary voltage is regulated at no load.

3) The device possesses the property of reversibility, that is, if the primary and secondary windings are interchanged, the principle of operation will not change as a result.

For the devices in Fig. 1, а, however, only the first two conditions are fulfilled, and for Fig. 1, б—not one of them.

But the difference between transformers with biased shunts is revealed not only in this. As will be shown below, TRPSh are original constructions which, depending on their design, possess the properties of one or another combination of an ordinary transformer and series-connected regulated saturable reactors. This leads to the fact that existing TRPSh differ from one another not only in their structural forms, operating modes, and economic indicators, but also have serious fundamental differences that do not permit describing them by general equations, drawing up for them a common equivalent circuit, or calculating them for identical control capabilities. It is necessary to bring all TRPSh, according to certain related features, into such a system that would make it possible, for definite groups of these devices, to develop a unified method for calculating their parameters and characteristics and to carry out the necessary analyses and syntheses.

In our view, this can be done if the fundamental distinction between biasable transformers with magnetic shunts is sought in the methods of creating secondary leakage fluxes, that is, fluxes closing through the magnetic cores of the shunt.

In transformer theory the phenomenon of double magnetic leakage is well known, i.e., when not only fluxes closing through air (primary leakage) are assigned to the category of leakage fluxes, but also fluxes closing through the magnetic shunt (secondary leakage) [6]. TRGSh are a vivid example of the practical use of such a phenomenon. In these devices, by biasing the magnetic shunt with a direct flux, one can control the magnitude of the secondary leakage flux and thereby regulate the secondary voltage. But the character of the secondary leakage flux is not the same for all TRGSh; it is determined by its association with the working windings of the device and may be created by the primary winding, by the secondary winding, or by both working windings at once.

Fig. 1.

In Fig. 1, a, b, v, g, these three possible methods of creating secondary leakage fluxes are shown using, as an example, simplified depictions of single-phase TRGSh with magnetic shunts arranged in various windows of their working windings. In the figures the following are denoted¹:

\(\Phi_{12}\)—the mutual-induction flux,
\(\Phi_{B_1}\) and \(\Phi_{B_2}\)—the air leakage fluxes,
\(\Phi_{S_1}\) and \(\Phi_{S_2}\)—the fluxes of the primary and secondary windings of the TRPSh, closing through the magnetic shunt.

The nature of the secondary leakage fluxes in a TRPSh determines their principal characteristics and control capabilities. If all TRPSh are divided into separate groups having the same character of secondary leakage fluxes, then in these groups all devices will have identical operating regimes, equivalent circuits, basic voltage-control equations, control capabilities and, all other things being equal, will differ from one another only in their structural forms and economic indicators.

According to the methods of creating secondary leakage fluxes, there will be three such groups:

a) TRPSh with secondary leakage of the primary winding (Fig. 1, a),
b) TRPSh with secondary leakage of the secondary winding (Fig. 1, b),
c) TRPSh with secondary leakage of both working windings (Fig. 1, c and 1, d).

In calculations and investigations of transformers regulated by shunt biasing, and of automatic-control systems using them, it is necessary to take into account the existence of the three listed varieties of these devices, which have substantial fundamental differences from one another.

TRPSh and combinations of an ordinary transformer with controllable saturation reactors

Regulated transformers with biased shunts may be represented as combinations of an ordinary transformer with series-connected controllable saturation reactors—magnetic amplifiers. Each variety of TRPSh will correspond to its own combination of an ordinary transformer and controllable saturation reactors. For brevity, in what follows these combinations will be called “TR—RDN.”

A TRPSh with secondary leakage of the primary winding will correspond to a combination of an ordinary transformer with an RDN connected in series in the primary circuit (Fig. 1, a and 2, a); a TRPSh with secondary leakage of the secondary winding will correspond to a combination of an ordinary transformer with an RDN connected in series in the secondary circuit (Fig. 1, b and 2, b); and a TRPSh with secondary leakage of both working windings will correspond to a combination of an ordinary transformer with controllable saturation reactors connected in series in the circuit of each transformer winding (Fig. 1, c, d and 2, c). The fundamental aspect of such a comparison is obvious and requires no special proof. A comparison of the TRPSh shown in Fig. 1 with the corresponding combinations in Fig. 2 shows that the TRPSh are original designs that make it possible to compactly combine these combinations in a single apparatus.

A transformer regulated by shunt biasing can always be decomposed into an equivalent “TR—RDN” combination and, conversely, from the parameters of this combination one can construct a TRPSh. The conditions for such transitions are fairly simple. First of all, the magnetic core of the transformer in the TR—RDN combination and the main magnetic core of the TRPSh, as well as the working windings of these apparatuses, must have identical parameters.

The parameters of the adjustable choke in the combination and the parameters of the magnetic core of the shunt of the replaced TRPSh must be related by the equality

\[ \frac{W_{\mathrm{t}}^{2}\cdot S_{\mathrm{sh}}}{l_{\mathrm{sh}}}\cdot \mu_{\sim \mathrm{sh}} = \frac{W_{\mathrm{d}}^{2}\cdot S_{\mathrm{d}}}{l_{\mathrm{d}}}\cdot \mu_{\sim \mathrm{d}}, \]

where \(W_{\mathrm{t}}\) is the number of turns of the TRPSh winding that creates the secondary-leakage flux,

\(S_{\mathrm{sh}}, l_{\mathrm{sh}}, \mu_{\sim \mathrm{sh}}\) are the cross section, length of the line of force, and magnetic permeability of the magnetic core of the shunt in the TRPSh;

\(W_{\mathrm{d}}\) is the number of turns of the working winding of the RDN;

\(S_{\mathrm{d}}, l_{\mathrm{d}}, \mu_{\sim \mathrm{d}}\) are the cross section, length of the line of force, and magnetic permeability of the choke.

This equality is drawn up on the basis of the condition that the reactive impedances of the adjustable saturation chokes in combinations—

Fig. 2.

—in TR–RDN combinations must be equal to the secondary-leakage impedances in the replaced TRPSh.

The parameters of the dc bias windings are calculated according to the law of equality of magnetomotive forces, and if the TRPSh has internal or external current feedback, then the chokes in the equivalent combinations must have these same feedback connections.

The TR–RDN combinations obtained in this way will have practically the same regulating capabilities as the TRPSh equivalent to them. The difference between the former and the latter, if one speaks of regulators actually made according to the conditions given above, will be

be determined only by economic indices—by the consumption of active materials, active losses, and air-leakage losses; that is, with the same nominal power factor and efficiency, the overall dimensions, etc., will be different.

The characteristics given in Fig. 3, a, b, c may serve as confirmation of what has been said.

Here the solid curves represent the external characteristics, obtained experimentally on single-phase TRPSh of three varieties, but having identical parameters, the same power (200 VA), and the same cross-section of the shunts, equal to the cross-section of the main magnetic core. In Fig. 3, a the characteristics are given for a TRPSh with secondary leakage of the primary winding; in Fig. 3, b—for a TRPSh with secondary leakage of the secondary winding; and in Fig. 3, c—for a TRPSh with secondary leakage of both working windings. The dashed curves in these same figures give the experimental external characteristics of TR–RDN combinations equivalent to the TRPSh under test. In order to bring the TR–RDN combinations as close as possible to the TRPSh under test, they were made according to the simplest conditions of mutual reduction:

\[ l_{\mathrm{d}} = l_{\mathrm{sh}},\quad S_{\mathrm{d}} = S_{\mathrm{sh}},\quad W_{\mathrm{d}_1} = W_{\mathrm{t}_1},\quad W_{\mathrm{d}_2} = W_{\mathrm{t}_2},\quad \mu_{\sim\mathrm{d}} = \mu_{\sim\mathrm{sh}} \]

(the grade of steel for all magnetic cores was taken the same—E-42).

Comparison of the solid and dashed curves in the figures shows that the external characteristics of the TRPSh and of the equivalent TR–RDN systems do not differ qualitatively from one another; that is, the regulating capabilities of these regulators are the same. But there is a noticeable quantitative difference in the characteristics, and this is quite natural, since the active resistances and air-leakage resistances of the working windings in the TRPSh and in the TR–RDN systems cannot be the same. The compactness of the TRPSh provides both appreciably lower copper consumption, that is, lower active resistances of the windings, and a sharp reduction of air-leakage losses. In this respect the TRPSh is a better apparatus than TR–RDN combinations.

Transformers regulated by premagnetization of shunts are independent, highly original apparatuses with a number of features peculiar to them, and equality, in the literal sense of the word, cannot be established between them and TR–RDN combinations. Nevertheless, reducing a TRPSh to TR–RDN combinations can be very useful theoretically.

First, this makes it possible conveniently to carry out a preliminary analysis and synthesis of TRPSh parameters.

Second, when there is a calculated and manufactured TRPSh intended for operation in one or another regulating system, then, by conditionally representing it in the form of an equivalent TR–RDN combination, one can, by methods already known in electrical engineering, carry out the calculation, analysis, and synthesis of its static and dynamic characteristics both in open-loop and in closed-loop regulating systems. The conditional TR–RDN combination will be the absolute principal counterpart of the TRPSh if it is assumed that, in it, the windings of the regulated saturable reactors have no active resistance or air-leakage resistance, and that for the transformer windings these parameters have been transferred from the TRPSh windings.

The equivalent circuit and its parameters for such a conditional TR–RDN combination will also be valid for the TRPSh replaced by it.

Third, reducing a TRPSh to an equivalent TR–RDN combination, and conversely—calculating a TRPSh in place of an available such combination—makes it possible, in a convenient and uniquely equivalent manner, to

Fig. 3.

Fig. 3.

compare these two magnetic regulators. This is very important, since before applying the TRPSh in a regulation system, it is necessary to substantiate sufficiently the advantages of this apparatus over possible TR–RDN combinations, which have almost equal regulating capabilities.

Visible labels in the graph: \(U_2(V)\); \(I_2/I_{2н}\); \(I_y=0\); \(I_y=0.3a\); \(I_y=0.5a\); \(I_y=0.75a\); \(I_y=1a\); \(I_y=1.5a\); b).

Fig. 3

Equivalent Circuits of the TRPSh

Earlier, three varieties of TRPSh were indicated. For each of these varieties there is its own equivalent circuit. These circuits are shown in Fig. 4, a, b, c: for a TRPSh with secondary leakage of the primary winding—Fig. 4, a; for a TRPSh with secondary leakage of the secondary winding—Fig. 4, b; and for a TRPSh with secondary leakage of both working windings—Fig. 4, c.

For these equivalent circuits one can write the voltage equations:

\[ \text{a)}\quad \dot U_1=\dot I_1(r_1+jx_{1в})+\dot I_1\cdot Z_{1s}+(\dot I_1+\dot I'_2)\cdot Z_{12}, \]

\[ \dot U'_2=\dot I'_2(r'_2+jx'_{2в})+(\dot I_1+\dot I'_2)\cdot Z_{12}; \]

\[ \text{b)}\quad \dot U_1=\dot I_1(r_1+jx_{1в})+(\dot I_1+\dot I'_2)\cdot Z_{12}, \]

\[ \dot U'_2=\dot I'_2(r'_2+jx'_{2в})+(\dot I_1+\dot I'_2)\cdot Z_{12}+\dot I'_2\cdot Z'_{2s}; \]

\[ \text{c)}\quad \dot U_1=\dot I_1(r_1+jx_{1в})+\dot I_1\cdot Z_{1s}+(\dot I_1+\dot I'_2)\cdot Z_{12}, \]

\[ \dot U'_2=\dot I'_2(r'_2+jx'_{2в})+\dot I'_2\cdot Z'_{2s}+(\dot I_1+\dot I'_2)\cdot Z_{12}. \]

Let us dwell on the parameters of the circuits and equations presented. We note at once that all parameters with subscript 1 will be assigned to the primary circuit, and those with subscript 2—to the secondary circuit. The parameters of the secondary circuit are taken as referred to the primary: \(r_1\) and \(r'_2\) are the active resistances of the working windings.

\(x_{1v}\) and \(x'_{2v}\) are leakage reactances through air. The method for determining these resistances is set forth in [1, 3].

\[ Z_{12}=\frac{r_{12} j x_{12}}{r_{12}+j x_{12}} \]

is the impedance of the magnetizing circuit of the working flux \(\Phi_{12}\).

Strictly speaking, this is not a constant quantity; it depends on the magnetic state of the shunts and, with a redistribution of the total flux between the main magnetic core and the shunt, will change somewhat. But since the flux \(\Phi_{12}\) passes through a section of the magnetic core that is not subjected to biasing by direct current, and the maximum value of this flux cannot exceed the value selected in advance within the knee of the magnetization curve of the main magnetic core (that is, within its practically linear part), the value \(Z_{12}\) may, without any particular error, be regarded as a constant parameter, just as the effective magnetic permeability for the flux \(\Phi_{12}\).

Fig. 4.

Under this assumption, the components of the parameter \(Z_{12}\)—\(r_{12}\) and \(x_{12}\)—are determined as the active and reactive resistances of the steel of the main magnetic core of the TRPSh at the maximum value of the flux \(\Phi_{12}\), i.e., with the shunts completely saturated.

In a TRPSh, for deeper regulation of the secondary no-load voltage, biasing of the main magnetic core by direct current may be provided. In this mode the impedance \(Z_{12}\) cannot be taken as a constant quantity, since it will already be expressed by a complicated variable-nonlinear function. We shall not dwell on this case, since biasing of the main magnetic core by a constant field is as yet practically not used.

Especially important parameters of a TRPSh are the secondary leakage impedances—\(Z_{1s}\) and \(Z'_{2s}\), where

\[ Z_{1s}=\frac{r_{1s}\cdot jx_{1s}}{r_{1s}+jx_{1s}} \quad \text{and} \quad Z'_{2s}=\frac{r'_{2s}\cdot jx'_{2s}}{r'_{2s}+jx'_{2s}} \]

(\(r_s\) and \(x_s\) are the active and reactive resistances of the steel of the shunt magnetic core).

\(Z_{1s}\) and \(Z'_{2s}\), even for a given design and parameters of the TRPSh, are a complex nonlinear-variable function of the load currents and the magnetizing current and, in general cases, do not admit a sufficiently accurate analytical expression.

If the TRPSh is replaced by an equivalent combination of an ordinary transformer with adjustable saturation chokes, then \(Z_{1s}\) and \(Z'_{2s}\) will precisely be the reactive resistances of the chokes in this combination. Consequently, in the equivalent circuits of the TRPSh, the resistances \(Z_{1s}\) and \(Z'_{2s}\) should be characterized as nonlinear-variable resistances of conditional saturation chokes controlled by magnetization, having magnetic-core parameters equal to the parameters of the magnetic core of the TRPSh shunt and numbers of alternating-current turns equal to \(W_1\) for \(Z_{1s}\) and \(W_2\) for \(Z_{2s}\), but having no active resistance and no leakage resistance of these windings. The values of \(Z_{1s}\) and \(Z'_{2s}\) can be determined with the same accuracy with which the reactance of an adjustable saturation choke can be determined. If the simultaneous magnetization curves and the steel-loss curves of the shunt magnetic core are given, then the functions \(Z_{1s}=f(I_1; I'_y)\) and \(Z'_{2s}=\varphi(I'_2; I_y)\) can be expressed graphically.

As a result we see that the equivalent circuits of the TRPSh are simple, T-shaped, but contain nonlinear elements. All the parameters of these circuits can be determined with sufficient accuracy by methods known in electrical engineering, which will make it possible to calculate the characteristics of the TRPSh as well. Calculations of external characteristics carried out by the graphical-analytical method for a number of TRPShs, when the simultaneous magnetization curves of the shunt, taken on manufactured TRPSh designs, were used to determine \(Z_{1s}\) and \(Z'_{2s}\), gave good agreement with experiments.

On the Application of TRPSh

At present, magnetic amplifiers are widely used in automatic-control systems, and their merits are well known. Magnetic amplifiers are often used together with unregulated transformers, which serve to match the voltage at the load with the line voltage. Such combinations of an ordinary transformer with magnetic amplifiers can be replaced, as was shown above, by transformers controlled by magnetization of the shunt. With equal regulating capabilities, the TRPSh, in comparison with TR–RDN combinations, will have a number of important advantages.

First of all, it is very valuable that the TRPSh constitutes a single structural unit. In comparison with TR–RDN combinations, this makes it compact, smaller in overall dimensions, and more convenient in installation. In addition, here the voltage of the working windings—and, in designs similar to Fig. 1, \(v\), \(g\), also the fluxes—is no longer redistributed between two units, which may be regarded as one of the fundamental distinguishing features of the TRPSh.

The fact that the working windings of this apparatus are not placed on different legs, while the shunts are located in the developed window of one of these windings, leads to a sharp reduction of air-leakage losses. These losses are much, sometimes several times, smaller not only in comparison with TRPSh of earlier designs, but also in comparison with TR-RDN combinations.

The compactness of TRPSh designs also reduces the consumption of copper (on average by 15–20%), and in some designs also the consumption of steel, which reduces active losses, i.e., increases efficiency. If the rated load voltage differs only slightly from the mains voltage, then biased autotransformers may be used, which provide an even greater saving of active materials.

Compactness, lower expenditures of active materials, and higher rated efficiency and $\cos\varphi$ already make it possible to regard TRPSh as serious competitors of TR-RDN combinations. It is true that the structural design of a TRPSh is more complex than that of a magnetic amplifier or of an ordinary transformer, and its manufacture is associated with a number of inconveniences, especially when the TRPSh has a small shunt cross section. The use here of magnetically soft materials even for the shunts entails much greater difficulties than for magnetic amplifiers. Therefore, in low-power installations, where the principal requirements are imposed on the sensitivity and regulating capabilities of the regulator, while the economic indicators are not decisive, the use of TRPSh will be ineffective. But in high-power installations, where magnetic regulators can be made of ordinary electrical steels and the economic indicators are among the decisive ones, it would be entirely expedient to use, instead of magnetic amplifiers with matching transformers, a transformer combining them structurally and regulated by biasing the shunt.

Of particular interest are TRPSh with secondary leakage of the primary and secondary working windings (see Fig. 1, c, d), since, other conditions being equal, in comparison with other varieties of these apparatuses it has greater sensitivity. If, in the two other varieties of TRPSh, the control signal controls the leakage reactance of one of the working windings, then here this same signal will simultaneously control the leakage reactance of both windings.

In Fig. 5, for comparison, the external characteristics are given of two regulators of equal power, when the shunt cross section is equal to the cross section of the main magnetic core: 1—the characteristics of a TRPSh with secondary leakage of the primary winding; 2—the characteristics of a TRPSh with secondary leakage of both working windings.

The principal parameters of these regulators were made identical.

Obviously, the most desirable of the characteristics presented will be characteristics 2. To obtain the same characteristics in regulator 1, it would have been necessary either to use higher-quality steel for the shunts or to increase the shunt cross section. These circumstances compel, for TRPSh designs with small leakage, the greatest preference to be given to TRPSh with secondary leakage of both working windings.

Transformers regulated by biasing the shunt may be recommended for a wide range of automatic-control systems. The use of

TRPSh in speed-control systems for DC traction motors of electric-locomotive drives, in systems for regulating the power of electric smelting furnaces, the temperature in salt heating baths, the heating current in copper-wire annealing installations under continuous drawing, the voltage on the electrodes of electrolytic-heating baths, etc.

At the Department of Electrification of Industrial Enterprises of the Tomsk Polytechnic Institute, a number of installations have been carried out in which, for comparison, TRPSh and various combinations of TR-RDN were used as regulators. In most cases here preference had to be given to TRPSh.

Fig. 5.

LITERATURE

1. A. M. Bamdas, V. A. Somov, and O. A. Shmidt. Transformers and stabilizers regulated by shunt biasing. Gosenergoizdat, 1959.

2. V. A. Somov. Elements of the theory of a transformer regulated by shunt biasing. Proceedings of TPI, vol. XIII, issue 2, 1957.

3. A. O. Shmidt, M. P. Shvakov. Calculations of transformers and autotransformers regulated by shunt biasing. Proceedings of TPI, vol. XIII, issue 2, 1957.

4. G. N. Petrov, B. L. Kraiz. Smooth contactless regulation of a transformer under load. Vestnik elektropromyshlennosti, No. 3, 1958.

5. A. A. Bessonov. Electrical circuits with steel. Gosenergoizdat, 1948.

6. G. N. Petrov. Transformers. Energoizdat, 1934.

7. V. I. Sharov. On the theory of operation of a transformer with shunt biasing. Elektrichestvo, No. 11, 1961.

Received by the editors
in June 1962.

1. For simplicity, the biasing windings and the unsplit magnetic cores of the shunts are not shown in the figures (they are depicted by thin lines), as is done in actual designs to make it possible to suppress odd harmonics in the biasing windings. ↩