DETERMINATION OF THE PARAMETERS OF TRPSh SHUNTS OPTIMAL FOR A GIVEN OPERATING MODE

V. P. OBRUSNIK

(Presented by the scientific seminar of the Faculty of Electromechanics)

General Statement of the Problem

The semiempirical formulas known in the literature for calculating the cross sections of shunts of transformers regulated by shunt premagnetization (TRPSh) [1, 2, 3, 4] are satisfactory for designing these apparatuses, but they cannot serve as a theoretically justified criterion in determining shunt parameters that are optimal for a TRPSh when it operates in specifically prescribed control systems.

The shunts will be optimal when, on the one hand, they reliably provide the required regulation characteristic of the TRPSh and, on the other hand, do not lead to excessive expenditure of active materials, excessive increases in dimensions and weight, active and reactive losses, and other indicators of the regulator. A quantity linking these two requirements may be the reactive resistance of the secondary (magnetic) leakage of the TRPSh.

In a closed-loop control system, the TRPSh operates on the regulation characteristic \(U_2=f(I_2)\). Its beginning will correspond to the absence of a control signal, and its end to the maximum of this signal. The voltage, current, and load resistance at the beginning and end of the regulation characteristic will be denoted, respectively, by the indices \(A\) and \(B\) \((U_{2A}, I_{2A}, Z_{HA}\) and \(U_{2B}, I_{2B}, Z_{HB})\). These quantities are specified by the preliminary conditions. For example, Fig. 1 gives a regulation characteristic corresponding to the conditions for stabilizing the secondary voltage \(U_2=U_{2\text{nom}}=\text{const}\) when the load resistance changes from \(Z_{H\min}\) to \(Z_{H\max}\). Here \(U_{2A}=U_{2B}=U_{2\text{nom}}\), \(Z_{HA}=Z_{H\max}\), \(Z_{HB}=Z_{H\min}\),

\[ I_{2A}=\frac{U_{2A}}{Z_{HA}}, \qquad I_{2B}=\frac{U_{2B}}{Z_{HB}} . \]

The TRPSh will be able to provide the previously specified regulation characteristic only if, at \(U_1=U_{1\min}\), \(f_1=f_{1\max}\), and with maximum premagnetization of the shunts, it gives at the load \(Z_{HB}\) a voltage and current not lower than \(U_{2B}\) and \(I_{2B}\), while at \(U_1=U_{1\max}\), \(f_1=f_{1\min}\), and with unpremagnetized shunts, the voltage and current at the load will be no greater than \(U_{2A}\) and \(I_{2A}\).

The first condition is fulfilled independently of the shunts by an appropriate calculation of the parameters of the main magnetic core and of the transformer windings.

To satisfy the second condition it is necessary to determine correctly the parameters of the shunt: the grade of steel, the length of the flux path, and the cross section.

The possibility of regulating the secondary voltage in a TRPSh is provided by diverting part of the magnetic flux into the shunt. In equivalent circuits of this regulator, the effect of the shunt is represented in the form of the leakage reactance \(X_s\) [5]. It can always be referred to the primary circuit and expressed through the shunt parameters

Fig. 1.

\[ X_s = 2\pi f_1 \cdot \frac{W_1^2 \cdot S_{\mathrm{sh}}}{l_{\mathrm{sh}}}\cdot \kappa_\mu \cdot \frac{B_{\mathrm{sh}}}{H_{\mathrm{sh}}}\ [\mathrm{ohm}], \tag{1} \]

where \(S_{\mathrm{sh}}, l_{\mathrm{sh}}, B_{\mathrm{sh}}\), and \(H_{\mathrm{sh}}\) are the cross section, flux-path length, induction, and field strength of the shunt;

\(W_1\) is the number of turns of the primary winding of the TRPSh;

\(\kappa_\mu\) is a dimensional coefficient.

From the standpoint of regulation, the shunt will be optimally calculated when, in the unsaturated state and at the load impedance \(Z_{\mathrm{HA}}\), it creates a leakage reactance no greater and no smaller than is sufficient to ensure the beginning of the regulating characteristic at \(U_1 = U_{1\max}\) and \(f_1 = f_{1\min}\).

If the value of such a reactance is denoted by \(X_{sA}\), then, once it is known, using equation (1) one can proceed to determine \(S_{\mathrm{sh}}\) and \(l_{\mathrm{sh}}\), for the selected steel grade, which are optimal for the specified operating mode.

Determination of the quantity \(X_{sA}\).

In [5] it is recommended that all TRPSh be subdivided, according to the nature of the secondary leakage fluxes, into three types.

There as well, for each type of TRPSh, complete equivalent circuits are given. For the calculations of \(X_{sA}\) these circuits, in Figs. 2, 3, 5, are given in simplified form. Here the active losses in the shunt steel are neglected, while leakage losses through air and in the active resistance of the working windings are taken into account by the coefficient \(\kappa\).

\[ \kappa = 1 - \delta \cdot \frac{I_{2A}}{I_{\mathrm{nom}}}. \tag{2} \]

For transformers of 1 kva and higher one may take \(\delta = 0.1 \div 0.15\); for low-power transformers, \(\delta = 0.15 \div 0.20\).

In the subsequent calculations all TRPSh parameters not associated with the shunt parameters are considered known. All quantities of the secondary circuit are taken as referred to the primary through the turns ratio

\[ \kappa_\omega = \frac{W_1}{W_2} \]

and are denoted by a prime.

It should be noted that it is not advisable to make the cross section of the TRPSh shunts larger than the cross section of the main magnetic core, since already at \(S_{\mathrm{sh}} = S_0\) the entire magnetizing flux, with the secondary circuit closed, can pass through the shunt, and a further increase of \(S_{\mathrm{sh}}\) has little effect on \(U'_2\) and \(I'_2\). From the equivalent circuits of the TRPSh it is evident that

\[ \frac{S_{\mathrm{sh}}}{S_0} \gg 1 \]

will occur (approximately) when \(I'_{2A} \to 0\) and \(U'_{2A} < 0.5\,U_{1\max}\), or when \(U'_{2A} \to 0\) and \(I'_{2A} < I_{10}\), where \(I_{10}\) is the magnetizing current of the TRPSh in the no-load mode with the shunts biased to full saturation. Therefore, for such values of \(U'_{2A}\) and \(I'_{2A}\), calculation of the shunt by \(X_{SA}\) is excluded. It is better to choose \(S_{\mathrm{sh}} = (0.8 \div 1)S_0\) and, if necessary, to apply additional biasing of the main magnetic core.

Fig. 2.

a) \(X_{SA}\) for a TRPSh with secondary leakage of the secondary winding

The equivalent circuit in Fig. 2 makes it possible, in order to determine \(X_{SA}\), to write expression (3)

\[ X_{SA} = X'_{2SA} = \frac{ \sqrt{(U_{1\max}\cdot \kappa)^2 - (U'_{2A}\cdot \cos \varphi_{HA})^2} - U'_{2A}\cdot \sin(\pm \varphi_{HA}) }{ I'_{2A} }. \tag{3} \]

The minus sign before the load angle \(\varphi_{HA}\) is used when its character is capacitive.

A TRPSh with secondary leakage of the secondary winding is not regulated at no load; therefore determination of the value \(X_{SA}\) here will be meaningful when \(I_{2A} \ne 0\).

b) \(X_{SA}\) for a TRPSh with secondary leakage of the primary winding

For the circuit in Fig. 3, vector diagrams are given for inductive (Fig. 4b) and capacitive (Fig. 4c) load characteristics. From the triangles \(cov\) one can write the general equation for determining \(X_{SA}\)

\[ U_{1\max}^{2}\kappa^{2} = \left[ I_{1A}\cdot X_{SA} + U'_{2A}\cdot \sin(\pm \varphi_{HA} \pm \alpha) \right]^2 + \left[ U'_{2A}\cdot \cos(\pm \varphi_{HA} \pm \alpha) \right]^2 . \tag{4} \]

Here the angle \(\alpha\) is always obtained as positive, since the active component of the magnetizing current is many times smaller than the inductive component.

After transformations of expression (4), we obtain

\[ X_{S A}=X_{1S A}=\frac{1}{I_{1A}}\sqrt{(U'_{2A})^2[\sin^2(\alpha\pm\varphi_{HA})-1]+(U_{1\max}\cdot k)^2} \]

\[ -\frac{U'_{2A}\cdot \sin(\alpha\pm\varphi_{HA})}{I_{1A}} . \tag{5} \]

The “plus” sign has been chosen before the radical, since the sought resistance is inductive in character. In the right-hand side of equation (5), the unknowns are \(I_{1A}\) and \(\alpha\). The methods for determining these quantities follow from the vector diagrams:

Fig. 3.

Fig. 4.

\[ I_{1A}=\sqrt{(I'_{2A})^2+2I'_{2A}\cdot I_0\cdot \cos\psi+I_0^2}, \tag{6} \]

\[ \psi=90-(\varphi_0\pm\varphi_{HA}), \tag{7} \]

\[ \varphi_0=\operatorname{Ark\ tg}\frac{I_{a0}}{I_{\mu0}}, \tag{8} \]

\[ I_0=\sqrt{I_{a0}^{2}+I_{\mu0}^{2}}, \tag{9} \]

\[ I_{a0}=\frac{P_{\mathrm{st.}\,0}}{U_{2A}^{\prime}}, \tag{10} \]

\[ I_{\mu0}=\frac{H_0 l_0}{W_1}, \tag{11} \]

\[ P_{\mathrm{st.}\,0}=G_{\mathrm{st.}\,0}\cdot p_{10}\cdot \left(\frac{B_{m0}}{10^{4}}\right)^{2.1}\cdot \left(\frac{f_{1\max}}{50}\right)^{1.4}, \tag{12} \]

where \(P_{\mathrm{st.}\,0}\) is the active loss in the steel of the main magnetic core;

\(p_{10}\) is the specific loss in the steel;

\(G_{\mathrm{st.}\,0}, B_{m0}, H_0, l_0\) are the weight, induction, field strength, and length of the line of force of the main magnetic core;

\(H_0\) is found from the magnetization curve \(B=f(H)\) from \(B_{m0}\) and is reduced to the rms value in A/cm.

\[ B_{m0}=\frac{U_{2A}^{\prime}\cdot 10^{8}}{4.44 f_{1\max}\cdot W_1\cdot S_0}\;[\mathrm{Gs}] \quad \text{— amplitude value.} \tag{13} \]

With a high-quality magnetic core, the losses in the steel may be neglected and it may be assumed that

\[ I_{a0}=0,\qquad \varphi_0=90^\circ,\qquad I_0=I_{\mu0}. \]

From the triangle formed by the vectors \(I_0\), \(I_{2A}^{\prime}\), and \(I_{1A}\), one can determine the angle \(\alpha\)

\[ \tg \frac{\alpha}{2}=\frac{r}{p-I_0}, \tag{14} \]

where

\[ p=\frac{1}{2}(I_{1A}+I_{2A}^{\prime}+I_0) \]

is the semiperimeter of the triangle;

\[ r=\sqrt{\frac{(p-I_0)(p-I_{2A}^{\prime})(p-I_{1A})}{p}} \]

is the radius of the inscribed circle.

The calculation of the quantity \(X_{SA}\) is considerably simplified when \(I_0 \ll I_{2A}^{\prime}\) (since then one may take \(I_0=0\), \(\alpha=0\), \(I_{1A}=I_{2A}^{\prime}\)), and in the no-load mode, when \(I_{2A}^{\prime}=0\).

c) \(X_{SA}\) for a TRPSh with secondary leakage of both working windings

For a TRPSh of this type, always \(X_{1s}=X_{2s}^{\prime}\). Indeed,

\[ X_{1s}=2\pi f_1\cdot \frac{W_1^{2}\cdot S_{\mathrm{sh}}}{l_{\mathrm{sh}}}\cdot k_\mu\cdot \frac{B_{\mathrm{sh}}\cdot l_{\mathrm{sh}}}{I_1W_1+I_2W_2} = \]

\[ =2\pi f_1\cdot W_1\cdot S_{\mathrm{sh}}\cdot \frac{B_{\mathrm{sh}}}{I_1+j_2^{\prime}}, \tag{15} \]

\[ X_{2s}^{\prime}=2\pi f_1\cdot \frac{W_2^{2}\cdot S_{\mathrm{sh}}}{l_{\mathrm{sh}}}\cdot k_\mu\cdot \frac{B_{\mathrm{sh}}\cdot l_{\mathrm{sh}}}{I_1W_1+I_2W_2}\cdot \left(\frac{W_1}{W_2}\right)^2 = \]

\[ =2\pi f_1\cdot W_1\cdot S_{\mathrm{sh}}\cdot \frac{B_{\mathrm{sh}}}{I_1+j_2^{\prime}}. \tag{16} \]

Consequently, \(X_{1s}=X_{2s}^{\prime}\).

For the equivalent circuit shown in Fig. 5, when the values \(I_{2A}^{\prime}\), \(U_{2A}^{\prime}\), \(\cos\varphi_{HA}\), \(U_1\) are specified and \(X_{SA}\) is unknown, the vector diagrams can be constructed only by the method of successive approximations.

To explain the analytical expressions given below, these diagrams are shown in Fig. 6: b, a—for an inductive load, and b, б—for a capacitive load.

If they are arranged in a rectangular coordinate system in such a way that the current vector \(I'_{2A}\) coincides with the \(y\)-axis, then for the point of intersection of the vectors \(U_{1\max}\) and \(I_{1A}\cdot X_{1SA}\), denoted by \(c\), one may write

\[ x_c^2+y_c^2=U_{1\max}^2\kappa^2, \tag{17} \]

\[ x_c=U'_{2A}\cdot\sin(\pm\varphi_{HA})+I'_{2A}\cdot X'_{2SA}+I_{1A}\cdot X_{1SA}\cdot\cos\alpha, \tag{18} \]

\[ y_c=U'_{2A}\cdot\cos(\pm\varphi_{HA})+I_{1A}\cdot X_{1SA}\cdot\sin\alpha. \tag{19} \]

Here \(\alpha\) is regarded as positive; otherwise two signs—“plus” and “minus”—would have to be placed before it.

Solving these equations, assuming \(X_{1SA}=X'_{2SA}=X_{SA}\), gives

Figure 5.

Figure 6.

\[ X_{SA}=\frac{1}{I_{\Sigma}}\cdot \left[ \sqrt{(B^2-1)(U'_{2A})^2+U_{1\max}^{2}\cdot\kappa^2} -U'_{2A}\cdot B \right], \tag{20} \]

where

\[ I_{\Sigma}=I_{1A}+I'_{2A} =\sqrt{I_{1A}^{2}+(I'_{2A})^{2}+2I_{1A}\cdot I'_{2A}\cdot\cos\alpha}, \]

\[ B=\frac{1}{I_{\Sigma}} \left[ I_{1A}\cdot\sin(\alpha\pm\varphi_{HA}) +I'_{2A}\cdot\sin(\pm\varphi_{HA}) \right]. \]

However, it is not possible to determine the value \(X_{SA}\) from expression (20) by a single solution, since \(I_{1A}\) and \(\alpha\) are functions of \(X_{SA}\):

\[ \psi = (90^\circ - \varphi_0 - \operatorname{Arctg}\varepsilon), \tag{21} \]

\[ \operatorname{tg}\varepsilon = \frac{U'_{2A}\cdot \sin(\pm\varphi_{HA}) + I'_{2A}\cdot X_{SA}} {U'_{2A}\cdot \cos(\pm\varphi_{HA})}. \tag{22} \]

\[ E'_{2A} = \sqrt{(U'_{2A})^2 + 2U'_{2A}\cdot I'_{2A}\cdot X_{SA}\cdot \sin(\pm\varphi_{HA}) + X_{SA}^{2}\cdot (I'_{2A})^{2}}, \tag{23} \]

\[ B_{m0} = \frac{E'_{2A}\cdot 10^8}{4.44\cdot f_{1\min}\cdot W_1\cdot S_0}, \qquad I_{a0}=\frac{P_{\mathrm{st}.0}}{E'_{2A}} \]

and so on; see expressions (6–14) in the preceding section. From expression (20) the required \(X_{SA}\) can be found by the method of successive approximations (assigning values of \(X_{SA}\), calculate the right-hand side of the expression until equality is obtained).

This solution can be accelerated if one writes

\[ \frac{1}{I_\Sigma} \left[ \sqrt{(B^2-1)(U'_{2A})^2 + U^2_{1\max}\cdot k^2} - U'_{2A}\cdot B \right] - X_{SA} = \Theta(X_{SA}) \tag{24} \]

and constructs this function in the coordinates \(X_{SA}\) and \(\Theta(X_{SA})\). Its intersection with the \(X_{SA}\) axis gives the required value of this resistance. The construction of the function \(\Theta(X_{SA})\) may be started from the largest value \(X_{SA}=X_{(SA)1}\), obtained for \(\alpha=0\),

\[ X_{(SA)1} = \frac{1}{2I'_{2A}} \left[ \sqrt{[\sin^2(\pm\varphi_{HA})-1](U'_{2A})^2 + U^2_{1\max}\cdot k^2} - U'_{2A}\cdot \sin(\pm\varphi_{HA}) \right], \tag{25} \]

then assigning a series of values of \(X_{SA}\) smaller than \(X_{(SA)1}\).

As can be seen, the calculations are rather cumbersome, but in particular cases equation (20) is considerably simplified:

a) In the no-load mode, or in a mode close to it, when \(I'_{2A}=0\), \(U'_{2A}=E'_{2A}\), and \(Z'_{HA}=\infty\), one may take \(\alpha=90^\circ\), and \(X_{SA}\) is equal to

\[ X_{SA}=\frac{U_{1\max}-U'_{2A}}{I_0}. \tag{26} \]

b) In the short-circuit mode, or in a mode close to it, when \(U'_{2A}=0\), \(I'_{2A}\gg I_0\), \(Z'_{HA}=0\), the quantity \(I_0\) may be neglected and one may take \(I_{1A}=I'_{2A}\), \(\alpha=0\).

Then

\[ X_{SA}=\frac{U_{1\max}\cdot k}{2I'_{2A}}. \tag{27} \]

c) In the load mode, when \(U'_{2A}\ne 0\) and \(I'_{2A}\gg I_0\), one may take \(\alpha=0\) and use formula (25) to determine \(X_{SA}\).

As can be seen, equations (20) and (25) have to be used only in cases when the specified value \(I'_{2A}\), with \(U'_{2A}\ne0\), proves to be greater than \(I_0\) and commensurate with it.

Determination of the shunt parameters from a known \(X_{SA}\)

In equation (1), after \(X_{SA}\) has been determined, three unknown quantities remain: \(S_{\mathrm{sh}}\), \(l_{\mathrm{sh}}\), and \(\mu_{\mathrm{sh}}=\dfrac{B_{\mathrm{sh}A}}{H_{\mathrm{sh}A}}\). The solution of this equation is still many-

significantly, because one and the same value of the shunt leakage resistance can be provided with the most varied combinations of the length of the line of force, the cross section, and the grade of steel. A specific variant of the steel grade, cross section, and length of the line of force embodied in the specific geometrical dimensions of the shunt can be chosen, for example, by seeking to ensure the minimum weight or the minimum overall dimensions of the TRPSh. The literature already considers questions concerning optimal types of magnetic cores and optimal designs of TRPSh [4]. We shall assume that, for the operating mode specified to us, the TRPSh has been chosen in terms of design and that the parameters of its main magnetic core (geometrical dimensions, steel grade, method of lamination, etc.) are known. Such a formulation of the problem makes it possible to determine part of the shunt dimensions from the dimensions of the main magnetic core.

The TRPSh designs known in the literature allow one to assume that the shunt dimensions are most favorable when the height of its limbs is equal to the height of the limbs of the main magnetic core, and the thickness and width of the packet do not exceed the thickness and width of the limbs of the main magnetic core. If this requirement is observed, then the width of the shunt window may also be regarded as known.

\[ b=\frac{\sum W_i\cdot S_i\cdot \kappa_{zi}}{h-c}, \tag{28} \]

where \(W_i,\ S_i,\ \kappa_{zi}\) are the number of turns, the cross section, and the filling factors of the windings located in the shunt window;

\[ h \]
is the total height of the shunt limb;

\[ c \]
is the width of the yoke.

Now the problem of determining the dimensions of the shunts reduces to determining the width and thickness of their limbs. For this it is necessary, first of all, to determine the grade of steel of the shunts. We shall assume that, for one consideration or another, it too has already been determined. Let us merely recall that the greater the saturation induction and the magnetic permeability in the shunt, the smaller the overall dimensions of the shunt and the higher the amplification factor of the TRPSh as a regulator. This must be taken into account when choosing the steel.

With this specification of the problem, the remaining unknowns—the width and thickness of the shunt limbs—are no longer difficult to determine. We shall show this by an example for the designs presented in Fig. 7a, b.

We shall consider all quantities denoted in the figures, except \(a\) and \(d\), to be known. Moreover, for economic and design considerations it is necessary that \(d\) be no greater than \(d_0\), and that \(a\) be no less than \(0.4d\) and no greater than \(a_0\).

Let us express the length of the line of force and the cross section of the shunt through its dimensions:

\[ l_{\text{sh}}=2(h-c+b+2a), \tag{29} \]

\[ S_{\text{sh}}=d\cdot a\cdot \kappa_{\text{st}}. \tag{30} \]

where \(\kappa_{\text{st}}\) is the coefficient of filling of the cross section with steel.

Substituting these values into equation (1) and expressing the field strength through ampere-turns, we obtain:

a) for the design in Fig. 7a

\[ X_{SA}=2\pi f_{1\min}\cdot W_1\cdot a\cdot d\cdot \kappa_{\text{st}}\cdot \kappa_{\mu}\cdot \frac{B_{\text{sh}A}}{I_{1A}+I'_{2A}}, \tag{31, а} \]

\[ H_{\text{sh}A}=\frac{(I_{1A}+I'_{2A})\cdot W_1}{2(h-c+b+2a)}. \tag{31, б} \]

Fig. 7.

b) for the construction in Fig. 7, b, when the shunts are embraced by the primary winding,

\[ X_{SA}=2\pi f_{1\min}\cdot W_1\cdot a\cdot d\cdot \kappa_{\text{st.}}\cdot \kappa_\mu\cdot \frac{B_{\text{sh}A}}{I_{1A}}, \tag{32,a} \]

\[ H_{\text{sh}A}=\frac{I_{1A}\cdot W_1}{2(h-c+b+2a)}. \tag{32,b} \]

c) for the construction in Fig. 7, b, when the shunts are embraced by the secondary winding,

\[ X_{SA}=2\pi f_{1\max}\cdot W_1\cdot a\cdot d\cdot \kappa_{\text{st.}}\kappa_\mu\cdot \frac{B_{\text{sh}A}}{I'_{2A}}, \tag{33,a} \]

\[ H_{\text{sh}A}=\frac{I'_{2A}\cdot W_1}{2(h-c+b+2a)}. \tag{33,b} \]

Everywhere \(B_{\text{sh}A}\) is found from the magnetization curve of the shunt through \(H_{\text{sh}A}\). The value of the coefficient \(\kappa_\mu\) is taken depending on the dimensions of the magnetization curve.

The value \(a\) from (31—33) may be determined either by the method of successive approximations, assigning a number of values of \(a\) until equality is obtained, or by an ordinary solution if the curve \(B_{\text{sh}}=f(H_{\text{sh}})\) is approximated by a known function. In the present case it is simpler to follow the first path. In determining \(a\), the value \(d\) must initially be taken equal to \(d_0\). If the found \(a\) turns out to be less than \(0.4d_0\), it is better to repeat the calculation, assigning a new value \(d<d_0\) (for \(a<0.4d\) an unfavorable shunt configuration is obtained).

The cross section of the shunt is practically inexpedient when it is greater than \((0.75\div 1)S_0\). Therefore, if such a result has appeared in correct calculations, it is better for the shunt to take a higher-grade steel or to apply additional magnetization of the main magnetic core. It would be possible to increase the window width \(b\); however, this is undesirable, since the larger \(b\) is, the greater is the overhang of the working winding, which increases leakage through air, the consumption of copper, etc.

Sometimes the lamination die for the shunt magnetic core can be selected ready-made (for example, for the constructions in Fig. 7,b and others). Then, of all its parameters, only the cross section \(S_{\text{sh}}=d\cdot a\cdot \kappa_{\text{st.}}\) remains unknown, which is found from equations (31—33) by a single solution, and from it the desired values of \(a\) and \(d\) are selected.

Formulas (31—33) and (3, 5, 20) can also be used for selecting a TRPSh in serial production of these apparatuses.

Use of the method for comparing TRPSh of three varieties

Let TRPSh of three varieties be given, with the same output power, operating in one mode, and having identical parameters of the main magnetic core. We shall determine whether, in this case, they will differ in their shunts. For simplicity of comparison we shall assume that the shunts in these TRPSh are made of one steel and have the same length of the line of force; that is, we shall carry out the comparison by cross section. Substituting into equation (1) the values of \(X_{SA}\) from (3, 5, 20) and expressing the field strength through ampere-turns, after transformations one may write:

a) for a TRPSh with secondary leakage of the primary winding,

\[ \sqrt{(U'_{2A})^2\cdot[\sin^2(\alpha\pm\varphi_{HA})-1]+U_{1\max}^2\kappa^2} -U'_{2A}\cdot \sin(\alpha\pm\varphi_{HA})= \]

\[ =2\pi f_{1\min}\cdot W_1\cdot B_{\text{sh}A}\cdot S_{\text{sh}}, \tag{34,a} \]

b) for a TRPSh with secondary leakage of the secondary winding

\[ \sqrt{U_{1\max}^{2} k^{2} - \left(U'_{2A}\cdot \cos \varphi_{HA}\right)^{2}} - U'_{2A}\cdot \sin(\pm \varphi_{HA}) = \]

\[ = 2\pi f_{1\min}\cdot W_{1}\cdot B_{\mathrm{sh}A_{2}}\cdot S_{\mathrm{sh}_{2}}. \tag{34,b} \]

c) for a TRPSh with secondary leakage of both working windings

\[ \sqrt{(U'_{2A})^{2}(B^{2}-1)+U_{1\max}^{2}k^{2}} - U'_{2A}\cdot B = 2\pi f_{1\min}\cdot W_{1}\cdot B_{\mathrm{sh}A_{3}}\cdot S_{\mathrm{sh}_{3}}. \tag{34,c} \]

Let us now analyze these equalities. If the magnetic cores of the TRPSh were made of steel having a rectangular magnetization curve, then in equalities \((34, a, b, c)\) \(B_{\mathrm{sh}A_{1}} = B_{\mathrm{sh}A_{2}} = B_{\mathrm{sh}A_{3}} = B_s\) and \(a=0\). We obtain

\[ S_{\mathrm{sh}_{1}}=S_{\mathrm{sh}_{2}}=S_{\mathrm{sh}_{3}} = \frac{ \sqrt{U_{1\max}^{2}\cdot k^{2}-\left(U'_{2A}\cdot \cos \varphi_{AH}\right)^{2}} - U'_{2A}\cdot \sin(\pm \varphi_{HA}) }{ 2\pi\cdot f_{1\min}\cdot W_{1}\cdot B_s }. \tag{35} \]

Consequently, when the TRPSh is made of high-quality steel having high magnetic permeability and sharply expressed saturation, the possibilities for all varieties of these devices in power-regulation modes are practically identical.

But if ordinary hot-rolled electrical steels are used (E-42, E-41, etc.), whose magnetization curve differs substantially from a rectangular shape, then under the comparison conditions indicated above the cross sections of the TRPSh shunts will not be identical. For example, at \(I'_{2A}\) close to \(I'_{2\mathrm{nom}}\), \(S_{\mathrm{sh}_{1}}\approx S_{\mathrm{sh}_{2}}\approx (1.10 \div 1.20)S_{\mathrm{sh}_{3}}\), although for the no-load mode \(S_{\mathrm{sh}_{1}}=S_{\mathrm{sh}_{3}}\).

The greatest difference in cross sections will occur in modes where \(I'_{2A}\) is commensurate with \(I_0\). Here the smallest will be \(S_{\mathrm{sh}_{3}}\) and the largest \(S_{\mathrm{sh}_{2}}\). The difference is the greater, the larger the angle and the steeper the rise of the saturated section of the magnetization curve. However, at \(I'_{2A}\gg I_{10}\), even for steels E-42 and E-41 this difference does not exceed \(25\%\).

Among all varieties of TRPSh, special mention should be made of the transformer with magnetic leakage of both working windings, since it possesses not only the properties of voltage regulation at no load, but also, in comparison with the other varieties of TRPSh, under otherwise equal conditions has a greater steepness of the falling section of the external characteristics, which should be regarded as one of its valuable properties [5].

On the accuracy of the method

If the error of calculating shunts by the method proposed here is measured by the deviation of the voltage actually obtained under load from the calculated voltage at a given \(I'_{2A}\), that is,

\[ \Delta = \frac{U_{2A\,\mathrm{fact}} - U_{2A\,\mathrm{set}}}{U_{2A\,\mathrm{set}}}\cdot 100\%. \]

then this error can sometimes reach \(15\%\). The greatest error is obtained when calculating the shunt parameters from the condition of ensuring a specified minimum no-load voltage. The accuracy of the calculations is mainly determined by the extent to which the magnetization curves of the steel used for the calculations differ from the real magnetization curves of the manufactured TRPSh, since the method does not take into account the influence of air gaps, the quality of assembly, the configuration

of the magnetic core, the method of lamination, and other factors, and this may substantially affect the accuracy of the calculations. Therefore, one must use magnetization curves obtained on magnetic cores which, in shape and lamination, are close to the magnetic cores of the selected TRPSh design. To a much lesser extent, the accuracy is affected by an inaccurate allowance for the active resistance of the working windings and for leakage losses.

The method was tested by check calculations of the cross sections of shunts made of TRPSh, using magnetization curves and voltage-loss coefficients \(k\) obtained experimentally on these devices. The error in this case was insignificant and was determined mainly by an inaccurate allowance for the actual filling of steel in the core.

In principle, the method is suitable for calculating the shunts of both single-phase and three-phase TRPSh of various designs.

REFERENCES

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

2. A. M. Bamdas, V. A. Suchkov. Transformer regulated by biasing of a shunt for rectifier installations with voltage cut-off. Transactions of the Gorky Polytechnic Institute, vol. XVII, issue 2, 1961.

3. A. M. Bamdas, M. A. Arkelov, S. V. Shapiro. Application of TRPSh for a single-phase rectifier-stabilizing installation with a storage battery. Transactions of the Gorky Polytechnic Institute, vol. XVII, issue 2, 1961.

4. A. M. Bamdas, S. V. Shapiro, O. D. Getmanenko. Determination of optimal designs of transformers and autotransformers regulated by biasing, and their calculation. Transactions of the Gorky Polytechnic Institute, vol. XVIII, issue 1, 1962.

5. A. I. Zaitsev, V. P. Obrusnik. Questions of the theory and application of transformers regulated by biasing of shunts. Izvestiya TPI, vol. 117, 1963.

Received by the editors
in June 1962.