A SIMPLIFIED METHOD FOR CALCULATING THE OPTIMAL GEOMETRY OF TRANSFORMERS REGULATED BY SHUNT BIASING

V. P. OBRUSNIK

(Presented by the scientific and technical seminar of the Department of EPGP of the Electromechanical Faculty)

It is known that the volume and weight of the active materials of electromagnetic apparatus depend not only on the physical quantities that determine the area of the window and of the core of the magnetic circuit, but also on the geometry of the latter. In transformers regulated by shunt biasing (TRSh), the influence of geometry on their volume and weight is quite appreciable, since the designs of these apparatus have an increased number of windings and

Fig. 1.

cores. Below, a simplified method is proposed for determining the ratios of the principal dimensions of an SBT under otherwise equal conditions, ensuring minimum expenditures of active materials for this apparatus.

In the discussion, the following notation is adopted for the principal dimensions of the SBT (Fig. 1):

\(C_n, C_v\) — thickness of the coils of the outer (subscript \(n\)) and inner (subscript \(v\)) operating AC windings;

\(C_d\) — thickness of the coil of the biasing windings;

\(a_0, b_0\) and \(a_{\mathrm{sh}}, b_{\mathrm{sh}}\) — width and thickness of the core of the main magnetic circuit (subscript \(0\)) and of the shunt core (subscript \(\mathrm{sh}\));

\(h_0, h_{\mathrm{sh}}\) — window height of the main magnetic circuit and of the shunt;

\(\delta_0\) — air gap in the window of the main magnetic circuit;

\(\delta\) — air gap between the coils of the operating and biasing windings;

\(S_0 = a_0 b_0\) — cross section of the main magnetic circuit;

\(S_{\mathrm{sh}} = a_{\mathrm{sh}} b_{\mathrm{sh}}\) — cross section of the shunt core.

We shall assume that all dimensions of the SBT can be expressed, by means of constant coefficients, through the dimensions of the main magnetic circuit \(a_0, b_0, h_0\) and the base coil thickness \(c_k\), and that the latter are related to the cross section of the main magnetic circuit \(S_0\) through variable coefficients \(K_c, K_h\), and \(K_s\):

\[ c_k=\frac{c_n}{\kappa_1}=\frac{c_v}{\kappa_2}=\frac{c_d}{\kappa_3}=\frac{\delta}{\kappa_4}, \qquad h_{\mathrm{sh}}=K_5 h_0, \]
\[ a_{\mathrm{sh}}=\kappa_6 a_0,\qquad b_{\mathrm{sh}}=\kappa_7 b_0,\qquad a_{\mathrm{sh}} b_{\mathrm{sh}}=\beta a_0 b_0; \tag{1} \]

\[ a_0=\sqrt{\frac{S_0}{K_s}}, \qquad b_0=\sqrt{S_0 K_s}, \tag{2} \]

where

\[ K_s=\frac{b_0}{a_0}, \]

\[ h_0=K_h\sqrt{S_0}, \qquad c_k=K_c\sqrt{S_0}. \]

In turn, solving the expressions

\[ I_{1\mathrm{nom}}\cdot W_1=c_k h_k n_k \kappa_{\mathrm{zk}} j = S_0 n_k \kappa_{\mathrm{zk}} j K_c K_h, \tag{3} \]

\[ W_1=\frac{U_1\cdot 10^8}{4.44 f_1\cdot B_{m0}\cdot S_0\cdot \kappa_{\mathrm{zc}}}, \tag{4} \]

\[ P_{1\phi}=U_1\cdot I_{1\mathrm{nom}} \tag{5} \]

with respect to \(S_0\), we obtain

\[ S_0=\sqrt{\frac{P_{1\phi}\cdot 10^8} {4.44 f_1 B_{m0}\cdot j\cdot \kappa_{\mathrm{zc}}\cdot \kappa_{\mathrm{zk}}\cdot n_k\cdot K_c\cdot K_h}} = D\cdot \sqrt{\frac{P_{1\phi}}{n_k K_c K_h}}, \tag{6} \]

where

\[ D=\frac{10^4}{\sqrt{4.44 f_1\cdot B_{m0}\cdot j\kappa_{\mathrm{zc}}\cdot \kappa_{\mathrm{zk}}}}, \tag{7} \]

\(h_k, n_k\) — height and number of coils of the primary winding;

\(\kappa_{\mathrm{zk}}, \kappa_{\mathrm{zc}}\) — averaged fill factors of the coils and cores with active materials;

\(j\) — averaged current density of the windings;

\(B_{m0}\) — operating induction of the main magnetic circuit;

\(P_{1\phi}\) — overall power of one phase.

Writing the expressions for the total volumes of the coils and cores of shunt-controlled transformers in general form—for example, for the design in Fig. 1 they will have the form:

\[ V_{\mathrm{k}}=4h_0\left[(a_0+a_{\mathrm{sh}}+b_0+2c_b)c_b+(a_0+a_{\mathrm{sh}}+b_0+2c_d+4c_b+ \right. \]
\[ \left. +2c_{\mathrm{n}}+2\delta)c_{\mathrm{n}}\right]+4h_{\mathrm{sh}}(a_{\mathrm{sh}}+b_{\mathrm{sh}}+2c_d)c_d; \tag{8,a} \]

\[ V_{\mathrm{c}}=2S_0(h_0+2c_{\mathrm{n}}+2c_b+2a_0+\delta_0)+4S_{\mathrm{sh}}(2a_{\mathrm{sh}}+c_d+c_{\mathrm{n}}+h_{\mathrm{sh}}), \tag{8,b} \]

and then transforming them with allowance for (1–2,6), in general form for all designs we obtain

\[ V_{\mathrm{k}}=\sqrt{D^3}\cdot \sqrt[4]{P_{1\phi}^{3}}\; A\sqrt[4]{\frac{K_hK_c}{n_{\mathrm{k}}^3}} \left(\frac{m}{\sqrt{K_s}}+n\sqrt{K_s}+K_c\right); \tag{9,a} \]

\[ V_{\mathrm{c}}=\sqrt{D^3}\cdot \sqrt[4]{P_{1\phi}^{3}}\, \frac{B}{\sqrt[4]{(K_hK_cn_{\mathrm{k}})^3}} \left(\frac{m'}{\sqrt{K_s}}+n'\sqrt{K_s}+q'K_c+pK_h\right), \tag{9,b} \]

where \(A, B, m', m, n, n', q, q', p\) are constant coefficients, whose values are not the same for different designs and depend substantially on the magnitude of the relative shunt cross section \(\beta=S_{\mathrm{sh}}/S_0\).

For example, if for the design in Fig. 1, in an implementation with external current coupling or without it, we take

\[ c_{\mathrm{n}}=c_b=0.8c_d=4\delta=c_{\mathrm{k}},\qquad h_0=h_{\mathrm{sh}}=h_{\mathrm{k}}; \]

\[ a_{\mathrm{sh}}=\beta a_0,\qquad b_{\mathrm{sh}}=b_0,\qquad \delta_0=0, \]

then, after transforming expressions (8,a,b) and reducing them to the form (9,a,b), we obtain

\[ V_{\mathrm{k}}=\sqrt{D^3}\sqrt[4]{P_{1\phi}^{3}}\cdot 4\sqrt[4]{\frac{K_hK_c}{n_{\mathrm{k}}^3}} \left(\frac{2+3.5\beta}{\sqrt{K_s}}+2.5\sqrt{K_s}+16K_c\right), \tag{10,a} \]

\[ V_{\mathrm{c}}=\sqrt{D^3}\sqrt[4]{P_{1\phi}^{3}}\cdot 4\frac{1}{\sqrt[4]{(K_hK_cn_{\mathrm{k}})^3}} \left[ \frac{1+2\beta^2}{\sqrt{K_s}}+ \right. \tag{10,b} \]

\[ \left. +(2+2.25\beta)K_c+\left(\frac{1}{2}+\beta\right)K_h \right]. \]

Whence \(A=4,\quad m=2+3.5\beta,\quad n=2.5,\quad q=16,\)

\[ B=4,\qquad m'=1+2\beta^2,\qquad n'=0,\qquad q'=2+2.25\beta, \]

\[ p=\frac{1}{2}+\beta. \]

In specific calculations the value of \(\beta\) is usually known and lies within the limits \(0.2\div1\).

Expressions (9,a,b) are convenient in that they are readily separated into two variable functions which, in particular solutions, do not depend on each other:

a) the function associated with the physical quantities \(B_{m0}, j, f_1, k_{\mathrm{zc}}, k_{\mathrm{zk}}\), and \(P_{1\phi}\),

\[ M=\sqrt{D^3}\sqrt[4]{P_{1\phi}^{3}}; \tag{11} \]

b) functions associated only with the geometry of the transformer,

\[ v_k=A\sqrt{\frac{K_h K_c}{n_k^3}}\left(\frac{m}{\sqrt{K_s}}+n\sqrt{K_s}+qK_c\right), \tag{12,a} \]

\[ v_c=\frac{B}{\sqrt[4]{(K_hK_c n_k)^3}} \left(\frac{m'}{\sqrt{K_s}}+n'\sqrt{K_s}+qK_c+pK_h\right). \tag{12,b} \]

We shall call the quantity \(v_k\) the dimensionless volume of the windings of the device, and \(v_c\) the dimensionless volume of its cores. For the weight of the windings and cores of an TRPSh one may write

\[ G_k=M\cdot g\,\xi\cdot v_k; \tag{13,a} \]

\[ G_c=M\cdot g\cdot v_c, \tag{13,b} \]

where \(g\) is the weight per unit volume of the cores;
\(\xi\) is a coefficient equal to the ratio of the weight per unit volume of the windings to the weight per unit volume of the cores.

The functions (12, a, b and 13, a, b) make it possible to find the optimal ratios of the geometrical dimensions of an TRPSh by analyzing only the expressions for the dimensionless volumes.

Assuming in particular solutions

\[ M=\sqrt{D^3}\cdot \sqrt[4]{P_{1\phi}^3}=\mathrm{const}, \]

since this function contains variables independent of \(v_k\) and \(v_c\), for the condition of minimum volume or weight we obtain the equalities

\[ \frac{\partial(\xi v_k+v_c)}{\partial K_s}=0, \tag{14} \]

\[ \frac{\partial(\xi v_k+v_c)}{\partial K_c}=0 \tag{15} \]

(for the minimum volume of the device one should take \(\xi=1\)).

Substituting (12,a, 12,b) into (14, 15) and solving the latter with respect to \(K_s\) and \(K_c\), we obtain

\[ K_{s\mathrm{op}}= \frac{ m+m'\dfrac{B}{A\xi K_hK_c} }{ n+n'\dfrac{B}{A\xi K_hK_c} }, \tag{16} \]

\[ K_{\mathrm{op}}^2+\frac{1}{5q} \left( \frac{m+nK_s}{\sqrt{K_s}}+ \frac{Bq'}{A\xi K_{h\mathrm{op}}} \right)K_{\mathrm{op}} -\frac{3B}{5A\xi q} \left( \frac{m'+n'K_s}{\sqrt{K_s}\,K_{h\mathrm{op}}}+p \right)=0. \tag{17} \]

Expression (16) is approximated with sufficient accuracy by the dependence

\[ K_{s\mathrm{op}}^{*}= \frac{ m+m'\dfrac{B}{A\xi} }{ n+n'\dfrac{B}{A\xi} }, \tag{18} \]

since the product \(K_h\cdot K_c\) is practically close to unity, and a change in the value of \(K_s\) within the range from \(K_{s\min}=\dfrac{m}{n}\) to \(1.5K_{s\mathrm{op}}^{*}\) has almost no eff—

affects the volume and weight of the device (\(K_{s\min}\) corresponds to the minimum volume or weight of the coils and is found from the solution of the expression \(\partial v_k/\partial v_c = 0\)).

The weak influence of the shape coefficient of the magnetic-core cross section on the total volume and weight of the device is seen from the curves in Fig. 2, constructed for the TRPSh in Fig. 1 at \(\xi = 1\) (the character of these curves is the same for all TRPSh designs).

Fig. 2. Dependence of \(\xi v_k + v_c\) on \(K_s\). The labels on the curves are:
\(K_h = 4,\ K_c = 0.12,\ \beta = 1\);
\(K_h = 4,\ K_c = 0.25,\ \beta = 1\);
\(K_h = 2,\ K_c = 0.25,\ \beta = 1\);
\(K_h = 4,\ K_c = 0.25,\ \beta = 0.2\).
The abscissa is \(K_s\); marked on it are \(K_s = \dfrac{m}{n}\) and \(K_s = K_{s\mathrm{op}}^{*}\).

A noticeable increase in volume (weight) is observed here for \(K_s < K_{s\min}\), since then both the volume of the coils and the volume of the cores increase; but for \(K_s > K_{s\min}\) the curves have a very flat character, with a weakly pronounced minimum. Within the limits from \(K_{s\min}\) to \((1.25 \div 1.5)K_{s\mathrm{op}}^{*}\), the increase in coil volume is almost compensated by the decrease in core volume. Therefore, as the optimal value of the coefficient \(K_s\), one may take its numerical value calculated by formula (18) and rounded, for one reason or another, to values limited by the bounds

\[ K_{s\min} = \frac{m}{n} \quad \text{and} \quad K_{s\max} = 1.25K_{s\mathrm{op}}^{*}. \]

The coefficients \(K_c\) and \(K_h\) are closely related by expression (17), which, in the region of positive values of \(K_c\), gives curves of hyperbolic character that have no minimum (Fig. 3). Therefore, to determine the optimal values of \(K_c\) and \(K_h\), it is necessary to construct the curve

\[ \xi v_k + v_c = f[K_h(K_c)], \tag{19} \]

where the values of \(K_h\) are taken arbitrarily (practically within the range \(1 \div 7\)), and \(K_c\) is found from expression (17) for each selected \(K_h\) (here the coefficients \(K_s = K_{s\mathrm{op}}\) and \(\xi\) are already considered known). For the TRPSh design in Fig. 1, this curve is presented in Fig. 4 (here \(\xi = 0.5,\ \beta = 1\)).

Function (19) in most cases has a very flat minimum, which gives some freedom in the choice of \(K_{h\mathrm{op}}\). For example, according to Fig. 4 one may take \(K_{h\mathrm{op}} = 2 \div 3\).

After determining the dimensionless coefficients \(K_{s\mathrm{op}},\ K_{h\mathrm{op}},\ K_{c\mathrm{op}}\), it is not difficult to find the optimal ratios of the principal dimensions of the shunt-bias-controlled transformer:

\[ \left. \begin{aligned} \frac{b_0}{a_0} &= K_{s\mathrm{op}}, \qquad \frac{h_0}{a_0} = K_{h\mathrm{op}}\sqrt{K_{s\mathrm{op}}};\\[4pt] \frac{c_0}{a_0} &= 2n_k \varepsilon_\delta K_{c\mathrm{op}}\sqrt{K_{s\mathrm{op}}}, \end{aligned} \right\} \tag{20} \]

Fig. 3.

where \(c_0\) is the window width of the main magnetic core;

\(\varepsilon_\delta\) is a coefficient that takes into account the air gap in the window of the main magnetic core. Depending on the construction of the shunt-bias-controlled transformer, \(\varepsilon_\delta = 1.1 \div 1.25\).

Fig. 4.

The linear dimensions can be obtained by means of expressions \((1 \div 2, 6)\).

Thus, to determine the optimal geometry of a TRPSh, the following scheme may be proposed:

1. The values of the fill factors and the specific weight of the active materials planned for manufacture of the TRPSh being designed are first established, after which the quantity \(\xi\) is calculated. The coefficients relating the thickness of the coils of the secondary and biasing windings to the thickness of the primary winding, taken as the base, are selected, as are the coefficients relating the dimensions of the main magnetic core to the dimensions of the shunt (the value of \(\beta\) must be specified). Check calculations have shown that deviations of the listed quantities by \(\pm 15\%\) from their true values have practically no effect on the results of the subsequent calculations.

2. General expressions are written for the volumes of the coils and cores and, by means of substitutions \((1 \div 2, 6)\), are reduced to the form \((9,a,b)\), from which the values of the coefficients \(A, B, m, n, q, m', n', q', p\) are written out.

3. Within the limits

\[ K_s = \frac{m}{n} \div 1.25 \frac{m + m' \dfrac{B}{A\xi}} {n + n' \dfrac{B}{A\xi}} \]

the value \(K_{\mathrm{sop}}\) is selected.

1. From expressions \((12, a,b; 17)\), curve \((19)\) is constructed, and in the region of its minimum \(K_{hop}\) is found.

2. From expression \((17)\), for the found \(K_{\mathrm{sop}}\) and \(K_{hop}\), \(K_{\mathrm{cop}}\) is calculated.

3. By formulas \((20, 1)\), the required ratios of the TRPSh dimensions are determined.

It should be noted that the proposed method for calculating the optimal geometry does not take into account the influence of the latter on the heat transfer of the apparatus and requires a certain concretization of the problem when choosing the values \(\xi\), \(\beta\), and \(k_1 \div k_7\); therefore, it should be regarded as auxiliary in solving the general problem of calculating the optimal parameters of a TRPSh. However, in a number of specific cases of TRPSh design this method is sufficiently rigorous and quite effective.

Since expressions \((9,a,b; 13,a,b)\) are valid not only for any TRPSh design but also for other electromagnetic apparatuses, the method of calculating the geometry by dimensionless volumes can be checked by calculating ordinary transformers or saturable reactors, the optimal ratios of the principal dimensions of which are considered established. For example, a check calculation of the geometry of a core-type transformer at \(\xi = 0.5\) gave the ratios presented in Table 1.

Table 1

The data of Table 1 fall within the limits of the ratios recommended for core-type transformers in [1, pp. 201–202].

The method of calculating the geometry of a TRPSh by its dimensionless volumes successfully replaces the approximate calculations by the formulas recommended in [2], which, in the notation adopted here, have the form:

\[ \left. \begin{aligned} S_0 &= \frac{3.5 \div 4.2}{\kappa_{\mathrm{sc}}} \sqrt{\frac{P_{1\phi}}{f_1 n_k}}, &\qquad b_0 &= (1.2 \div 1.5)\sqrt{S_0},\\[4pt] a_0 &= \frac{S_0}{b_0}, &\qquad h_0 &= (2 \div 2.8)\sqrt{S_0 n_k}. \end{aligned} \right\} \tag{21} \]

To confirm what has been said, Table 2 is given, in which the calculated data for the principal dimensions and volumes of the active materials of the TRPSh in Fig. 1 are summarized, designed for minimum weight with the following initial data:

\[ P_{1\phi}=500\ \mathrm{VA}, \qquad B_{m0}=15\ \mathrm{kG}, \qquad f_1=50\ \mathrm{Hz}, \]

\[ \kappa_{\mathrm{zk}}=0.36, \qquad \kappa_{\mathrm{sc}}=0.85, \qquad j=200\ \mathrm{A/cm^2}, \]

\[ \xi=0.5, \qquad \beta=1, \qquad c_b=c_{\mathrm{n}}=0.8C_d=48. \]

The tabulated data in column I were calculated by the method described above; in column II, by formulas (21), using expression (6) to determine the coil thickness.

As is seen from Table 2, the dimensions of the TRPSh calculated by the proposed method provide appreciably lower expenditures of active materials than the dimensions found from formulas (21).

Table 2

It is interesting to note that expressions (6, 16), after a certain transformation, can be used in calculations of TRPSh whose magnetic cores are made of standard stamped laminations intended for other devices (the most suitable are Π- and Ш-shaped laminations used in the manufacture of magnetic amplifiers). For known lamination dimensions, for \(K_h\) and \(K_c\) we obtain the formulas:

\[ K_h=\frac{h_0}{a_0\sqrt{K_s}}, \tag{22} \]

\[ K_c=\frac{c_0}{2n_k \varepsilon \delta a_0\sqrt{K_s}}. \tag{23} \]

Substitution of the values \(K_h\) and \(K_c\) from (22), (23) into (6) and (16), with subsequent transformation, gives the expressions

\[ K_s=-\frac{b_0^2 h_0 c_0}{D^2 \cdot P_{1\phi}\cdot 2\varepsilon_\delta}, \tag{24} \]

\[ K_{\mathrm{sop}}^{**}=\frac{m}{n-m'N}, \tag{25} \]

where

\[ N=\frac{B}{A\xi}\cdot \frac{2\varepsilon_\delta\cdot n_k} {\dfrac{c_0}{a_0}\cdot \dfrac{h_0}{a_0}} . \tag{26} \]

When calculating a TRPSh for minimum volume or weight of active materials, it is necessary that the value, obtained from formula (24), of the thickness coefficient of the stack of the core of the main magnetic circuit be equal to \(K_{\mathrm{sop}}^{**}\) from expression (26), or not go beyond the limits imposed by the condition

\[ \frac{m}{n}<k_s\leq K_{\mathrm{sop}}^{**}. \]

One can solve the problem inverse to the preceding one: to find the rated power of a TRPSh for a rational cross section of its main magnetic circuit. Then

\[ P_{1\phi}=\frac{b_0 h_0 c_0}{k_{\mathrm{sop}}^{**}2\varepsilon_\delta\cdot D^2}. \tag{27} \]

Solving the questions of correctly calculating the configuration of TRPSh magnetic circuits, which ensures for these devices, all other conditions being equal, minimum expenditure of active materials, is of great practical importance. First, only with the optimal geometry of TRPSh is it possible to realize their potential capability of providing, compared with equivalent combinations of a transformer with regulated saturation chokes, a savings of active materials of \(10 \div 15\%\). Second, the theoretical substantiation of the optimal ratios of the principal dimensions of TRPSh magnetic circuits will make it possible to manufacture stamped laminations for the latter, which will improve the electromagnetic characteristics and simplify the manufacturing technology of TRPSh. Therefore, a method for determining the optimal geometry of TRPSh by analyzing the dimensionless volumes of coils and cores, combined with other criteria for the optimal parameters of the device, may be very useful for the design of biased transformers.

LITERATURE

1. R. Kh. Bal’yan. Low-Power Transformers. Sudpromgiz, 1961.

2. A. M. Bamdas, S. V. Shapiro, O. D. Getmanenko. Determination of optimal designs of transformers and autotransformers regulated by shunt biasing, and their calculation. Proceedings of GPI, vol. XVIII, issue 1, 1962.