CALCULATION OF THE GEOMETRY AND PARAMETERS OF HIGH-POWER LOW-VOLTAGE TRPSh

V. P. OBRUSNIK, E. G. ZAVADSKII, V. G. KISELEV

(Presented by the scientific and technical seminar of the Department of Static Converters of the Scientific Research Institute of AEM at TPI)

In industry there exists a whole range of automated installations where it is necessary not only to regulate the voltage at the load, but also to match it and provide galvanic isolation with respect to the line voltage. Such installations include various types of thermal and melting furnaces, heating baths, hardening or electroplating baths, etc. From the standpoint of reliability, simplicity, and technical-and-economic indicators, the most acceptable systems here prove to be those in which the power regulating element is a transformer regulated by magnetization of a magnetic shunt (TRPSh). The merits of these devices have been sufficiently discussed in the technical literature [1, 2, 3, etc.].

The theory for calculating a TRPSh is well presented in [2]. However, in developing high-power regulators with a low-voltage output, a number of features arise in calculating their geometry and parameters, associated with the specifics of the constructive design of the secondary winding.

Below we present theoretical developments that make it possible to design such devices with minimum weight or dimensions, and give some practical recommendations for their structural implementation.

The secondary winding of low-voltage transformers regulated by magnetization, as a rule, should be made external and enclosing the magnetic shunt. This improves its cooling conditions and simplifies the construction of the apparatus. It is usually made in one, less often in two layers, of a copper or aluminum busbar, or cast from these same materials. To avoid large energy losses in current-carrying circuits, the load circuits are also connected by busbars of large cross section, with the contact connections often made with water cooling.

To determine the optimal parameters of the TRPSh and its structural dimensions, we shall use the method set forth in [3]. In this case, we express all TRPSh dimensions, by means of constant coefficients, through the dimensions of the main magnetic core \(a_0, b_0, h_0\) and the base thickness of the coil \(C_k\), and relate the latter to the cross section of the main magnetic core \(S_0\) through variable coefficients \(K_c, K_h\), and \(K_s\), where

\[ K_s=\frac{b_0}{a_0}; \quad a_0=\sqrt{\frac{S_0}{K_s}}; \quad b_0=\sqrt{S_0\cdot K_s}; \quad K_h=\frac{h_0}{S_0}; \quad K_c=\frac{C_k}{\sqrt{S_0}} \tag{1} \]

Expressions for determining the cross-sectional area of the main magnetic core \(S_o\) have the form

\[ S_o = D \sqrt{\frac{P_{2\phi}}{n_k \cdot K_c \cdot K_h}}, \tag{2} \]

where

\[ D = \frac{1}{\sqrt{4.44 \cdot f \cdot B_{mo} \cdot K_{zc} \cdot K_{zk}}}, \]

\[ \begin{aligned} P_{2\phi} &= U_2 \cdot I_{2\mathrm{nom}} &&\text{— power of one phase of the transformer;}\\ B_{mo} &&&\text{— maximum induction of the main magnetic core;}\\ I &&&\text{— average current density in the windings;}\\ K_{zk},\ K_{zc} &&&\text{— average filling factors of the coils and magnetic-core legs with active materials;}\\ n_k &&&\text{— number of coils of the secondary winding of the transformer;}\\ a_o,\ b_o,\ h_o,\ S_o &&&\text{— dimensions and cross section of the main magnetic core;}\\ C_k &&&\text{— base thickness of the coils.} \end{aligned} \]

Substituting \(S_o\) from (2) into the expression

\[ W_2 = \frac{U_{2\max}}{4.44 \cdot f \cdot B_{mo} \cdot S_c \cdot K_{zc}}, \tag{3} \]

we obtain

\[ W_2 = \frac{U_{2\max}} {4.44 \cdot f \cdot B_{mo} \cdot K_{zk} \cdot D \sqrt{\dfrac{U_{2\max} \cdot I_2}{n_k \cdot K_c \cdot K_h}}}. \tag{4} \]

Replacing \(K_c \cdot K_h = L\) and substituting the value of \(D\) into (4), after transformations we obtain

\[ W_2 = D_w \sqrt{\frac{j U_2 \cdot L}{I_2}}, \tag{5} \]

where

\[ D_w = \sqrt{\frac{n_k \cdot K_{zk}}{4.44 \cdot f \cdot B_{mo} \cdot K_{zc}}}. \]

For this type of ferromagnetic device, the value of the maximum induction is determined within the limits of the linear section of the magnetization curve. The filling factors of the steel and coils are taken in accordance with transformer-building practice. Consequently, in (5) the number of turns of the secondary winding will depend on the value of the current density, the load voltage and current, and the coefficient \(L\), which is related to the geometrical dimensions of the device. The voltage and current are usually specified, while the current density can be determined from the expression

\[ j = \sqrt[7]{\frac{M_j^4 \cdot n_k \cdot L}{D_j^2 \cdot P_{2\phi} \cdot K_c^4}}, \tag{6} \]

where

\[ M_j = \frac{\alpha_k \cdot \Delta \tau_k \cdot 10^6} {\rho_k \cdot K_{zk}}; \qquad D_j = \frac{1} {\sqrt{4.44 \cdot f \cdot B_{mo} \cdot K_{zk} \cdot K_{zc}}}, \]

\[ \begin{aligned} \alpha_k &\text{— specific heat-transfer coefficient of the coils;}\\ \rho_k &\text{— specific resistance of the coil material;}\\ \Delta \tau_k &\text{— established permissible temperature rise of the coils.} \end{aligned} \]

To select the value of \(L_{\mathrm{opt}}\) at which the TRPSh will have the lowest weight, it is necessary to construct the dependence of the total dimensionless volume of the apparatus on \(L\), in accordance with [4]. From the curve \(V_\Sigma=f(i)\), which has a clearly pronounced minimum, the region \(L_{\mathrm{opt}}\) is determined, in which the change in the apparatus weight as \(L\) changes remains within reasonable limits. In this case the expression for the total dimensionless volume of the ferromagnetic device as a function of \(L\) has the form [4]:

\[ V_\Sigma= \frac{1.4\,(a_1+pz)\,B} {\sqrt[7]{\left[\frac{2z\,(a_1+pz)}{5(a_2z+a_3L)}\right]^2 (n_k^2\cdot z\cdot L)^3}}, \tag{7} \]

and the curves \(V_\Sigma=f(L)\), for example, for the design shown in Fig. 1, at different values \(z=\dfrac{h_o}{a_o}\), have the form shown in Fig. 2.

Fig. 1

Fig. 2

Having determined the value \(L_{\mathrm{opt}}\), we find the coefficient

\[ K_c=\frac{L_{\mathrm{opt}}}{z}\sqrt{\frac{2z\,(a_1+pz)}{5(a_2z+a_3L)}} , \tag{7} \]

where

\[ a_1=a\cdot m\cdot L+m',\qquad a_2=a\cdot n\cdot L+n',\qquad a_3=aqL+q', \]

\(p\) is a constant coefficient depending on the selected design of the apparatus.

Knowing \(L_{\mathrm{opt}}\) and \(K_c\) and specifying the permissible temperature rise of the apparatus coils, and also taking, in accordance with transformer-design practice, the value of the specific heat-transfer coefficient \(\alpha_k\), one can determine from (6) the average current density in the windings.

In this case all quantities in expression (5) are known, and the optimum number of turns of the transformer secondary winding can be found. It should be noted that, when determining \(W_2\) from (5), a fractional result may be obtained. In this case it is recommended to round the value of \(W_2\) to the nearest integer corresponding to the region \(L_{\mathrm{opt}}\). From expression (3), \(S_o\) is determined and all dimensions of the transformer are found from (1).

In the practice of designing ferromagnetic devices, in addition to the general case considered above, particular cases may also occur, namely:

a) the number of turns of the secondary winding is specified,
b) the current density \(j\) in the windings is specified,
c) the number of turns \(W_2\) and the current density \(j\) in the windings are known.

In these cases the calculations are considerably simplified.

If the number of turns of the transformer secondary winding \(W_2\) is known, then the steel cross section of the main magnetic core \(S_o\) can be determined from the expression:

\[ S_o=\frac{U_{2\max}}{4.44 f\cdot B_{mo}\cdot W_2\cdot K_{\mathrm{zc}}}. \tag{9} \]

Having described the design by the method of [3], we construct the curves \(V_\Sigma=f(L)\) and from them find the region \(L_{\mathrm{opt}}\). This makes it possible to determine the variable coefficients of the apparatus geometry

\[ K_s=\frac{2z(a_1+pz)}{5(a_2z+a_3L)},\quad K_h=\frac{z}{\sqrt{K_s'}},\quad K_c=\frac{L\sqrt{K_s'}}{z} \tag{10} \]

provided that \(z\) is chosen simultaneously with \(L\). The most technological designs are obtained for \(z=5\text{—}8\). From the coefficients \(k_s\), \(k_h\), \(k_c\) and \(S_o\) obtained, we determine all geometrical dimensions of the transformer. To select the copper cross section of the windings, the current density is found from (6).

In the case when the current density in the apparatus windings is specified, the calculation is carried out in the following order.

As in the preceding case, the dependence \(V_\Sigma=f(L)\) is constructed, \(L\) is chosen in the zone \(L_{\mathrm{opt}}\), and the steel cross section of the main magnetic core is found from (2), where \(K_c\cdot K_h=L\) has already been chosen, and then \(W_2\) is found from (3). As in the preceding case, specifying the temperature rise \(\Delta t\) and the specific heat-transfer coefficient of the coils \(\alpha_k\), we find the coefficients

\[ K_c=\sqrt[4]{\frac{M_j^{4}\cdot n_k\cdot L}{D_j^{2}\cdot P_{2\phi}\cdot j^{7}}}. \tag{11} \qquad K_s=\left(\frac{z}{K_h}\right)^2, \]

where

\[ z=5\div 7. \]

Knowing the coefficients \(K_c\), \(K_h\), \(K_s\), from (1) one can determine all geometrical dimensions of the apparatus.

If the turns of the transformer secondary winding and the current density \(j\) are specified, then the cross section of the main magnetic core is found from (3), and from (2) the value \(L\) is found. As in the preceding case, from (11) the value \(K_c\) and all dimensions of the apparatus are determined.

The geometrical dimensions of the shunt are determined, as a rule, in accordance with the equalities

\[ a_{\mathrm{sh}}=a_o,\quad b_{\mathrm{sh}}=\beta b_o;\quad h_{\mathrm{sh}}=h_o, \]

where the value \(\beta = \dfrac{S_{\mathrm{sh}}}{S_{0}}\) depends on the regulation depth, the required minimum short-circuit current, and other specific conditions.

Thus, the proposed calculation method makes it possible to calculate, relatively simply, the geometrical dimensions of a transformer regulated by premagnetization of a magnetic shunt for optimum weight or overall dimensions. Despite its simplicity, it is based on relatively rigorous mathematical relationships describing the electromagnetic and thermal operating conditions of the apparatus.

REFERENCES

1. V. P. Obrusnik, E. G. Zavatskii. Prospects for the application of high-power premagnetized regulators in thermal installations. Proceedings of the VI Conference on Problems of Production Automation, vol. 2, Tomsk, 1969.

2. A. M. Bamdas, S. V. Shapiro. Transformers Regulated by Premagnetization. “Energiya,” 1965.

3. V. P. Obrusnik. A simplified method for calculating the optimal geometry of transformers regulated by premagnetization of shunts. Izvestiya TPI, vol. 132, Tomsk, 1965.

4. V. P. Obrusnik, V. G. Kiselev, Z. N. Soboleva. Analysis of structural variants of single-phase TRPSh and the possibilities of their application in automatic installations. Izvestiya TPI, vol. 243, Tomsk, 1971.