The Basis Section Method for Calculating the Geometry of Magnetically Biased Transformers
Abstract Generated abstract
The paper proposes a “basic section” method for calculating the geometry of magnetically biased transformers and related ferromagnetic devices. It expresses all principal dimensions through the main core cross section and a small set of dimensionless coefficients, then derives formulas for coil and core volumes, cooling surfaces, and economic indicators under different design criteria such as minimum volume, cost, or weight. The method reduces optimization to determining coefficients such as winding thickness, core height, and core aspect ratio from simple analytic relations and curves of the geometry index versus a selected parameter. The authors argue that this approach simplifies analysis and synthesis for varied biased transformer designs, supports manufacturable dimensional choices within a broad optimum region, and requires less computation than existing variational methods.
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BASIC SECTION METHOD FOR CALCULATING THE GEOMETRY OF BIASED TRANSFORMERS
V. P. OBRUSNIK, V. G. KISELEV, G. A. KHAIDUROVA
(Presented by the scientific-technical seminar of the Scientific Research Institute of Electromechanics.)
A method is proposed for calculating geometry, tested in the design of many constructions of conventional transformers, magnetic amplifiers, and transformers with biasing. In principle, it is also suitable for calculating other ferromagnetic devices (FD), for example, frequency and phase-number converters, etc. The known methods, which have given practically exhaustive answers to questions concerning the geometry of known FDs [2, 3, etc.], prove to be cumbersome and sometimes of little acceptability when new design variants of FDs of one type or another appear. In particular, difficulties arise in the study
Fig. 1
of the geometry of transformers with biasing, the great variety of whose design variants (several tens) requires, when variational methods are used, a large amount of computational work and turns the problem of analysis and synthesis based on calculation results into a serious problem.
The authors propose expressing all dimensions of the TRPSh by means of constant coefficients through the dimensions of the main magnetic core \(a_0, b_0, h_0\) (Fig. 1) and the base thickness of the coils \(C_k\), and then relating
with the cross section of the main magnetic core \(S_o\) through the variable coefficients \(K_c, K_h, K_b\), and \(K_s\) [1]:
\[ \left. \begin{gathered} a_{\text{sh}}\cdot b_{\text{sh}}=\beta\cdot b_o\cdot a_o,\quad a_{\text{sh}}=K_1 a_o,\quad b_{\text{sh}}=K_2 b_o,\quad h_{\text{sh}}=K_3 h_o,\\ C_k=\frac{C_h}{K_4}=\frac{C_b}{K_5}=\frac{C_d}{K_6}=\frac{\delta}{K_7}. \end{gathered} \right\} \tag{1} \]
\[ \left. \begin{gathered} a_o=\frac{\sqrt{S_o}}{K_b},\quad b_o=K_b\cdot\sqrt{S_o},\quad \text{where } K_b=\sqrt{\frac{b_o}{a_o}},\quad h_o=K_h\sqrt{S_o},\\ C_k=K_c\sqrt{S_o},\quad K_s=\frac{h_o\cdot C_k}{S_o}=K_c\cdot K_h,\quad S_{\text{sh}}=\beta S_o \end{gathered} \right\} \tag{2} \]
Here the values of the coefficients \(K_1, K_2, K_3, K_7\) are established depending on the design features of the TRPSh. The values of the coefficients \(K_4, K_5, K_6\) are found from the ratios between the ampere-turns of the individual windings and the ratios of the current densities in them, as well as taking into account the electrical circuit for connecting the windings. Recommendations for determining the listed coefficients are well known [1, 2, 3].
Expressing in the usual way the volumes and cooling surfaces of the cores and coils by means of linear dimensions and using relations (1), (2), one can obtain equations of the form
\[ V_k=\sqrt{S_o^3}\cdot K_c\cdot K_h A\left(\frac{m}{K_b}+nK_b+qK_c\right), \tag{3} \]
\[ \Pi_k=S_o\cdot A(K_h+K_c)2\varphi_{pk}\cdot\left(\frac{m}{K_b}+nK_b+qK_c\right), \tag{4} \]
\[ V_c=\sqrt{S_o^3}\cdot B\left(\frac{m'}{K_b}+n'K_b+q'K_c+pK_h\right), \tag{5} \]
\[ \Pi_c=S_o'2\varphi_{pc}B\left(\frac{m'}{K_b}+n'K_b+q'K_c+pK_h\right). \tag{6} \]
Here \(S_o\) is the basic cross section of the magnetic core,
\[ S_o=D_o^2\cdot\frac{1}{\sqrt{j\cdot B_m}}\cdot\frac{1}{\sqrt{K_c\cdot K_n}} =D_o^2\cdot\frac{1}{\sqrt{j\cdot B_m}}\cdot\frac{1}{\sqrt{K_s}}, \tag{7} \]
where
\[ D_o=\sqrt[4]{\frac{P_n\cdot k_p}{m\cdot4\cdot K_f\cdot K_{zc}\cdot K_{zk}\cdot n_k}}, \tag{8} \]
\[ k_p=\frac{1}{\cos\varphi_n\cdot\eta_n}; \]
\(A, B, m, m', n, n', q, q', p\) are constant coefficients, completely determined for the given design,
\(\varphi_{pk}, \varphi_{pc}\) are coefficients excluding the areas of the surfaces of coils and cores that do not participate in cooling,
\(P_n\) is the rated load power,
\(\eta_n, \cos\varphi_n\) are the rated values of the efficiency and load power factor,
\(K_{zc}, K_{zk}\) are the filling factors of the cores and coils,
\(n_k\) is the number of coils of the primary winding,
\(j\) is the current density in the basic winding,
\(B_{\mathrm m}\) is the maximum value of the magnetic induction in the main core;
\(m, f, k_f\) are the number of phases, the frequency, and the waveform coefficient of the line voltage.
\[ E=\frac{K_{\mathrm{ek}}V_{\mathrm k}+K_{\mathrm{ec}}V_{\mathrm c}}{P_{\mathrm n}} =\frac{K_{\mathrm{ec}}(\alpha V_{\mathrm k}+V_{\mathrm c})}{P_{\mathrm n}}, \tag{9} \]
\[ j=\sqrt{\frac{\sigma_{\mathrm k}\Delta\tau_{\mathrm k}\Pi_{\mathrm k}} {\rho_{\mathrm k}K_{\mathrm{ek}}V_{\mathrm k}}}, \qquad B=\sqrt{\frac{\sigma_{\mathrm c}\Delta\tau_{\mathrm c}\Pi_{\mathrm c}} {\gamma_{\mathrm c}K_{\mathrm{ec}}\rho_{\mathrm c}V_{\mathrm c}}}, \]
Solving (3–7) together with the expressions known from [2, 3], we obtain expressions for the specific economic indicators of a frequency-converter transformer for various design conditions, containing very simple formulas for the geometry coefficients, which cannot be achieved with other basic quantities.
Below is one of the formulas for the geometry coefficient, determining the dimensional ratios of the biased transformer for calculations under the known conditions of “independent geometry.”
\[ K_{\mathrm r} =aA\sqrt[4]{K_s}\left(\frac{m}{K_b}+nK_b+qK_c\right) +\frac{B}{\sqrt[4]{K_s^3}} \left(\frac{m'}{K_b}+n'K_b+q'K_c+pK_h\right). \tag{10} \]
As is evident, expression (10) makes it possible to determine easily the optimum values of \(K_s\), \(K_c\), \(K_h\), and \(K_b\) for the minimum of \(K_{\mathrm r}\) by finding the partial derivatives with respect to these quantities:
\[ K_c=\sqrt{\frac{pK_s}{aqK_s+q'}}, \qquad K_h=\sqrt{(aqK_s+q')\frac{K_s'}{p}}, \tag{11} \]
\[ K_s^3a_3+K_s^2a_2+K_sa_1+a_0=0, \qquad K_b=\sqrt{\frac{amK_s+m'}{anK_s+n'}}, \]
where \(a_3, a_2, a_1, a_0, a\) are constant numerical coefficients expressed through \(A, B, m, m', n,\ldots\), etc.
The minimum of the specific economic indicators of all frequency-converter transformers, including biased transformers, has a very extensive domain. Therefore it is best to determine the optimum values of \(K_c\), \(K_h\), and \(K_b\) from (11) and similar formulas in combination with plotting the curves \(K_{\mathrm r}=f(K_s)\), from which the entire region \(K_{s\mathrm{op}}\) can be found. This gives great freedom in designing biased transformers, taking into account requirements of structural feasibility and manufacturability. For example, substituting \(K_b\), \(K_c\), and \(K_h\) from (11) into (10), we obtain
\[ K_{\mathrm r}^* =\frac{2B}{\sqrt[4]{K_s^3}} \left[ \sqrt{(amK_s+m')(anK_s+n')} +\sqrt{pK_s(aqK_s'+q')} \right]. \tag{12} \]
The curves \(K_{\mathrm r}=f(K_s)\) according to expression (12) for a shell-type biased transformer (Fig. 1) are shown in Fig. 2. Here the values of \(K_{s\mathrm{op}}\), calculated from (11), are marked by points. The curves for \(\alpha=1\) correspond to the condition of minimum volume, for \(\alpha<1\) to minimum cost, and for \(\alpha>1\) to minimum weight.
In [4], expressions are given for the geometry coefficients for all typical cases of frequency-converter transformer design, and on their basis it is not difficult to analyze the optimum geometric ratios of biased transformers of any design.
In brief, the method for calculating the geometry of biased transformers is as follows:
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On the basis of the specified design conditions, the quantities \(\beta\), \(\alpha\) are determined and the design of the TPPSh is selected.
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The values of the coefficients \(K_1 \div K_7\) are established, expressions are compiled for the volume of the coils and cores; on their basis the numerical values of the coefficients \(A\), \(B\), \(m\), \(m'\), \(n\), \(n'\), \(q\), \(q'\), \(p\) are established.
Fig. 2
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From the expression corresponding to the selected condition of the economic index, a graph of the dependence \(K_{\mathrm{r}} = f(K_s)\) is constructed.
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From the curve \(K_{\mathrm{r}} = f(K_s)\), a zone of values of \(K_{s\mathrm{op}}\) is selected in which the function \(K_{\mathrm{r}}\) is minimal.
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For the selected value of \(K_{s\mathrm{op}}\), the values of the dimensionless coefficients \(K_b\), \(K_h\), \(K_c\) are calculated. The values of these coefficients may be varied in the optimum zone of \(K\) so that the shapes of the apparatus are obtained as structurally feasible and manufacturable.
After determining the values of the dimensionless coefficients, the designer finds the quantities \(B_m\) and \(j\), calculates the basic section \(S_0\) from (9, 10), and determines all dimensions of the ferromagnetic device from (1) and (2).
The existing experience in calculating biased transformers and a number of other ferromagnetic devices by the proposed method has shown that it is easy to master, makes it possible to carry out analysis and synthesis of the geometry of ferromagnetic devices clearly and simply, and does not require large expenditures of computation time.
REFERENCES
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V. P. Obrusnik. A simplified method for calculating the optimal geometry of transformers regulated by shunt biasing. Izvestiya TPI, vol. 132, Publishing House of TTU, Tomsk, 1965.
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R. Kh. Bal’yan. Low-Power Transformers. Sudpromgiz, Leningrad, 1961.
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I. I. Belopol’skii, L. G. Pikalova. Calculation of Low-Power Transformers and Chokes. Moscow, Gosenergoizdat, 1963.
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V. P. Obrusnik. Theory of the Optimal Geometry of Ferromagnetic Devices. Technical report of the Research Institute of AEM at TPI, Tomsk, 1972.