CALCULATION OF THE ELECTROMAGNETIC PARAMETERS OF LOW-POWER HIGH-FREQUENCY POWER TRANSFORMERS

V. P. OBRUSNIK, G. A. KHAIDUROVA

(Presented by the scientific and technical seminar of the Research Institute of AEM)

In modern installations for industrial, household, and special purposes, the need for power transformers operating at increased frequency is sharply increasing. This is due to the fundamental properties of many devices in automation, telemechanics, industrial electronics, electric drives, etc. It is also important that increasing the frequency substantially reduces the weight and overall dimensions of transformers, while the latter account for a large proportion by weight in any electrical system. It is known that frequency directly or indirectly affects almost all calculated parameters of ferromagnetic devices, and this influence is most appreciable at frequencies above 100 Hz, since losses in magnetic cores begin to increase sharply.

For transformers with a power of up to 10 kilovolt-amperes operating at low frequency (its upper limit has not been precisely established and lies within 50–100 Hz), choosing the operating induction at the knee of the saturation curves of electrical steels does not lead to significant heating of the magnetic cores. But such a simple choice of induction for transformers with a power above several tens of volt-amperes operating at increased frequency (above 100 Hz), and for transformers with a power of more than several kilovolt-amperes operating even at low frequency, may lead to substantial overheating of the magnetic cores, large magnetizing currents, and other phenomena. These circumstances have led to the need to subdivide power transformers into low-frequency and high-frequency ones [1, 3]; a tendency toward such subdivision also exists in methods for calculating transformer parameters. In the authors’ opinion, the most correct judgments on this matter were expressed in the fundamental work [2] by R. H. Balyan, where it is noted that the calculation of (low-power) power transformers will be general, independent of the frequency range, if, taking the latter into account, the induction, current density, and specific losses in the steel are optimally selected for the apparatus.

On the basis of the foregoing and of the classical propositions of works [1, 2, 3, and others], the authors have obtained and propose below general expressions acceptable for calculating the electromagnetic parameters of all transformers that have wound, laminated, or pressed magnetic cores, row winding of the windings without division into cooling ducts, and without thermal insulation from the magnetic core.

In full notation these expressions have the form:

for the calculated maximum permissible induction,

\[ \begin{aligned} B={}&\sqrt[12]{\left(\frac{\alpha_k \tau_k}{\rho_{c1}\left(\dfrac{f_1}{f_{16}}\right)^{\gamma} q_c K_{zc}}\right)^7 \cdot \frac{\alpha_k \tau_k \cdot 10^{-5}}{\rho_k K_{zk}} \left(\frac{4K_f n_k K_{zc} K_{zk} f_1}{P_{gf}}\right)^2} \\ &\times \sqrt[12]{2^8 \varphi_{pk}\varphi_{pc}^{\,7} \left(z+\frac{x}{K_o}\right) \cdot \frac{x \cdot z \cdot (1+y)^7}{y^5 K_o^{\,5}} \cdot \frac{1}{\beta^7}} \sqrt[12]{\nu^7\left(\frac{1+\beta\sqrt{\nu}}{1+\nu}\right)^8}; \end{aligned} \tag{1} \]

for the calculated maximum permissible current density,

\[ \begin{aligned} j={}&\sqrt[12]{\left(\frac{\alpha_k \tau_k}{\rho_k K_{zk}}\right)^7 \cdot \frac{\alpha_k \tau_k \cdot 10^{61}}{\rho_{c1}\left(\dfrac{f_1}{f_{16}}\right)^{\gamma} q_c K_{zc}} \cdot \left(\frac{4K_f n_k K_{zc} K_{zk} f_1}{P_{gf}}\right)^2} \\ &\times \sqrt[12]{2^8 \varphi_{pk}^{\,7}\varphi_{pc} \left(z+\frac{x}{K_o}\right)^7 \frac{y \cdot K_o^{\,5}\cdot(1+y)}{x^5 \cdot z^5} \cdot \frac{1}{\beta}} \cdot \sqrt[12]{\nu \cdot \left(\frac{1+\beta\sqrt{\nu}}{1+\nu}\right)^8}, \end{aligned} \tag{2} \]

Here the following notation is used, according to [2, 5]:

\[ \rho_k,\ \alpha_k,\ K_{zk} \]
— the specific resistance, heat-transfer coefficient, and fill factor of the coils;

\[ \tau_k \]
— the permissible temperature rise of the coils relative to \(20^\circ\mathrm{C}\);

\[ \gamma \]
— the frequency coefficient of losses in the steel;

\[ \nu \]
— the ratio of losses in the magnetic cores to losses in the windings;

\[ \beta=\frac{\Pi_{oc}}{\Pi_{ok}} \]
— the ratio of the cooling surface of the magnetic cores \(\Pi_{oc}\) to the cooling surface of the coils \(\Pi_{ok}\);

\[ g_c,\ K_{zc},\ \rho_{c1} \]
— the specific weight, fill factor, and specific losses at \(B=1\ \mathrm{T}\) and base frequency \(f_{16}\) for the magnetic core;

\[ f_1 \]
— the frequency of the voltages of the supply network;

\[ K_f \]
— the waveform factor of the line voltage;

\[ n_k \]
— the number of coils of the primary winding;

\[ P_{gf} \]
— the apparent power of one phase;

\[ \varphi_{pk},\ \varphi_{pc} \]
— the relative cooling surface of the coils and cores;

\[ K_o \]
— the ratio of the width of the magnetic-core window to the thickness of one coil of the primary winding;

\[ x,\ y,\ z \]
— the geometric ratios of the dimensions of the magnetic core.

Expressions (1, 2) have been obtained from the condition of ensuring the permissible overheating of the windings; therefore, the final choice of the calculated quantities should be checked against additional conditions:

\[ B_{\text{calc}}\leq B_s;\qquad \Delta U\%_{\text{calc}}\leq \Delta U\%_{\text{specified}}. \]

It should also be noted that the current density according to (2) must be taken for the inner winding of the transformer, for example, the primary. For the outer windings, according to [2], it should be taken into account that

\[ j_{\text{out}}=\varepsilon j_{\text{in}},\quad \text{where } \varepsilon=0.5\div 0.8. \]

If the design configuration of the transformers is known, the material for their magnetic cores and coils has been selected, and the cooling conditions are known, then the values of most of the quantities in expressions (1, 2) are quickly established from [1, 2, 3, 6], and the expressions themselves are sharply simplified.

It is necessary to dwell in more detail only on the quantities \(\beta\), \(\varphi_{\mathrm{pk}}\), \(\varphi_{\mathrm{ps}}\), \(\nu\). The first three of these are expressed in terms of the cooling surfaces and the total surfaces of the coils and magnetic cores by the formulas:

\[ \varphi_{\mathrm{pk}} = \frac{\Pi_{\mathrm{ok}}}{\Pi_{\mathrm{k}}} \, \frac{ A_{0}\left(m_{0}+n_{0}y+q_{0}\dfrac{x}{K_{0}}\right) }{ 2A\left(m+ny+q\dfrac{x}{K_{0}}\right) }. \tag{3} \]

\[ \varphi_{\mathrm{ps}} = \frac{\Pi_{\mathrm{os}}}{\Pi_{\mathrm{s}}} = 1- \frac{ z\cdot n_{\mathrm{k}} }{ B\left(m' + n'y + q'\dfrac{x}{K_{0}} + pz\right) }, \tag{4} \]

\[ \beta = \frac{\Pi_{\mathrm{os}}}{\Pi_{\mathrm{ok}}} = \frac{1+y}{z+\dfrac{x}{K_{0}}} \cdot \frac{2B}{A_{0}} \cdot \frac{ m' + n'y + q'\dfrac{x}{K_{0}} + z\left(p-\dfrac{n_{\mathrm{k}}}{B}\right) }{ m_{0}+n_{0}y+q_{0}\dfrac{x}{K_{0}} }. \tag{5} \]

Here the coefficients \(A\), \(A_{0}\), \(B\), \(m\), \(m'\), \(m_{0}\), \(n\), \(n'\), \(n_{0}\), \(q\), \(q'\), \(q_{0}\), \(p\) are, for each design of ferromagnetic devices, quite de-

Fig. 1.

by definite numerical values, which are easy to find by expressing the volumes of the coils and cores through the geometric dimensions, as is done in [1, 3, etc.]. For the transformer designs in Fig. 1, the values of these coefficients are given in Table 1.

Table 1

As is clear from (3, 4, 5), with known geometric ratios of the dimensions \(x, y, z\), it is not difficult to determine the quantities \(\varphi_{\mathrm{mk}}, \varphi_{\mathrm{nc}}, \beta\). Moreover, they are little affected by the actual limits of variation of \(x, y, z\).

Special attention should be paid to the quantity \(\nu\), which determines the ratio of power losses in the cores and coils. As is clear from (1, 2), increasing \(\nu\) always leads to an increase in induction and a decrease in current density. No extrema are observed here. The question of the proper choice of the value of \(\nu\) is resolved by analyzing the influence of this quantity on the specific economic indicators (s.e.i.) of the transformer. It is known [3, 4, 5] that, when designing ferromagnetic devices from the condition of permissible heating with unspecified values of induction and current density, an inversely proportional dependence is always observed between the s.e.i. and the quantities \(j, B\), namely:

\[ \text{s.e.i.}=\mathcal{E}=\frac{1}{\sqrt{j^3 \cdot B^3}} . \tag{6} \]

Substituting into (6) the values of \(j\) and \(B\) from (1, 2), after transformations we obtain

\[ \mathcal{E}=\left[\frac{1+\nu}{\sqrt{\nu+\beta \nu}}=\varphi,\right] \tag{7} \]

with a minimum for \(\mathcal{E}\) with respect to \(\nu\).

The curves \(\varphi=f(\nu)\) for \(\beta=\mathrm{const}\) are shown in Fig. 2.

Equating to zero the partial derivative of \(\varphi\) with respect to \(\nu\), we shall have

\[ 0=\frac{\nu_0-2\beta\sqrt{\nu_0}-1}{1+2\beta\sqrt{\nu_0}+\beta^2\nu_0}. \tag{8} \]

Since \(\nu_0>0,\ \beta>0\), then

\[ \nu''-2\beta\sqrt{\nu_0}-1=0, \]

whence

\[ \nu_0=\left(\beta+\sqrt{\beta^2+1}\right)^2 . \tag{9} \]

As can be seen, the optimum value of \(v\), equal to \(v_0\), depends only on the coefficient \(\beta\), which determines the ratio of the cooling surfaces of the cores and coils of the ferromagnetic device (in our case, the transformer). The smallest value of \(v_0\) is equal to 1 and is observed for \(\beta=0\) (toroidal devices with continuous winding of the coils), and \(v_0\) is always greater than 1 when \(\beta>0\).

For transformer designs in Fig. 1, the value of \(\beta\) does not go beyond \(0.25 \div 0.75\), and therefore \(v=2 \div 3\).

Here the choice of the value \(v\) within the range \(2 \div 3\) has practically no effect on the specific electromagnetic parameter (Fig. 2) and only provides a certain freedom in calculating \(B\) and \(j\) from a purely subjective standpoint, bearing in mind that as \(v\) increases the volume of the magnetic cores increases and the volume of the coils decreases.

Fig. 2.

We shall give a practical explanation of the calculations by expressions (1, 2) for a specific example. Let us determine the values of current density and induction admissible under heating conditions for a power single-phase low-voltage transformer of core construction as shown in Fig. 1, \(a\), designed on laminated magnetic cores of type PL made of steel E 350 with a thickness of 0.15 mm and with coils made of copper wire. The supply-network frequency is 400 Hz, apparent power 600 VA, natural cooling, admissible overheating of the coils \(t=50^\circ\mathrm{C}\).

According to reference data [2], for the specified conditions we shall have

\[ \rho_c = 9 \frac{\mathrm{W}}{\mathrm{kg}}, \quad q_c = 7.65 \frac{\mathrm{g}}{\mathrm{cm}^3}, \quad \rho_{k(70)} = 0.021 \frac{\Omega \cdot \mathrm{mm}^2}{\mathrm{m}}, \]

\[ K_{zc}=0.85,\quad K_{zk}=0.35,\quad \alpha_k = 1 \cdot 10^{-3}\frac{\mathrm{W}}{\mathrm{cm}^2 \cdot \mathrm{deg}}. \]

We also specify the average values of the quantities \(x, y, z\) of the standardized series of PL cores: \(x=1.6,\ y=2,\ z=3.5\).

The quantities \(\varphi_{\mathrm{pk}}, \varphi_{\mathrm{ps}}, \beta\) are calculated from (3, 4, 5) using the values of the coefficients \(A, A^\circ, B, m, m', m_0, n, n', n_0, q, q', q_0, p\) from Table 1:

\[ \varphi_{\mathrm{pk}} = \frac{4}{2\cdot 4} \frac{ 1+1\cdot 2+7.85\,\frac{1.6}{6} }{ 2.5+2.5\cdot 2+9.82\,\frac{1.6}{6} } =0.252, \]

\[ \varphi_{\mathrm{ps}} = 1- \frac{3.5\cdot 2}{ 2\left(1.57+0\cdot 2+6\cdot \frac{1.6}{6}+1.1\cdot 3.5\right) } =0.502, \]

\[ \beta = \frac{1+2}{3.5+\frac{1.6}{6}} \cdot \frac{2\cdot 2}{4} \cdot \frac{ 1.57+6\cdot \frac{1.6}{6}+\left(1.1-\frac{2}{2}\right)3.5 }{ 1+1\cdot 2+7.85\cdot \frac{1.6}{6} } =0.55. \]

The optimum value of \(\nu\) may be selected from the curves \(\varphi_\nu=f(\nu)\) in Fig. 2 for \(\beta=0.6\), or calculated from (9)

\[ \nu_0=\left(0.55+\sqrt{0.55^2+1}\right)^2=2.86\simeq 3. \]

Now \(j\) and \(B\) are calculated from (1, 2):

\[ j= \sqrt[12]{ \left(\frac{1\cdot 50}{2.1\cdot 0.35}\right)^7 \frac{1\cdot 50\cdot 10^{61}}{ 9\left(\frac{400}{400}\right)^{1.6}\cdot 7.65\cdot 0.85 } \cdot \left( \frac{4\cdot 1.1\cdot 2\cdot 0.35\cdot 0.85\cdot 400}{600} \right)^2 } \times \]

\[ \times \sqrt[12]{ 2^8\cdot 0.252\cdot 0.502 \left(3.5+\frac{1.6}{6}\right)^7 \frac{2\cdot 6^5(1+2)}{1.6^5\cdot 3.5^5} \cdot \frac{1}{0.6} } \times \]

\[ \times \sqrt[12]{ 3\cdot \left( \frac{1+0.6\sqrt{3}}{1+3} \right)^8 } = 1.96\cdot 10^6\ \frac{\mathrm{A}}{\mathrm{m}^2}, \]

\[ B= \sqrt[12]{ \frac{1\cdot 50}{ 9\left(\frac{400}{400}\right)^{1.6}\cdot 7.65\cdot 0.85 } \cdot \frac{1\cdot 50\cdot 10^{-5}}{2.1\cdot 0.35} \left( \frac{4\cdot 1.1\cdot 2\cdot 0.35\cdot 0.85\cdot 400}{600} \right)^2 } \times \]

\[ \times \sqrt[12]{ 2^8\cdot 0.252\cdot 0.502^7 \left(3.5+\frac{1.6}{6}\right) \cdot \frac{1.6\cdot 3.5(1+2)^7}{2^5\cdot 6} \cdot \frac{1}{0.6^7} } \times \]

\[ \times \sqrt[12]{ 3^7 \left( \frac{1+0.6\sqrt{3}}{1+3} \right)^8 } = 1.06\ \mathrm{T}. \]

Knowing the calculated values of \(j\) and \(B\), one can next determine the main parameter of the transformer—the cross section of the magnetic core—which determines all the remaining parameters. Thus, according to [4] we shall have:

\[ S_c= \sqrt{ \frac{P_{\mathrm{gf}}\cdot y\cdot K_0}{ 4\cdot K_f\cdot B\cdot f_1\cdot j\cdot K_{\mathrm{zk}}\cdot K_{\mathrm{zc}}\cdot \eta_k\cdot x\cdot z } } = \]

\[ = \sqrt{ \frac{600\cdot 10^8\cdot 2\cdot 6}{ 4\cdot 1.1\cdot 1.06\cdot 400\cdot 1.96\cdot 10^6\cdot 0.35\cdot 0.85\cdot 2\cdot 1.6\cdot 3.5 } } = 7.7\ \mathrm{cm}^2 \]

For the found cross section \(S=7.7\ \mathrm{cm}^2\), with \(x=1.6,\ y=2,\ z=3.5\), the most suitable magnetic core will be PL \(20\times 40\times 50\).

The proposed method for calculating the induction and current density of power transformers is based on simpler than in [1, 2, 3] expres-

… and, while providing high accuracy of the results, requires a smaller amount of initial data.

REFERENCES

1. R. Kh. Balyan. Low-Power Transformers. Leningrad, Sudpromgiz, 1961.

2. R. Kh. Balyan. Transformers for Radio Electronics. Moscow, “Sovetskoe radio,” 1971.

3. I. I. Belopol’skii, L. G. Pikalova. Calculation of Low-Power Transformers and Chokes. Moscow, Gosenergoizdat, 1963.

4. V. P. Obrusnik. A simplified method for calculating the optimal geometry of transformers controlled by magnetization of shunts. Izvestiya TPI, vol. 132, Tomsk, TTU Publishing House, 1964.

5. V. P. Obrusnik, V. G. Kiselev, G. A. Khaidurova. Methodology for the analysis and synthesis of the geometry of magnetized transformers. Proceedings of the VII Scientific and Technical Conference on Problems of Production Automation. Tomsk, TTU Publishing House, 1971.

6. A. M. Bambas, Yu. A. Savinovskii. Chokes for Radio Equipment. Moscow, “Sovetskoe radio,” 1969.