TOWARD AN ENGINEERING CALCULATION OF THE MAIN PARAMETERS OF ELECTROMAGNETIC DEVICES ON STANDARDIZED ROD CORES WITH GAPS

L. V. Traut, E. I. Goldstein

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In designing linear chokes, as well as transreactors, inductive sensors, and certain other electromagnetic devices with air (nonmagnetic) gaps in the magnetic circuit, one has to deal with two types of computational problems. In the so-called direct problems, the purpose of the calculation is to determine the inductance \(L\) of a system of known configuration for a specified number of turns \(w\):

\[ L = w^{2}G_{\mathrm{p}}. \tag{1} \]

A rigorous determination of the total permeance of the system presents great difficulty; therefore the magnetic resistance of the ferromagnetic portions of the magnetic core is often neglected in comparison with the resistance of the gaps. However, even in this case only the principal permeance of the air gaps \(G_0\) is determined comparatively simply:

\[ G_0 = \mu_0 \frac{ab}{\delta}, \tag{2} \]

where \(a\) and \(b\) are the transverse dimensions of an air gap of length \(\delta\).

The remaining permeances—the fringing permeance and the leakage permeance—can be determined by one of the widely known methods of Roters, Schmidel, and others. The most accurate results can be obtained by using the graph-analytical method of B. K. Bul’ for determining the individual components of leakage and fringing permeances [1]. A common shortcoming of all the methods mentioned is the laboriousness of the calculations and, most importantly, the difficulty of ascertaining the influence of the individual dimensions of the system on its total permeance.

Even greater difficulties arise in solving the so-called inverse problems, when it is required to select the magnitude of the air gap that provides a specified inductance of the system for a given number of turns. In practice the inverse problem is solved by the method of successive approximations; that is, repeated, very laborious calculations are required.

In this article, as applied to electromagnetic devices on standardized cores, the possibility is shown of using graphical dependences that substantially simplify the solution of the problems formulated above (with account taken only of the resistance of the air gaps).

The studies carried out by the authors have shown that the most universal dependence is

\[ K_{\phi}=f(\delta'), \]

where \(K_{\phi}\) is the fictitious-gap coefficient;
\(\delta'\) is the relative gap.

\[ K_{\phi}=\frac{G_{\mathrm{p}}}{G_{0}}, \tag{3} \]

\[ \delta'=\frac{\delta}{l_{c}}. \tag{4} \]

In expression (4), \(l_{c}\) is the length of the mean line of force. In calculating the dependences \(K_{\phi}=f(\delta')\), dimensionless geometry coefficients \(x, y,\) and \(z\) and the basic dimension \(a\) are used:

\[ x=\frac{b}{a}, \qquad y=\frac{c}{a}; \qquad z=\frac{h}{a}. \tag{5} \]

The basic dimensions of cores \(a, b, c,\) and \(h\) are shown in Fig. 1.

Fig. 1. Basic dimensions of P-shaped cores. 1—winding; 2—core of type P; 3—core of type PL.

For all standardized cores, the length of the mean line of force depends only on the basic dimension \(a\), remaining unchanged for cores of a definite series group:

\[ l_{c}=a(2y+2z+3.14). \tag{6} \]

Similarly, through the coefficients \(x, y, z\) and the dimension \(a\), all the remaining calculated quantities characterizing a given magnetic system can be expressed.

Since the permeances depend on the geometric dimensions of the system, all cores of the given series can be characterized by a single family of graphs \(K_{\phi}=f(\delta')\) for specified values of the coefficient \(x\), \(y\), or \(z\).

In principle, to obtain the indicated dependences, both graphoanalytical and analytical methods for determining the total permeance of the system may be used.

Taking into account the considerable laboriousness of the calculations, it is expedient to use an electronic digital computer (EDC), which also predetermined the choice, as the initial ones, of analytical expressions for the total permeance obtained by I. I. Pekker [2--3].

The Appendix gives the final calculation expressions for determining \(K_{\phi}\), obtained from the corresponding formulas [2--3] with allowance for the notation of the present article and expressions (3)--(6).

Calculations of \(K_{\phi}\) from expressions (11)--(19) were carried out on an EDC. Programmers of the computing center of the Tomsk Polytechnic Institute, E. I. Sadovskaya and V. P. Timoshchenko, as well as students S. K. Sazanbaev and A. K. Sovetov, took part in the work. The results obtained are given in Figs. 2 and 3 for cores of type \(P\), respectively for \(y=1,\ z=3\) and \(y=1.5,\ z=4\) (Fig. 2), and \(y=2,\ z=3\) and \(y=2,\ z=5\) (Fig. 3). In Figs. 4 and 5 the results are given for cores of type PL.

Let us consider the plan for solving the most characteristic problems in the calculation of electromagnetic devices with gaps when using the dependences

\[ K_{\phi}=f(\delta') \quad \text{and} \quad \frac{\delta'}{K_{\phi}}=f(\delta'). \]

a) Determination of \(L\) for known dimensions \((a,\ b,\ c,\ h\) and \(\delta)\) and number of turns \((w)\).

1) from expressions (5) we determine the coefficients \(x,\ y\), and \(z\);

2) from expressions (4) and (6) we determine the relative gap \(\delta'\);

3) from the dependence \(K_{\phi}=f(\delta')\) for the given \(x,\ y\), or \(z\), we determine the fictitious-gap coefficient \(K_{\phi}\);

4) we determine the permeance of the main (calculated) air gap from expression (2);

5) we determine the value of the inductance from expression (7):

\[ L=w^{2}G_{0}K_{\phi}. \tag{7} \]

b) Determination of the required number of turns to ensure a prescribed value of the inductance of a specified system.

The problem is solved in the same way as the preceding one (items 1--4), and the number of turns is determined from formula (8):

\[ w=\sqrt{\frac{L}{G_{0}K_{\phi}}}. \tag{8} \]

c) Determination of the gap value at which a specified inductance of a system with known core dimensions is ensured for a specified number of turns.

To solve the stated problem we use the relations:

\[ G_{\mathrm{n}}=\frac{L}{w^{2}}=G_{0}K_{\phi}=K_{\phi}\mu_{0}\frac{ab}{\delta}, \tag{9} \]

\[ \frac{\delta'}{K_{\phi}}=\frac{w^{2}\mu_{0}ab}{L\cdot l_{c}}. \tag{10} \]

1) determine the ratio \(\dfrac{\delta'}{K_{\phi}}\) from expression (10);

2) from the dependence \(\dfrac{\delta'}{K_{\phi}}=f(\delta')\) for the given \(x\), \(y\), or \(z\), determine the relative gap \(\delta'\);

3) determine the required gap \(\delta\) from expression (4).

Verification of the proposed method

For verification, specimens were made, the basic data of which are given in Table 1.

Tables 2 and 3 show the main stages in the calculation and verification of the above-mentioned specimens by the proposed method.

In all the cases considered, the calculation error by the method presented does not exceed 18%.

For comparison, Table 4 gives a number of values of \(K_{\phi}\) obtained by the method considered above \((K_{\phi3})\), experimentally \((K_{\phi2})\), and by using the graphoanalytical method of Prof. Buhl \((K_{\phi1})\) according to [4].

Appendix

\[ K_{\phi \mathrm{p}}=\delta'\kappa'\sqrt{\frac{AБ}{x}}\cdot \operatorname{cth}2\delta'\kappa'\sqrt{\frac{Б}{x\cdot A}} +\frac{\delta'\kappa'}{2x} \left\{\left[\frac{x+2}{\pi}+\frac{\delta'\kappa'}{2}+1\right] \right. \]
\[ \left. {}\times \ln\left(\frac{2}{\delta'\kappa'}+1\right)-1 \right\} \left(1+\frac{1}{2zy}\right) +\frac{2}{3}\frac{Б\cdot \kappa'\cdot \delta'}{x}; \tag{11} \]

\[ A=1+\frac{2\delta'\kappa'}{\pi}\left(1+\frac{1}{x}\right) +\frac{(\delta'\kappa')^2}{2x}; \tag{12} \]

\[ Б=\frac{1}{\ln\left(\frac{2y}{x}+1\right)} +\frac{1}{\pi}\ln\left(1+\frac{2}{y}\right); \tag{13} \]

\[ \kappa'=2z+2y+\pi; \tag{14} \]

\[ K_{\phi \mathrm{pl}}= \frac{(x+2)}{8\left(\dfrac{z}{\delta'\kappa'}+1\right)x} \left\{ (2\delta'\kappa'+y) \left[ \ln\left(1+\frac{y}{\delta'\kappa'}\right)-\frac{1}{2} \right] \right. \]
\[ \left. {}-\frac{\pi\left[(\delta'\kappa')^2+y\delta'\kappa'+\dfrac{y^2}{y}\right]}{4(z+\delta'\kappa')} \right\} +\frac{B\cdot \Gamma\cdot \delta'\kappa'}{4(z+\delta'\kappa')^2(\operatorname{sh}2\delta'\kappa' m)\cdot x} \]
\[ +\frac{2\cdot Б\cdot E}{\left[\dfrac{z^2}{(\delta'\kappa')^2}+\dfrac{2z}{\delta'\kappa'}+1\right]x\delta'\kappa'}; \tag{15} \]

Table 1

b) \(K_{\varphi n}\)

Graph annotations:

• \(K_{\varphi n}\)
• \(\dfrac{\delta'}{K_{\varphi n} l_i}\cdot 10^{-3}\)
• \(K_{\varphi n}\)
• \(\dfrac{\delta'}{K_{\varphi n}}\)
• \(x = 0.5\)
• \(x = 1\)
• \(x = 1.2\)
• \(x = 1.5\)
• \(x = 2\)
• \(x = 2.5\)
• \(y = 1.5;\quad z = 4\)
• \(\delta'\cdot 10^{-3}\)

Fig. 2. Calculated dependences for rod cores of type T:
a) for \(y = 1.5;\ z = 4;\ x = 0.5;\ 1;\ 1.2;\ 1.5;\ 2;\ 2.5;\)
b) for \(y = 1.5;\ z = 4;\ x = 0.5;\ 1;\ 1.2;\ 1.5;\ 2;\ 2.5;\)
c) for \(y = 1;\ z = 3;\ x = 0.5;\ 0.75;\ 1;\ 1.25;\)

(a) \(K_{\phi p}\)

Graph with axes and families of curves labeled as follows:

• Vertical axis: \(K_{\phi p}\)
• Tick marks: \(.05,\ .07,\ .09,\ .11,\ .13,\ .15,\ .17,\ .19\)
• Upper/left internal scale: \(\dfrac{\delta'}{K_{\phi p}}\)
• Tick marks: \(2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14\)
• Horizontal scale:
• Tick marks: \(2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20\)
• Lower/right annotation: \(\dfrac{\delta'}{K_{\phi p}}\)
• Right-side annotation: \(\delta' \cdot 10^{-3}\)
• Parameter annotation: \(y = 2;\ z = 3\)

Curve labels visible on the graph:

\[ \chi = 0.5 \]

\[ \chi = 0.75 \]

\[ \chi = 1 \]

\[ \chi = 1.25 \]

\[ \chi = 1.5 \]

\[ \chi = 1.6 \]

\[ \chi = 2 \]

\[ \chi = 2.5 \]

Additional repeated curve labels visible:

\[ \chi = 0.5,\quad \chi = 0.75,\quad \chi = 1,\quad \chi = 1.5,\quad \chi = 2 \]

Fig. 3. Design dependences for rod cores of type II.
a) for \(y=2\), \(z=5\), \(x=0.5;\ 0.75;\ 1;\ 1.25;\ 1.6;\ 2;\ 2.5\)
b) for \(y=2\), \(z=3\), \(x=0.5;\ 0.75;\ 1;\ 1.25;\ 1.6;\ 2;\ 2.5\)
c) \(x=0.5;\ 0.75;\ 1;\ 1.25;\)

![Graph with families of curves; visible annotations transcribed below.]

Visible graph annotations:

• Vertical-axis label: \( \sigma',\ K_{\phi m} \)
• Scale label: \( \dfrac{\delta'}{K_{\phi m}}\cdot 10^{-3} \)
• Horizontal-axis label: \( K_s^{\phi m} \)

Curve and parameter labels:

\[ x = 2;\quad y = 2.5 \]

\[ \frac{z = 123;\ 154;\ 193}{z = 246} \]

\[ x = 193;\quad y = 1.23 \]

\[ \frac{\delta'}{K_{\phi m}} \]

Axis and scale values visible:

\[ 0.5,\ 0.6,\ 0.7,\ 0.8,\ 0.9,\ 1.0,\ 1.1 \]

\[ 0,\ 4,\ 8,\ 12,\ 16,\ 20,\ 24 \]

\[ 2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20 \]

Fig. 4. Calculated dependences for rod cores of type PL:
a) PL6.5; \(x = 1.93;\ y = 1.23;\ z = 1.23;\ z = 1.54;\ 1.93;\ 2;\ 2.46;\ x = 2;\ y = 2;\ z = 5.\)
b) PL8; \(x = 1.56;\ y = 1.25;\ z = 1.56;\ 2;\ 2.5;\ 3.14.\)

b) \(K_{\phi m}\)

Visible graph labels and annotations:

• Vertical axis: \(K_{\phi m}\)
• Vertical scale: \(0.5,\ 0.6,\ 0.7,\ 0.8,\ 0.9,\ 1.0,\ 1.1\)
• Additional vertical-axis label: \(\dfrac{\delta'}{K_{\phi m}}\cdot 10^3\)
• Corresponding scale: \(0,\ 4,\ 8,\ 12,\ 16,\ 20,\ 24\)
• Horizontal scale: \(2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20\)

Curve-family labels:

• \(\dfrac{\delta'}{K_{\phi m}}\)
• \(K_{\phi m}\)
• \(z=2\)
• \(z=2.5;\ 2.56\)
• \(z=32\)
• \(x=40\)
• \(m=10\div 12.5,\ x=13[[unclear]],\ q=128,\ [[unclear]]=10^3\)

Fig. 5. Calculated relationships for rod cores of type PL:
a) PL10–12.5; \(x = 1.34\); \(y = 1.28\); \(z = 2;\ 2.5\);
b) PL12.5–40; \(x = 2\); \(y = 1.6\); \(z = 2.4;\ 2.5;\ 3.0;\ 3.1;\ 3.2;\ 4.0;\ 4.8;\ 5\).

Visible graph labels: \(\delta/K_{\mathrm{fpl}}\), \(\delta' \cdot 10^{3}/K_{\mathrm{fpl}}\), \(K_{\mathrm{fpl}}\), \(z = 30;\ 3.1;\ 3.2\), \(z = 10;\ 4.1\), \(z = 4.8;\ 5\), \(z = 2.4;\ 2.5\), PL12.5–40; \(x = 2;\ y = 1.6\).

Table 2

Table 3

Table 4

Type P specimen

Type PL specimen

\[ M= \sqrt{ \frac{B}{ x+\left[\frac{2}{\pi}\delta' k'(x+1)+\frac{(\delta' k')^{2}}{2}\right] \left[1-\frac{\pi}{\dfrac{2zy}{(\delta' k')^{2}}+\dfrac{2y}{\delta' k'}}\right] } }; \tag{16} \]

\[ B=\left[z+\frac{(\operatorname{sh} 2\delta' k' M)}{M} +z(\operatorname{ch} 2\delta' k' M)\right]; \tag{17} \]

\[ \Gamma= \left\{ x+\left[\frac{2}{\pi}\delta' k'(x+1)+\frac{(\delta' k')^{2}}{2}\right] \left[1-\frac{\pi}{\dfrac{2zy}{(\delta' k')^{2}}+\dfrac{2y}{\delta' k'}}\right] \right\} \times \]

\[ \times\{(\operatorname{sh} 2\delta' k' M)+Mz[1+(\operatorname{ch} 2\delta' k' M)]\}; \tag{18} \]

\[ E= \left[ \frac{z^{3}}{12} +\frac{\delta' k'+z}{4M^{2}} -\left(\frac{1}{M^{2}}+z^{2}\right) \frac{(\operatorname{sh} 2\delta' k' M)}{8M} -\frac{z}{4M^{2}}(\operatorname{ch} 2\delta' k' M) \right]. \tag{19} \]

The coefficients \(B\) and \(k'\) are the same as for system P.

REFERENCES

1. B. K. Bul'. Determination of the errors and limits of applicability of formulas for specific magnetic permeance. Elektrichestvo, No. 4, 1960.

2. I. I. Pekker. Calculation of the inductance of Sh- and P-shaped electromagnets with a flat armature. IVUZ, Elektromekhanika, No. 8, 1963.

3. I. I. Pekker. Calculation of the inductance of electromagnets with a yoke and an armature of Sh- and P-shaped form. IVUZ, Elektromekhanika, No. 10, 1964.

4. E. I. Goldshtein. Certain questions in the calculation and design of smoothing chokes. Dissertation submitted for the degree of Candidate of Technical Sciences, 1964.