Influence of the Ratios of the Main Dimensions of Valve-Type Electromagnets on the Energy of the Stray Magnetic Field

A. V. Kurnosov

(Presented by the scientific seminar of the departments “Electrical Machines and Apparatus” and “General Electrical Engineering”)

The widespread use of electromagnets in automation and telemechanics equipment, radio engineering, instrument making, and other fields brings to the fore the task of raising the theoretical calculation of electromagnets in all respects to the same level at which, for example, the calculation of electrical machines currently stands. The solution of this problem must proceed along the path of systematizing the accumulated production experience in the manufacture of electromagnets and of finding conditions corresponding to the rational use of active materials.

The consumption of active materials depends first of all on the correct choice of the principal geometric ratios of the dimensions of the electromagnet, which have a direct influence on the stray magnetic field.

The presence of leakage fluxes in an electromagnetic mechanism greatly complicates its calculation. In practically all encountered dimensional ratios, the leakage fluxes are commensurate with the main flux, and they must be taken into account when calculating electromagnets. Thus, in [1 and 2] values are given for leakage coefficients for certain electromagnetic mechanisms, where $\delta = 1.8—3$ with the armature of the electromagnet in the open position. Therefore it is of interest to investigate the influence of the ratios of the principal geometric dimensions on the nature of the variation of the leakage flux.

Fig. 1. Sketch of an electromagnet.

Like the main flux, determined by the magnetic energy stored in the working air gap, the stray field can be characterized by the magnetic leakage energy $W_{\mathrm{mp}}$, which in turn depends on the ratios of the principal geometric dimensions of the electromagnet.

The stray magnetic field is characterized by the conductance of the leakage flux $\Phi_y$. Figure 1 shows an electromagnetic system in which—

the difference of magnetic potentials between the frame and the core varies with increasing coordinate \(x\). In this case, if saturation of the magnetic circuit is neglected, the difference of magnetic potentials between points \(a\) and \(в\) is zero, while at the top, between points \(a^1\) and \(в^1\), there is the full potential difference determined by the m.m.f. of the coil. In connection with this, in calculating \(W_{\mathrm{mp}}\) it is necessary to find an expression for the leakage flux linkage \(\Psi_p\) through the permeance of the leakage flux, taking into account the nature of the distribution of the difference of magnetic potential along the length \(l_k\).

The energy of the stray magnetic field may be represented as

\[ W_{\mathrm{mp}}=\int_0^I \Psi_p\,di . \tag{1} \]

The leakage flux linkage on an elementary segment \(dx\) at a distance \(x\) from the beginning of the coil may be expressed as [3]

\[ d\Psi_{px}=d\Phi_{\mathrm{ux}}W_x, \tag{2} \]

where the leakage flux is

\[ d\Phi_{\mathrm{ux}}=\frac{IW}{l_k}xg\,dx . \]

The number of turns over the length \(x\) may be determined as

\[ W_x=W\cdot x/l_k, \]

then

\[ \Psi_p=\int_0^{l_k} IW\cdot x/l_k\cdot gW\cdot x/l_k\,dx =IW^2\frac{gl_k}{3}. \tag{3} \]

Substituting (3) into (1), we obtain

\[ W_{\mathrm{mp}}=(IW)^2\frac{gl_k}{6}. \tag{4} \]

To find the dependence of the energy of the stray magnetic field on the ratios of the principal geometrical dimensions of the electromagnet, it is necessary in expression (4) to express \(IW\) through the geometrical parameters of the electromagnet. This can be done if it is assumed that:

a) the steel cross section \(Q_{fe}\) is constant along the length of the magnetic circuit;

b) with the armature in the unoperated position, the entire m.m.f. \(IW\) of the winding is expended in conducting the magnetic flux through the working air gap (saturation of the steel is neglected);

c) with the armature in the operated position, the entire m.m.f. of the winding is expended in conducting the magnetic flux along the length \(l_{fe}\) of the magnetic circuit;

d) hysteresis over the incomplete operating cycle of the electromagnet is neglected;

e) after operation the electromagnet remains under current for a long time and reaches a steady temperature rise above the surrounding medium \(\tau\);

f) the current \(I\) in the winding during motion of the electromagnet armature remains constant and equal to the steady-state value.

\(IW\) may be expressed as \(H\cdot l_{fe}\), where \(H\) is the magnetic-field strength of the magnetic circuit in the closed position of the armature. The magnetic-field strength in a homogeneous magnetic circuit corresponding to the starting ampere-turns \(AW_{\mathrm{tr}}\) is equal to

\[ H=\frac{jK_zQ_0}{1{,}65\,l_{fe}}, \tag{5} \]

where \(j\) is the current density in the winding;
\(K_z\) is the winding fill factor;
\(Q_0\) is the cross-sectional area of the winding;
\(1.65 = AW_{\mathrm{hot}}/AW_{\mathrm{tr}}\) [4].
\(AW_{\mathrm{hot}}\) are the ampere-turns of the winding permissible with respect to heating. The permissible current density in the winding can be determined by the equality:

\[ j^2 \rho V_{\mathrm{ok}} K_z = K_{\mathrm{to}} S_0 \tau, \tag{6} \]

where \(K_{\mathrm{to}}\) is the heat-transfer coefficient from the surface;
\(S_0 = (S_{\mathrm{n}} + \alpha S_{\mathrm{v}})\) [4] is the heat-dissipating surface of the winding;
\(S_{\mathrm{n}}\) is the outer surface of the winding;
\(S_{\mathrm{v}}\) is the inner surface of the winding;
\(\alpha\) is a coefficient;
\(\rho\) is the specific resistance of the winding wire.

Substituting the value from (6) into (5), we obtain

\[ H = \frac{K_0 \sqrt{S_0 V_{\mathrm{ok}}}}{l_{fe} l_{\mathrm{m.sr.}}}, \tag{7} \]

where

\[ K_0 = \sqrt{\frac{K_{\mathrm{to}} K_z \tau}{1.65^2 \rho}} . \]

Substituting \(H \cdot l_{fe}\) in place of \(IW\) in (4), we obtain an expression for the energy of the stray field in terms of the electromagnetic and geometrical parameters of the electromagnet:

\[ W_{\mathrm{mr}} = \frac{K_0^2 S_0 V_{\mathrm{ok}} l_k g}{6 l_{\mathrm{m.sr.}}^2}, \tag{8} \]

where \(V_{\mathrm{ok}}\) is the volume of the winding,
\(l_{\mathrm{m.sr.}}\) is the mean length of a winding turn.

For a valve-type electromagnet in which the yoke width is approximately equal to \((1.25—2.5)D/2\) (in our case the yoke width is equal to \(2D/2\)), one may use the approximate formula [2] for determining the specific conductance of the leakage flux between a cylinder and a parallel plane, which is defined as

\[ g = \frac{K_a 2 \pi \mu_0}{\ln\!\left(2n + \sqrt{4n^2 - 1}\right)}, \tag{9} \]

where \(K_a = 0.85—0.92\) is a correction factor [2],

\[ n = \frac{D/2}{d}. \]

Substitute expression (9) into (8).

To find the dependence of the energy of the stray magnetic field on the ratios of the principal geometrical dimensions of the electromagnet, we relate the expression obtained to the sum of the volumes of the active materials, i.e., we shall find the dependence of the specific energy of the stray magnetic field on the ratios of the principal geometrical dimensions \(X = d/D\) and \(Y = l_k/D\). We shall carry out the analysis for a valve-type electromagnet (Fig. 1) in the case where \(\alpha = 1\), since this somewhat simplifies the expression for the specific energy of the stray field without significantly affecting the character of the curves \(\Gamma(X, Y)\).

Expressing, in the right-hand side of the newly obtained expression, all geometrical parameters in terms of the relative quantities \(X\) and \(Y\), and taking the absolute electromagnetic parameters and the base dimension outside...

in the left-hand side of the expression, we obtain on the right-hand side a geometric factor \(\Gamma(X,Y)\), characterizing the dependence of the electromagnetic energy stored in the stray magnetic field of the electromagnet on the ratios of the principal geometric dimensions

\[ \frac{W_{\mathrm{mp}}}{K_{01}D\left(V_{fe}+V_{\mathrm{ok}}\right)} = \frac{Y^3(1-X)} {\left(2.36\cdot X^4+X^2Y+X^2+Y\right) \ln\left(\frac{1+\sqrt{1-X^2}}{X}\right)}, \tag{10} \]

here

\[ K_{01}=\frac{4K_a\rho_0K_0^2}{3}. \]

The curves \(\Gamma(X,Y)\) were plotted as a function of \(X\) for various constant values of \(Y\) (Fig. 2). From the curves \(\Gamma(X,Y)\), and also from (10), it is evident that

Fig. 2. Geometric factor characterizing the energy of the stray magnetic field of a valve-type electromagnet.

to a large extent the character of the variation of the geometric factor is influenced by \(Y\). For \(Y>3\), the sharp increase in the geometric factor of the stray magnetic field, and consequently the increase in the specific magnetic energy stored in the stray field of the valve-type electromagnet—

of this type suggests the inadvisability of using \(Y \geqslant 3\). In addition, the maximum of the geometric factor of the stray magnetic field occurs at \(X = 0.25\) for \(\alpha = 1\).

The curves (Fig. 2) are of value for a visual representation of the nature of the dependence of the specific energy of the stray magnetic field on the ratios of the principal geometric dimensions of a valve-type direct-current electromagnet.

Fig. 3. Model of an electromagnet with \(X = 0.39;\ Y = 1.18\).

Fig. 4. Model of an electromagnet with \(X = 0.59;\ Y = 1.4\).

To verify the theoretical conclusions, a number of electromagnet models were fabricated with different values of \(X\) and \(Y\). The models were calculated taking into account the performance of a definite mechanical

Fig. 5. Distribution of magnetic flux along the core length \(l_c\) for model No. 1 at an initial air gap \(\delta_{\mathrm{n}} = 6\) mm. 1—\(IW = 1580\) ampere-turns; 2—\(IW = 2380\) ampere-turns; 3—\(IW = 3160\) ampere-turns; 4—\(IW = 3960\) ampere-turns.

operation. As an example, in Fig. 3 and Fig. 4 two models are shown in which the coil height is the same and equal to 6.5 cm, while the core diameters are different. Thus \(d_1 = 2.2\) cm, \(d_2 = 2.6\) cm.

Figures 5 and 6 show, respectively, the curves of flux distribution along the core lengths for the first model with \(X = 0.39\) and \(Y = 1.18\), and for the second—with \(X = 0.59\) and \(Y = 1.4\). For the same work,

Fig. 6. Distribution of the magnetic flux along the core length \(l_c\) for model No. 2 at an initial air gap \(\delta_n = 6\) mm.
\(1\)—\(IW = 978\) A-turns; \(2\)—\(IW = 1210\) A-turns; \(3\)—\(IW = 1485\) A-turns; \(4\)—\(IW = 2000\) A-turns; \(5\)—\(IW = 2150\) A-turns.

\(\sigma_1 \cong 2.7\), and \(\sigma_2 \cong 2\). Moreover, for the first model the copper consumption was 30% greater than for the second, while the steel consumption was 10% less.

Conclusions

a) The use of \(X = 0.25\) and values close to it in a valve-type electromagnet, as follows from Fig. 2, is inadvisable. In designing an electromagnet, the value of \(X\) should be chosen to the right of \(X = 0.25\), since choosing it to the left leads to a sharp increase in the consumption of such an expensive and scarce material as copper. The value of \(Y\) should be limited to the range 2–3, since any further increase leads to a sharp increase in the stray field.

b) Experimental verification confirmed the correctness of the theoretical conclusions. Thus, electromagnets in which \(X\) was chosen the same as in existing industrial electromagnets (\(X = 0.32\text{–}0.45\)) have larger \(\sigma\) than electromagnet models with \(X = 0.55\text{–}0.6\). Moreover, models with \(X = 0.32\text{–}0.45\) have increased consumption of active materials.

REFERENCES

1. F. A. Stupel’. Calculation and Design of Electromagnetic Relays. Gosenergoizdat, 1950.

2. A. V. Gordon and A. G. Slivinskaya. DC Electromagnets. Gosenergoizdat, 1960.

3. N. E. Lysov. Calculation of Electromagnetic Mechanisms. Gosenergoizdat, 1949.

4. M. G. Koblenz. Choice of optimal copper and steel ratios in DC apparatus. Vestnik elektropromyshlennosti, No. 11, 1961.