OPTIMAL RATIOS OF THE MAIN DIMENSIONS OF A VALVE-TYPE DC ELECTROMAGNET

A. V. KURNOSOV

(Submitted by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The choice of the design of a direct-current electromagnet according to the design factor [1] \(K=\dfrac{\sqrt{F_{\mathrm{n}}}}{\delta_{\mathrm{n}}}\), where \(F_{\mathrm{n}}\) is the initial tractive force of the electromagnet and \(\delta_{\mathrm{n}}\) is the initial working air gap, does not provide grounds for judging the degree of rational use of active materials or the approach to the most advantageous ratios of the principal geometric dimensions of the electromagnet, since variations in \(K\) lie within a wide range. Thus, for a valve-type electromagnet, \(K\) varies within the limits \((2.6—26)\) [2]. The ratios of the core diameter to the outer diameter of the coil, \(d/D\), for existing designs of valve-type electromagnets in DC contactors usually lie within the limits \((0.32—0.45)\), but these ratios are far from optimal [3].

As a first approximation, the basis for selecting the ratios between the main dimensions may be taken as the ratio

\[ \frac{A_y}{V_{\mathrm{ok}}+V_{fe}} \tag{1} \]

and the values of \(d/D\) and \(l_k/D\) (\(l_k\) is the length of the coil, Fig. 1) corresponding to the maximum of expression (1) may be found, where
\(A_y=F_{\mathrm{n}}\delta_{\mathrm{n}}\) is the conventional work [1, 2];
\(V_{\mathrm{ok}}\) is the volume of the coil winding;
\(V_{fe}\) is the volume of the steel of the magnetic circuit.

Having determined the maximum of expression (1), we thereby determine the optimal geometric ratios at which the electromagnet develops the greatest force for a given working air gap.

In deriving the basic ratios, we adopt the following assumptions:

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

b) when the armature of the electromagnet is in the unpulled position, saturation of the steel is neglected;

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

d) after operation, the electromagnet remains energized for a long time and reaches a steady temperature rise \(\tau\) above the ambient medium;

e) the current in the winding remains unchanged during the motion of the armature.

Under the adopted assumptions, the pull-in ampere-turns \(AW_{\mathrm{tr}}\) in the open position of the armature are determined as

\[ \frac{IW_{\mathrm{heat}}}{1.65}=\frac{\Phi_\delta}{G_{\mu\delta}}=\frac{B_\delta \delta_{\mathrm{n}}}{\sigma_{\mathrm{v}}\mu_0}, \tag{2} \]

where

\[ \frac{IW_{\mathrm{heat}}}{1.65}=AW_{\mathrm{tr}} \,[3], \]

\(IW_{\mathrm{heat}}\) — ampere-turns permissible with respect to heating;

\(\Phi_\delta\) — magnetic flux in the working air gap;

\(B_\delta\) — induction in the working air gap;

\(\mu_0\) — magnetic permeability of air;

\[ \sigma_{\mathrm{v}}=\frac{G_{\mu\delta}}{G_{\mu\mathrm{t}}} \]

— a coefficient taking into account the bulging of the magnetic lines of force. In existing electromagnet designs its value is taken as \((1.2—1.5)\) [4];

\(G_{\mu\mathrm{t}}\) — permeance to the magnetic flux of the end part of the working air gap.

The losses in the winding of any direct-current electromagnet under steady-state conditions can be found as

\[ P=I^2\rho\,\frac{l_{\mathrm{m.avg}}}{q_{\mathrm{m}}}, \tag{3} \]

where \(I\) — steady-state value of the current in the coil;

\(W\) — number of turns of the coil;

\(\rho\) — resistivity of the winding copper at the steady heating temperature of the winding;

\(l_{\mathrm{m.avg}}\) — mean length of a winding turn;

\(q_{\mathrm{m}}\) — cross section of the copper conductor.

Expression (3) can be written as

\[ P=\frac{(IW_{\mathrm{heat}})^2 l_{\mathrm{m.avg}}}{K_{\mathrm{fill}}Q_0}, \tag{4} \]

where \(K_{\mathrm{fill}}\) — winding fill factor;

\(Q_0\) — cross section of the winding window.

On the other hand, the losses in the winding at a steady temperature rise \(\tau\) of the winding above the ambient medium can be expressed as

\[ P=K_{\mathrm{to}}S_0\tau, \tag{5} \]

where \(K_{\mathrm{to}}\) — heat-transfer coefficient from the surface being blown;

\(S_0\) — external surface of the electromagnet washed by air.

From expression (2) we find \(\delta_{\mathrm{n}}\) as

\[ \delta_{\mathrm{n}}=IW_{\mathrm{heat}}\frac{\sigma_{\mathrm{v}}\mu_0}{1.65B_\delta}. \tag{6} \]

Equating (4) and (5), we find an expression for \(IW_{\mathrm{heat}}\) as

\[ IW_{\mathrm{heat}}=\sqrt{\frac{K_{\mathrm{to}}S_0\tau K_{\mathrm{fill}}Q_0}{\rho l_{\mathrm{m.avg}}}}. \tag{7} \]

We substitute the value for \(IW_{\mathrm{gor}}\) into (6); the value found for \(\delta_{\mathrm{n}}\) is substituted into the expression for \(A_y\); the value \(F_{\mathrm{n}}\) is found in the first approximation by Maxwell’s formula; then \(A_y\) is found as

\[ A_y=\frac{B_\delta Q_{fe}}{2\cdot 1.65\sigma_{\mathrm{v}}} \sqrt{\frac{K_{\mathrm{to}}K_{\mathrm{z}}S_0Q_0}{\rho l_{\mathrm{m.av}}}} =KQ_{fe}\sqrt{\frac{S_0Q_0}{l_{\mathrm{m.av}}}}, \tag{8} \]

where

\[ K=\frac{B\delta}{2\cdot 1.65\sigma_{\mathrm{v}}} \sqrt{\frac{K_{\mathrm{to}}K_{\mathrm{z}}\tau}{\rho}}. \]

Expression (8) may be written as

\[ \frac{A_y}{K}=Q_{fe}\sqrt{\frac{S_0Q_0}{l_{\mathrm{m.av}}}}. \tag{8,a} \]

Expression (8,a) gives the dependence of the conditional work on the geometrical parameters of the electromagnet.

To evaluate electromagnets by the limiting loading of active materials, let us consider the ratio of the conditional work to the sum of the volumes of the active materials. To this end, we divide the left- and right-hand sides of expression (8,a) by the sum of the volumes of the active materials:

\[ \frac{A_y}{K\left(V_{fe}+V_{\mathrm{ok}}\right)} = \frac{ Q_{fe}\sqrt{\dfrac{S_0Q_0}{l_{\mathrm{m.av}}}} }{ V_{fe}+V_{\mathrm{ok}} }. \tag{9} \]

Fig. 1. Sketch of the electromagnet.

Expression (9) is suitable for analyzing any type of traction electromagnet. Let us apply (9) to the analysis of a valve-type electromagnet (Fig. 1).

In expression (9), \(S_0\) may be represented as \(S_0=S_{\mathrm{n}}+\alpha S_{\mathrm{v}}\), where \(S_{\mathrm{n}}\) is the outer surface of the coil and \(S_{\mathrm{v}}\) is the inner surface of the coil. We neglect the end cooling surface of the coil. On the basis of investigations of electromagnet coils, M. G. Koblents [3] proposed an empirical formula for determining the permissible losses in dc coils:

\[ P=K_{\mathrm{to}}\left(S_{\mathrm{n}}+\alpha S_{\mathrm{v}}\right), \tag{10} \]

where \(\alpha\) is an empirical coefficient depending on the type of coil manufacture and varying within the limits \((0.9\text{--}2.4)\). Expressing, on the right-hand side of expression (9), all geometrical parameters of the electromagnet through its principal dimensions \((d,D,l_k)\) and referring them to some one dimension (in our case \(D\)), we obtain

\[ \frac{A_y}{K\left(V_{fe}+V_{\mathrm{ok}}\right)\sqrt{D}} = \frac{ X^2y\sqrt{\dfrac{(1-X)(1+\alpha X)}{1+X}} }{ 2.36X^4+X^2(1+y)+y }, \tag{11} \]

where \(X=d/D;\quad y=l_k/D\) (Appendix 1).

We shall denote the right-hand side of expression (11) by \(\Gamma\) and call it the geometric factor. When the coil is wound on a metal sleeve, \(\alpha\) is found to be equal to 1.7 [4]. Substituting \(\alpha = 1.7\) into (11) and assigning different values to \(X\) and \(y\), we obtain the curves of the dependence \(\Gamma = f(X,y)\) (Fig. 2). The maximum value of the geometric factor

Fig. 2. Curves of the dependence \(\Gamma = f(X,y)\) for \(\alpha = 1.7\).

\(\Gamma_{\max}\) lies within the range \((0.55 \div 0.7)\) of variation of \(X\). Figure 3 gives the curves of the dependence \(\Gamma_{\max} = f(X_{\mathrm{opt}})\) for different values of \(\alpha\). There is no clearly expressed maximum with respect to \(y\); however, as is evident from Fig. 2, the growth of \(\Gamma(X,y)\) for \(y > 3\) slows sharply. The value of \(y\) may be recommended within the range \((1-3)\). Figure 4 presents the curves of the dependence \(X_{\mathrm{opt}} = f(y)\).

In order to proceed to the direct design of the electromagnet, it is necessary to find one of its geometric dimensions. Having specified \(X_{\mathrm{opt}}\) from the curves in Fig. 3, we find \(\Gamma_{\max}\); from the curves in Fig. 4 we find the value of \(y\). The sum of the volumes of the active materials in the denominator of expression (11) must be expressed through \(X\) and \(y\).

\[ V_{fe} + V_{\mathrm{ok}} = \frac{\pi d^2}{4}\, l_{fe} + \frac{\pi l_k}{4}\left(D^2 - d^2\right), \]

\[ l_{fe} = 2l_k + D + \frac{3\pi d^2}{4D}, \]

\[ \frac{3\pi d^2}{4D} = 3\Delta, \]

where \(\Delta\) is the thickness of the yoke (Appendix 1).

\[ V_{fe} + V_{\mathrm{ok}} = aD^3, \quad \text{where } a = \frac{\pi}{4}\left[X^2\left(\frac{3\pi}{4X^2} + y + 1\right) + y\right]. \]

Fig. 3. Curves of the dependence \(F' = f(X_{\mathrm{opt}})\) for various values of \(a\).

Fig. 4. Curves of the dependence \(X_{\mathrm{opt}} = f(Y)\) for various values of \(a\).

Substituting the value \((V_{fe}+V_{ok})\) into (11), we find the expression for \(D\) as

\[ D=\sqrt[7]{\frac{A_y^2}{\Gamma^2 a^2 K^3}} . \tag{12} \]

To verify the conclusions obtained above and the calculated values, two models of electromagnets were investigated. Both models were calculated for one and the same initial force of 1 kg. The first model was made in accordance with the calculations set forth above. For the second model, the ratios between the principal dimensions were taken to be the same as in the existing KP 21/33 contactor. The magnetic circuit of both models was made of ARMCO steel. For comparison, the KP21/33 contactor was tested. The calculated and experimental data are summarized in Table 1.

Table 1

\(F_n\) is the initial tractive force of the electromagnet in the absence of a pole shoe;

\(F_{np}\) is the initial tractive force of the electromagnet in the presence of a pole shoe; for all three electromagnets the diameter of the pole shoe is 35 mm.

The copper consumption for the first model, in comparison with KP 21/33, was 70%; the copper consumption for the second model, in comparison with KP 21/33, was 110%; the steel consumption for the first model, in comparison with KP 21/33, was 115%; the steel consumption for the second model was the same as for KP 21/33.

The test results confirmed the correctness of the theoretical conclusions. The comparative tractive characteristics of the electromagnets are shown in Fig. 5.

Conclusions

1. On the basis of the considerations presented above and the construction of the curves \(\Gamma=f(d/D,\ l_k D)\), the optimal geometric ratios in valve-type electromagnets should be taken as \(d/D=(0.55\text{--}0.7)\), \(l_k/D=(1\text{--}3)\).

Fig. 5. Curves of the dependence of the tractive forces of electromagnets:
1 — \(F=f(\delta)\) for \(d=28\) mm;
\(1'\) — for \(d_{\mathrm{p}}=35\) mm;
2 — \(F=f(\delta)\) for \(d=21.4\) mm;
\(2'\) — for \(d_{\mathrm{p}}=35\) mm;
3 — \(F=f(\delta)\) for \(d_{\mathrm{p}}=35\) mm.

1. The fulfillment of these ratios will make it possible to save a significant amount of scarce winding copper and to increase the utilization of active materials.

Appendix 1

\[ \frac{A_y}{K\left(V_{fe}+V_{ok}\right)\sqrt{D}} = \frac{Q_{fe}\sqrt{\dfrac{S_0Q_0}{l_{\mathrm{m.avg}}}}}{\left(V_{fe}+V_{ok}\right)} . \tag{9} \]

We take the width of the electromagnet yoke to be equal to \(D\) (Fig. 1).

\[ Q_{fe}=\pi d^2/4,\qquad Q_0=l_k/2\,(D-d),\qquad l_{\mathrm{m.avg}}=\pi/2\,(D+d), \]

\[ V_{fe}=Q_{fe}l_{fe}=\pi d^2/4\,(2l_k+D+3\Delta),\quad \text{where }\Delta=\pi d^2/4D, \]

\[ V_{ok}=Q_0l_{\mathrm{m.avg}}=\pi/4\,l_k(D^2-d^2), \]

\[ S_0=S_n+\alpha S_v=\pi Dl_k+\alpha\pi dl_k=\pi l_k(D+\alpha d). \]

Substituting the expressions found into (9) and referring them to \(D\), we obtain expression (11).

REFERENCES

1. Roters. Electromagnetic Mechanisms. Gosenergoizdat, 1949.
2. A. V. Gordon, A. G. Slivinskaya. Direct-Current Electromagnets. Gosenergoizdat, 1960.
3. M. G. Koblenz. Selection of optimal ratios of copper and steel in direct-current apparatus. Bulletin of the Electrical Industry, No. 11, 1961.
4. F. A. Stupel. Calculation and Design of Electromagnetic Relays. Gosenergoizdat, 1950.