V. M. KHRUSHCHOV.

On the Calculation of Single-Phase Commutator Motors.

In the calculation of commutator motors of alternating current, in addition to the usual general requirements, there is also one specific requirement, namely, that the electromotive force of pulsation in the short-circuited coil, in order to avoid severe sparking at starting, should not exceed certain values. This requirement imposes a limitation on the magnetic flux; whereas in other motors, with an increase in their power, a correspondingly larger magnetic flux can be chosen, here this is impossible by virtue of the above. In this particular feature of alternating-current commutator motors one should also seek the reason why they cannot be built for unlimitedly large powers. Already in the calculation of motors of medium power this difficulty makes itself felt, and indeed more strongly if the motor is to operate at a speed close to synchronism, and less if it is independent of synchronism. Therefore, for example, a motor with series excitation can be easier to construct for high power than a repulsion motor.

The mechanical power of the motor in watts is equal to:

\[ W_m = I_2 \cdot E_a \cos(E_a, I_2) = \frac{p}{a} \cdot \frac{n}{60} \cdot N \cdot I_2 \frac{\Phi_{\max}}{\sqrt{2}} \cos \varphi_2 \cdot 10^{-8} = \]

\[ 4.44 \cdot c_2 \, D \cdot AS_2 \cdot \Phi_{\max} \cdot \cos \varphi_2 \cdot 10^{-8} \qquad\qquad\qquad 1) \]

The electromotive force of pulsation in the short-circuited coil at starting, for the same flux, is equal to

\[ e_k = 4.44 \cdot c_1 w_k \cdot \Phi_{\max} \cdot 10^{-8} \ \text{V}. \]

In these equations the symbols denote:

\(I_2\)—current in the armature,

\(E_a\)—electromotive force of rotation,

\(p\)—number of pairs of poles,

\(a\)— “ ” branches of the armature,

\(n\)—number of revolutions of the armature,

\(N\)—number of armature bars,

\[ c_2 = \frac{p \cdot n}{60} \]
—number of periods of rotation,

\(D\)—diameter of the armature in cm.,

\[ AS_2=\frac{N\cdot I_2}{\pi D} \]

—the linear loading of the armature,

\(c_1\)—number of periods of the network,

\(w_k\)—the greatest number of series-connected short-circuited turns.

From equations 1 and 2 the armature diameter is obtained

\[ D=\frac{c_1}{c_2}\cdot\frac{w_k}{e_k}\cdot\frac{W_m}{AS_2\cos\varphi_2} =\frac{1}{U}\cdot\frac{736}{\eta_m\cos\varphi_2}\cdot\frac{P}{AS_2}\ldots\ldots\ldots 3) \]

Here:

\[ U=\frac{c_2}{c_1}\frac{e_k}{w_k}, \]

\(\eta_m\)—mechanical efficiency,

\(P\)—power in horsepower.

Equation 3 shows that the armature diameter increases with increasing motor power and decreases with increasing \(AS_2\) and \(U\).

Substituting in eq. 2

\[ \Phi_{\max}=\frac{\pi D}{2p}\alpha_i l_i B_v \]

we obtain:

\[ D l_i= \frac{e_k\cdot 2p\cdot 10^8}{4.44\pi\alpha_i c_1 w_k B_v} = \frac{e_k\cdot 2\cdot c_2\cdot 60\cdot 10^8}{4.44\pi\alpha_i c_1 w_k B_v n} = \frac{8.6\cdot 10^8}{\alpha_i}\cdot\frac{U}{B_v n} \ldots\ldots\ldots 4) \]

here:

\(l_i\)—ideal armature length,

\(\alpha_i\)—field form factor,

\(B_v\)—density of the lines of force in the air.

From equation 4 it follows that, if \(U\), \(B_v\), and \(n\) are unchanged, then the product \(D l_i\) (independently of the power) is constant. And since, according to eq. 3, \(D\) increases with increasing power, then \(l_i\), consequently, in this case must decrease.

The weight of the copper lying in the active layer of the rotor is equal to

\[ G'_{m.r}=8.9\frac{AS_2\cdot\pi D\cdot l_i}{s_a}\cdot 10^{-5}, \]

substituting \(D l_i\) from equation 4, we obtain

\[ G'_{m.r}=\frac{240.2}{\alpha_i}\,10\cdot\frac{U\cdot AS_2}{B_v\cdot n\cdot s_a}, \ldots\ldots\ldots 5) \]

here:

\(s_a\)—current density per \(1\text{ mm}^2\)

The cross-section of the iron of the teeth over the length \(\alpha_i\tau\)

\[ b_z=\frac{\alpha_i\tau l_i B_v}{B_z}. \]

Cross-section of the slots over the length \(\alpha_i \tau\)

\[ b_{\mathrm{sl}}=\alpha_i i_i \tau-\frac{k_1}{k_2}\cdot \frac{\alpha_i i_i \tau B_{\mathrm{v}}}{B_{\mathrm{z}}} =\alpha_i i_i \tau\left(1-\frac{k_1}{k_2}\cdot \frac{B_{\mathrm{v}}}{B_{\mathrm{z}}}\right), \]

\(k_2\) is a coefficient allowing for the insulation between the iron sheets.

\(\tau\)—pole pitch;

\[ k_1=\frac{l_i}{l} \]

—the ratio of the ideal armature length to the actual one;

\(B_{\mathrm{z}}\)—mean induction in the teeth.

Tooth height

\[ h_{\mathrm{z}}= \frac{\alpha_i \tau AS_2 l_i}{f_{\mathrm{m}} s_{\mathrm{a}} b_{\mathrm{sl}}} = \frac{AS_2}{f_{\mathrm{m}}s_{\mathrm{a}} \left(1-\frac{k_1}{k_2}\frac{B_{\mathrm{v}}}{B_{\mathrm{z}}}\right)}, \]

where:

\(f_{\mathrm{m}}\)—copper space factor.

Weight of the iron of the rotor teeth

\[ G_{\mathrm{Fe,r}}= 7.8\cdot 2p\frac{b_{\mathrm{z}}h_{\mathrm{z}}}{\alpha_i} = 7.8\frac{\pi D l_i}{k_3}\, \frac{AS_2 B_{\mathrm{z}}\cdot 10^{-5}} {f_{\mathrm{m}}s_{\mathrm{a}} \left(1-\frac{k_1}{k_2}\frac{B_{\mathrm{v}}}{B_{\mathrm{z}}}\right)} = \]

\[ = \frac{210.5\cdot 10^{-3}}{\alpha_i}\, \frac{U\cdot AS_2} {f_{\mathrm{m}}s_{\mathrm{a}} \left(B_{\mathrm{z}}-\frac{k_1}{k_2}B_{\mathrm{v}}\right)n} \ldots\ldots\ldots\ldots\ldots\ldots (6) \]

Equations 4–6 show that the dimensions of the active layer do not depend directly on the power and may be changed chiefly by changing \(U\). But since \(U\) cannot exceed certain values, it is evident that, for alternating-current commutator motors, in contrast to other motors, there is a power limit above which they cannot be constructed.

The weight of the copper and iron of the active layer of the stator may be represented by formulas analogous to 5 and 6.

Volume of stator iron (without teeth)

\[ V_{\mathrm{Fe,s}}= \frac{\pi}{4}\left(D_1^2-D^2\right)l = \frac{\pi}{4}\frac{D^2l_i}{k_1}\left(k_3^2-k_4^2\right). \]

Substituting the values of \(Dl_i\) and \(D\) from Eqs. 3 and 4, we obtain

\[ V_{\mathrm{Fe,s}}= \frac{27\cdot 10^{-8}} {4\alpha_i k_1 \cos\varphi^2 \eta B_{\mathrm{v}} AS_2} \cdot W\cdot \left(k_3^2-k_4^2\right), \]

where:

\[ k_3=\frac{D_1}{D}=\frac{D+2a+h_{\mathrm{z}}}{D}. \]

\[ k_4=\frac{D_2}{D}\cdot\frac{D+2h_3}{D} \]

\(D_1,\ D_2,\ a\), and \(h_3\), see Fig. 1.

Since

\[ a=\frac{\Phi_{\max}}{2k_2 l B_{ac}} =\frac{\alpha_i\pi k_1}{4k_2}\cdot\frac{D}{p}\cdot\frac{B_v}{B_{ac}}, \]

then

\[ V_{ac}=\frac{27\cdot 10^8\pi}{4k_2\cos\varphi_2}\cdot \frac{W_m}{p\,n\,B_{ac}\,AS_2} \left(1+\frac{\alpha_i\pi k_1}{4k_2}\cdot \frac{B_v}{p\,B_{ac}}+\frac{h_{ac}}{D}\right), \]

where

\(B_{ac}\)—the induction in the stator iron.

The volume of the rotor iron may be represented by a similar formula:

\[ V_{ir}=\frac{27\cdot 10^8\pi}{4\cdot k_2\cos\varphi_2}\cdot \frac{W_m}{p\,n B_{ar}AS_2} \left(1-\frac{\alpha_i\pi k_1}{4\cdot k_2p}\cdot\frac{B_v}{B_{ar}}-\frac{h_{zr}}{D}\right). \]

If \(B_{ac}\) does not differ appreciably from \(B_{ar}\), and \(h_{zc}\) from \(h_{zr}\), then the weight of the connecting iron can be represented by the formula:

\[ G''_{ir}=\frac{5.52\cdot 10^5}{k_2\cos\varphi_2}\cdot \frac{W_m}{c_2 B_v AS_2}\ldots\ldots\ldots\ldots 7) \]

Fig. 1

This formula, even with a relatively large difference between \(B_{ac}\) and \(B_{ar}\), and also between \(h_{zc}\) and \(h_{zr}\), has sufficient accuracy, since the sum of the last two terms in the expressions \(V_{ac}\) and \(V_{ir}\) is always considerably less than unity. Formula 7 shows that the weight of the connecting iron does not depend on \(U\). The weight of the connecting iron per force is

\[ g''_{ir}=\frac{4.06\cdot 10^8}{k_2\eta_m\cos\varphi_2}\cdot \frac{1}{c_2B_v\,AS_2}\ldots\ldots\ldots\ldots 7a) \]

it does not directly depend either on the power or on the number of revolutions and decreases with an increase in \(c_2\), \(B_v\), and \(AS_2\).

Expression 7a makes it possible easily to determine the percentage losses in the connecting iron and shows that they do not directly depend on the power.

The weight of the connecting copper of the rotor is equal to

\[ G'''_{mr}=8.9\frac{AS_2 lc}{s_a}\cdot\pi D\cdot 10^{-5} =43.9\cdot 10^{-5}\cdot k_c\cdot\frac{AS_2D^2}{ps_a} \]

\[ =\frac{43.9\cdot 10^{-5}k_c}{\cos^2\varphi_2}\cdot \frac{W_m^2}{p\,s_aAS_2U^2}\ldots\ldots\ldots\ldots 8) \]

where:

\[ k_c=\frac{l}{\tau} \]

is the ratio of the length of the connecting part of the conductor to the pole pitch.

Expression 8 shows that the weight of the connecting copper increases in proportion to the square of the power, and that its reduction is achieved by increasing \(AS_2\) and still more by increasing \(c\).

In any motor in which the material is to be well utilized, the ratio of the amount of connecting copper to that lying in the active layer must not exceed a certain value. Let us denote this ratio by \(k_6\); then

\[ \frac{43.9 \cdot 10^{-5} k_c}{\cos^2 \varphi_2}\, \frac{W_m^2}{p \cdot s_a \cdot AS_2 \cdot U^2} = k_6\, \frac{240.2 \cdot 10^3 \, U \cdot AS_2}{\alpha_i \cdot B_v \cdot n \cdot s_a} \]

\(U\), for which, for given \(W_m, AS_2, B_v, n, s_a\), and \(p\), \(k_6\) will have the assigned value, is determined from the formula:

\[ U = 2.546 \cdot 10^2 \sqrt[3]{\frac{k_c}{k_6}\, \frac{\alpha_i}{\eta_m^2 \cos^2 \varphi_2}\, \left(\frac{P \cdot n}{AS_2}\right)^2 \frac{B_v}{c_2}} \qquad \ldots\ldots\ldots\ldots 9 \]

The total weight of the rotor copper, with the aid of Eqs. 9 and 5, is represented by the formula

\[ G_{m.r.} = (1+k_6)\, \frac{80}{s_a}\, \sqrt[3]{ \frac{5}{\cos^2 \varphi_2}\, \frac{k_c}{k_6}\, \left(\frac{\pi W_m}{\alpha_i B_v \cdot n}\right)^2 \frac{AS_2}{p} } \qquad \ldots\ldots\ldots\ldots 10) \]

This expression, with variable \(k_6\), reaches a minimum for \(k_6 = 0.5\). The weight of copper per horsepower is expressed as

\[ g_{m.r.} = \frac{6120(1+k_6)}{s_a} \sqrt[3]{ \frac{k_c}{k_6}\, \frac{AS_2}{P \cdot c_2 n} \left( \frac{1}{\alpha_i \eta_m \cos \varphi_2 \cdot B_v} \right)^2 } \qquad \ldots\ldots 10a) \]

for example, for \(P = 75,\quad n = 750,\quad AS_2 = 250,\quad B_v = 5000,\quad \alpha_i = 0.8;\)

\[ \eta_m \cos \varphi_2 = 0.95,\quad s_a = 3,\quad k_c = 1.5,\quad k_6 = \frac{k_c}{0.8} \]

\[ g_{mr} = 1.04 \ \text{kg/hp}. \]

The weight of copper per horsepower is inversely proportional to the cube root of the power and of the number of revolutions, and can be reduced by increasing \(c_2\) and \(B_v\) and by decreasing \(AS_2\).

As formulas 7 and 10 show, the weight of the connecting iron decreases with increasing \(AS_2\), while the weight of copper increases; substituting in Eq. 6 the value of \(U\) from 9, it can be shown that the weight of the iron of the active layer is represented by the formula:

\[ g'_z = \frac{0.88}{ f_m(1+k_6) \left( \frac{B_z}{B_v} - \frac{k_1}{k_2} \right) } \cdot g_{mp} \qquad \ldots\ldots\ldots\ldots 6a) \]

and, consequently, also increases with increasing \(AS_2\). Therefore there exists some value of \(AS_2\) at which the cost of the active material of the motor will be the least. By means of uncomplicated transformations one can find the mathematical expression for this \(AS_2\), which shows that this minimum is attained…

is reached at high values of \(AS_2\). For a motor of 100 horse power and 750 revolutions (with the cost of iron 50 kopecks per kilogram and copper 2 rubles per kilogram), the following cost of the active material \(P_a\) is obtained as a function of \(AS_2\).

As is evident from this table, the minimum is reached at about \(AS_2 = 400\) and is very flat; thus, at \(AS_2 = 200\) the cost of the active material is only 0.65% greater than the minimum.

For three-phase asynchronous motors one can derive formulas analogous to \(6_a\), \(7_a\), and \(10_a\), differing from them only in the coefficients. With the aid of these formulas it can be shown that, for medium powers, under identical conditions, the weight of the connecting iron in a series motor is approximately 57%, and in repulsion and Latour-Winter Eichberg motors 62%, greater than in an asynchronous motor. The weight of copper and iron of the active layer in a series motor is approximately 12%, in a repulsion motor 15%, and in a W. E. L. motor 35% greater than in an asynchronous motor.

The current in the bar is equal to

\[ I_a = \frac{\pi \cdot D \cdot AS_2}{N} = \frac{\beta \cdot AS^2}{2 z \cdot k_7}, \ldots \ldots \ldots \ldots \ldots \ldots 11) \]

where:

\[ \beta \]
— the width of a commutator segment with insulation (commutator pitch),

\(z\)—the number of turns in the element coil,

\[ k_7 = \frac{D_k}{D}, \]
\(D_k\)—the diameter of the commutator.

The current in the bar does not depend directly on the power.

The length of the commutator

\[ l_k = k_8 l_{kn} = k_8 \frac{2 I_2}{s_u \cdot m \cdot \beta \cdot p_k} = \frac{k_8 \cdot 4 a \cdot I_a}{s_u \cdot m \cdot \beta \cdot p_k} = \frac{k_8}{k_7} \cdot \frac{2 \cdot AS_2}{z \cdot s_u \cdot m \cdot k_k}, \]

where:

\(l_k\)—the total length of the commutator,

\(l_{kn}\)—the useful length of the commutator,

\[ k_s = \frac{l_k}{l_{kn}}, \]

\(s_u\)—the current density under the brush per 1 cm.;

\(m\)—the number of commutator segments covered by the brush,

\(p_b\)—the number of brush rows,

\[ k_k=\frac{p_b}{a}. \]

This expression shows that the length of the commutator does not depend directly either on the power or on the number of revolutions.

The surface of the commutator

\[ O_k=\pi D_k l_k=\frac{2\pi k_8 AS_2}{z\,s_u\,m\,k_k}\,D = \frac{2\pi k_8}{\cos\varphi_2\,s_u\,z\,m\,k_k}\cdot\frac{W_m}{U}. \]

The weight of the commutator

\[ G_{kl}=8.9\cdot 10^3\, \frac{\gamma\,2\pi\,k_8 h_k}{\cos\varphi_2\,s_u\,z\,m\,k_k}\cdot \frac{W_m}{U^2}, \ldots\ldots\ldots\ldots\ldots 12) \]

where

\(\gamma\)—the ratio of the width of the segment to the commutator pitch,

\(h_k\)—the height of the commutator segment.

The weight of the commutator per 1 horsepower

\[ g_{kl}=41.1\, \frac{k_8\gamma}{\cos\varphi_2\,\eta_m}\cdot \frac{1}{s_u\,z\,m\,k_k}\cdot \frac{h_k}{U} \ldots\ldots\ldots\ldots\ldots 12a) \]

for example, with \(U=3.5,\ s_u=13,\ z=1,\ m=2,\ k_k=2,\ h_k=3.5,\ k_8=1.25,\)

\[ \cos\varphi_2\,\eta_m=0.92 \]

\[ g_{kl}=0.91. \]

Expression \(12_a\) shows that, from the standpoint of cheapness of the commutator, the highest possible values of \(U\) are desirable and, consequently, as is seen from formula 9, the lowest possible values of \(AS_2\).

The electromotive force of rotation of the rotor \(E_a\) may be represented by the formula

\[ E_a=U\cdot\frac{Kz}{\pi a} = U\frac{D_k z}{\beta a}, \ldots\ldots\ldots\ldots\ldots 13) \]

where

\[ K=\frac{N}{2z} \]

is the number of commutator segments.

The percentage losses in the commutator transition are equal to

\[ V_{kl}= \frac{200\,\triangle P}{E_a\cos\varphi_2} = \frac{200}{\cos\varphi_2}\cdot \frac{\triangle P}{U}\cdot \frac{\beta a}{zD_k}, \ldots\ldots\ldots\ldots\ldots 14) \]

where

\(\triangle P\)—the voltage drop at the transition between the brush and the commutator.

Equations 12 and 13 show that, in order to increase the electromotive force of rotation of the rotor and to reduce the losses in the commutator transition, it is necessary to make the diameter of the commutator as large as possible and the width of the commutator segment and the number of parallel branches of the armature as small as possible.

From Eqs. 13 and 14, with the aid of expression 3, we obtain

\[ E_a = k_7 \frac{W_m z}{AS_2 \beta a \cos \varphi_2}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 13) \]

and

\[ V_{\mathrm{cl}} = \frac{200 \cdot \triangle P}{\cos^2 \varphi_2 \cdot k_7}\, \frac{AS_2 \beta a}{W_m z} \ldots\ldots\ldots\ldots\ldots\ldots\ldots 14a) \]

These equations show that, for unchanged \(k_7\), \(AS_2\), \(\beta\), \(a\), and \(z\), with increasing power \(E_a\) increases, while \(V_{\mathrm{cl}}\) decreases. For example, for \(P = 100\) h.p., \(n = 750\) rpm, \(AS_2 = 300\), \(z = 1\), \(\beta = 0.5\), \(k_7 = 0.8\), \(a = 4\), \(\triangle P = 1.3\),

\[ E_a = 102 \quad \text{and} \quad V_{\mathrm{cl}} = 2.57\%. \]

Since \(a = k \cdot P\), where \(k\) is some coefficient, with increasing \(n\), and consequently (for a given \(n\)) \(c_2 - E_a\), decreases, while \(V_{\mathrm{cl}}\) increases. For a given \(c_2\), engines with a high number of revolutions will have larger \(E_a\) and smaller \(V_{\mathrm{cl}}\).

In the same way it can be shown that the percentage losses in the copper of the rotor may be represented by the formula:

\[ V_{\mathrm{m.r}} = k_{\mathrm{p}} \frac{54 \cdot 10^6(1+k_{\mathrm{b}})}{z} \cdot \frac{U \cdot AS_2 \cdot s_a}{W_m \cdot B_{\mathrm{v}} \cdot n} = k_{\mathrm{p}} \frac{54 \cdot 10^6(1+k_{\mathrm{b}})}{\alpha_i} \cdot \frac{s_a \cos \varphi_2}{D \cdot B_{\mathrm{v}} \cdot n}, \ldots\ldots 15) \]

where \(k_{\mathrm{p}}\) is the ratio of the resistance under alternating current to the resistance under direct current. The percentage losses in copper decrease with increasing power (in inverse proportion to the diameter of the armature). For example, for \(P = 100\) h.p., \(n = 750\), \(AS_2 = 300\), \(B_{\mathrm{v}} = 5000\), \(U = 4.35\), \(s_a = 3\), \(\alpha_i = 0.8\), \(k_{\mathrm{p}} = 1.15\), \(k_{\mathrm{b}} = 1.5\),

\[ V_{\mathrm{mr}} = 2.75\%. \]

As formulas 13a, 14a, and 15 show, in order to obtain larger values of \(E_a\) and small \(V_{\mathrm{cl}}\) and \(V_{\mathrm{mr}}\), the linear loading must be as small as possible.

We can now compare with each other two types of single-phase commutator motors: 1) with a number of periods of rotation independent of synchronism, such as, for example, a series motor; 2) with a number of revolutions close to synchronism, such as, for example, the repulsion motor and the Latour-Winter-Eichberg motors. The distinctive property of the first is that they permit much larger values of \(U\) than the second; this property is above all decisive in the question of the limit of power. Equation 9 may be represented in the form

\[ P = 246 \cdot \eta_m \cos \varphi_2 \cdot \frac{AS_2}{n} \sqrt{\frac{k_{\mathrm{b}}}{k_{\mathrm{c}}}\, \frac{U^3 c_2}{\alpha_i B_{\mathrm{v}}}} \ldots\ldots\ldots\ldots\ldots 16) \]

In order that the weight of copper not be too great, \(k_{\mathrm{b}}\), as Eq. 10 shows, must not exceed certain values; we shall take as the limiting value

\[ k_{\mathrm{b}} = \frac{k_{\mathrm{c}}}{0.8}, \]

substitute,

in Eq. 16, \(\eta_m \cos \varphi_2 = 0.95\), \(AS_2 = 500\), \(n = 250\), \(c_2 = 66.67\), \(\alpha_i = 0.8\), \(B_v = 6000\); then for motors not tied to synchronism, i.e., with

\[ U = \frac{c_2}{c_1}\cdot \frac{e_k}{w_k} = 4.5 \text{–} 20, \]

we obtain the limiting power

\[ P_{\mathrm{gr}} = 5500 \text{ hp}. \]

And for motors tied to synchronism, with \(U = 5\) and \(c_2 = 50\),

\[ P_{\mathrm{gr}} \simeq 470 \text{ hp}, \]

while for \(n = 750\), \(AS_2 = 400\), \(B_v = 5000\), \(c_2 = 50\), \(U = 3.5\), \(\eta_m \cos \varphi_2 = 0.93\), \(\alpha_i = 0.8\),

\[ P_{\mathrm{gr}} \simeq 100 \text{ hp}. \]

These figures do not give the actual practical limits of the power of single-phase commutator motors, since the reason given above for limiting their power is not the only one that is decisive in this question; they indicate only the great difference in this respect that exists between motors tied to synchronism and those independent of it. On the other hand, in the latter, under special conditions, by reducing \(w_k\) to 1, it is possible to take \(U = 7\); then the power limit is pushed still farther back. Since, with a simple parallel winding, the brush width under these conditions is equal to the width of a commutator segment, which from the standpoint of good contact is undesirable, one usually resorts for this purpose to a double parallel winding (with \(a = 2p\)), which of course causes a reduction of \(E_a\) and an increase of \(V_k\).

Eq. 16 shows that the limiting power is inversely proportional to the number of revolutions of the motor, that motors with a high number of periods of rotation have a greater limiting power, and that the latter can be increased by increasing the linear loading.

The ratio \(\dfrac{c_2}{c_1}\) in motors not tied to synchronism is equal to 2–4; in motors tied to it, it is somewhat less than 1. Therefore, for a given \(U\), the number of short-circuited turns \(w_k\) in the former may be 2–4 times greater than in the latter. This circumstance in some cases makes it possible, in motors independent of synchronism, to obtain a smaller commutator weight and smaller losses in it.

The weight of iron and copper does not depend on \(U\), and therefore, under identical conditions, is approximately the same for all commutator motors (the difference is caused only by the difference in \(\alpha_i\) for different kinds of motors). The number of periods of rotation of the mo-

of motors independent of synchronism is usually close to 50; therefore the entire copper and iron weight in motors connected with synchronism is only approximately equal to the weight of those independent of it when they operate on a network with high periodicity; otherwise, here too they are inferior to the latter. If, for example, a network with number of periods \(=25\) is given, then the series motor, under otherwise identical conditions, will have approximately 25% less weight of copper and iron of the teeth, and half the weight of connecting iron, than the repulsion motor.

In calculating single-phase commutator motors, the author considers it most convenient to proceed as follows. From the data \(P\) and \(n\), choose \(c_2\), having specified \(B_v\), \(AS_2\), \(k_b\), and having estimated \(\alpha_i\), \(\cos \varphi_2\), \(\eta_m\), and \(k_c\); from Eq. 9 determine \(U\). Substituting this value of \(U\) into Eq. 3, we obtain the armature diameter, and from Eq. 4 the length. For example: \(P=80\) h.p., \(n=750\), with \(B_v=5000\), \(AS_2=250\), \(k_b=k_c\), \(\alpha_i=0.8\), \((\eta_m \cos \varphi_2)^2=0.9\), \(c_2=50\),

\[ U=2.36\cdot 10^{-2}\sqrt[3]{\left(\frac{80\cdot 750}{250}\right)^2 \frac{5000}{50}}=4.23 \]

\[ D=58.5\ \text{cm.}\quad \text{and}\quad l_i=21\ \text{cm.} \]

From the determined \(U\), for the given \(c_1\), and depending on the starting conditions, the selected \(e_k\) gives \(w_k\), which determines the type of winding. From the selected commutator diameter and the width of the commutator plate, the number of commutator plates and the number of armature bars are obtained. The further calculation proceeds in the usual way.

Vs. Khrushchev.

Tomsk, August 1914.

MISPRINTS.