On the Efficiency of Shunt DC Motors

L. I. Gandzha

One of the most widespread methods for determining the efficiency of shunt DC motors is the no-load method. While sufficiently accurate, this method has, over others, the undoubted advantage of possessing the greatest simplicity.

If a shunt motor rotates at no load, then all the power supplied to it (Fig. 1) is expended on losses in it, equal to the no-load losses:

\[ P_{xx} = U_n \cdot I_0. \tag{1} \]

On the other hand, the no-load losses, or the power supplied to the motor at no load, may be expressed as follows:

\[ P_{xx} = P_{mex} + P_{zhel} + U_n i_{vn} + I_{a0}^{2} r_0, \tag{2} \]

where

\(P_{mex}\) — mechanical losses,

\(P_{zhel}\) — the same, iron losses,

\(U_n i_{vn}\) — losses in the field winding at the rated field current and rated line voltage,

\(I_{a0}^{2} r_0\) — Joule losses at no load in the armature circuit of the motor, the resistance of which is equal to the total internal resistance of the motor.

Since the natural mechanical characteristic of a shunt motor is stiff, it may be assumed that, with the motor speed changing little when the load changes, the sum of the first three terms in expression (2) practically remains constant. Denoting this sum by

\[ P_c = P_{mex} + P_{zhel} + U_n \cdot i_{vn} = \text{const}, \tag{3} \]

where \(P_c\) denotes the constant losses, we obtain from expression (2):

\[ P_{xx} = P_c + I_{a0}^{2} r_0. \tag{4} \]

For any other load, different from no load, the losses in the motor will be equal to:

\[ P_{nagr} = P_c + I_a^{2} r_0. \tag{5} \]

Then the efficiency of the shunt motor will be equal to:

\[ \eta = \frac{U_n I - P_{nagr}}{U_n I} = \frac{U_n I - (P_c + I_a^{2} r_0)}{U_n I}; \tag{6} \]

Fig. 1

here:

$U_n I$ — the power supplied to the motor under load and at the nominal mains voltage,

$I_a$ — the current in the armature under motor load.

The calculation of the efficiency according to expression (6) requires knowledge of the constant losses $P_c$, which are determined in the no-load test from expression (4):

\[ P_c=P_{xx}-I_{a0}^2 r_a=U_n I_0-I_{a0}^2 r_a . \tag{7} \]

In a number of cases it is necessary to obtain the value of the motor efficiency quickly.

By means of slight transformations of expression (6), a formula for the motor efficiency can be obtained which has the same degree of accuracy and makes it possible to construct the curve of the motor efficiency as a function of the armature current, requiring knowledge only of the ammeter readings. In this case the test circuit shown in Fig. 1 can be simplified by using only one ammeter to measure the armature current and the excitation current, and a double-pole switch instead of the knife switch (Fig. 2). It is not difficult to see that in one position of the switch the current in the motor armature is measured, and in the other—the current in its excitation winding. Let us transform expression (6), replacing $P_c$ in it according to equation (7).

Fig. 2

\[ \eta=\frac{U_n I-(P_c+I_a^2 r_a)}{U_n I} = \frac{U_n I-\left[P_{xx}-I_{a0}^2 r_a+I_a^2 r_a\right]}{U_n I}; \]

then substituting, instead of $P_{xx}$, its value from equation (1), and bearing in mind that

\[ I_0=I_{a0}+i_{vn}, \]

we obtain:

\[ \eta= \frac{U_n I-\left[U_n(I_{a0}+i_{vn})-I_{a0}^2 r_a+I_a^2 r_a\right]} {U_n I} \quad \text{or, since} \]

\[ I=I_a+i_{vn}, \]

\[ \eta= \frac{ U_n(I_a+i_{vn})-U_n I_{a0}-U_n i_{vn}+I_{a0}^2 r_a-I_a^2 r_a } {U_n(I_a+i_{vn})} = \]

\[ = \frac{ U_n I_a+U_n i_{vn}-U_n I_{a0}-U_n i_{vn}+I_{a0}^2 r_a-I_a^2 r_a } {U_n(I_a+i_{vn})} = \frac{ U_n(I_a-I_{a0})-(I_a^2-I_{a0}^2)r_a } {U_n(I_a+i_{vn})} = \]

\[ = \frac{ U_n(I_a-I_{a0})-(I_a-I_{a0})(I_a+I_{a0})r_a } {U_n(I_a+i_{vn})}, \]

or finally:

\[ \eta= \frac{(I_a-I_{a0})\left[U_n-(I_a+I_{a0})r_a\right]} {U_n(I_a+i_{vn})}. \tag{8} \]

Let us give formula (8) still another form, expressing all quantities entering it in percent; then

\[ \eta= \frac{(I_a-I_{a0})\left[U_n-(I_a-I_{a0})r_a\right]} {U_n(I_a+i_{vn})} = \frac{ (I_a-I_{a0})\left[1-\dfrac{I_a+I_{a0}}{U_n}r_a\right] } {I_a+i_{vn}} . \]

Substituting into the expression obtained the nominal resistance, taken as 100% and equal to

\[ R_n=\frac{U_n}{I_{an}}, \]

we obtain:

\[ \eta = \frac{(I_a-I_{a0})\left[1-\frac{(I_a+I_{a0})}{U_n}\,r_0\frac{R_n}{R_n}\cdot\frac{100}{100}\right]} {I_a+i_{vn}} = \]

\[ = \frac{(I_a-I_{a0})\left[1-\frac{(I_a+I_{a0})}{I_n}\frac{r_0^{\%}}{100}\right]} {I_a+i_{vn}} = \frac{(I_a-I_{a0})\left[1-\frac{(I_a+I_{a0})}{I_n}\frac{r_0^{\%}}{100}\cdot\frac{100}{100}\right]} {I_n+i_{vn}} = \]

\[ = \frac{\frac{I_a-I_{a0}}{I_n}\cdot\frac{100}{100} \left[1-(I_a^{\%}+I_{a0}^{\%})\frac{r_0^{\%}}{10^4}\right]} {\frac{I_a+i_{vn}}{I_n}\cdot\frac{100}{100}} = \]

\[ = \frac{(I_a^{\%}-I_{a0}^{\%})\frac{1}{10^6} \left[10^4-(I_a^{\%}+I_{a0}^{\%})r_0^{\%}\right]} {\left(\frac{I_a+i_{vn}}{I_n}\right)\frac{100}{100}} = \]

\[ = \frac{(I_a^{\%}-I_{a0}^{\%})\left[10^4-(I_a^{\%}+I_{a0}^{\%})r_0^{\%}\right]} {(I_a^{\%}+i_{vn}^{\%})10^4} \]

or finally:

\[ \eta^{\%}= \frac{(I_a^{\%}-I_{a0}^{\%})\left[10^4-(I_a^{\%}+I_{a0}^{\%})r_0^{\%}\right]} {(I_a^{\%}+i_{vn}^{\%})\cdot 10^4}. \tag{9} \]

Expression (9) is more convenient to use in calculating an electric drive, where usually all quantities are expressed as percentages of nominal values. The quantity \(r_0^{\%}\) can be measured, taken from catalog data, or, finally, can be determined from the natural characteristic of the motor (Fig. 3), as

\[ r_0^{\%}=\Delta n^{\%} \]

at nominal torque. To confirm the correctness of the formula obtained, Fig. 4 gives the curve \(\eta^{\%}=f(I_a^{\%})\), taken from the no-load test for the machine:

The internal resistance of this machine is equal to \(r_d = 0.154\) ohm. The curve (Fig. 4) was obtained by the method described at the beginning of the present work and in the literature [1], which involves determining the constant losses \(P_c\). Calculations carried out according to (9) gave the same curve as in Fig. 4.

Fig. 3

Fig. 4

Obviously, formula (9) can also be used in the case where the motor operating condition differs from the nominal one. In this case, instead of \(U_n\) and \(i_{an}\), their actual values must be substituted.

REFERENCES

1. Filippov M. F.—A Brief Guide to Laboratory Work in the Course on Electrical Machines, Tomsk, 1940.