On the Calculation of the Field Rheostat of an Adjustable-Speed Motor with Shunt and Separate Excitation

L. I. Gandzha

Since the speed of a direct-current motor with shunt or separate excitation can be regulated by changing the magnetic flux, whose dependence on the excitation current is expressed by the magnetization curve, the calculation of the regulating rheostat inserted in the circuit of the shunt or separate field winding reduces to determining the values of the excitation current at all speed steps of the motor. The resistances of the steps (or sections) of the rheostat are determined by the magnitude of the voltage applied to the field-winding circuit and by the above-mentioned values of the excitation current at the specified speed steps of the motor.

The field rheostat must be calculated so that, on the one hand, the specified series of speeds is maintained and, on the other, the law of speed regulation specified by the driven mechanism is maintained. The latter may be different: regulation may be carried out at constant torque, at constant power, or in the presence of some more complex dependence of power or torque on the speeds of the specified regulating series; it is determined solely by the form of the dependence characterizing the driven mechanism:

\[ M_{\mathrm{st}} = f(n), \]

where

\(M_{\mathrm{st}}\) is the static torque referred to the motor shaft, and \(n\) is the number of revolutions per minute of the driven motor.

Below is given the most general method for calculating a field rheostat, suitable for any type of characteristic \(M_{\mathrm{st}} = f(n)\).

In Fig. 1 are shown the natural mechanical characteristic of the motor at \(\Phi_{n}^{\delta} = 100\%\), the artificial characteristic at \(\Phi_{i.m}^{\delta} < \Phi_{n}^{\delta}\), and the characteristic \(M_{\mathrm{st}} = f(n^{\delta})\). It is evident that the speeds \(n_{i}^{\delta}, \ldots, n_{i.m}, \ldots\) belong to the specified regulating series (here and below the subscript \(m\) denotes the correspondence of the given quantity to the \(m\)-th step of regulation).

The artificial characteristic corresponding to the weakened magnetic flux \(\Phi_{i.m}\) is expressed by the equation

\[ n_i = \frac{U_n - I_i \cdot r_0}{c_e \Phi_{i.m}}, \tag{1} \]

which, when written for the case of steady operation at speed \(n_{i.m}\) on the considered \(m\)-th step of regulation, has the form:

\[ n_{i.m} = \frac{U_n - I_{i.m} \cdot r_0}{c_e \cdot \Phi_{i.m}}, \tag{2} \]

where

\(r_0\) is the total internal resistance of the motor armature circuit,
\(U_n\) is the rated voltage of the motor,

\(I_{\mathrm{i.m}}\) is the armature current when the motor operates on the \(m\)-th artificial characteristic at speed \(n_{\mathrm{i.m}}\), and

\(c'_e\) is the coefficient of proportionality between the counter-emf of the motor and the product \(n_{\mathrm{i.m}}\Phi_{\mathrm{i.m}}\), defined as

\[ c'_e=\frac{U_n}{\Phi_n\cdot n_{0e}} . \tag{3} \]

Fig. 1

When the motor is transferred from speed \(n_e\) to speed \(n_{\mathrm{i.m}}\), its current will change, according to Fig. 1, from the value \(I_e\) to the value \(I_{\mathrm{i.m}}\) in the ratio

\[ \frac{M_{\mathrm{i.m}}}{M_e} \]

due to the change in the static torque according to the characteristic \(M_{\mathrm{st}}=f(n)\), and in the ratio

\[ \frac{\Phi_n}{\Phi_{\mathrm{i.m}}} \]

due to the change in magnetic flux. Thus we shall have:

\[ I_{\mathrm{i.m}}=I_e\frac{M_{\mathrm{i.m}}}{M_e}\cdot\frac{\Phi_n}{\Phi_{\mathrm{i.m}}}, \tag{4} \]

and after substituting (3) and (4) into (2) we obtain

\[ n_{\mathrm{i.m}}= \frac{ U_n-I_e\cdot\frac{\Phi_n}{\Phi_{\mathrm{i.m}}}\cdot\frac{M_{\mathrm{i.m}}}{M_e}\cdot r_0 }{ \Phi_{\mathrm{i.m}}\cdot\frac{U_n}{\Phi_n\cdot n_{0e}} }, \]

whence

\[ \frac{n_{\mathrm{i.m}}}{n_{0e}}= \frac{ \frac{U_n}{U_n} -\frac{I_e}{U_n}\cdot\frac{\Phi_n}{\Phi_{\mathrm{i.m}}}\cdot\frac{M_{\mathrm{i.m}}}{M_e}\cdot r_0 }{ \frac{\Phi_{\mathrm{i.m}}}{\Phi_n} }. \tag{5} \]

Expression (5) makes it possible to obtain the final result in a simpler form if all the quantities entering into it are expressed in percentage values. To do this, let us take as 100% the quantities \(n_{0e}, U_n, I_n, \Phi_n, M_n\), and \(R_n\), and represent (5) in the form:

\[ 100 \cdot \frac{n_{u.m}}{n_{0e}} = \frac{ \left( \frac{U_n}{U_n} - \frac{I_e}{U_n}\cdot \frac{I_n}{I_n}\cdot \frac{\Phi}{\Phi_{u.m}}\cdot \frac{M_{u.m}}{M_n}\cdot \frac{M_n}{M_e}\cdot r_0\cdot \frac{100}{100}\cdot \frac{100}{100} \right)\cdot 100 }{ \frac{\Phi_{u.m}}{\Phi_n}\cdot 100 } \cdot 100, \]

then

\[ n_{u.m}^{\%} = \frac{ 10^4 - I_e^{\%}\cdot r_0^{\%}\cdot \frac{M_{u.m}^{\%}}{\Phi_{u.m}^{\%}}\cdot \frac{10^2}{M_e^{\%}} }{ \Phi_{u.m}^{\%} }. \tag{6} \]

Expression (6) may be represented in the form of the following quadratic equation, if one takes into account that at \(\Phi=\Phi_n\), \(I_e^{\%}=M_e^{\%}\):

\[ \left(\Phi_{u.m}^{\%}\right)^2 - \frac{10^4}{n_{u.m}^{\%}}\cdot \Phi_{u.m}^{\%} + \frac{r_0^{\%}\cdot M_{u.m}^{\%}}{n_{u.m}^{\%}}\cdot 100 =0, \tag{7} \]

whence

\[ \Phi_{u.m}^{\%} = \frac{5\cdot 10^3}{n_{u.m}^{\%}} \left( 1 \pm \sqrt{ 1 - 4\frac{r_0^{\%}\cdot M_{u.m}^{\%} n_{u.m}^{\%}}{10^6} } \right). \tag{8} \]

Equation (8) makes it possible, by calculation, to find the value of the magnetic flux corresponding to the speed and load of the motor at a given stage of regulation.

The general nature of the dependence \(\Phi_{u.m}^{\%}=f(n_{u.m}^{\%})\) is shown graphically in Fig. 2. As is evident from (8) and Fig. 2, in the region of speeds from 0 to \(n_{u.\max}\) the magnetic flux may assume two values; at point \(a\), for which \(n_{u.m}^{\%}=n_{u.\max}^{\%}\), the magnetic flux has a single-valued value determined from (8) on the basis of the fact that, at

\[ 1 - 4\frac{r_0^{\%}\cdot M_{u.\max}^{\%}\cdot n_{u.\max}^{\%}}{10^6}=0, \]

\[ n_{u.\max}^{\%} = \frac{0.25\cdot 10^6}{r_0^{\%}\cdot M_{u.\max}^{\%}} \tag{9} \]

and

\[ \Phi_{u.\max}^{\%} = \frac{5\cdot 10^3}{n_{u.\max}^{\%}}, \tag{10} \]

where the index \(\max\) characterizes the maximum speed and the values of the other quantities corresponding to it.

Fig. 2

Fig. 2 and expression (8) make it possible to conclude that, as the magnetic flux is weakened, the speed of the motor increases not without bound, but only up to a certain limit, determined by point \(a\) and by equations (9) and (10). With further weakening of the magnetic flux, the speed begins to decrease. However, this limit of speed increase lies outside the range of speeds permissible for the motor, and in practice it cannot be reached without detriment to the machine.

Thus the branch \(ab\) of the curve in Fig. 2 characterizes the region of actually permissible operating conditions of the motor, while the branch \(ac\) characterizes the region of conditions not permissible for it.

In accordance with what has been said, the sign before the radical in (8) must be chosen so that the values of the magnetic fluxes obtained in the calculations correspond to the branch \(ab\), which determines the region of permissible conditions for the motor. This condition can evidently be represented as

\[ \Phi_n > \Phi_{n.m} > \Phi_{n.\max}. \tag{11} \]

Assuming, on the basis of (11),

\[ \Phi_{n.m} = \Phi_n \]

and bearing in mind that in this case (Fig. 1)

\[ n_{n.m} = n_e, \]

\[ M_{n.m} = M_e, \]

we represent (8) in the form

\[ \Phi_n^{\%} = \frac{5 \cdot 10^3}{n_e^{\%}} \left( 1 \pm \sqrt{ 1 - 4 \frac{r_0^{\%} \cdot M_e^{\%} \cdot n_e^{\%}}{10^6} } \right). \tag{12} \]

If, in the general case, the motor on the natural characteristic operates in a condition different from the nominal one, then, according to Fig. 1,

\[ \frac{M_e^{\%}}{M_n^{\%}} = \frac{n_{0.e}^{\%} - n_e^{\%}}{n_{0.e}^{\%} - n_n^{\%}} = k, \]

whence

\[ \left. \begin{aligned} M_e^{\%} &= k \cdot M_n^{\%} = k \cdot 100,\\ n_e^{\%} &= n_{0.e}^{\%} - k\left(n_{0.e}^{\%} - n_n^{\%}\right) = 100 - k\left(100 - n_n^{\%}\right) \end{aligned} \right\} \tag{13} \]

Substituting (13) into (12) and bearing in mind that

\[ r_0^{\%} = 100 - n_n^{\%}, \]

we obtain

\[ \Phi_n^{\%} = \frac{5 \cdot 10^3}{100 - k\left(100 - n_n^{\%}\right)} \left\{ 1 \pm \sqrt{ 1 - 4 \frac{ \left(100 - n_n^{\%}\right) \cdot k \cdot \left[100 - k\left(100 - n_n^{\%}\right)\right] }{10^4} } \right\}, \]

or

\[ \Phi_n^{\%} = \frac{5 \cdot 10^3}{100 - k\left(100 - n_n^{\%}\right)} \left\{ 1 \pm \frac{1}{10^2} \sqrt{ \left[10^2 - 2 \cdot k \cdot \left(100 - n_n^{\%}\right)\right]^2 } \right\}, \]

or

\[ \Phi_n^{\%} = \frac{5 \cdot 10}{100 - k\left(100 - n_n^{\%}\right)} \left\{ 100 \pm \left[100 - 2 \cdot k \cdot \left(100 - n_n^{\%}\right)\right] \right\}, \tag{14} \]

from which it is seen that, when the \(+\) sign is chosen, equation (14) becomes the identity

\[ \Phi_n^{\%} = 100\%. \]

Thus, in calculations by (8), for all real cases of speed regulation of an electric drive, the \(+\) sign should be taken before the radical.

In the particular case, if the drive operates on the natural characteristic under nominal conditions,

\[ k=1, \]

and expression (14) becomes

\[ \Phi_k^{\%}=\frac{5\cdot 10}{n_{\mathrm{н}}^{\%}} \left\{100\pm\left[100-2\left(100-n_k^{\%}\right)\right]\right\}, \tag{15} \]

whence it also follows that, when the sign \(+\) is chosen, (15) becomes the identity

\[ \Phi_{\mathrm{н}}^{\%}=100\%. \]

For calculating the steps or sections of the rheostat, it is necessary to have the magnetization curve of the given machine (Fig. 3)

\[ \Phi=f(i_{\mathrm{в}}). \]

The latter is obtained experimentally quite simply if, as before, percentage values of the quantities are used.

Fig. 3

Fig. 4

If the no-load characteristic of the machine under separate excitation is taken (Fig. 4)

\[ E=f(i_{\mathrm{в}}) \]

while rotating it at speed \(n_0\), then, bearing in mind that

\[ E=c_e\Phi n_0, \]

\[ U_{\mathrm{н}}=c_e\Phi_{\mathrm{н}}n_0, \]

we obtain

\[ 100\frac{E}{U_{\mathrm{н}}}=100\frac{\Phi}{\Phi_{\mathrm{н}}}, \]

or

\[ E^{\%}=\Phi^{\%}. \]

If the no-load characteristic is taken at

\[ n=k_1 n_0, \]

then

\[ E=c'_e\cdot \Phi \cdot n, \]

\[ U_n=c'_e\cdot \Phi_n \cdot n_0, \]

\[ \frac{E}{U_n}\cdot 100=\frac{\Phi}{\Phi_n}\cdot \frac{n}{n_0}\cdot 100, \]

or

\[ E^{\%}=k_1\cdot \Phi^{\%}. \]

The relations presented make it possible to pass from the no-load characteristic (Fig. 4) to the magnetization curve (Fig. 5)

\[ \Phi^{\%}=f(i_v). \]

If, further, the values of the magnetic flux have been determined from (8) for the specified regulation series

\[ n_{e}^{\%},\ n_{u_1}^{\%},\ n_{u_2}^{\%},\ n_{u_3}^{\%},\ldots, \]

with

\[ \Phi_n^{\%},\ \Phi_{u_1}^{\%},\ \Phi_{u_2}^{\%},\ \Phi_{u_3}^{\%},\ldots, \]

then from the curve of Fig. 5 the corresponding values of the excitation current are determined

\[ i_{v.n}^{\%},\ i_{v.u_1}^{\%},\ i_{v.u_2}^{\%},\ i_{v.u_3}^{\%},\ldots, \]

where the nominal excitation current is taken as 100%. Knowing the nominal voltage of the excitation winding \(U_{v.n}\), we obtain the resistances of the rheostat steps in ohms

\[ R_1=\frac{U_{v.n}\cdot 100}{i_{v.u_1}^{\%}\cdot i_{v.n}}-r_{o.v}, \]

\[ R_2=\frac{U_{v.n}\cdot 100}{i_{v.u_2}^{\%}\cdot i_{v.n}}-r_{o.v}, \]

\[ R_3=\frac{U_{v.n}\cdot 100}{i_{v.u_3}^{\%}\cdot i_{v.n}}-r_{o.v} \]

and so on.

The resistances of the rheostat sections are determined (Fig. 6) as

\[ r_1 = R_1, \]

\[ r_2 = R_2 - R_1, \]

\[ r_3 = R_3 - R_2, \]

and so on.

Fig. 6

Conclusions

1. As a result of the study, a general method is given for calculating regulating rheostats for shunt- and separately excited motors for any form of the dependence \(M_{\mathrm{st}} = f(n)\).

2. It is shown that, when the magnetic flux is weakened, the speed does not increase without bound, but only up to a certain limit \(n_{\mathrm{sh.max}}\), after which it begins to decrease with further reduction of the magnetic flux.

3. The application of the method of expressing the values of all quantities in percentages made it possible to obtain simple working formulas necessary for calculating regulating rheostats.