ON CALCULATING THE CHARACTERISTICS OF A SERIES-EXCITED MOTOR SUPPLIED BY A PWM CONVERTER

A. I. Sapozhnikov, V. A. Beinarovich, A. I. Alifirov

(Presented by the scientific and technical seminar of the Scientific Research Institute of AEM)

Pulse regulation of speed is finding ever wider application in DC electric drives, especially in installations with autonomous power supply (battery-powered electric locomotives, electric carts, electric automobiles, electric loaders, etc.) and in traction installations with trolley power supply (electric locomotives, trolleybuses, trams, etc.). In such installations, as a rule, series-excited motors (DPS) are used, which provide large starting torques and smoothness of transient processes, and are supplied from power pulse-width converters (PWM converters) built with thyristors.

The calculation of the electromechanical characteristics of a DPS, representing the dependence of the rotational speed \(\omega\) on the motor current \(I\), \(\omega(I)\), when supplied from a PWM converter with allowance for the nonlinear dependence of the excitation flux \(\Phi\) on the motor current \(\Phi(I)\), is the subject of works [1, 2, 3, 4, 5], in which graphical or grapho-analytical calculation methods are used.

Common to these calculation methods is the use of the natural electromechanical characteristic of the motor \(\omega_e(I)\), the magnetization curve \(\Phi(I)\) [1, 2, 5], and the assumptions [2, 3, 4, 5] that the commutation period of the power circuit is considerably shorter than the electromechanical and electromagnetic time constants of the electric drive, that the commutating elements are inertialess, that the rotational speeds do not change during the commutation period, that the motor emf depends only on rotational speed and motor current (through the magnetization curve), and that the processes in the extinction circuit of the main thyristor do not affect the motor circuit.

Taking these assumptions into account, when a DPS is supplied from a PWM converter, the expression—common to all calculation methods—for the artificial electromechanical characteristic of the DPS as a function of the relative on-time \(\gamma\) and the motor current \(I\) can be written in the following form:

\[ \omega_i = \frac{\gamma (U - \Delta U) - I R_\mathrm{e}} {U - I R_\mathrm{dv}} \, \omega_e , \tag{1} \]

where \(U\) is the supply-source voltage;
\(\Delta U\) is the voltage drop across the power thyristor;
\(\omega_e\) is the angular rotational speed of the DPS corresponding to the motor current \(I\) on the natural electromechanical characteristic at supply voltage \(U\);

\(R_{\mathrm{e}}\) — total resistance of the power circuit of the PWM converter–series-excited motor;
\(R_{\mathrm{dv}}\) — resistance of the motor circuit.

Fig. 1.

Figure 1 shows the dependences \(\omega_i(\gamma, I)\), calculated by the methods set forth in [1, 2, 4, 5]. The different calculation results for the same values of the relative on-duration \(\gamma\) and motor current \(I\) arise because of the different value of \(R_{\mathrm{e}}\) that is taken:

in [1]
\[ R_{\mathrm{e}}=R_{\mathrm{dv}}+R_{\mathrm{d}}(1-\gamma)+(R_0+R_{\mathrm{t}})\gamma; \]

in [2]
\[ R_{\mathrm{e}}=R_{\mathrm{dv}}, \]
a special case;

in [4]
\[ R_{\mathrm{e}}=R_{\mathrm{dv}}+\gamma(R_{\mathrm{t}}+R_0); \]

in [5]
\[ R_{\mathrm{e}}=R_{\mathrm{dv}}+R_0; \]

where

\[ R_0,\ R_{\mathrm{t}},\ R_{\mathrm{d}} \]
are the resistances, respectively, of the power supply, the thyristor, and the shunting diode.

This makes it possible to divide the proposed calculation methods into three groups, in which:

1) the resistance of the motor circuit is considered constant over the pulse and pause intervals [2];

2) the resistance of the motor circuit is different over the pulse and pause intervals and differs by the value of the internal resistance of the power supply [2, 5];

3) the resistance of the motor circuit changes in the same way as in the second group, but in addition the volt-ampere characteristic of the power switch [4] or the resistance of the power switch and shunting diode [1] is taken into account. This calculation group may also include the graphical method [3], which, in addition, makes it possible to take into account the external characteristic of the converter.

The purpose of the present work is to determine the influence of the parameters of the electric motor, the power supply, and the volt-ampere characteristic of the power thyristor on the results of calculating the electromechanical characteristics of the series-excited motor and the conditions under which various factors may be neglected.

To determine the influence of the above-listed factors on the accuracy of calculating the electromechanical characteristic of a series-excited motor when supplied from a PWM converter, let us write expression (1) in relative units. As the base values of current, voltage, speed, and flux we take the nominal values of these quantities. In doing so, we take into account that

\[ U_{\mathrm{n}}-I_{\mathrm{n}}R_{\mathrm{dv}}=E_{\mathrm{n}}=c\Phi_{\mathrm{n}}\omega_{\mathrm{n}}, \tag{2} \]

where

\[ c=\frac{p\cdot N}{2\pi\cdot a} \]

is a coefficient determined by the design parameters of the machine [6].

Then dependence (1), in relative units, takes the form

\[ \gamma_i=\frac{\gamma(\upsilon-\Delta\upsilon)\alpha-i\beta_e}{\alpha-i}\,\gamma_e, \tag{3} \]

where

\[ \gamma_i=\frac{\omega_i}{\omega_n};\quad \gamma_e=\frac{\omega_e}{\omega_n} \]

are the relative values of the rotational speed of the motor on the artificial and natural characteristics, respectively;

\[ \upsilon=\frac{U}{U_n};\quad \Delta\upsilon=\frac{\Delta U}{U_n} \]

are the relative values of the supply voltage and of the voltage drop across the thyristor;

\[ \beta_e=\frac{R_e}{R_{\mathrm{dv}}} \]

is the relative value of the resistance of the motor power circuit;

\[ i=\frac{I}{I_n} \]

is the relative value of the armature-circuit current;

\[ \alpha=\frac{U_n}{I_n R_{\mathrm{dv}}} \]

is the ratio of the rated supply voltage to the rated voltage drop in the motor circuit.

Electromechanical characteristics calculated by any method described in [1, 2, 3, 4, 5] can be converted into mechanical characteristics; for this purpose the transition characteristic \(\dfrac{M}{I}=f(I)\) is needed, or they can be calculated if the catalog characteristic \(\omega_e=f(M)\) is available and, by analogy with the limiting electromechanical characteristic \(\omega_{\mathrm{gr}}=f_1(I)\), the limiting mechanical characteristic of the DPV, \(\omega_{\mathrm{gr}}=f(M)\), has been constructed. Then, for the natural mechanical characteristic, the following expression is valid:

\[ \omega_e=\frac{U}{c\Phi}-M\frac{R_{\mathrm{dv}}}{c^2\Phi^2} =\omega_{\mathrm{gr}}-M\frac{R_{\mathrm{dv}}}{c^2\Phi^2} \tag{4} \]

and for the artificial mechanical characteristic

\[ \omega_i=\frac{(U-\Delta U)\gamma}{c\Phi} -M\frac{R_e}{c^2\Phi^2} =\omega_{\mathrm{gr}}\left(1-\frac{\Delta U}{U}\right)\gamma -M\frac{R_e}{c^2\Phi^2}, \tag{5} \]

where

\(M\) is the torque on the motor shaft;

\[ \omega_{\mathrm{gr}}=\frac{U}{c\Phi} \]

is the value of the limiting rotational speed on the characteristic \(\omega_{\mathrm{gr}}=f(M)\), corresponding to the torque \(M\);

\(\omega_e\) is the value of the rotational speed on the catalog characteristic \(\omega_e=f(M)\), corresponding to the torque \(M\);

\(\Phi\) is the magnetic flux of the motor, corresponding to the torque \(M\).

Expressions (4) and (5) can be written in the form

\[ \omega_e-\omega_{\text{gr}}=-M\frac{R_{\text{dv}}}{c^2\Phi^2}; \]

\[ \omega_i-\omega_{\text{gr}}\left(1-\frac{\Delta U}{U}\right)\gamma = -M\frac{R_e}{c^2\Phi^2} \]

and, if their ratio is taken, after simple transformations one obtains

\[ \omega_i=\frac{R_e}{R_{\text{dv}}}\omega_e- \left[ \frac{R_e}{R_{\text{dv}}} - \left(1-\frac{\Delta U}{U}\right)\gamma \right]\omega_{\text{gr}}, \tag{6} \]

or, in relative units, taking \(U=U_n\),

\[ \nu_i=\rho_e\nu_e-\left[\rho_e-(1-\Delta \nu)\gamma\right]\nu_{\text{gr}} . \tag{7} \]

If, however, \(R_0=0\) and the power commutator is ideal, the expression takes the form

\[ \nu_i=\nu_e-(1-\gamma)\nu_{\text{gr}} . \tag{8} \]

Naturally, in passing from the limiting and natural characteristics to the artificial one, the recalculation of the rotational speed is performed for one and the same value of the torque on the motor shaft. To obtain the dependence of the rotational speed on the electromagnetic torque, it is necessary to have the initial dependences \(\omega_e\) and \(\omega_{\text{gr}}=f_1(M_{\text{em}})\).

In order to extend the results of investigating the influence of the parameters \(\alpha\), \(\rho_e\), and \(\Delta \nu\) on the accuracy of calculating electromechanical characteristics to all types of DPT, an expression is introduced for the reference speed, which is obtained when supplied from idealized power sources and a PWM converter, that is, \(\rho_e=1\), \(\Delta\nu=0\), and \(\nu=1\). Expression (3) then takes the form

\[ \nu_{\text{э}}=\frac{\gamma\lambda-1}{\alpha-1}\nu_e . \tag{9} \]

The relative deviation of the calculated speed value, determined by the nonidealities of the power source \((R_0>0)\) and the PWM converter \((R_t>0;\ \Delta U>0)\), is obtained by dividing (3) by (9), and this makes it possible not to take into account the form of the magnetization curve, which is different for different machines:

\[ \frac{\nu_i}{\nu_{\text{э}}} = \frac{\gamma\alpha\nu-1}{\gamma\lambda-1} - \frac{\Delta\rho_e\, i}{\gamma\alpha-1} - \frac{\Delta\omega\gamma}{\gamma\lambda-1} = \frac{\gamma\alpha\nu-1}{\gamma\alpha-1} - \frac{\Delta\nu_p}{\nu_{\text{э}}} - \frac{\Delta\nu_k}{\nu_{\text{э}}}; \tag{10} \]

where \(\Delta\nu_p\) is the relative magnitude of the speed drop caused by additionally inserted resistances;

\(\Delta\nu_k\) is the relative magnitude of the speed drop caused by the voltage drop in the power switch.

\[ \Delta\rho_e=\rho_e-1 = \gamma\frac{R_t+R_0}{R_{\text{dv}}} + (1-\gamma)\frac{R_d}{R_{\text{dv}}}, \tag{11} \]

if \(R_t=R_d\) is assumed, then

\[ \Delta\rho_e=\frac{\gamma R_0+R_t}{R_{\text{dv}}}. \tag{12} \]

For some types of traction motors, Table 1 gives, as an example, the basic data and the values of the coefficient \(\alpha\).

From an analysis of expression (10), proceeding from the specified accuracy of calculating the drive characteristics, it is possible to determine the admissibility of idealizing the PWM converter with respect to neglecting the voltage drop across the thyristor switch \((\Delta U=0)\) and neglecting the magnitudes of the source resistances (for \(R_0=0\),

Table 1

of the thyristor (for \(R_t=0\)) and of the shunting diode (for \(R_d=0\)), i.e., taking \(\Delta \nu_\rho=0\) (for \(\Delta \rho_z=0\)).

The idealization may be considered permissible if the relative deviation of the speed

\[ \frac{\Delta \nu_\Sigma}{\nu_e} = \frac{\Delta \nu_\rho+\Delta \nu_k}{\nu_e} = \frac{\Delta \rho_z i}{\gamma \alpha-i} + \frac{\gamma \lambda \Delta \nu}{\gamma \alpha-i} \tag{13} \]

from expression (10) does not exceed the specified calculation accuracy.

Analysis of expression (13), taking into account the value of \(\alpha\) (Table 1), makes it possible to obtain the dependence of the limiting values \(\Delta \rho_z\) on the relative on-time \(\gamma\) and the motor current \(i\), at which the divergence of the calculated characteristics in comparison with the reference characteristic according to expression (9) does not exceed the specified accuracy, taking \(\Delta \nu_k=0\),

\[ \Delta \rho_z = \frac{\Delta \nu_\rho}{\nu_e}\cdot \frac{\gamma \alpha-i}{i}. \tag{14} \]

Naturally, the greatest reduction in speed will occur at the maximum value of the current \(i=\lambda\), where \(\lambda\) is the factor of the maximum current relative to the rated current. With this taken into account, expression (9) takes the form

\[ \Delta \rho_z \leq \frac{\Delta \nu_z}{\nu_e}\cdot \frac{\gamma \alpha-\lambda}{\lambda} = \delta \frac{\gamma \alpha-\lambda}{\lambda} \tag{15} \]

If inequality (15) is satisfied for a real electric-drive circuit, calculation methods of the first group may be used, taking \(\Delta \rho_z=0\). In this case the accuracy of the calculations will be no worse than \(\delta\) over the entire range of variation of the motor current. Otherwise, methods of the second group must be used.

Fig. 2.

The second component of the speed reduction in expression (13) is determined by the voltage drop in the thyristor switch

\[ \frac{\Delta \nu_k}{\nu_e} = \frac{\gamma \lambda \Delta \nu}{\gamma \alpha-i}. \tag{16} \]

Figure 2 shows the dependences of the relative decrease in speed $\dfrac{\Delta \nu_k}{\nu_e}$ on the value of $\gamma$ for various values of $\alpha$ and for a voltage drop in the thyristor switch $\Delta U = 0.8\ \mathrm{V}$, calculated from expression (16).

The results presented are equally suitable for assessing the influence of parameters on the accuracy of calculations of mechanical characteristics.

REFERENCES

1. V. D. Nagorskii. Control of a dc motor by means of high-frequency pulses. Izvestiya AN SSSR. Energetika i avtomatika, No. 2, 1960.

2. A. D. Dinkel, S. P. Vasilevskii, A. S. Yurin. Calculation of the characteristics of series-excited motors under pulse control of rotational speed. Elektrotekhnika, 1968, No. 6.

3. N. I. Dzhus. Calculation of the characteristics of a series-excited motor. Izvestiya vuzov. Energetika, 1971, No. 6.

4. V. A. Beinarovich, A. I. Sapozhnikov. Pulse control of series-excited motors in in-plant transport mechanisms. “Automated electric drive in the national economy.” Proceedings of the Fifth All-Union Conference on Automated Electric Drive. Vol. II, “Energiya,” 1971.

5. A. S. Yurin. Static characteristics of a series-excited motor under pulse control. “Special Electric-Drive Systems.” Issue 2, Perm, Rotaprint of the Perm Polytechnic Institute, 1967.

6. V. P. Andreev, Yu. A. Sabinin. Fundamentals of Electric Drive. Moscow–Leningrad, Gosenergoizdat, 1963.