ERRORS OF THE R–C DIFFERENTIATOR OF AN INSTRUMENT FOR RECORDING THE MECHANICAL CHARACTERISTICS OF ELECTRIC MOTORS

O. G. MAKARCHENKO

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

In works [1, 2] an instrument, the “mechanoscope,” is described for recording the mechanical characteristics (dynamic and static) of an asynchronous motor, with the possibility of their continuous registration and visual observation on the screen of a cathode-ray tube. There is experience in the successful use of “mechanoscopes” in educational, research, and factory laboratories.

Like any instrument, the “mechanoscope” must have its own measurement accuracy class, which above all determines the limits of its practical application. At present there are only indirect methods for determining the error in recording mechanical characteristics [4]. Knowledge of the errors of the “mechanoscope” is of great importance for the correct evaluation of the dynamic properties of the electric drives under study. However, these questions have not been investigated sufficiently.

The present paper considers the error introduced by the \(R-C\) differentiator, which has a large specific weight in determining the angular acceleration of the motor.

The rotational speed of an alternating-current machine is determined by the rotating torques of the machine and the load, as well as by the moments of inertia of the rotating masses. In low-inertia drives the transient torques that arise when alternating-current motors are started cause nonuniformity in the change of speed.

Integrating the equation of motion of the drive (3) with the static torque equal to zero, we obtain an expression for the instantaneous value of the angular rotational speed of the motor for the initial instant of starting, assuming that the acceleration process proceeds very slowly in comparison with the speed of the electromagnetic transient processes:

\[ \begin{aligned} \omega(t) &= \omega_{\text{п}} + \omega_1(t) + \omega_2(t) - \omega_3(t) - \omega_4(t) = \omega_{\text{п}} + \frac{1}{J} M_{\text{п}} \cdot t + \\ &\quad + \frac{1}{J} M_{\text{п}} \frac{1}{\alpha_1 + \alpha_2} \cdot e^{(\alpha_1+\alpha_2)t} - \frac{1}{J} M_{\text{п}} \left[ A \sin(\omega_1 t + \varphi_1)\cdot e^{\alpha_1 t} + \right. \\ &\quad \left. + B \sin(\omega_1 t + \varphi)\cdot e^{\alpha_2 t} \right], \end{aligned} \tag{1} \]

where \(J\) is the moment of inertia of the drive;

\(\alpha_1; \alpha_2\) are the damping coefficients of the free components of the magnetizing and short-circuit currents;

\(\varphi_{\text{к}}\) is the phase-shift angle between current and voltage in the short-circuit regime;

\(M_{\text{п}}\) is the starting torque, calculated by the refined Kloss formula for the slip at which the motor is connected to the network;

\(\omega_{\text{p}}\) is the angular velocity at the moment it is connected to the network,

\[ A=\sqrt{\frac{1+t_{\text{d}}^{2}\varphi_{\text{k}}}{\alpha_{1}^{2}+\omega_{1}^{2}}}, \]

\[ B=\sqrt{\frac{1+t_{\text{d}}^{2}\varphi_{\text{k}}}{\alpha_{2}^{2}+\omega_{1}^{2}}}, \]

\[ \varphi_{1}=\operatorname{arctg}\frac{\alpha_{1}-\omega_{1}\operatorname{tg}\varphi_{\text{k}}}{\omega_{1}+\alpha_{1}\operatorname{tg}\varphi_{\text{k}}}, \]

\[ \varphi_{2}=\operatorname{arctg}\frac{\alpha_{2}+\omega_{1}\operatorname{tg}\varphi_{\text{k}}}{\omega_{1}-\alpha_{2}\operatorname{tg}\varphi_{\text{k}}}. \]

From expression (1) it is evident that the speed curve contains components of three types:

1. The speed component \(\omega_{1}(t)\), arising under the action of the motor starting torque and varying according to a linear law, relatively slowly.

2. The aperiodic speed component \(\omega_{2}(t)\), decaying practically at the same rate as the leakage fluxes, since \(\alpha_{2} \gg \alpha_{1}\).

Fig. 1

1. Alternating-sign components pulsating at the network frequency: the slowly decaying speed component \(\omega_{3}(t)\), and \(\omega_{4}(t)\), which decays rapidly.

Let us consider the action of voltages proportional to these speed components on the passive capacitive-resistive differentiating circuit of the “mechanoscope” (Fig. 1).

The differential equation for the output voltage, for an input voltage varying in accordance with \(\omega_{1}(t)\), is written in operator form as

\[ U_{\text{out}}(p)= \frac{U}{p^{2}R_{i}C_{\text{p}}+p\left(\frac{C_{\text{p}}}{C}+\frac{R_{i}}{R}+1\right)+\frac{1}{RC}}. \]

The solution of this equation will contain a free and a forced component,

\[ U_{\text{out}}(t)=URC+Ae^{p_{1}t}+B\cdot e^{p_{2}t}. \tag{3} \]

It follows from equation (3) that the differentiation error will be equal to zero in the absence of the free component, i.e., if the roots of the characteristic equation are real and equal.

Taking in equation (3) \(C_{\text{p}}=0,\ R_{i}\ne 0\) and \(C_{\text{p}}\ne 0;\ R_{i}=0\), we obtain expressions for the output voltage, respectively:

\[ U_{\text{out}}(t)=URC\left[1-e^{-\frac{t}{(R+R_{i})C}}\right] \tag{4} \]

\[ U_{\text{out}}(t)=URC\left[1-e^{-\frac{t}{R(C+C_{\text{p}})}}\right] \tag{5} \]

\[ U_{\text{out}}(t)=URC\left(1-e^{-\frac{t}{RC}}\right) \tag{6} \]

Equations (4), (5), (6) show that the internal resistance of the source of the applied voltage and the parasitic capacitance lead to an increase in the time constant of the differentiating circuit and thereby to an increase in the differentiation error.

Since the amplitude of the components of the motor speed \(\omega_3(t)\) and \(\omega_4(t)\) decreases relatively slowly over the time interval \(\pi/2\), we shall regard it as constant when considering the action of these components on the \(R\)—\(C\) circuit.

When a voltage varying according to the law

\[ U_{\text{in}}=R_eU\cdot e^{j(\omega_1 t+\varphi)} \tag{7} \]

acts on the \(R\)—\(C\) circuit, the differential equation for the output voltage is written in the form

\[ U_{\text{out}}(p)=U\cdot e^{j\varphi}\frac{p^2}{(p-j\omega_1)\left(\frac{1}{RC}+p\right)}, \tag{8} \]

and will have the solution

\[ U_{\text{out}}(t)=\frac{UR}{Z}\left[\sin(\omega_1 t+\varphi-\gamma)-\frac{1}{\omega_1RC}\cdot \cos(\varphi-\gamma)\cdot e^{-\frac{t}{RC}}\right]. \tag{9} \]

For the condition of zero phase shift, from (9) we obtain

\[ \gamma=\frac{\pi}{2}. \tag{10} \]

Having specified the phase error \(\Delta\varphi\), one can also determine from equation (9) the time constant of the differentiating circuit

\[ T=\frac{\operatorname{tg}\Delta\varphi}{\omega_1}. \tag{11} \]

When a voltage proportional to the speed component \(\omega_2(t)\) acts on the \(R\)—\(C\) circuit, the equation for the output voltage will have the form

\[ U_{\text{out}}(t)=\frac{U}{\left(\frac{1}{RC}-\alpha\right)} \left(-\alpha\cdot e^{-\alpha t}+\frac{1}{RC}\cdot e^{-\frac{t}{RC}}\right). \tag{12} \]

It follows from expressions (3), (9), and (12) that, in order to reduce the differentiation error, it is necessary to reduce the time constant of the differentiating circuit. However, the latter leads to a decrease in the output voltage and an increase in the derivative scale \(\mu\)

\[ \mu=\frac{U'_{\text{out}}(t)}{U_{\text{out}}(t)}, \tag{13} \]

where \(U'_{\text{out}}(t)\) is the output voltage of ideal differentiation;
\(U_{\text{out}}(t)\) is the output voltage of real differentiation.

The derivative scale for the output voltages (3), (9), and (12) is determined, respectively, by the expressions

\[ \mu_1=\frac{1}{T}, \tag{14} \]

\[ \mu_2=\frac{Z\omega_1}{R}\approx\frac{1}{T}, \tag{15} \]

\[ \mu_3=\frac{1-\alpha T}{T}\approx\frac{1}{T}. \tag{16} \]

The error in differentiating voltages proportional to the velocity components is a variable quantity and decreases with the differentiation time. However, with an appropriate choice of the time constant of the \(R—C\) circuit, the amplitude errors of the output voltage can be reduced to a minimum.

The relative error of the differentiation operation, in accordance with expressions (3), (9), and (12), will be equal to

\[ \delta_{\Sigma}=\delta_1+2\delta_2+\delta_3, \tag{17} \]

\[ \delta_1=e^{-\frac{t}{T}}, \tag{18} \]

\[ \delta_2=\frac{1}{\omega_1 T}\cdot \frac{\cos(\varphi-\gamma)}{\sin(\omega_1 t+\varphi-\gamma)} \cdot e^{-\frac{t}{T}}, \tag{19} \]

\[ \delta_3=\frac{1}{\alpha T}\cdot e^{\left(\alpha-\frac{1}{T}\right)t}. \tag{20} \]

Figure 2 shows the dependence of the relative error of the \(R—C\) circuit on time, calculated for the case: \(T=0.0001\) sec; \(\alpha=660\ \frac{1}{\text{sec}}\).

Fig. 2.

From the graph (Fig. 2) it is seen that the relative error does not exceed \(1\%\) after a time of \(0.0008\) sec has elapsed. The phase shift from (11) in this case is equal to \(1^\circ 48'\).

Conclusions

1. The electrical \(R—C\) circuit differentiates voltages proportional to the velocity components with one and the same derivative scale.

2. For voltages proportional to the velocity components \(\omega_2(t)\) and \(\omega_4(t)\), the differentiation error increases, respectively, by \(\frac{1}{\alpha T}\) and \(\frac{1}{\omega_1 T}\) times.

1. Under the actual operating conditions of the “mechanoscope,” the mean differentiation error does not exceed 1%.

References

1. A. I. Zaitsev, M. P. Tabinskii, A. S. Bobrovskii. An installation for recording the dynamic mechanical characteristics of an asynchronous motor. Izvestiya TPI, vol. 117, 1963.

2. Report on the research topic “Development, study, and manufacture of an instrument for recording the mechanical characteristics of electric machines.” Tomsk, 1968.

3. I. I. Petrov, A. M. Meistel. Special Operating Modes of an Asynchronous Electric Drive. “Energiya,” 1968.

4. A. I. Zaitsev, M. P. Tabinskii, O. G. Makarchenko. Study of electromagnetic transient processes in asynchronous electric motors. Proceedings of the Fifth Scientific and Technical Conference on Problems of Production Automation, vol. 1, Tomsk, 1967.