Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 117
1963

Energy Performance of Steady-State Operating Modes of an Electric Drive Based on a Magnetic Amplifier–DC Motor System

M. A. Borovikov

(Presented by the scientific seminar of the Electromechanical Faculty)

In practical calculations of electric drives it is often necessary to determine the energy performance indicators of drive operation in steady-state modes.

The principal energy indicators of a drive based on a magnetic amplifier (MA)–DC motor system are: the useful mechanical power \(P_{\mathrm{m}}\) delivered from the shaft, the electrical power \(P_1\) consumed from the alternating-current network, the efficiency \(\eta\), and the power factor \(\cos \varphi_c\) of the drive. DC drives based on the MA–motor system have recently found very wide application and are produced for a large range of powers [1, 2]; therefore, knowledge of the efficiency and power factor and the development of methods for determining them are very important for the design of such electric-drive systems.

In the present work, a method is proposed for determining the indicated energy indicators, using as an example an MA–motor system constructed according to the PMA-type drive circuit [1] in a single-phase design.

Figure 1 shows the drive system under investigation. Here \(Y\) is the armature of the separately excited DC motor; \(w\) are the working windings of the magnetic amplifier, connected in a self-saturation circuit; B1–B4 are valves for rectifying the alternating current and producing the self-saturation effect; \(w_{\mathrm{cm}}\) is the bias winding of the magnetic amplifier. In the system, in addition to the internal positive current feedback, there are also external feedbacks with respect to motor current and voltage. Positive external current feedback is implemented by means of the amplifier winding \(w_2\); negative voltage feedback is introduced into the MA control winding \(w_1\), which combines the functions of the reference winding and the voltage feedback winding.

The determination of the required energy indicators of the drive is carried out by means of the equivalent circuit of the MA operating with an active load (Fig. 1 of the preceding article in this collection), under the assumption that the actual nonsinusoidal voltages and current in the working circuit of the magnetic amplifier can be replaced by sinusoidal ones, ensuring

providing the same rms values as the real ones [3, 4]. To simplify the calculations, we shall regard the active resistance of the working circuit of the magnetic amplifier, \(R_{\mathrm{mu}}=r_{\mathrm{mu}}+r_{\mathrm{v}}\), as constant, since it varies within comparatively small limits [4], and allowing for its nonlinearity, without appreciably increasing the accuracy of the calculations, only complicates them.

As the initial data for calculating the energy indices, as in the case of calculating the mechanical characteristics, use is made of the input-output load characteristics of the magnetic amplifier \(I_{\mathrm{n}}=f(Iw)_y\), operating into various active load resistances \(R_{\mathrm{n}}=\mathrm{const}\) (see the preceding article in the collection). For a specified value of the load resistance \(R_{\mathrm{n}}\), the mean value of the voltage across it is determined from the expression

\[ U_{\mathrm{n}}=I_{\mathrm{n}}\cdot R_{\mathrm{n}}. \tag{1} \]

Fig. 1.

The rms values of the current and voltage at the output of the magnetic amplifier are proportional to the corresponding mean values

\[ I_{\sim}=\gamma_1\cdot I_{\mathrm{n}}, \tag{2} \]

\[ U_{\mathrm{n}\sim}=\gamma_2\cdot U_{\mathrm{n}}. \tag{3} \]

By expressions (2) and (3), the mean values of the load voltage and current are referred to the alternating-current side. The magnitudes of the conversion coefficients \(\gamma_1\) and \(\gamma_2\) depend on the rectifier circuit and on the nature of the load. For a single-phase bridge circuit with an active load, \(\gamma_1=\gamma_2=1.11\) [3]. Since the magnetic amplifier distorts the current curve, \(\gamma_1\ne 1.11\). When the magnetic amplifier is loaded by active resistances, \(\gamma_1=\gamma_2\), and they vary, as the load changes, within the range \(1.11 \div 1.21\). When the magnetic amplifier is loaded by counter-emf, in the general case \(\gamma_1\ne\gamma_2\). Considering only the continuous-current mode, it may also be assumed that \(\gamma_1=\gamma_2=1.11 \div 1.21\). For simplicity we shall

consider the transformation coefficients unchanged and equal to 1.15.

After reducing the load to the AC side, we shall have the values \(I_{\sim}\), \(U_{\mathrm{n}\sim}\), \(I_{\sim}R_{\mathrm{mu}}\). In this case the power factor of the drive can be found (Fig. 1 of the preceding article in the collection) from the expression

\[ \cos \varphi_c = \frac{U_{\mathrm{n}\sim}+I_{\sim}R_{\mathrm{mu}}}{U_c}. \tag{4} \]

Knowing \(\cos \varphi_c\), one can find the power \(P_1\) consumed from the network,

\[ P_1 = U_c \cdot I_{\sim}\cos \varphi_c . \tag{5} \]

The shaft power is determined by the usual expression

\[ P_{\mathrm{m}}=\frac{(M-\Delta M)\cdot n}{975}, \tag{6} \]

where \(M\) is the electromagnetic torque of the motor;
\(\Delta M\) is the loss torque.

The efficiency of the drive will be determined by the expression

\[ \eta=\frac{P_{\mathrm{m}}}{P_1+\Delta P} = \frac{(M-\Delta M)n}{975\left(U_c\cdot I_{\sim}\cos \varphi_c+\Delta P\right)}. \tag{7} \]

where \(\Delta P\) denotes additional losses in the motor field winding, in the control windings of the magnetic amplifier, and in potentiometer P.

The essence of the method for determining the indicated quantities for the circuit of Fig. 1 consists in constructing the locus of the operating point of the magnetic amplifier, i.e., the dependence of the control ampere-turns \((Iw)_y\) on the load current \(I_{\mathrm{n}}\), \((Iw)_y=f(I_{\mathrm{n}})\), for a definite position of the slider of potentiometer P. The locus of the operating point of the magnetic amplifier for any specified load current \(I_{\mathrm{n}}=I_{\mathrm{a}}\) determines the value of the load resistance \(R_{\mathrm{n}}=R_{\mathrm{e}}\), and consequently the speed and voltage (see the preceding article in the collection) of the motor. Having determined the speed and voltage of the motor, all the necessary energy indices can be determined from expressions (4)--(7).

To construct the locus of the operating point of the magnetic amplifier, we construct a series of load characteristics \(I_{\mathrm{n}}=f(Iw)_y\) for different values \(R_{\mathrm{n}}=\mathrm{const}\) (Fig. 2). From the current characteristics, using (1) and (3), we construct curves of the rms voltage \(U_{\mathrm{n}\sim}=f(Iw)_y\) for the same resistances. In Fig. 2, three load characteristics of current and voltage are shown for each, so as not to darken the drawing too much.

Next it is necessary to construct the characteristics of the feedbacks and forward connections. For the circuit of Fig. 1, the resultant ampere-turns of the magnetic amplifier are composed of the sum of the ampere-turns of windings \(w_1\) and \(w_2\),

\[ (Iw)_y=(Iw)_1+(Iw)_2 . \tag{8} \]

The ampere-turns of winding \(w_1\) may be conditionally divided into two components: \((Iw)_z\), the setting ampere-turns, which determine the forward-connection characteristic, and \((Iw)_u\), the ampere-turns of negative voltage feedback. In this case the resultant ampere-turns will be determined as the algebraic sum of the setting ampere-turns, the ampere-turns of the current-feedback winding \((Iw)_r=(Iw)_2\), and the ampere-turns of voltage feedback,

\[ (Iw)_y=(Iw)_z+(Iw)_r-(Iw)_u . \tag{9} \]

Fig. 2.

The ampere-turns of current feedback \((Iw)_{\mathrm{t}}\) are directly proportional to the load current

\[ (Iw)_{\mathrm{t}}=\beta\cdot I_{\mathrm{n}}= \frac{R_{\mathrm{sh}}}{R_{\mathrm{sh}}+R_z}\cdot I_{\mathrm{n}}. \tag{10} \]

Here \(R_z\) is the resistance of the current winding.

To determine the ampere-turns \((Iw)_z\) and \((Iw)_u\), let us consider the loop \(1—2—3—4—5\) of the circuit in Fig. 1. Applying the superposition principle to determine the current in winding \(w_1\), we obtain the required expressions

\[ (Iw_z)=c_1\cdot U_y= \frac{R_b\cdot w_1}{\left(\dfrac{R_b\cdot R_a}{R_1}+R_a+R_b\right)\cdot R_1}\,U_y, \tag{11} \]

\[ (Iw)_u=\alpha\cdot U_{\mathrm{n}}= \frac{(R_a+R_b)\cdot w_1}{R_a\cdot R_b+R_1\cdot R_a+R_1\cdot R_b}\,U_{\mathrm{n}}, \tag{12} \]

where \(R_1\) is the resistance of the circuit of winding \(w_1\);
\(R_a\) and \(R_b\) are the resistances of the arms of potentiometer P;
\(U_y\) is the voltage on potentiometer P.

Thus, \(c_1=\mathrm{const}\) and \(\alpha=\mathrm{const}\) for a definite position of the slider of potentiometer P.

Using now (11), we construct in Fig. 2 the vertical \(AB\) of the setting ampere-turns for a definite position of the slider of potentiometer P. From (12) we determine the magnitude of the ampere-turns \((Iw)_u\) as a function of the rms voltage and, converting the rms voltage into the effective one, construct the dependence of the ampere-turns of voltage feedback on the effective load voltage \((Iw)_u=f(U_{\mathrm{n}\sim})\), plotting it to the left of the vertical \(AB\), since the feedback is negative. This dependence is represented in Fig. 2 by ray \(AD\). Next, from (10), we construct the characteristic of positive current feedback, plotting it from the origin of coordinates \(I_{\mathrm{n}}, (Iw)_y\) (ray \(O_2C\)).

After constructing all the feedback characteristics, one can construct the trajectory of the operating point of the magnetic amplifier. In order to find, for example, the operating point of the magnetic amplifier for the resistance \(R_{\mathrm{n}}=R_1\), it is necessary to project point \(1\), the intersection of ray \(AD\) with the curve \(U_{\mathrm{n}\sim}=f(Iw)_y\), onto the abscissa axis, and from the obtained point \(2\) draw a straight line parallel to the characteristic of current feedback \(O_2C\) until it intersects at point \(3\) with the curve \(I_{\mathrm{n}}=f(Iw)_y\) for \(R_{\mathrm{n}}=R_1\). It is not difficult to verify that, for point \(3\), condition (9) is satisfied, i.e., it is the required point of the trajectory \((Iw)_y=f(I_{\mathrm{n}})\) for the given position of the slider of potentiometer P. By analogous constructions, the other points of the trajectory are found (points \(7\ldots 9\)). From the current trajectory of the operating point, by shifting it upward along the vertical until it intersects with the corresponding voltage curves, we obtain the curve of variation of the voltage at the output of the magnetic amplifier as the load changes, or the trajectory of the amplifier operating point in the axes \(U_{\mathrm{n}\sim}, (Iw)_y\), \((Iw)_y=f(U_{\mathrm{n}\sim})\) (curve \(4—8—10\)).

From the dependence \((Iw)_y=f(I_{\mathrm{n}})\), specifying the values of the current \(I_{\mathrm{n}}=I_{\mathrm{a}}\), and using the known values of the load resistance \(R_{\mathrm{n}}=R_z\), we find, with the aid of the auxiliary characteristics \(I_{\mathrm{a}}=f(n)\) for \(R_z=\mathrm{const}\) (see the preceding article in the collection), the motor speed \(n\). From the current, the value of the electromagnetic torque is found:

\[ M=c_{\mathrm{m}}I_{\mathrm{a}}=\frac{1}{1.03}\,\frac{E}{n}\,I_{\mathrm{a}}. \tag{13} \]

Thus, the mechanical characteristic of the motor in the circuit of Fig. 1 will be obtained for a definite position of the slider of potentiometer P, that is, for a definite value of the resistance \(R_b\). Figure 3 shows characteristics for two values of \(R_b\), with \(R_{b1} > R_{b2}\).

To determine the power factor \(\cos \varphi_c\) of the power section of the circuit, in the left quadrant of Fig. 2 we construct a circle of radius \(R_u\), equal, in scale, to the rms value of the supply voltage. This circle will represent the locus of the ends of the supply-voltage vector, provided that it is constant. We draw the auxiliary ray \(O_1E\) at an angle of \(45^\circ\) in order to transfer the values \(U_{n\sim}\) from the right quadrant to the abscissa axis of the left quadrant. In the left quadrant we also construct the dependence of the rms value of the voltage drop across the active resistance \(R_{\mathrm{mu}}\) on the load current, that is, the dependence \((I_{\sim}R_{\mathrm{mu}})=f(I_n)\) (ray \(O_1F\)). Under the assumptions adopted \((R_{\mathrm{mu}}=\mathrm{const}, \gamma_1=\mathrm{const})\), this dependence will be linear.

Suppose, for example, that it is required to determine the value of the power factor when the system operates at point 7 of curve 3—7—9. We determine the voltage \(U_{n\sim}\) at the load for this point (point 8) and transfer the obtained value to the left quadrant (horizontal 8—13). The value of the current \(I_n\) at point 7 is also transferred to the left quadrant as far as ray \(O_1F\), and at point 11 we obtain the magnitude of the voltage drop \((I_{\sim}R_{\mathrm{mu}})\). We project point 11 onto the abscissa axis and, from the obtained point 12, draw a straight line parallel to ray \(O_1E\) until it intersects the horizontal 8—13 at point 14. Projecting points 13 and 14 onto the abscissa axis, we obtain the voltage triangle \(baO_1\), in which the side \(O_1b\) represents the rms load voltage \(U_{n\sim}\), and the side \(ba\) the voltage \(U_{\mathrm{mu}}\), since the segment \(bc\) represents the voltage drop across the resistance \(R_{\mathrm{mu}}\), while \(O_1a\) is the supply voltage. Consequently, the angle \(aO_1b\) is the angle \(\varphi_c\) for the specified MA load.

Since

\[ \cos \varphi_c=\frac{O_1c}{O_1a}=\frac{O_1c}{R_u}, \]

in order to obtain the values of \(\cos \varphi_c\) it is sufficient to construct on the abscissa axis a scale of its values, taking the radius of the circle as unity. In this case the segment \(O_1c\) will simultaneously represent the value of the drive power factor for the given current \(I_n=I_a\), that is, also for the motor torque \(M\). Carrying out analogous constructions for the entire trajectory 3—7—9 and calculating, in this process, the values of the torque on the shaft, we obtain the dependence of the power factor on the load torque, \(\cos \varphi_c=f(M_v)\). The dependences \(\cos \varphi_c=f(M_v)\) for two positions of the slider of potentiometer P are shown in Fig. 3.

Using, further, expressions (5), (6), and (7), from the mechanical characteristic and the dependence \(\cos \varphi_c=f(M_v)\), one can determine the values of \(P_1\), \(P_m\), and \(\eta\), and construct the dependences of the drive efficiency on the torque on the motor shaft for any position of the slider of potentiometer P. The dependences \(\eta=f(M_v)\) for two cases are shown in Fig. 3.

All the constructions and graphs presented in the work relate to a real motor of type MI-12F, \(U_{\mathrm{nom}}=110\ \mathrm{V}\); \(P_{\mathrm{nom}}=0.12\ \mathrm{kW}\); \(I_{\mathrm{nom}}=1.46\ \mathrm{A}\); \(n_{\mathrm{nom}}=2000\ \mathrm{rpm}\); \(R_a=2.2\ \Omega\); \(R_{\mathrm{ov}}=600\ \Omega\); \(\eta_{\mathrm{nom}}=0.74\), connected according to the circuit of Fig. 1. The magnetic amplifier is made of steel grade E-42, stamping Sh-40U, stack height \(36\ \mathrm{mm}\); \(w_{\sim}=600\); \(w_1=1000\); \(w_2=40\); \(w_{\mathrm{cm}}=1000\); rectifiers of type D-304.

The dependences \(\cos \varphi_c=f(M_v)\) and \(\eta=f(M_v)\) constructed by the methods described differed from those obtained experimentally by no more than

Fig. 3.

Graph labels: \(\cos \varphi\); \(M_b,\ \mathrm{kg\cdot m}\); \(M_{b\,\mathrm{nom}}\); \(R_{b1}\); \(R_{b2}\); \(\cos \varphi\); 1; 2; 0; 0.5; 1.0; 50; 90.

Fig. 4.

Graph labels: \(n,\ \mathrm{rpm}\); \(M,\ \mathrm{kg\cdot m}\); \(M_{\mathrm{nom}}\); \(M_{\mathrm{p.o.}}\); \(R_{b1}\); \(R_{b2}\); \(\Delta N\); 0; 500; 1000; 1; 2.

by 10%, which in most engineering calculations proves acceptable.

Conclusions

1. The use of load input–output characteristics of magnetic amplifiers makes it possible to apply simple graphical-analytical methods for calculating the energy-performance indices of drives based on an MA–motor system with various feedbacks.

2. The power factor of the drive is proportional to the rotational speed.

3. Drives of the PMU type have a sufficiently high efficiency, but a low power factor under deep speed regulation.

4. The described method is also applicable to three-phase magnetic-amplifier circuits and may be recommended for engineering calculations that do not require high accuracy.

References

1. Ya. B. Rozman. Industrial series of adjustable electric drives PMU and PMS. TsINTIE, 1960.

2. H. Winkler. A new range of drives with power ratings from 2.8 to 187 kW, regulated by means of magnetic amplifiers. AEG Mitteilungen, vol. 49, No. 10/11, 1959.

3. M. A. Rosenblat. Magnetic Amplifiers. “Sovetskoe Radio,” 1955.

4. D. A. Alenchikov. Calculation of static characteristics of systems with choke control. Avtomatika i telemekhanika, vol. XX, No. 5, 1959.

5. D. A. Alenchikov, V. S. Kulebakin. A method for calculating the characteristics of direct-current motors with choke control. Avtomatika i telemekhanika, vol. XX, No. 7, 1959.

Received by the editors
in May 1962.