SOME ISSUES OF ENERGY RELATIONS IN A CONVERTER WITH ADJUSTABLE FREQUENCY

A. I. Skorospeshkin, E. F. Obertan, K. A. Khorkov
Sh. S. Roiz

One of the most important problems of the alternating-current electric drive at the present time is the creation of an adjustable electric drive. Frequency control of AC motors (asynchronous and synchronous), as the most economical method, is beginning to be widely introduced into industry. Therefore, the creation of alternating-current sources with adjustable frequency corresponds to the modern requirements of the development of electrical machine engineering.

We have designed, built, and are studying a model of a converter with adjustable frequency. The schematic diagram of the converter, explaining its principle of operation, is presented in Fig. 1.

Fig. 1. Schematic diagram of a converter with adjustable frequency.

It should be noted that converters of fixed increased frequency exist [1, 2, 3].

A distinctive feature of the converter described is the possibility of regulating the frequency by means of a semiconductor commutator located at the converter input.

This converter is two-stage and contactless. Windings \(W_1\) and \(W_2\) belong to the first stage, and windings \(M_3\) and \(W_4\) to the second. Both stages are combined in one magnetic core,

In this case all the conditions for combining two electrical machines in one magnetic circuit are fulfilled. Rotation of the converter is effected by an asynchronous motor.

The rotating magnetic field of winding \(W_1\) is obtained by means of a semiconductor commutator; in this case the speed of rotation of the field is set by the required output frequency.

When current \(I_1\) flows through winding \(W_1\), an m.m.f. \(F_1\) is produced. The latter, with the aid of the semiconductor commutator, creates a rotating magnetic flux \(\Phi_1\) with the number of poles \(2P_1\). The converter rotor rotates at speed \(n\), and in winding \(W_2\), with the number of poles \(2P_1\), an e.m.f. is induced with frequency

\[ f_2=\frac{P_1\left(n\pm n_1\right)}{60}, \tag{1} \]

where \(n_1\) is the speed of rotation of flux \(\Phi_1\) relative to the stator. The sign \((+)\) refers to the counter-rotation of flux \(\Phi_1\) and the rotor, and \((-)\) to rotation in the same direction.

The load of the rotor winding \(W_1\) is winding \(W_3\) of the second cascade, with the number of poles \(2P_2\). The alternating current, flowing through winding \(W_1\), creates a flux \(\Phi_3\) rotating relative to the rotor at speed

\[ n_2=\frac{60 f_2}{P_2}. \tag{2} \]

Depending on the phase-sequence order of winding \(W_3\), flux \(\Phi_3\) relative to the stator will rotate at speed

\[ n_3=n\pm n_2. \tag{3} \]

The sign \((+)\) refers to rotation of the flux and rotor in the same direction; \((-)\), to counter-rotation.

Taking (1), (2), and (3) into account, the expressions for the output frequencies may be written in the following form:

\[ f_{1\text{out}}=\frac{(P_2+P_1)n}{60}\pm \frac{P_1 n_1}{60}, \tag{4a} \]

\[ f_{2\text{out}}=\frac{(P_2-P_1)n}{60}\pm \frac{P_1 n_1}{60}. \tag{4б} \]

The frequency of the e.m.f. of the output winding will thus be determined by expressions (4a) and (4б).

The plus sign in the parentheses in (4a) corresponds to the reverse phase-sequence order of windings \(W_2\) and \(W_3\), and the minus sign in (4б) to the direct order.

Of the signs \((\pm)\) in (4a), \((+)\) refers to the counter direction of rotation of the field of winding \(W_1\) and the rotor, and \((-)\) to the same direction; in (4б), \((-)\) refers to the counter direction, and \((+)\) to the same direction.

To analyze the operation of the converter, it is necessary to determine the base and critical output frequencies.

As the base output frequencies one may take the frequencies with stationary flux \(\Phi_1\). From expressions (4a) and (4б), for \(n_1=0\), we find:

\[ f_{1b\,\text{out}}=\frac{(P_2+P_1)n}{60}, \tag{5a} \]

\[ f_{2b\,\text{out}}=\frac{(P_2-P_1)n}{60}. \tag{5б} \]

As the critical output frequencies we take the frequencies at which the output power of the converter is equal to zero. One critical output frequency \(f_{1\text{cr out}}=0\), with output power \(P_{\text{out}}=0\), is obtained—

… occurs with the direct phase sequence of windings \(W_2\) and \(W_3\) and the speed of rotation of field \(\Phi_1\), \(n_1=2n\), with counter-rotation of the field of winding \(W_1\) and of the rotor (according to 46, \(\frac{p_2}{p_1}=3\)).

Another critical speed, at which \(P_{\text{out}}=0\), is obtained with co-rotation of the field of winding \(W_1\) and the rotor (according to 46) when \(n_1=n\):

\[ f_{2\,\text{cr out}}=\frac{(P_2-P_1)n}{60}+\frac{P_1 n_1}{60}=\frac{P_2 n}{60}. \tag{6} \]

An analogous expression is obtained from (4a) with co-rotation of the field of winding \(W_1\) and the rotor when \(n=n_1\).

For analysis of the operating modes, let us construct the dependences of the output frequencies and output \(f_{\text{out}}=f(f_{\text{in}})\). We shall take the ratio \(\frac{P_2}{P_1}\) equal to 3. The change in the speed of rotation of the flux of winding \(W_1\) is determined by the formula

\[ n=\pm(0\div 1)n. \]

The dependences \(f_{1\,\text{out}}, f_{2\,\text{out}}=f(f_{\text{in}})\) are presented in Fig. 2. From the figure one can determine the operating mode of the converter for various frequency ranges (the figure corresponds to the model that was built). For the range \(ab\) the first cascade operates in the converter mode (the flux of winding \(W_1\) rotates opposite to the direction of the rotor); the second cascade operates in the generator mode (the direction of the flux of winding \(W_3\) and of the rotor is opposite). In the range \(bc\) the first and second cascades operate in the generator mode. In the range \(cd\) the first cascade operates in the generator mode, the second as an electromagnetic brake. In the range \(de\) both cascades operate in converter modes. For the ranges \(ab\) and \(bc\) the direct phase sequence of windings \(W_2\) and \(W_3\) is adopted; for \(cd\) and \(de\), the reverse sequence.

Fig. 2. Dependences \(f_{1\,\text{out}}, f_{2\,\text{out}}=f(f_{\text{in}})\) for the fabricated model.

For the constructed model, having \(2P_1=2\) and \(2P_2=6\), the following values of the critical and base frequencies were obtained:

\[ f_{1\,\mathrm{crit\,out}}=0,\quad f_{2\,\mathrm{crit\,out}}=15\ \mathrm{Hz},\quad f_{1\,\mathrm{base\,out}}=100\ \mathrm{Hz},\quad f_{2\,\mathrm{base\,out}}=200\ \mathrm{Hz}. \]

Next, a study was made of the dependence of the input power on the output frequency \(P_1=f(f_{\mathrm{out}})\), the useful power of the drive motor \(P_{\mathrm{dv}}=f(f_{\mathrm{out}})\), and the efficiency \(\eta=f(f_{\mathrm{out}})\).

Analytical expressions were obtained for \(P_1\), \(P_{\mathrm{dv}}\), and \(\eta\). A comparison with experimental data was carried out.

For the range \(ab\)

\[ P_1=P_{\psi4}\left[-\frac{\omega_1P_1}{\omega(P_2-P_1)-\omega_1P_1}\right] +\left(\sum P_3-P_{\mathrm{Cu}2}\right)\frac{\omega_1}{\omega_1-\omega} +\sum P_1, \tag{7} \]

\[ P_{\mathrm{dv}}=P_{\psi4}\left[1+\frac{\omega_1P_1}{\omega(P_2-P_1)-\omega_1P_1}\right] +\left(\sum P_3-P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega+\omega_1}. \tag{8} \]

For the range

\[ P_1=P_{\psi4}\left[\frac{\omega_1P_1}{\omega(P_2-P_1)+\omega_1P_1}\right] +\left(\sum P_3-P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega_1-\omega} +\sum P_1, \tag{9} \]

\[ P_{\mathrm{dv}}=P_{\psi4}\left[1-\frac{\omega_1P_1}{\omega(P_2-P_1)+\omega_1P_1}\right] +\left(\sum P_3+P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega+\omega_1}. \tag{10} \]

For the range \(cd\)

\[ P_1=P_{\psi4}\left[-\frac{\omega_1P_1}{\omega(P_2-P_1)-\omega_1P_1}\right] +\left(\sum P_3+P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega-\omega_1} +\sum P_1, \tag{11} \]

\[ P_{\mathrm{dv}}=P_{\psi4}\left[1+\frac{\omega_1P_1}{\omega(P_2-P_1)-\omega_1P_1}\right] +\left(\sum P_3+P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega-\omega_1}, \tag{12} \]

For the range \(de\)

\[ P_1=P_{\psi4}\left[\frac{\omega_1P_1}{\omega(P_2+P_1)+\omega_1P_1}\right] +\left(\sum P_3+P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega+\omega_1} +\sum P_1, \tag{13} \]

\[ P_{\mathrm{dv}}=P_{\psi4}\left[1-\frac{\omega_1P_1}{\omega(P_2+P_1)+\omega_1P_1}\right] +\left(\sum P_3+P_{\mathrm{Cu}2}\right)\frac{\omega}{\omega+\omega_1}, \tag{14} \]

where

\[ P_{\psi4} \]

is the electromagnetic power of output winding \(W_4\);

\[ \sum P_3=P_{\mathrm{Cu}3}+P_{Ee3} \]

are the losses in winding \(W_3\);

\[ P_{\mathrm{Cu}1} \]

are the losses in winding \(W_1\);

\[ \omega \]

is the angular velocity of rotation of the rotor;

\[ \omega_1 \]

is the angular velocity of rotation of the winding field.

For any range, the efficiency of the converter is equal to

\[ \eta=\frac{P_{\mathrm{out}}}{P_1+P_{\mathrm{dv}}}. \tag{15} \]

where

\[ P_{\mathrm{out}}=P_{\psi4}\cdot\eta_4 \]

is the output power of the converter;

\[ \eta_4 \]

is the efficiency of winding \(W_4\).

Figure 3 shows the experimental (solid lines) and calculated (dashed lines) dependences \(P_1=f(f_{\mathrm{out}})\) at different output powers.

Curves 1 correspond to an output power of \(210\ \mathrm{W}\); 2—\(180\ \mathrm{W}\); 3—\(120\ \mathrm{W}\); 4—\(60\ \mathrm{W}\). The figure shows the satisfactory convergence of the calculated and experimental curves. From the figure, the expedient frequency range of the converter can be determined. For the model under investigation it lies within \(80\text{–}120\ \mathrm{Hz}\) and \(180\text{–}220\ \mathrm{Hz}\).

Fig. 3. Dependence \(P_1=f(f_{\mathrm{out}})\) for various values of output power.

The input power of the converter can be reduced by means of capacitors connected at the output. For the model under study, with a capacitance \(C=5\ \mu\mathrm{F}\), the input power is reduced by more than a factor of 2 (the capacitance was determined by selection).

Under these conditions, the power of the semiconductor switch is 7–10 times less than the power of the converter.

In the ranges 120–150 cps and 150–180 cps, the input power may turn out to be negative. This can be explained by the fact that, in the \(bc\) and \(cd\) ranges, the first stage operates in generator mode.

The experimental (solid lines) and analytical (dashed lines) dependences \(P_{\mathrm{dv}}=f(f_{\mathrm{out}})\) at various output powers—1—120 W, 2—180 W, 3—120 W, 4—60 W—are presented in Fig. 4.

Fig. 4. Dependence \(P_{\mathrm{dv}}=f(f_{\mathrm{out}})\) for various values of output power.

It follows from the figure that operation of the motor is most expedient in the ranges 80–120 Hz and 180–220 Hz. In these modes the power of the drive motor is somewhat higher than the power when the motor operates at the basic frequencies (for our case, by approximately 15–20 percent).

At frequencies of 120–150 Hz and 150–180 Hz the motor power increases rather sharply. This is explained by the fact that in these ranges the first stage operates in generator mode.

An important indicator of any electric machine is the magnitude of its efficiency. In Fig. 5 the calculated (dashed lines) and experimental (solid lines) dependences \(\eta=f(f_{\text{out}})\) are given. It is seen from the figure that in the frequency range most expedient for operation the efficiency is approximately 48 percent.

Fig. 5. Dependence \(\eta=f(f_{\text{out}})\).

In general, as a result of the investigation of the converter with adjustable frequency, the frequency range most expedient for operation is established. It lies within 80–120 Hz and 180–220 Hz, which, of course, will affect some limited possibilities for its application.

The required power of the semiconductor commutator is determined. It is approximately 15–20 percent of the converter power.

It should be noted that the circuit under investigation is very interesting. Under certain conditions it can operate as a power amplifier. Work on investigating its properties and characteristics is continuing.

REFERENCES

1. V. S. Novokshenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1960.

2. M. S. Mikhailov-Mikulinskii. Calculation of magnetic circuits of electrical machines with two rotating fields. Elektromekhanika, No. 10, 1962.

3. V. S. Nersesyan. Elements of the calculation of single-machine contactless frequency converters. Proceedings of the Gorky Polytechnic Institute, Vol. XIX, Issue 3, 1963.