EQUIVALENT CIRCUIT, VECTOR AND CIRCLE DIAGRAMS OF BRUSHLESS VARIABLE-FREQUENCY AC EMAs

A. I. Skorospelkin, Sh. S. Roiz, B. I. Kostylev

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

A brushless variable-frequency AC EMA is a two-stage electric machine, the stages of which are combined in a single magnetic circuit with the conditions for combination satisfied. The schematic diagram of such a BEMA is shown in Fig. 1. Windings A and B constitute the first stage, with the number of poles \(2p_1\), and windings C and D constitute the second stage, with the number of poles \(2p_2\). Windings A and D are located on the stator; windings B and C are interconnected and located on the rotating rotor. In this arrangement, winding A, which is the input winding, is made in the manner of the windings of dc machines, and rotation of its field is achieved by means of a controlled semiconductor commutator (UPK). The remaining windings are made in the manner of stator windings of ac machines.

Assuming that the mutual inductances vary according to a sinusoidal law and taking into account only the fundamental harmonics of the currents in the windings, for the circuits of the diagram in Fig. 1, in the steady state, on the basis of the Lagrange–Maxwell equations, the following voltage-balance equations can be obtained [1].

For the first phases of windings B and C,

\[ 0 = I_{mB}\left[(r_B + r_C)\sin(\omega_B t + \alpha_1) + 1.5(L_B + L_C)\cos(\omega_B t + \alpha_1)\right] + \]

\[ + I_A M_{AB}\omega_B \cos(\omega_B t) + \frac{3}{2} I_{mD}\cos(\omega_B t \pm \alpha_2) M_{CD}\omega_D \tag{1} \]

Fig. 1. Schematic diagram of a brushless variable-frequency EMA.

For the first phase of winding D,

\[ U_{D1}=-I_{mD}\left[r_D\sin(\omega_D t\pm \alpha_2)+1.5L_D\omega_D\cos(\omega_D t\pm \alpha_2)\right]- \]

\[ -\frac{3}{2}I_{mBC}M_{CD}\cos(\omega_D t\pm \alpha_1)\omega_D, \tag{2} \]

where \(\omega_B=\pm\omega_A-\omega_p\) is the angular frequency of variation of the rotor currents and mutual inductances of the first stage;

\(\omega_D=\omega_m\pm\omega_B\) is the angular frequency of variation of the currents of the output winding;

\(\omega_A\) and \(\omega_p\) are the components of the angular frequency \(\omega_B\), caused respectively by the rotation of the field of winding A and of the rotor;

\[ \omega_m=\omega_p\frac{P_2}{P_1} \]

is the angular frequency of variation of the mutual inductances of the second stage.

The positive direction is taken to be the direction of rotation of the rotor. The signs \(\pm\) before \(\alpha_1\) and \(\alpha_2\) in (1) and (2) correspond to the aiding and opposing rotation of the field of winding C and of the rotor.

Equations (1) and (2), in symbolic form with transition to rms values of currents and voltages, are written as:

\[ 0=\dot I_{BC}\left[(r_B+r_C)+j1.5\omega_B(L_B+L_C)\right]+j\dot I_A\omega_{M1}M_{AB}+j\frac{3}{2}\dot I_D\omega_BM_{CD}. \tag{3} \]

\[ U_{D1}=-\dot I_D(r_D+j1.5\omega_DL_D)-j\frac{3}{2}\dot I_{BC}\omega_DM_{CD}. \tag{4} \]

Denoting the ratio \(\dfrac{\omega_B}{\omega_D}\) by \(S\) and carrying out certain transformations, we obtain

\[ 0=\dot I_B\left[\frac{r_B}{S}+j(1.5L_B-M_{AB})\omega_D\right]+(\dot I_A+\dot I_B)jM_{AB}\omega_D+ \]

\[ +\dot I_{BC}\left[\frac{r_C}{S}+j1.5(L_C-M_{CD})\omega_D\right]+(\dot I_{BC}+\dot I_D)jM_{CD}\,1.5\omega_D, \tag{5} \]

\[ U_{D1}=-\dot I_D\cdot\left[r_D+j1.5(L_D-M_{CD})\omega_D\right]-(\dot I_D+\dot I_{BC})jM_{CD}\cdot1.5\omega_D, \tag{6} \]

where

\[ Z_B=\frac{r_B}{S}+jx_{\sigma B} \]
is the total impedance of winding B,

\[ Z_C=\frac{r_C}{S}+jx_{\sigma C} \]
is the total impedance of winding C,

\[ Z_D=r_D+jx_{\sigma D} \]
is the total impedance of winding D,

\[ Z_{01}=r_{01}+jx_{01} \]
is the total impedance of the magnetizing circuit of the first stage;

\[ X_{01}=M_{AB}\omega_D \]
is the mutual-inductance reactance,

\[ Z_{02}=r_{02}+jx_{02} \]
is the total impedance of the magnetizing circuit of the second stage,

\[ X_{02}=M_{CD}\cdot1.5\omega_D \]
is the mutual-inductance reactance,

\[ \dot I_{01}=(\dot I_A+\dot I_{BC}) \]
is the magnetizing current of the first stage;

\[ \dot I_{02}=(\dot I_{BC}+\dot I_D) \]
is the magnetizing current of the second stage.

Equations (5) and (6) correspond to the equivalent circuit and vector diagram shown in Figs. 2 and 3.

Fig. 2. Equivalent circuit of the BEMУ-RCh.

Fig. 3. Vector diagram of the BEMУ-RCh.

For a BEMУ of adjustable frequency it is also possible to construct a circle diagram, which makes it possible to determine the necessary parameters as the frequency changes.

The construction of the circle diagram is carried out as follows.

Denoting the ratio \(\dfrac{\omega_B}{\omega_p}\) by \(S_1\) and \(\dfrac{\omega_D}{\omega_p} = \dfrac{P_2}{z_1} \pm S_1'\), equations (3) and (4) are written in the following form:

\[ 0 = \dot I_{BC}\left[\frac{r_B + r_C}{S_1} + j1{,}5\omega_p(L_B + L_C)\right] + jI_A\omega_p M_{AB} + jI_D\omega_p M_{CD}. \tag{7} \]

\[ U_{D1} = -\dot I_D\left[\frac{r_D}{\alpha \pm S_1} + j1{,}5\omega_p L_D\right] - j\dot I_{BC}\omega_p M_{CD}\cdot 1{,}5 \tag{8} \]

Transforming (7) and (8),

\[ \dot E_{01} = \dot I_{BC}\left[\frac{R_{BC}}{S_1} + jX_{\sigma BC}\right] + (\dot I_{BC} + \dot I_D)Z_{02} + \dot I_{BC}\cdot Z_{01} \tag{9} \]

\[ \dot U_{D1} = -\dot I_D\left[\frac{r_D}{\alpha \pm S_1} + jX_{\sigma D}\right] - (\dot I_{BC} + \dot I_D)Z_{01}, \tag{10} \]

where

\[ E_{01} = -I_A\omega_p M_{AB}, \]

\[ \frac{R_{BC}}{S_1'} = \frac{r_B + r_C}{S_1}, \]

\[ X_{\sigma BC} = X_{\sigma B} + X_{\sigma C}. \]

According to (9) and (10), the equivalent circuit shown in Fig. 4, a, is constructed. The transformed equivalent circuit with the circuit brought out is shown in Fig. 4, b.

Fig. 4. a) Transformed equivalent circuit of the BEMU-RCh;
b) equivalent circuit of the BEMU-RCh

The expression for the current \(I_D\) from the equivalent circuit is written as

\[ I_D = E_{01} \frac{ S_1(a \pm S_1) }{ \left[ \sigma_1 R_{BC}a + (r_D\sigma_1^2 + r_{01}\sigma_1 a + r_H\sigma_1^2 a \pm R_{BC}\sigma_1)S_1 + (r_{01}\sigma_1 + r_H\sigma_1^2)S_1^2 \right] + } \]

\[ \frac{ }{ +j\left(\sigma_1 X_{01}+\sigma_1 X_{\delta BC}X_H\sigma_1^2+X_{\delta D}\sigma_1^2\right) } = E_{01} \frac{\dot{B}\,S_1' + \dot{C}\,S_1^2}{\dot{D}+\dot{E}\,S+\dot{F}S^2}, \tag{11} \]

where

\[ \begin{aligned} B &= a,\\ C &= \pm 1,\\ D &= \sigma_1 R_B a,\\ E &= r_D\sigma_1^2 + r_{01}\sigma_1 a + r_H\sigma_1^2 a \pm R_{BC}\varepsilon_1,\\ F &= r_{01}\sigma_1 + r_H\sigma_1^2. \end{aligned} \]

Diameter of the circle

\[ D_1= \frac{E_{01}} {\sigma_1 X_{01}+B_1X_{BC}+X_H\sigma_1^2+X_{\delta D}\sigma_1^2}. \tag{12} \]

Equation (11) is the equation of a bicircular curve having a circle with double values of the parameter.

The expression for the current \(I_{02}\)

\[ I_{02}= \frac{E_{01}} { \left[ (r_{01}+r_{02})+\frac{R_{BC}}{S_1'} \right] +j(X_{01}+X_{\delta BC}+X_{02}) }. \tag{13} \]

Diameter of the circle

\[ D_2= \frac{E_{01}} {X_{01}+X_{\delta BC}+X_{02}}. \tag{14} \]

Equation (13) is the equation of a circle.
Using (11)—(13), we construct the circle diagram. The latter is presented in Fig. 5.

The constructed equivalent circuits, vector diagram, and circle diagrams make it possible to determine the required parameters under any operating mode of the amplifier.

Fig. 5. Circle diagram of the BEMU-RCh.

Equation (6) is the equation of the external characteristic. After carrying out certain transformations, we obtain the equation of the external characteristic in a more acceptable form

\[ \dot U_{D1}=\dot E_D-\dot I_D\left(Z_D+Z'_{02}\right), \tag{15} \]

where

\[ \dot E_{D1}=\frac{\dot I_A Z_{01} Z_{02}}{\left(Z_{01}+Z_{02}+Z_B+Z_C\right)}, \tag{16} \]

\[ Z'_{02}=Z_{02}\cdot\frac{Z_{01}+Z_B+Z_C}{Z_{01}+Z_{02}+Z_B+Z_C}. \tag{17} \]

The gain coefficient of the BEMU is equal to the ratio of the output and input powers, i.e.

\[ K_y=\frac{P_D}{P_A}, \tag{18} \]

where \(P_D\) is the power of the three phases of winding \(D\),

\(P_A\) is the power of the control winding.

The power of one phase of winding \(D\) is equal to

\[ P_{D1}=U_{D1}\cdot I_D=E_{D1}\cdot I_D-I_D^2\left(Z'_{02}+Z_D\right). \tag{19} \]

The maximum value of the power will be obtained if \(\dfrac{\partial P}{\partial I_D}\) is set equal to zero.

In this case

\[ I_D=\frac{E_{D1}}{2\left(Z_D+Z'_{02}\right)}, \tag{20} \]

\[ U_{D1}=\frac{E_{D1}}{2}. \tag{21} \]

The maximum power of the three phases of winding \(D\) is

\[ P_{D\max}=3P_{D\max}=3\,\frac{E_{D1}^{2}}{4\left(Z_D+Z'_{02}\right)} . \tag{22} \]

The control power is

\[ P_A=I_A U_A . \tag{23} \]

The voltage of the control winding, in general form, is written as

\[ U_A=I_A r_A-E_A . \tag{24} \]

The component \(E_A\) is due to the fact that the resultant flux rotates relative to the stationary winding \(A\) with frequency \(\omega_A\) and shifts relative to the connection points as the load changes.

The amplification factor, taking into account (22), (23), and (24), is

\[ K_y=\frac{3}{4}\, \frac{Z_{01}^{2}Z'_{02}} {\left(Z_D+Z'_{02}\right)\left(Z_{01}+Z_{02}+Z_B+Z_C^{2}\right)^2\left(r_A-\dfrac{E_A}{I_A}\right)} \tag{25} \]

Thus, considering a brushless variable-frequency AC EMA from the standpoint of the general theory of electrical machines, one can obtain all the necessary equations and parameters characterizing the operating modes of the amplifier.

REFERENCES

1. A. I. Skoryuspeshkin. Problems of the general theory of electrical machines as applied to brushless AC electric-machine amplifiers (BEMA) of alternating current. Izvestiya TPI, vol. 145, 1966.

2. A. G. Kaplyansky. Introduction to the General Theory of Electrical Machines. GEI, 1941.

3. G. N. Petrov. Electrical Machines. GEI, 1963.

4. M. P. Kostenko. Electrical Machines. Special Part. GEI, 1949.