Brushless Electric Machine Amplifier (BEMA) of Variable Frequency with an Induction Generator
Abstract Generated abstract
This paper proposes a three-stage brushless electric-machine amplifier of regulated frequency in which an asynchronous generator is connected in parallel with the first stage to reduce the counter-emf induced in the control winding under active-inductive loading. It derives frequency relations, energy diagrams, voltage-balance equations, an equivalent circuit, vector and circle diagrams, and expressions for the external characteristic and gain coefficient. The analysis indicates that the preferable operating mode uses opposing rotation of the rotor fields of the input and output stages, enabling a comparatively wide frequency regulation range. Conditions are also obtained for selecting the asynchronous generator parameters so that it delivers active power most effectively and supports the desired amplifier characteristics.
Full Text
BRUSHLESS ELECTRIC-MACHINE AMPLIFIER (BEMA) OF VARIABLE FREQUENCY WITH AN ASYNCHRONOUS GENERATOR
A. I. Skorospeleshkin, Sh. S. Roiz, E. F. Obertan, B. I. Kostylev
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
Theoretical and experimental studies of a two-stage BEMA of variable frequency [1] showed that, when the amplifier operates into an active-inductive load (an asynchronous motor), a counter-emf arises in the control winding, which substantially reduces the amplification factor over the entire frequency range of the amplifier. The counter-emf is due to the transverse component of the resultant magnetic flux of the first stage of the amplifier.
In order to reduce the counter-emf, it is proposed to supplement the two-stage circuit of a brushless EMA [1] with an asynchronous generator connected in parallel with the first stage of the amplifier.
The asynchronous generator, being a purely inductive load with respect to the first stage, generates active power into the load. The magnitude of the counter-emf induced in the control winding is thereby substantially reduced.
The principle of operation of the variable-frequency amplifier is explained in Fig. 1. The amplifier consists of three stages. Windings A and B constitute the first stage with number of poles \(2p_1\), windings E and F the second stage with number of poles \(2p_2\), and windings C and D the third stage with number of poles \(2p_3\). Windings A, F, and D are located on the stator; windings B, E, and C are electrically interconnected and located on the rotating rotor.
At the input of control winding A, a semiconductor—
Fig. 1. Circuit of a brushless EMA of variable frequency with an asynchronous generator.
… commutator (UPC), whose circuit is assembled on thyristors with artificial capacitive commutation [2].
When the control winding A is supplied with direct current, an m.m.f. \(F_a\) is produced. The magnetizing force \(F_a\) creates a magnetic flux \(\Phi_a\) rotating in space (the rotation of the magnetic flux is effected by means of the UPC).
When the rotor rotates at speed \(n\), the e.m.f. of winding B varies with frequency
\[ f_B = p_1 (n + n_A), \tag{1} \]
where \(n_a\) is the rotational speed of the magnetic flux \(\Phi_a\) relative to the stator, in rev/sec,
\(n\) is the rotor speed in rev/sec.
The loads of rotor winding B are windings E and C.
The alternating current flowing through winding C creates an m.m.f. \(F_c\). The latter creates a magnetic flux \(\Phi_c\), rotating relative to the rotor at the speed
\[ n_c = \frac{p_1}{p_3}(n + n_A). \tag{2} \]
Depending on the order of phase sequence of winding C, the rotational speed of the flux \(\Phi_c\) relative to the stator (winding D) will be determined by the formula
\[ n_D = n \pm n_c. \tag{3} \]
The signs \(\pm\) in (3) correspond to the aiding and opposing rotation of the flux \(\Phi_c\) and the rotor.
On the basis of the purpose of the third cascade, which must operate in the asynchronous-generator mode, the rotation of the flux \(\Phi_c\) and of the rotor must be opposing.
Simultaneously with the processes occurring in the third cascade, the alternating current flowing through winding E creates an m.m.f. \(F_e\). The magnetizing force \(F_e\) creates a magnetic flux \(\Phi_E\), rotating relative to the rotor at the speed
\[ n_E = \frac{p_1}{p_2}(n + n_A). \tag{4} \]
The output e.m.f. of winding F varies with frequency
\[ f_F = p_2(n \pm n_E). \tag{5} \]
The signs \(\pm\) in (5) correspond to the aiding and opposing rotation of the field \(\Phi_E\) and the rotor.
Thus, in a variable-frequency amplifier with an asynchronous generator, regulation of the output frequency is possible from certain base frequencies \(f_{F1\delta}\) and \(f_{F2\delta}\).
The latter are expressed by the following formulas:
\[ f_{F1\delta} = n(p_2 - p_1), \tag{5a} \]
\[ f_{F2\delta} = n(p_2 + p_1). \tag{5б} \]
The upper frequency limit is restricted by the pole number of the third cascade. The maximum output frequency is equal to
\[ f_{F \max} = 13 \cdot n. \tag{6} \]
Using equation (5), frequency characteristics were constructed for the case when \(2p_1 = 2p_2 = 2;\ 2p_3 = 16;\ n = 50\) rev/sec (Fig. 2).
Frequency characteristic 1 corresponds to the opposing direction
Fig. 2. Frequency characteristics of a brushless EMU of variable frequency
rotation of the field of winding E and of the rotor; frequency characteristic 2 corresponds to the same direction.
The frequency characteristics presented show that a brushless EMU of variable frequency with an asynchronous generator can operate on two frequency characteristics. The most appropriate frequency characteristic will be determined when considering the energy relations.
We carry out the analysis of the energy relations using the equation for angular velocities [3]
\[ \sum \omega_s = \sum \omega_r, \tag{7} \]
where
\[
\sum \omega_s
\]
is the algebraic sum of the angular velocities of the fields of the stator windings,
\[
\sum \omega_r
\]
is the algebraic sum of the angular velocities of the fields of the rotor windings and of the rotor itself.
Let us consider the energy relations for each cascade separately.
For the first cascade
\[ \omega_B = \omega + \omega_A . \tag{8} \]
Multiplying the right- and left-hand sides of equation (8) by \(M_1\), where \(M_1\) is the electromagnetic torque due to the interaction of the control flux \(\Phi_A\) and the rotor current \(I_B\), we obtain
\[ P_{\psi B} = P_{m_1} + P_{\psi A}, \tag{9} \]
where
\[
P_{\psi B} = M_1 \cdot \omega_B
\]
is the electromagnetic power of winding B;
\[
P_{\psi A} = M_1 \cdot \omega_A
\]
is the electromagnetic power of winding A;
\[
P_{m_1} = M_1 \cdot \omega
\]
is the mechanical power on the shaft of the first cascade.
For the second cascade
\[ \omega_F = \omega \pm \omega_E . \tag{10} \]
The signs \(\pm\) correspond to the same and opposite directions of rotation of the field \(\Phi_E\) and of the rotor.
Electromagnetic power of winding F
\[ P_{\psi E}=P_{m2}\pm P_{\psi E} \tag{11} \]
where
\[ P_{\psi E}=M_2\omega_2 \]
is the electromagnetic power of winding E,
\[ P_{m2}=M_2\omega \]
is the mechanical power on the shaft of the second cascade,
\(M_2\) is the electromagnetic torque caused by the interaction of the flux \(\Phi_F\) and the current \(I_E\).
For the third cascade
\[ \omega_D=\omega-\omega_C. \tag{12} \]
The minus sign before \(\omega_C\) means that the third cascade operates in the mode of an asynchronous generator.
The electromagnetic power of winding D
\[ P_{\psi D}=P_{m3}-P_{\psi C}, \tag{13} \]
where
\[ P_{\psi C}=M_3\cdot\omega_C \]
is the electromagnetic power of winding C,
\[ P_{m3}=M_3\omega \]
is the mechanical power on the shaft of the third cascade,
\(M_3\) is the electromagnetic torque caused by the interaction of the flux \(\Phi_D\) and the current \(I^C\).
According to (9), (11), and (13), two energy diagrams have been constructed, presented in Fig. 3. The energy diagram in Fig. 3a corresponds to frequency characteristic 1, and Fig. 3b to characteristic 2 (Fig. 2).
Fig. 3. Energy diagrams of a brushless BEMA of variable frequency
Analysis of the energy diagrams shows that the most expedient operating mode of the amplifier is the mode in which the field \(\Phi_E\) and the rotor rotate in opposite directions. In this case the electromagnetic output power is equal to
\[ P_{\psi F}=P_{\psi B}+P_{\psi C}+P_{m2}. \tag{14} \]
The electromagnetic powers \(P_{\psi C}\), \(P_{\psi B}\), \(P_{\psi E}\), and \(P_{\psi F}\) can be represented in the following form:
\[ P_{\psi C}=P_{m3}\frac{(\omega+\omega_A)p_1}{\omega p_1}, \tag{15} \]
\[ P_{\psi B}=P_{m_1}\frac{(\omega+\omega_A)}{\omega}, \tag{16} \]
\[ P_{\psi E}=P_{\psi F}\frac{(\omega+\omega_A)p_1}{\omega(p_2+p_1)+\omega_Ap_1}, \tag{17} \]
\[ P_{\psi F}=\left[ P_{m_3}\frac{(\omega+\omega_A)p_1}{\omega\cdot p_3} - P_{m_1}\frac{(\omega+\omega_A)}{\omega} \right]\cdot \frac{\left[\omega(p_2+p_1)+\omega_A\cdot p_1\right]}{(\omega+\omega_A)\cdot P_1}. \tag{18} \]
Theoretical analysis of the equations of the BEMA of adjustable frequency
We carry out the theoretical analysis of the amplifier operation on the basis of the Lagrange–Maxwell equations.
In the most general form, the voltage-balance equations are written as follows:
\[ \begin{aligned} U_A={}&i_A\cdot r_A+L_A\frac{di_A}{dt} +\frac{d(m_{1AB}\cdot i_{B_1})}{dt} +{}\\ &+\frac{d(m_{2AB}\cdot i_{B_2})}{dt} +\frac{d(m_{3AB}\cdot i_{B_3})}{dt}; \\[6pt] U_B={}&-i_{B_1}\cdot r_B-L_B\frac{di_{B_1}}{dt} -0.5L_B\frac{di_{B_2}}{dt} -0.5L_B\frac{di_{B_3}}{dt} -\frac{d(m_{AB}\cdot i_A)}{dt}; \\[6pt] U_C={}&i_{C_1}\cdot r_C+L_C\frac{di_{C_1}}{dt} +0.5L_C\frac{di_{C_2}}{dt} +0.5L_C\frac{di_{C_3}}{dt} +{}\\ &+\frac{d(m_{1C1D}\cdot i_{D_1})}{dt} +\frac{d(m_{1C2D}\cdot i_{D_2})}{dt} +\frac{d(m_{1C3D}\cdot i_{D_3})}{dt}; \\[6pt] 0={}&i_{D_1}\cdot r_D+L_D\frac{di_{D_1}}{dt} +0.5L_D\frac{di_{D_2}}{dt} +0.5L_D\frac{di_{D_2}}{dt} +{}\\ &+\frac{d(m_{1D1C}\cdot i_{C_1})}{dt} +\frac{d(m_{1D2C}\cdot i_{C_2})}{dt} +\frac{d(m_{1D3C}\cdot i_{C_3})}{dt}; \\[6pt] U_E={}&i_E\cdot r_E+L_E\frac{di_{E_1}}{dt} +0.5L_E\frac{di_{E_2}}{dt} +0.5L_E\frac{di_{E_3}}{dt} +{}\\ &+\frac{d(m_{1E1F}\cdot i_{F_1})}{dt} +\frac{d(m_{1E2F}\cdot i_{F_2})}{dt} +\frac{d(m_{1E3F}\cdot i_{F_3})}{dt}; \\[6pt] U_F={}&-i_{F_1}\cdot r_F-L_F\frac{di_{F_1}}{dt} -0.5L_F\frac{di_{F_2}}{dt} -0.5L_F\frac{di_{F_3}}{dt} -{}\\ &-\frac{d(m_{1F1E}\cdot i_{E_1})}{dt} -\frac{(m_{1F2E}\cdot i_{E_2})}{dt} -\frac{d(M_{1F3E}\cdot i_{E_3})}{dt}. \end{aligned} \tag{19} \]
In the steady-state operating mode of the amplifier, the variation of the currents and mutual inductances is determined by the equations:
\[ i_{B_1}=I_{mB}\cdot \sin(\omega_Bt+\alpha_1); \]
\[ i_{B_2}=I_{mB}\cdot \sin\left(\omega_Bt+\alpha_1+\frac{2\pi}{3}\right); \]
\[ i_{B_3}=I_{mB}\cdot \sin\left(\omega_Bt+\alpha_1+\frac{2\pi}{3}\right); \]
\[ \begin{aligned} i_{C_1} &= I_{mC}\cdot \sin(\omega_C t+\alpha_2);\\ i_{C_2} &= I_{mC}\cdot \sin\left(\omega_C t+\alpha_2+\frac{2\pi}{3}\right);\\ i_{C_3} &= I_{mC}\cdot \sin\left(\omega_C t+\alpha_2-\frac{2\pi}{3}\right);\\[6pt] i_{D_1} &= I_{mD}\cdot \sin(\omega_D t+\alpha_3);\\ i_{D_2} &= I_{mD}\cdot \sin\left(\omega_D t+\alpha_3-\frac{2\pi}{3}\right);\\ i_{D_3} &= I_{mD}\cdot \sin\left(\omega_D t+\alpha_3-\frac{2\pi}{3}\right);\\[6pt] i_{E_1} &= I_{mE}\cdot \sin(\omega_E t+\alpha_4);\\ i_{E_2} &= I_{mE}\cdot \sin\left(\omega_E t+\alpha_4-\frac{2\pi}{3}\right);\\ i_{E_3} &= I_{mE}\cdot \sin\left(\omega_E t+\alpha_4-\frac{2\pi}{3}\right);\\[6pt] i_{F_1} &= I_{mF}\cdot \sin(\omega_F t+\alpha_5);\\ i_{F_2} &= I_{mF}\cdot \sin\left(\omega_F t+\alpha_5+\frac{2\pi}{3}\right);\\ i_{F_3} &= I_{mF}\cdot \sin\left(\omega_F t+\alpha_5-\frac{2\pi}{3}\right); \end{aligned} \tag{20} \]
\[ \begin{aligned} m_{1AB} &= M_{AB}\cdot \sin(\omega_{M_1}t);\\ m_{2AB} &= M_{AB}\cdot \sin\left(\omega_{M_1}t+\frac{2\pi}{3}\right);\\ m_{3AB} &= M_{AB}\cdot \sin\left(\omega_{M_1}t-\frac{2\pi}{3}\right);\\[6pt] m_{1C1D} &= M_{CD}\cdot \sin(\omega_{M_3}t);\\ m_{1C12D} &= M_{CD}\cdot \sin\left(\omega_{M_3}t-\frac{2\pi}{3}\right);\\ m_{1C3D} &= M_{CD}\cdot \sin\left(\omega_{M_3}t+\frac{2\pi}{3}\right);\\[6pt] m_{1D1C} &= M_{CD}\cdot \sin(\omega_{M_3}t);\\ m_{1D1C} &= M_{CD}\cdot \sin\left(\omega_{M_3}t-\frac{2\pi}{3}\right);\\ m_{1D2C} &= M_{CD}\cdot \sin\left(\omega_{M_3}t+\frac{2\pi}{3}\right);\\[6pt] m_{1E1F} &= M_{EF}\cdot \sin(\omega_{M_2}t);\\ m_{1F2F} &= M_{EF}\cdot \sin\left(\omega_{M_2}t+\frac{2\pi}{3}\right);\\ m_{1F3F} &= M_{EF}\cdot \sin\left(\omega_{M_2}t-\frac{2\pi}{3}\right). \end{aligned} \]
\[ \begin{gathered} m_{1FE}=M_{EF}\cdot \sin(\omega_{M2}t);\\ m_{1F2E}=M_{EF}\cdot \sin\left(\omega_{M2}t+\frac{2\pi}{3}\right);\\ m_{1F3E}=M_{EF}\cdot \sin\left(\omega_{M2}t-\frac{2\pi}{3}\right);\\ \omega_B=\omega_C=\omega_E=\omega+\omega_A;\\ \omega_{M1}=\omega_B=\omega+\omega_A;\\ \omega_{M2}=\omega\frac{p_1}{p_2}\quad \text{where}\\ \omega_{M3}=\omega\frac{p_1}{p_3};\\ \omega_F=\omega\frac{p_1}{p_2}+\omega_E;\\ \omega_D=\omega\frac{p_1}{p_3}-\omega_C. \end{gathered} \]
Taking (20) into account, the voltage-balance equations take the form:
\[ \begin{gathered} U_B=-I_{mB}\left[r_B\cdot \sin(\omega_B t+\alpha_1)+1.5L_B\omega_B\cdot \cos(\omega_B t+\alpha_1)\right]- I_A\cdot M_{AB}\cdot \omega_B\cdot \cos\omega_B t;\\ U_C=I_{mC}\left[r_C\cdot \sin(\omega_C t+\alpha_2)+\right.\\ \left.+1.5L_C\omega_C\cdot \cos(\omega_C t+\alpha_2)\right] -\frac{3}{2}I_{mD}\cdot \omega_C M_{CD}\cdot \sin(\omega_C t-\alpha_3);\\ U_E=I_{mE}\cdot \left[r_E\cdot \sin(\omega_E t+\alpha_4)+1.5L_E\omega_E\cdot \cos(\omega_E t+\alpha_4)\right]-\\ -\frac{3}{2}I_{mF}\cdot M_{EF}\cdot \omega_E\cdot \sin(\omega_E t-\alpha_5);\\ U_F=I_{mF}\left[r_F\cdot \sin(\omega_F\cdot t+\alpha_5)+1.5L_F\omega_F\cos(\omega_F t+\alpha_5)\right]-\\ -\frac{3}{2}I_{mE}\cdot M_{EF}\omega_F\sin(\omega_F t-\alpha_4); \end{gathered} \tag{21} \]
\[ \begin{gathered} 0=I_{mD}\left[r_D\cdot \sin(\omega_D t+\alpha_3)+1.5L_D\omega_D\cos(\omega_D t+\alpha_3)\right]-\\ -\frac{3}{2}I_{mC}\cdot M_{CD}\cdot \omega_D\cdot \sin(\omega_D t-\alpha_2). \end{gathered} \]
Another notation, in symbolic form and rms values:
\[ \begin{gathered} \dot U_B=-\dot I_B\left[r_B+j(L_B-M_{AB})\omega_B\cdot 1.5\right] -j(\dot I_A+\dot I_B)\cdot \omega_B M_{AB}\cdot 1.5;\\ \dot U_C=\dot I_C\left[r_C+j(L_C-M_{CD})\omega_B\cdot 1.5\right] +j(\dot I_C+\dot I_D)\omega_C\cdot M_{CD}\cdot 1.5;\\ \dot U_E+\dot I_E\left[r_E+j(L_E-M_{EF})\omega_E 1.5\right] +j(\dot I_F+\dot I_E)\omega_E M_{EF}\cdot 1.5;\\ \dot U_F=-\dot I_F\left[r_F+j(L_F-M_{EF})\omega_F\cdot 1.5\right] -j(\dot I_F+\dot I_E)\cdot \omega_F\cdot M_{EF}\cdot 1.5;\\ 0=\dot I_D\left[r_D+j(L_D-M_{CD})\omega_D\cdot 1.5\right] +j(\dot I_C+\dot I_D)\omega_D\cdot M_{CD}\cdot 1.5. \end{gathered} \tag{22} \]
Denote the ratio \(\dfrac{\omega_B}{\omega}=\alpha\), and write equations (22) in the form:
\[ \frac{\dot U_B}{\alpha} = -\dot I_B\left[\frac{r_B}{\alpha}+jx_{sB}\right] +\dot I_{01}\cdot Z_{01}; \]
\[ Z_{01}=Z_{01}+jM_{AB}\cdot \omega\cdot 1.5; \]
\[ \frac{\dot U_C}{\alpha}=\dot I_C\left[\frac{r_C}{\alpha}+jX_{\sigma B}\right]+\dot I_{03}\cdot Z_{03}; \]
\[ Z_{03}=r_{03}+jM_{CD}\cdot\omega\cdot1.5; \tag{23} \]
\[ \frac{\dot U_E}{\alpha}=\dot I_E\left[\frac{r_E}{\alpha}+jX_{\sigma E}\right]+\dot I_{02}\cdot Z_{02}; \]
\[ Z_{02}=r_{02}+jM_{EF}\cdot\omega\cdot1.5; \]
\[ \frac{\dot U_F}{\alpha+p_2} =-\dot I_F\left[\frac{r_F}{\alpha+p_2}+jX_{\sigma F}\right]+\dot I_{02}\cdot Z_{02}; \]
\[ 0=\dot I_D\left[\frac{r_D}{\alpha-p_3}+jX_{\sigma D}\right]+\dot I_{03}\cdot Z_{03}, \]
where
\[
\dot I_{01}=\dot I_A+\dot I_B
\]
is the magnetizing current of the first cascade,
\[
\dot I_{02}=\dot I_E+\dot I_F
\]
is the magnetizing current of the second cascade,
\[
\dot I_{03}=\dot I_C+\dot I_D
\]
is the magnetizing current of the third cascade.
On the basis of (23) we construct the equivalent circuit and the vector diagram (Figs. 4, 5), from which the necessary parameters can be determined and then the circle diagram can be constructed (Fig. 6).
Fig. 4. Equivalent circuit of a brushless BEMA of regulated frequency
The construction of the circle diagram is carried out as follows:
- The circle diagram of the third cascade is constructed with the corresponding slip scale \(S_3\) at \(\alpha=1\):
a) the diameter of the circle
\[ D_3=\frac{U_0}{X_{\sigma C}+X_{\sigma D}}, \]
b) the current at \(S_3=\pm\infty\)
\[ I_{03}=\frac{U_0}{X_{\sigma C}+X_{03}}, \]
c) the point \(S_3=1\).
\[ I_C=\frac{U_0}{(r_D+r_C)+j(X_{\sigma C}+X_{\sigma D})}. \]
Fig. 5. Vector diagram of a brushless electric-machine amplifier of variable frequency
Visible labels in the diagram: $\dot I_D$, $\dot I_{01}$, $\dot I_{02}$, $\dot I_{03}$, $\dot I_C$, $-\dot I_D$, $\dot I_B$, $\dot I_E$, $\dot E_B$, $-\dot E_C$, $-\dot E$, $\dot U_C = \dot U_E = \dot U_B$.
Fig. 6. Circle diagram of a brushless EMU of adjustable frequency
- The circle diagram of the second cascade is constructed with the corresponding scale \(S_2\) at \(\alpha = 1\):
a) diameter of the circle
\[ D_2 = - \frac{U_0}{X_{\sigma E} + X_{\sigma F}}, \]
b) point \(S_2 = \pm \infty\)
\[ I_{02} = \frac{U_0}{X_{\sigma E} + X_{02}}, \]
c) point \(S_3 = 1\).
\[ \dot I_E=\frac{U_0}{(r_E+r_F)+j(X_{\sigma F}+X_{\sigma E})}. \]
-
Specifying different values of \(\alpha\), we construct the hodograph of the current \(I_B\).
-
Knowing the direction and magnitude of the current \(I_B\), one can determine the value of the control current \(I_A\) for any value of the parameter \(\alpha\).
The circle diagram makes it possible to analyze the various operating modes of the amplifier.
From the system of equations (23), the equation of the external characteristic can be obtained:
\[ \frac{\dot U_F}{a+p_2}=\dot E_F-I_H\left(Z_{\sigma F}+Z_{02}+Z''\right), \tag{24} \]
where
\[ \dot E_F=\frac{K(E_{B0}\cdot Z_{02})}{K(Z_{01}+Z_{\sigma B})+Z_{\sigma E}+Z_{02}}; \]
\[ Z''=\frac{Z_{02}^{2}}{K(Z_{01}+Z_{\sigma B})+Z_{\sigma E}+Z_{02}}; \]
\[ K=\frac{Z_{03}+Z_C\left(1+\dfrac{Z_{03}}{Z_D}\right)} {\dfrac{Z_{03}\cdot Z_{01}}{Z_D}+(Z_{01}+Z_{03})+(Z_B+Z_C)\left(\dfrac{Z_{03}}{Z_D}\right)}; \]
\[ E_{B0}=\dot I_A\cdot Z_{01}. \]
Expression (24) shows that the voltage at the output of the brushless EMA depends to a considerable extent on the coefficient \(K\), which is determined by the parameters of the asynchronous generator.
On the basis of the available equations, an expression has been obtained for the gain coefficient:
\[ K_y=\frac{3}{4} \left[ \frac{KZ_{02}}{K(Z_{01}+Z_{\sigma B})+Z_{\sigma E}+Z_{02}} \right]^2 \cdot \frac{I_A\cdot Z_{01}^{2}} {(Z_{\sigma F}+Z_{02}+Z'')(\dot I_A\cdot r_A-E_A)}, \tag{25} \]
where \(E_A\) is the counter-emf in the control winding.
Next we shall carry out a certain analysis of the operation of the third cascade of the amplifier. For this purpose we pass from the equivalent circuit (Fig. 4) to an equivalent circuit with an external magnetizing circuit (Fig. 7).
Fig. 7. Equivalent circuit of a brushless variable-frequency EMA with an external circuit.
The parameters in the equivalent circuit of Fig. 7 are as follows:
\[ \sigma_1 = 1 + \frac{Z_{\sigma C}}{Z_{03}}; \]
\[ \sigma_2 = 1 + \frac{Z_{01} + Z_{\sigma B}}{Z_{92}}; \]
\[ Z_{92} = \frac{Z_{91}\cdot (Z_{03} + Z_{\sigma C})}{Z_{91} + Z_{03} + Z_{\sigma C}}; \]
\[ Z_{91} = Z_{\sigma E} + \frac{(Z_H + Z_{\sigma F})Z_{02}}{Z_H + Z_{\sigma F} + Z_{02}}. \]
On the basis of this equivalent circuit we write expressions for the active components of the currents \(I_1\) and \(I_2\).
\[ I_{1a} = \frac{E_{BO}\cdot R_1}{R_1^2 + X_1^2}, \tag{26} \]
\[ I_{2a} = \frac{E_{BO}\cdot R_2}{R_2^2 + X_2^2}, \tag{27} \]
where
\[ R_1 = \sigma_2\left(r_a + \frac{r_B}{a} + \frac{r_C}{a}\sigma_1\sigma_2 + \frac{r_D}{a - p_3}\sigma_1^2\sigma_2\right), \]
\[ X_1 = \sigma_2\left(X_{01} + X_{\sigma B} + X_{\sigma C}\sigma_1\sigma_2 + X_{\sigma D}\sigma_1^2\sigma_2\right), \]
\[ R_2 = r_{01} + \frac{r_B}{a} + r_{92}, \]
\[ X_2 = X_{01} + X_{\sigma B} + X_{92}. \]
From expression (26) we obtain the condition under which the value of the current \(I_{1a}\) is maximum. This condition is written in the form
\[ \frac{r_D}{a - p_3}\sigma_1^2\cdot \sigma_{2\mathrm{cp}} \approx X_{01} + X_{\sigma B} + X_{\sigma C}\cdot \sigma_1\sigma_{2\mathrm{cp}} + X_{\sigma D}\cdot \sigma_1^2\sigma_{2\mathrm{cp}}. \tag{28} \]
The parameter \(\sigma_{2\mathrm{cp}}\) is equal to
\[ \sigma_{2\mathrm{cp}} = \frac{\sigma_{2x} + \sigma_{2k}}{2}, \]
where
\[ \sigma_{2x} = 1 + \frac{(Z_{01} + Z_{\sigma B})(Z_{\sigma E} + Z_{\sigma C} + Z_{02} + Z_{03})} {(Z_{\sigma E} + Z_{02})(Z_{\sigma C} + Z_{03})} \;— \]
at no load of the amplifier,
\[ \sigma_{2k} = 1 + \frac{(Z_{01} + Z_{\sigma B})\left(Z_{\sigma E} + Z_{\sigma C} + Z_{03} + \dfrac{Z_{02}\cdot Z_{\sigma F}}{Z_{02} + Z_{\sigma F}}\right)} {\left(Z_{\sigma E} + \dfrac{Z_{02}\cdot Z_{\sigma F}}{Z_{02} + Z_{\sigma F}}\right)(Z_{\sigma C} + Z_{03})} \;— \]
at short circuit of the amplifier.
Using (28), curves were plotted for the variation of the current \(I_{1a}\) over the entire frequency range of operation of the amplifier at no load, for different values of the resistance \(Z_D\) (Fig. 8).
Fig. 8. Change in current
Data for construction:
\[
r_{01}=r_{02}=r_{03}=0
\]
— losses in the steel are not taken into account;
\[ r_{\mathrm{B}}=r_{\mathrm{E}}=r_{\mathrm{C}}=r_{\mathrm{F}}=1.1;\qquad X_{02}=40\ \Omega;\qquad X_{01}=X_{03}=10\ \Omega; \]
\[ X_{\sigma F}=X_{\sigma C}=X_{\sigma E}=0.9\ \Omega;\qquad X_{\sigma D}=0.5\ \Omega;\qquad I_A=5\ \mathrm{A};\qquad \alpha=1\div4. \]
It is evident from the figure that the maximum value of the current \(I_{1a}\) is maintained over a considerable range of \(\alpha\) as the resistance \(r_D\) increases.
Consequently, in this case the asynchronous generator delivers the greatest active power. The condition under which the asynchronous generator operates most effectively is written as
\[ r_D\cdot\sigma_1^2\sigma_2 \approx \frac{r_{01}+\dfrac{r_{\mathrm{B}}}{\alpha}-r_{э2}}{2} + \frac{X_{01}+X_{\sigma \mathrm{B}}+X_{э2}} {2\left(r_{01}+\dfrac{r_{\mathrm{B}}}{\alpha}+r_{э2}\right)} . \tag{29} \]
As a result, the findings of the work performed may be formulated as follows:
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The proposed circuit of a brushless electric-machine amplifier of regulated frequency with an asynchronous generator makes it possible to obtain the basic characteristics required for the amplifier and a comparatively wide range of frequency regulation.
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The frequency characteristics and energy diagrams obtained show that the most expedient operating mode of the amplifier is the mode with opposing connection of the rotor windings of the input and output stages.
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The obtained equations of the equivalent circuit, vector diagrams, and circle diagrams make it possible to analyze all operating modes of the BEMA and to determine the necessary characteristics.
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On the basis of analysis of the circuit and of the parameters of the asynchronous generator, condition (29) was obtained for its most expedient operation.
REFERENCES
- A. I. Skorospeshkin, M. L. Kostyrev, E. F. Obergan. Electric-machine amplifier of regulated frequency. Izvestiya TPI, vol. 160, 1966.
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A. I. Skorospeshkin, M. L. Kostyrev, É. F. Obergán. Semiconductor commutator for excitation of asynchronized machines. The state and prospects for the development of semiconductor excitation of synchronous motors, abstracts for a scientific-technical conference, Sverdlovsk, 1966.
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A. I. Skorospeshkin, É. F. Obergán, Sh. S. Roiz. Energy relationships in brushless EMU of direct and alternating current. Izvestiya TPI, vol. 160, 1966.
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A. I. Skorospeshkin. Questions of the general theory of electric machines as applied to brushless electric-machine amplifiers (BEMA) of alternating current. Izvestiya TPI, vol. 145, 1966.