Issues of the General Theory of Electrical Machines as Applied to Brushless AC Electromechanical Amplifiers (BEMA)

A. I. Skorospelkin

(Recommended by the seminar of the departments of electrical machines and general electrical engineering.)

The development of the general theory of electrical machines promotes the further development of the theory of electrical machines and the solution of a number of the most important practical problems in electrical machine building. Among the questions that can be solved on the basis of the general theory should be included the possibility of comparing different classes of electrical machines and developing new machines.

It should be noted that the development of the general theory of electrical machines is usually carried out as applied to machines of normal design—synchronous, asynchronous, and DC machines [1, 2, 3]. In this case the machine is reduced to an idealized two-pole and two-phase machine, and then an analysis is made of its operating modes. Such an approach has its own advantages and disadvantages. The advantages consist in the fact that the analytical relationships are simplified and the number of equations is reduced. The disadvantages are a certain artificiality, in which the real machine is a reflection of the initial idealized machine.

Combined electrical machines of the type of fixed- and variable-frequency converters, brushless electromechanical DC and AC amplifiers of fixed and variable frequency, as far as we know, are not considered from the standpoint of the general theory of electrical machines, although the necessity of developing a general approach for them is obvious. A general approach to combined machines, in our opinion, can be developed on the basis of Maxwell’s theory of electromagnetic phenomena and Lagrange’s equations [4, 5, 6]. In this case the general equations are derived from a direct consideration of the physical properties of these machines.

The brushless electromechanical amplifiers (BEMA) and converters that we are developing are two-stage. Both stages are combined in one magnetic core, provided that the conditions of combination are fulfilled. Fulfillment of such conditions makes it possible in investigations first to consider each stage as a separate machine, with subsequent establishment of the connection between the individual stages.

When using the Lagrange–Maxwell equations, it appears possible to consider an electrical machine consisting of two groups of windings: one of them is stationary and is located on the stator, the other is rotating and is located on the rotor. In this case each of the windings can be considered separately, and for it the Lagrange equations of the second kind are represented in the form of two

equations. One of them is written for electromechanical forces, the other for electromotive forces.

In final form these equations are written by the following formulas:

for electromechanical forces

\[ M_{\text{em}}=-\frac{1}{2} i_a i_b \frac{\partial m_{ab}}{\partial h}, \tag{1} \]

for electromotive forces

\[ e_k=i_k r_k+\frac{d(l_k i_k)}{dt}+\frac{d(m_{ka} i_a)}{dt}. \tag{2} \]

Here \(M_{\text{em}}\) is the generalized external electromechanical force,
\(e_k\) is the generalized e.m.f.,
\(h\) is the generalized geometric coordinate,
\(i_k\) is the loop current (generalized velocity),
\(r_k,\ l_k,\ m_{ab},\ i_a,\ i_b,\ i_k\) are the active resistance, inductances, mutual inductances, and currents of the windings under consideration.

In each cascade of a combined machine, for each of the windings, general equations of electromotive forces and electrical powers and equations of electromechanical forces and electromechanical powers may be written, taking into account the interaction of the stator and rotor windings.

These equations are written in the following form:

1. Equations of e.m.f.

\[ e_k=i_k r_k+\frac{d(l_k i_k)}{dt}+\sum \frac{d(m_{ka} i_a)}{dt}. \tag{3} \]

1. Equations of electrical powers

\[ P_{\text{el}}=e_k i_k=i_k^2 r_k+i_k\frac{d(l_k i_k)}{dt}+i_k\sum \frac{d(m_{ka} i_a)}{dt}. \tag{4} \]

1. Equations of electromechanical forces

\[ M_{\text{em}}=-\frac{1}{2} i_a i_b \frac{\partial m_{ab}}{\partial h}. \tag{5} \]

1. Equations of electromechanical powers

\[ P_{\text{em}}=M_{\text{em}}\frac{dh}{dt} =-\frac{1}{2} i_a i_b \frac{\partial m_{ab}}{\partial h}\frac{\partial h}{\partial t}. \tag{6} \]

As the generalized geometric coordinate \(h\) in electrical machines, the rotor angle of rotation \(\alpha\) is taken. Then, in steady-state operation,

\[ h=\alpha=\omega t. \tag{7} \]

In this case the generalized electromechanical force will be the torque.

Since the inductances depend only on the geometric coordinates, and an electrical machine is characterized by one geometric coordinate \(\alpha\), the partial derivatives \(\frac{\partial m_{ab}}{\partial \alpha}\) will be equal to the total derivatives \(\frac{dm_{ab}}{d\alpha}\). Therefore (5) and (6) take the form:

\[ M_{\mathrm{em}}=-\frac{1}{2} i_a i_b \frac{d m_{ab}}{d \alpha}; \tag{8} \]

\[ P_{\mathrm{em}}=-\frac{1}{2} i_a i_b \frac{d m_{ab}}{d t}. \tag{9} \]

It should be noted that in a nonsalient-pole machine the inductances and mutual inductances between phases may be regarded as constant; only the mutual inductances between the stator and rotor windings vary, which is a necessary condition for the conversion of mechanical energy into alternating-current electrical energy, or vice versa.

From the system of equations (3)—(6) one can determine the currents, powers, and torques, and thereby establish the interrelation between the individual stages.

Let us determine the necessary quantities as applied to an alternating-current BEMA, the circuit and operating principle of which are explained in Fig. 1. The following notation is used in the figure:

Fig. 1. Circuit of a brushless electromechanical alternating-current amplifier (BEMA).

$A$ — control winding. Located on the stator and supplied with direct current.

$B$, $C$ — rotor windings of the first and second stages of the amplifier.
$D$ — output winding. Located on the stator.
The numerals 1, 2, 3 denote the phase sequence in the three-phase rotor and stator windings.
$\omega_p$ — angular velocity of rotation of the rotor.
$\omega_1$; $\omega'_2$; $\omega_2$ — angular velocities of rotation of the fields in the windings.
The arrows show the directions of rotation of the rotor $B$, $C$, and $D$, and of the fields in stages I and II.
$P_1$ and $P_2$ — numbers of pole pairs in stages I and II. The direction of rotation of the field in winding $C$ depends on the phase sequence; correspondingly, $\omega_2$ also changes.

Let us write the emf equilibrium equations for control winding \(A\) and for one of the phases of windings \(B\), \(C\), and \(D\), assuming that the windings are made symmetrically. In the most general form they are written by the following expressions:

For winding \(A\)

\[ U_A=i_A r_A+L_A\frac{di_A}{dt}+i_1\frac{dm_{1AB}}{dt}+m_{1AB}\frac{di_1}{dt}+m_{2AB}\frac{di_2}{dt}+ \tag{10} \]

\[ +i_2\frac{dm_{2AB}}{dt}+m_{3AB}\frac{di_3}{dt}+i_3\frac{dm_{3AB}}{di}. \]

For the first phase of windings \(B\) and \(C\)

\[ 0=i_1(r_b+r_c)+L_B\frac{di_1}{dt}+L_C\frac{di_1}{dt}+m_{1AB}\frac{di_A}{dt}+i_A\frac{dm_{1AB}}{dt}- \]

\[ -M_B\frac{di_2}{dt}-M_B\frac{di_3}{dt}-M_C\frac{di_2}{dt}-M_C\frac{di_3}{dt}+m_{1C1D}\frac{di_{D1}}{dt}+ \tag{11} \]

\[ +i_{D1}\frac{dm_{1C1D}}{dt}+m_{1C2D}\frac{di_{D2}}{dt}+i_{D2}\frac{dm_{1C2D}}{dt}+ \]

\[ +m_{1C3D}\frac{di_{D3}}{dt}+i_{D3}\frac{dm_{1C3D}}{dt}. \]

For the first phase of winding \(D\)

\[ U_{D1}=i_{D1}r_D+L_D\frac{di_{D1}}{dt}-M_D\frac{di_{D2}}{dt}-M_D\frac{di_{D3}}{dt}+ \]

\[ +m_{1D1C}\frac{di_1}{dt}+i_1\frac{dm_{1D1C}}{dt}+m_{1D2C}\frac{di_2}{dt}+i_2\frac{dm_{1D2C}}{dt}+ \tag{12} \]

\[ +m_{1D3C}\frac{di_3}{dt}+i_3\frac{dm_{1D3C}}{dt}. \]

Taking into account that

\[ i_1+i_2+i_3=0, \]

\[ i_{D1}+i_{D2}+i_{D3}=0 \tag{13} \]

and adopting, according to [7],

\[ -M_B=0{,}5L_B, \]

\[ -M_C=0{,}5L_C, \]

\[ -M_D=0{,}5L_D, \]

one can respectively transform (10), (11), (12).

Notation in the formulas:

\(i_A\), \(i_1\), \(i_2\), \(i_3\), \(i_{D1}\), \(i_{D2}\), \(i_{D3}\) are the instantaneous values of the currents in windings \(A\), \(B\), \(C\), and \(D\);

\(r_A\), \(r_B\), \(r_C\), \(r_D\) and \(L_A\), \(L_B\), \(L_C\), \(L_D\) are the active resistances and inductances of windings \(A\), \(B\), \(C\), and \(D\), with \(r_D\) including the load resistance.

\(M_B\), \(M_C\), \(M_D\) are the mutual inductances between the phases of windings \(B\), \(C\), and \(D\).

The letter \(m\), with the corresponding subscripts, denotes the variable mutual inductances between the stator and rotor windings in the first and second stages.

\(U_A\)—the constant voltage applied to the control winding;
\(U_{D_1}\)—the instantaneous value of the phase voltage of the \(D\) winding.

Equations (10), (11), (12) are written for the instantaneous values of all quantities and are valid for any steady-state or transient mode.

For the time being we shall consider only the steady-state mode and shall take into account only the fundamental harmonics. In the circuit of Fig. 1, for the rotor we shall assume counter-rotation of the fields and the rotor in both stages, and in the stator—rotation in agreement with the rotor.

In this case the variations of currents and mutual inductances will be determined by the equations:

\[ \begin{aligned} i_1 &= I_{mBC}\sin(\omega_1 t+\alpha_1); & m_{1AB} &= M_{AB}\sin\omega_1 t; \\[4pt] i_2 &= I_{mBC}\sin\left(\omega_1 t+\alpha_1+\frac{2\pi}{3}\right); & m_{2AB} &= M_{AB}\sin\left(\omega_1 t+\frac{2\pi}{3}\right); \\[4pt] i_3 &= I_{mBC}\sin\left(\omega_1 t+\alpha_1-\frac{2\pi}{3}\right); & m_{3mB} &= M_{AB}\sin\left(\omega_1 t-\frac{2\pi}{3}\right); \end{aligned} \tag{15} \]

\[ \begin{aligned} i_{D1} &= I_{mD}\sin(\omega_2 t+\alpha_2); \\[4pt] i_{D2} &= I_{mD}\sin\left(\omega_2 t+\alpha_2-\frac{2\pi}{3}\right); \\[4pt] i_{D3} &= I_{mD}\sin\left(\omega_2 t+\alpha_2+\frac{2\pi}{3}\right); \\[8pt] m_{1C1D} &= M_{CD}\sin(\omega_M t); & m_{2C1D} &= M_{CD}\sin\left(\omega_M t+\frac{2\pi}{3}\right); \\[4pt] m_{1C2D} &= M_{CD}\sin\left(\omega_M t-\frac{2\pi}{3}\right); & m_{2C2D} &= M_{CD}\sin(\omega_M t); \\[4pt] m_{1C3D} &= M_{CD}\sin\left(\omega_M t+\frac{2\pi}{3}\right); & m_{2C3D} &= M_{CD}\sin\left(\omega_M t-\frac{2\pi}{3}\right); \\[4pt] m_{3C1D} &= M_{CD}\sin\left(\omega_M t-\frac{2\pi}{3}\right); \\[4pt] m_{3C2D} &= M_{CD}\sin\left(\omega_M t+\frac{2\pi}{3}\right); \\[4pt] m_{3C3D} &= M_{CD}\sin(\omega_M t); \end{aligned} \]

\(I_{mBC}, I_{mD}, M_{AB}, M_{CD}\)—the corresponding amplitude values of the currents and mutual inductances.

In these equations

\[ \begin{aligned} \omega_1 &= \omega_p P_1;\\ \omega_M &= \omega_p P_2;\\ \omega_2 &= \omega_p(P_2-P_1), \end{aligned} \tag{16} \]

where \(\omega_M\) is the angular velocity with which the mutual inductance changes in the second stage.

For the steady-state regime, equation (10) takes the form:

\[ U_A = I_A r_A \qquad \text{and} \qquad I_A = \frac{U_A}{r_A}. \tag{17} \]

Equations (11) and (12), taking (13), (16) into account, after certain transformations are written in the following form:

\[ \begin{aligned} 0 ={}& I_{mBC}\left[\sin(\omega_1 t+\alpha_1)(r_B+r_C) +(15L_B+1{,}5L_C)\omega_1\cos(\omega_1 t+\alpha_1)\right] + \\ &+ I_A M_{AB}\omega_1\cos\omega_1 t + I_{mD}M_{CD}\bigl[\sin\omega_M t\cdot\omega_2\cos(\omega_2 t+\alpha_2) + \\ &\qquad + \sin(\omega_2 t+\alpha_2)\omega_M\cos(\omega_M t) + \\ &\qquad + \sin\left(\omega_M t-\frac{\widetilde{2\pi}}{3}\right) \omega_2\cos\left(\omega_2 t+\alpha_2-\frac{\widetilde{2\pi}}{3}\right) + \\ &\qquad + \sin\left(\omega_2 t+\alpha_2-\frac{\widetilde{2\pi}}{3}\right) \omega_M\cos\left(\omega_M t-\frac{\widetilde{2\pi}}{3}\right) + \\ &\qquad + \sin\left(\omega_M t+\frac{\widetilde{2\pi}}{3}\right) \omega_2\cos\left(\omega_2 t+\alpha_2+\frac{\widetilde{2\pi}}{3}\right) + \\ &\qquad + \sin\left(\omega_2 t+\alpha_2+\frac{\widetilde{2\pi}}{3}\right) \omega_M\cos\left(\omega_M t+\frac{\widetilde{2\pi}}{3}\right)\bigr] ; \end{aligned} \tag{18} \]

\[ \begin{aligned} U_{mD}\sin\omega_2 t ={}& I_{mD}\left[\sin(\omega_2 t+\alpha_2)r_D +1{,}5L_D\omega_2\cos(\omega_2 t+\alpha_2)\right] + \\ &+ I_{mBC}M_{CD}\bigl[\sin(\omega_M t)\omega_1\cos(\omega_1 t+\alpha_1) +\sin(\omega_1 t+\alpha_1)\omega_M\cos(\omega_M t) + \\ &\qquad + \sin\left(\omega_M t+\frac{\widetilde{2\pi}}{3}\right) \omega_1\cos\left(\omega_1 t+\alpha_1+\frac{\widetilde{2\pi}}{3}\right) + \\ &\qquad + \sin\left(\omega_1 t+\alpha_1+\frac{\widetilde{2\pi}}{3}\right) \omega_M\cos\left(\omega_M t+\frac{\widetilde{2\pi}}{3}\right) + \\ &\qquad + \sin\left(\omega_M t-\frac{\widetilde{2\pi}}{3}\right) \omega_1\cos\left(\omega_1 t+\alpha_1-\frac{\widetilde{2\pi}}{3}\right) + \\ &\qquad + \sin\left(\omega_1 t+\alpha_1-\frac{\widetilde{2\pi}}{3}\right) \omega_M\cos\left(\omega_M t-\frac{\widetilde{2\pi}}{3}\right)\bigr] . \end{aligned} \tag{19} \]

After transformations, (18) and (19) take the form:

\[ \begin{aligned} 0 ={}& I_{mBC}\left[\sin(\omega_1 t+\alpha_1)(r_B+r_C) +(1{,}5L_B+1{,}5L_C)\omega_1\cos(\omega_1 t+\alpha_1)\right] + \\ &+ I_A M_{AB}\omega_1\cos(\omega_1 t) -\frac{3}{2}I_{mD}M_{CD}\omega_1\sin(\omega_1 t-\alpha_2); \end{aligned} \tag{20} \]

\[ \begin{aligned} U_{mD}\sin\omega_2 t ={}& I_{mD}\left[\sin(\omega_2 t+\alpha_2)r_D +1{,}5L_D\omega_2\cos(\omega_2 t+\alpha_2)\right] - \\ &-\frac{3}{2}I_{mBC}M_{CD}\omega_2\sin(\omega_2 t-\alpha_1). \end{aligned} \tag{21} \]

When the phase sequence in the rotor of the second stage is changed, i.e., when the direction of rotation of the field is changed, when

\[ \omega_2=\omega_p(P_2+P_1), \]

equations (20) and (21) are written in the form

\[ \begin{aligned} 0={}& I_{mBC}\left[\sin(\omega_1 t+\alpha_1)(r_B+r_C) +(1.5L_B+1.5L_C)\omega_1\cos(\omega_1 t+\alpha_1)\right]+\\ &+ I_A M_{AB}\omega_1\cos(\omega_1t) +\frac{3}{2}I_{mD}M_{CD}\omega_1\sin(\omega_1t+\alpha_2); \end{aligned} \tag{23} \]

\[ \begin{aligned} U_{mD}\sin\omega_2 t ={}& I_{mD}\left[\sin(\omega_2t+\alpha_2)r_D +1.5L_D\omega_2\cos(\omega_2t+\alpha_2)\right]+\\ &+\frac{3}{2}I_{mBC}M_{CD}\omega_2\sin(\omega_2t+\alpha_1). \end{aligned} \tag{24} \]

Equations (20)—(24) make it possible to carry out a certain analysis of the components and to determine the amplitude values of the currents \(I_{mBC}\) and \(I_{mD}\) at no load, short circuit, and any load. Then one can proceed to determining the powers and torques.

The required e.m.f.’s, respectively for windings \(B\), \(C\), and \(D\), are

\[ \begin{aligned} e_{B_1} &= I_A\omega_1M_{AB}\cos\omega_1t;\\ e_{C_1} &= \mp\frac{3}{2}I_{mD}M_{CD}\omega_1\sin(\omega_1t\mp\alpha_2);\\ e_{D_1} &= \mp\frac{3}{2}I_{mBD}M_{CD}\omega_2\sin(\omega_2t\mp\alpha_1). \end{aligned} \tag{25} \]

The signs \((\mp)\) in (25) correspond to \(\omega_2=\omega_p(P_2\mp P_1)\).

The electrical powers of winding \(A\) and of one phase of windings \(B\), \(C\), and \(D\) are:

\[ P_{\text{el. }A}=I_A^2r_A; \]

\[ P_{\text{el. }B_1}=i_1e_{B_1} =I_{mBC}\sin(\omega_1t+\alpha_1)I_A\omega_1M_{AB}\cos\omega_1t; \]

\[ P_{\text{el. }C_1}=i_1e_{C_1} =I_{mBC}\sin(\omega_1t+\alpha_1) \left[\mp\frac{3}{2}I_{mD}M_{CD}\omega_1\sin(\omega_1t\mp\alpha_2)\right]; \tag{26} \]

\[ P_{\text{el. }D_1}=i_1e_{D_1} =I_{mD}\sin(\omega_2t+\alpha_2) \left[\mp\frac{3}{2}I_{mBC}M_{CD}\omega_2\sin(\omega_2t\mp\alpha_1)\right]. \]

The electromechanical powers of stages I and II per phase, according to (1), are

\[ P_{\text{em }AB1} =-\frac{1}{2}I_AI_{mBC}\sin(\omega_1t+\alpha_1)\omega_1M_{AB}\cos\omega_1t; \tag{27} \]

\[ \begin{aligned} P_{\text{em }C1B} ={}&-\frac{1}{2}i_{C1}i_{D1}\frac{dm_{1C1D}}{dt} -\frac{1}{2}i_{C1}i_{D2}\frac{dm_{1C2D}}{dt} -\frac{1}{2}i_{C1}i_{D3}\frac{dm_{1C3D}}{dt}\\ ={}&-\frac{1}{2}I_{mBC}I_{mD} \left[\mp\frac{3}{2}\sin(\omega_1t\mp\alpha_2)\right] \sin(\omega_1t+\alpha_1)\cdot M_{CD}. \end{aligned} \]

The rotating torques for stages I and II, according to (8), are

\[ M_{\text{em }AB1} =-\frac{1}{2}I_AI_{mBC}M_{AB}\sin(\omega_1t+\alpha_1)\cos\omega_1t; \]

\[ M_{\mathrm{em}C1D} = -\frac{1}{2} I_{mBC} I_{mD}\sin(\omega_1 t+\alpha_1) \left[ \mp \frac{3}{2}\sin(\omega_1 t \mp \alpha_2) \right]. \tag{28} \]

Thus, on the basis of the general Lagrange–Maxwell equations, all the necessary data of a combined electrical machine of the BEMA type are determined. From them, the necessary analysis of all operating modes of the amplifier can be carried out.

In the amplifier, as a rule, feedback windings are provided in order to compensate for the armature reaction. The currents, powers, and torques produced by them are taken into account in an analogous manner.

If the field of the control winding rotates (a BEMA with adjustable frequency), then the general equations for the amplifier remain the same as those considered above; only the values of the angular velocities change accordingly, taking into account the direction of rotation of the fields and of the rotor.

REFERENCES

1. B. Adkins. General Theory of Electrical Machines. GEI, 1960.

2. D. White and G. Woodson. Electromechanical Energy Conversion. Energiya Publishing House, 1964.

3. M. I. Alyab’ev. General Theory of Shipboard Electrical Machines. Sudostroenie Publishing House, 1965.

4. A. G. Kaplyanskii. Introduction to the General Theory of Electrical Machines. GEI, 1941.

5. V. V. Bazilevich. On the question of the general theory of electrical machines. Elektrichestvo, No. 21, 1930.

6. I. M. Sadovskii. Electrodynamics of commutator electrical machines. Elektrichestvo, No. 4, 1949.

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