SOME ISSUES IN THE THEORY OF AN ASYNCHRONOUS BRUSHLESS FREQUENCY CONVERTER

V. S. Novokshonov

(Presented by the scientific-methodological seminar of the EMF of TPI)

Recently, in a number of branches of industry, there has been a great need for brushless converters of increased frequency. It is known [2] that the use of asynchronous electric motors of increased frequency makes it possible to considerably reduce the weight of hand-held electric tools (electric saws, electric drills, etc.), and to simplify the design and operation of high-speed machine tools for the woodworking, automotive, and aviation branches of industry.

The operation of asynchronous two-machine frequency converters and synchronous generators of increased frequency, when they work in damp, dusty rooms or in the open air, is difficult because these converters have a brush device, whose operation under the stated conditions is highly unreliable. In a number of branches of industry, the use of converters having brush contact is not permitted at all. The coal industry has a great need for brushless converters of increased frequency. Lightweight mine electric drills of increased frequency were designed and manufactured by plants of the Ministry of the Coal Industry as early as 1950. However, the absence of a brushless frequency converter having satisfactory characteristics did not allow them to be introduced into mines.

Static converters of increased frequency have not found wide application as power sources for asynchronous motors, because they have a low utilization of active materials, a sharply drooping external characteristic, and a low power factor [3, 9]. Rotating frequency converters based on the principle of distortion of the magnetic field also have not found wide application [5]. Although their characteristics are somewhat better than those of static converters, they nevertheless remain unsatisfactory.

In comparison with static frequency converters and with converters based on the principle of distortion of the magnetic field, the asynchronous brushless frequency converter under investigation (ABFC) has a number of valuable properties: 1) the possibility of obtaining a good external characteristic; 2) higher technical and economic operating indices; 3) higher utilization of active materials.

Under the author’s direction, a technical design of an ABFC for supplying mine electric drills of increased frequency was carried out at the Tomsk Electromechanical Plant. This technical design of the ABFC was approved by the Ministry of the Coal Industry of the USSR and put into production. The technical design was preceded by the manufacture and testing of an experimental model of the ABFC.

1. Description and principle of operation of the ABFC

The ABFC (Fig. 1) consists of an asynchronous driving motor AD, an asynchronous converter AP, and a synchronous machine SM. The pri-

drive motor and the asynchronous converter are combined in a single magnetic core. The combination of two magnetic fields in one magnetic core is possible only if the inequality is satisfied:

\[ p_1 \pm p_2 = \pm 1, \tag{1} \]

where \(p_1\) and \(p_2\) are the numbers of pole pairs of the first and second fields.

If, when two magnetic fields are superposed, inequality (1) is not fulfilled, then forces of unilateral magnetic attraction and vibrations [1] will appear in the air gap, which may cause mechanical vibrations of the entire machine, shaft deflection, and rapid wear of the bearings. Consequently, the combination of a drive motor having the number of pole pairs \(p_1\) and an asynchronous converter having the number of pole pairs \(p_2\) in one magnetic core is possible only if inequality (1) is satisfied.

Fig. 1.

Fig. 2.

Figure 2 shows the basic circuit diagram of the ABFC. Winding 1, having the number of pole pairs \(p_1\) and located in the slots of the stator stack, is the primary winding of the ABFC drive motor. Winding 3, having the number of pole pairs \(p_2\) and located in the same slots of the stator stack as winding 1, is the secondary winding of the asynchronous converter. There must be no transformer coupling between windings 1 and 3. The latter condition will be fulfilled if the winding factor \(k_{w1-2}\) of winding 1 with respect to the magnetic field of the asynchronous converter \(\Phi_2\) and the winding factor \(k_{w3-1}\) of winding 3 with respect to the magnetic field of the drive motor \(\Phi_1\) are equal to zero:

\[ k_{w1-2}=k_{y1-2} k_{\beta1-2} k_{\xi1-2}=0; \tag{2a} \]

\[ k_{w3-1}=k_{y3-1} k_{\beta3-1} k_{\xi3-1}=0, \tag{2б} \]

where \(k_{y1-2}\), \(k_{\beta1-2}\), \(k_{\xi1-2}\) are the coefficients of pitch shortening, distribution of sections, and distribution of the coils of winding 1 with respect to the field \(\Phi_2\);

\(k_{y3-1}\), \(k_{\beta3-1}\), \(k_{\xi3-1}\) are the coefficients of pitch shortening, distribution of sections, and distribution of the coils of winding 3 with respect to the field \(\Phi_1\).

Winding 2, located in the rotor slots, simultaneously performs the role of the secondary winding of the drive motor and the role of the primary winding of the asynchronous converter. With respect to the magnetic flux of the asynchronous converter, winding 2 is an ordinary three-phase

a two-layer winding with diametrically arranged coils, having a number of parallel branches \(a_2\) equal to the number of pole pairs. With respect to the magnetic flux of the drive motor, the system of parallel branches of each phase of such a winding constitutes an \(a_2\)-phase short-circuited winding with a phase displacement of \(\dfrac{p_1}{p_2} 2\pi\) (in electrical degrees of the magnetic field \(\Phi_1\)). Therefore, in those cases when this winding is considered as the primary winding of the asynchronous converter, it will be denoted by the index “2”; in those cases when this winding is considered as the secondary winding of the drive motor, it will be denoted by the index “2”. The corresponding indices will also denote the parameters of the primary winding of the asynchronous converter and of the secondary winding of the drive motor.

Winding \(C\), having a number of pole pairs \(p_C\), is the armature winding of the synchronous machine and is located in the slots of its rotor. \(OB\) is the excitation winding of the poles of the synchronous machine, which are fastened to the stator. The pole excitation winding is supplied with direct current from solid rectifiers \(B\); the latter receive power from transformer \(T\), provided with a high-frequency voltage stabilizer \(CH\).

Let us consider the principle of operation of the ABFC.

If the primary winding of the drive motor (winding 1) is connected to a supply network with frequency \(f_1\), then, under the action of the motor torque \(M_1\), the rotor of the ABFC will rotate in space with speed

\[ n = n_1(1 - s) = 60 f_1 \frac{1}{p_1}(1 - s), \tag{3} \]

where \(n_1 = \dfrac{60 f_1}{p_1}\) is the synchronous speed of rotation of the field \(\Phi_1\) of the drive motor;

\(s\) is the slip of the ABFC rotor relative to the field \(\Phi_1\).

In winding \(C\) of the armature of the synchronous machine an emf will be induced with frequency:

\[ f_C = \frac{n p_C}{60} = f_1 \frac{p_C}{p_1}(1 - s). \tag{4} \]

The magnetic field of the asynchronous converter \(\Phi_2\), produced by the voltage applied to the terminals of winding 2, will rotate relative to the rotor with speed:

\[ n_2 = \frac{60 f_C}{p_2} = 60 f_1 \frac{p_C}{p_1 p_2}(1 - s). \tag{5} \]

The speed of rotation of the field \(\Phi_2\) in space, \(n_3\), will be equal to:

\[ n_3 = n \pm n_2 = 60 f_1 \frac{1}{p_1}\left(1 \pm \frac{p_C}{p_2}\right)(1 - s). \tag{6} \]

The plus sign refers to the case of concordant rotation of the rotor and the field \(\Phi_2\), and the minus sign to the case of their counter-rotation. Then the frequency of the emf induced by the field \(\Phi_2\) in winding 3 will be equal to:

\[ f_3 = \frac{n_3 p_2}{60} = f_1 \frac{p_2 \pm p_C}{p_1}(1 - s). \tag{7} \]

In regimes close to the nominal one, \(s \ll 1\). Therefore the ratio of the increased frequency \(f_3\) to the frequency of the supply network \(f_1\) is determined by the ratio of the numbers of pole pairs of the drive motor, the asynchronous converter, and the synchronous machine comprising the ABFC, and by the direction of rotation of the field \(\Phi_2\) relative to the rotor.

2. Energy Diagrams of the Distribution of Powers and Losses in an ABFC

The distribution of powers and losses in an ABFC can be represented visually in the form of energy diagrams. In analyzing the distribution of powers, we take as the initial quantity the useful power of the ABFC, \(P_z\), which is equal to the electromagnetic power \(P_{\psi 3}\) transmitted to the stator (to winding 3) from the rotor by the field \(\Phi_2\), after subtracting the copper losses of winding 3, \(p_{\mathrm{Cu}3}\), and the iron losses of the stator, \(p_{\mathrm{Fe}3}\), caused by the field \(\Phi_2\):

\[ P_z = P_{\psi 3} - p_{\mathrm{Cu}3} - p_{\mathrm{Fe}3}. \tag{8} \]

The electromagnetic power \(P_{\psi 3}\) is equal to the product of the electromagnetic torque \(M_2\), which arises as a result of the interaction of the field \(\Phi_2\) and the current \(I_3\), and the angular velocity of rotation of the field \(\Phi_2\) relative to the stator, \(\omega_3\):

\[ P_{\psi 3} = M_2 \omega_3. \tag{9} \]

Since \(\omega_3\) is equal to the algebraic sum of the angular velocities of rotation of the rotor \(\omega\) and of the field \(\Phi_2\) relative to the rotor \(\omega_2\), the electromagnetic power \(P_{\psi 3}\) may be regarded as the algebraic sum of two components:

\[ P_{\psi 3} = M_2 \omega + M_2 \omega_2 = P_{ma} + P_{\psi 2}, \tag{10} \]

where \(P_{ma}\) is the power at the circumference of the rotor of the asynchronous converter, equal to the mechanical power on its shaft;
\(P_{\psi 2}\) is the electromagnetic power of winding 2.

\[ \frac{P_{\psi 2}}{P_{\psi 3}} = \frac{\omega_2}{\omega_3}. \]

The angular velocities of rotation \(\omega\), \(\omega_2\), \(\omega_3\), according to (3), (5), and (6), are equal to:

\[ \omega = \omega_1(1-s), \qquad \omega_2 = \pm \omega_1 \frac{p_C}{p_2}(1-s), \qquad \omega_3 = \omega_1\left(1 \pm \frac{p_C}{p_2}\right)(1-s). \tag{11} \]

From the equations given above it is not difficult to obtain the following expressions for the powers \(P_{\psi 2}\) and \(P_{ma}\):

\[ P_{\psi 2} = P_{\psi 3}\frac{p_C}{p_C \pm p_2} = P_{\psi 3}\frac{1}{s_3}; \tag{12} \]

\[ P_{ma} = \left(1 - \frac{1}{s_3}\right)P_{\psi 3}, \tag{13} \]

where the slip of winding 3 relative to the magnetic flux \(\Phi_2\) is

\[ s_3 = \frac{p_C \pm p_2}{p_C}. \tag{14} \]

The plus sign corresponds to the case of concordant rotation of the rotor and the field \(\Phi_2\) relative to the rotor, and the minus sign to the case of their opposite rotation.

It is seen from equations (12) and (13) that the distribution of powers in the ABFC depends both on the ratio of the numbers of pole pairs \(p_2\) and \(p_C\), and on the direction of rotation of the field \(\Phi_2\) relative to the rotor. In this connection, four essentially different cases should be distinguished, differing not only in the distribution of powers in the ABFC, but also in the operating modes of its individual windings. Let us consider these cases.

a) The direction of rotation of the field \(\Phi_2\) relative to the rotor coincides with the direction of rotation of the rotor itself

In this case, for any values of the numbers of pole pairs \(p_2\) and \(p_C\), the slip \(s_3 > 1\). Consequently, windings 2 and 3 may be regarded as windings of an asynchronous machine operating in the braking mode. The electromagnetic power \(P_{\psi 3}\) is equal, as is evident from equations (10), (12), and (13), to the arithmetic sum of the powers \(P_{ma}\) and \(P_{\psi 2}\):

\[ P_{\psi 3}=P_{ma}+P_{\psi 2}. \tag{15} \]

The power \(P_2\) at the terminals of winding 2 is equal to:

\[ P_2=P_{\psi 2}+p_{\mathrm{Cu}2}+p_{\mathrm{Fe}2}, \tag{16} \]

where \(p_{\mathrm{Cu}2}\) are the copper losses of winding 2 corresponding to the current with frequency \(\omega_2\);

\(p_{\mathrm{Fe}2}\) are the iron losses of the rotor.

The power at the terminals of the armature winding of the synchronous machine, which in this case operates in generator mode, is equal to the power \(P_2\):

\[ P_C=P_2. \tag{17} \]

The mechanical power on the shaft of the synchronous machine \(P_{mc}\) is equal to the sum of the power \(P_C\), the copper losses of the armature winding \(p_{\mathrm{Cu}c}\), and the iron losses of the armature \(p_{\mathrm{Fe}c}\):

\[ P_{mc}=P_C+p_{\mathrm{Cu}c}+p_{\mathrm{Fe}c}. \tag{18} \]

The mechanical power on the shaft of the driving motor is

\[ P_{md}=P_{ma}+P_{mc}. \tag{19} \]

And, finally, the power \(P_1\) consumed by the ABFC is equal to:

\[ P_1=P_{md}+p_{\mathrm{Cu}2'}+p_{\mathrm{Cu}1}+p_{\mathrm{Fe}1}+p_m, \tag{20} \]

where \(p_{\mathrm{Cu}1}\) and \(p_{\mathrm{Cu}2'}\) are the copper losses of winding 1 and the copper losses of winding \(2'\), corresponding to the slip current;

\(p_{\mathrm{Fe}1}\) are the iron losses of the ABFC caused by the field \(\Phi_1\);

\(p_m\) are the mechanical losses of the ABFC.

On the basis of the equations obtained, the energy diagram shown in Fig. 3 has been constructed.

b) The direction of rotation of the field \(\Phi_2\) is opposite to the direction of rotation of the rotor, and \(p_c>p_2\)

In this case, according to (14), the slip of winding 3 relative to the field \(\Phi_2\) is \(0<s_3<1\). Consequently, windings 2 and 3 may be regarded as windings of an asynchronous machine operating in motor mode. The synchronous machine will operate in generator mode.

A detailed consideration of the distribution of powers and losses in the ABFC for this case leads to the following formulas:

\[ P_{\psi 3}=P_{\psi 2}-P_{ma}, \qquad P_{md}=P_{mc}-P_{ma}. \tag{21} \]

The values \(P_2\), \(P_C\), \(P_{mc}\), \(P_1\) are determined by equations (16), (17), (18), and (20).

The energy diagram corresponding to the distribution of powers and losses in the ABFC for the case under consideration is shown in Fig. 4.

c) The direction of rotation of the field \(\Phi_2\) relative to the rotor is opposite to the direction of rotation of the rotor itself, and \(p_C=p_2\)

In this case, according to (14), (11), and (9), the slip \(s=0\), the angular velocity \(\omega=0\), the electromagnetic power transferred from the rotor to winding 3 is \(P_{\psi3}=0\), and the ABFC as a frequency converter loses all meaning.

d) The direction of rotation of the field \(\Phi_2\) relative to the rotor is opposite to the direction of rotation of the rotor itself, and \(p_C<p_2\)

In this case the slip \(s_3<0\). Windings 2 and 3 of the asynchronous converter operate in the generating mode of an asynchronous machine, while the synchronous machine operates in the motoring mode. A detailed

Fig. 3.

Fig. 4.

Fig. 5.

analysis of the distribution of powers and losses in the ABFC for this case leads to the following formulas:

\[ P_{\psi3}=P_{ma}-P_{\psi2}, \qquad P_2=P_{\psi2}-p_{\mathrm{Cu}2}-p_{\mathrm{Fe}2}; \]

\[ P_C=P_2, \qquad P_{mc}=P_C-p_{\mathrm{Cu}c}-p_{\mathrm{Fe}c}, \qquad P_{md}=P_{ma}-P_{mc}. \tag{22} \]

The value of the power \(P_1\) is determined by formula (20). The energy diagram of the ABFC for the case under consideration is shown in Fig. 5.

From a comparison of the formulas and energy diagrams presented, it is seen that in case “a” the electromagnetic power \(P_{\psi3}\) transferred to winding 3 is equal to the arithmetic sum of the powers \(P_{ma}\) and \(P_{\psi2}\), while in cases “b” and “d” it is equal to their difference. At the same time, in case “d” all the power passing through winding 2 and the armature winding of the synchronous machine, and in case “b” a considerable part of it, does not participate in the process of transferring power to the ABFC load, but circulates around a closed loop, causing useless losses. Thus, economically the most expedient operating mode of the ABFC is that in which the direction of rotation of the field \(\Phi_2\) relative to the rotor coincides with the direction of rotation of the rotor itself. In deriving the analytical relations given below, it is assumed that the ABFC operates precisely in this mode.

3. Equivalent circuits and analytical relationships

Windings \(1\) and \(2'\) are the primary and secondary windings of the drive induction motor, whose equivalent circuit does not differ in any way from the equivalent circuit of an ordinary induction motor.

The current of winding \(1\):

\[ \dot I_1=\dot I_{\mu 1}+\dot I''_2=I_1 e^{-j\varphi_1}. \tag{23} \]

The current of winding \(2'\), referred to the refined equivalent circuit, is equal to:

\[ -\dot I''_{2'}=\frac{U_1}{\left(r'_1+\dfrac{r''_{2'}}{s}\right)+j\left(x'_1+x'_{2'}\right)}. \tag{24} \]

The electromagnetic torque of the ABCF drive motor is

\[ M_1= \frac{ m_1 U_1^2 \dfrac{r''_{2'}}{s} }{ \omega_1 \left[ \left(r'_1+\dfrac{r''_{2'}}{s}\right)^2+ \left(x'_1+x'_{2'}\right)^2 \right] }. \tag{25} \]

Fig. 6.

Since the magnetic fluxes \(\Phi_1\) and \(\Phi_2\) are in a common magnetic circuit, the magnetizing current of winding \(1\), \(I_{\mu 1}\), depends on the magnitude of the magnetic inductions of both fields, which is a specific feature of the ABCF drive motor. The calculation of \(I_{\mu 1}\) is given below.

The equivalent circuit of the ABCF circuit—synchronous generator–asynchronous converter (windings \(2\) and \(3\))–load—is shown in Fig. 6. Here \(r_c, x_c, r_2, x_2, r'_3, x'_3, r'_n, x'_n\) are the parameters of windings \(C, 2, 3\) and of the ABCF load at the frequency \(f_c\), corresponding to \(s=0\). The parameters of winding \(3\) and of the load are referred to winding \(2\). In order that the mechanical characteristics of induction motors supplied from the ABCF be independent of the frequency \(f_3\), the voltage at the terminals of winding \(3\), \(U'_{3s}\), must satisfy the well-known equation of M. P. Kostenko [6], which, as applied to the ABCF, takes the form:

\[ \frac{U'_{3s}}{U'_3}=\frac{f_{3s}}{f_{30}}, \tag{26} \]

where \(f_{3s}\) and \(U'_{3s}\) are the current values of frequency and voltage, and \(f_{30}\) and \(U'_3\) are the frequency \(f_{3s}\) at \(s=0\) and the voltage \(U_{3s}\) at \(f_{3s}=f_{30}\); hence

\[ U'_{3s}=U'_3(1-s). \tag{27} \]

If the magnetizing branch is brought out to the terminals \(a\!-\!a\), we obtain a simplified equivalent circuit (Fig. 7), which makes it possible to derive the fol—

the following dependences for the load current \(I_{\mu}^{\prime\prime}\) and the emf of the synchronous generator \(E_c\):

\[ I_{\mu}^{\prime\prime} = U_3^{\prime\prime}(1-s) \left[ \frac{\dfrac{r_{\mu}^{\prime\prime}}{s_3}} {\left(\dfrac{r_{\mu}^{\prime\prime}}{s_3}\right)^2+x_{\mu}^{\prime\prime 2}(1-s)^2} - j\frac{x_{\mu}^{\prime\prime}(1-s)} {\left(\dfrac{r_{\mu}^{\prime\prime}}{s_3}\right)^2+x_{\mu}^{\prime\prime 2}(1-s)^2} \right]; \tag{28} \]

\[ \dot{E}_c^{\prime\prime} = U_3^{\prime\prime}(1-s) \left[ \frac{ \left( r_c' + r_2' + \dfrac{r_3^{\prime\prime}}{s_3} + \dfrac{r_{\mu}^{\prime\prime}}{s_3} \right) + j\left(x_c' + x_2' + x_3^{\prime\prime} + x_{\mu}^{\prime\prime}\right)(1-s) } { \dfrac{r_{\mu}^{\prime\prime}}{s_3} + jx_{\mu}^{\prime\prime}(1-s) } \right]. \tag{29} \]

Fig. 7.

Knowing the current \(I_{\mu}^{\prime\prime}\) and the generator emf \(E_c\), one can obtain expressions for the electromagnetic power of the synchronous generator

\[ P_{\psi c} = m_c U_3^{\prime\prime 2} \frac{ r_c' + r_2' + \dfrac{r_3^{\prime\prime}}{s_3} + \dfrac{r_{\mu}^{\prime\prime}}{s_3} } { \left[ \dfrac{r_{\mu}^{\prime\prime}}{s_3(1-s)} \right]^2 + x_{\mu}^{\prime\prime 2} } \]

and its electromagnetic torque

\[ M_c = \frac{P_{\psi c}}{\omega_c} = \frac{P_{\psi c}}{\omega} = m_c U_3^{\prime\prime 2} \frac{ \left( r_c' + r_2' + \dfrac{r_3^{\prime\prime}}{s_3} + \dfrac{r_{\mu}^{\prime\prime}}{s_3} \right)(1-s) } { \omega_1 \left[ \dfrac{r_{\mu}^{\prime\prime}}{s_3} + x_{\mu}^{\prime\prime 2}(1-s)^2 \right] }, \tag{30} \]

as well as the expression for the electromagnetic torque of the asynchronous converter:

\[ M_2 = \frac{ m_2 I_{\mu}^{\prime\prime 2} \left( \dfrac{r_3^{\prime\prime}}{s_3}+\dfrac{r_{\mu}^{\prime\prime}}{s_3} \right) } {\omega_2} = \frac{ m_2 U_3^{\prime\prime 2} \left( \dfrac{r_3^{\prime\prime}}{s_3}+\dfrac{r_{\mu}^{\prime\prime}}{s_3} \right)(1-s) } { \omega_1\dfrac{p_c}{p_2} \left[ \left(\dfrac{r_{\mu}^{\prime\prime}}{s_3}\right)^2 + x_{\mu}^{\prime\prime 2}(1-s)^2 \right] }. \tag{31} \]

As is evident from the expressions given above, the characteristics of the ABFC are a function of its parameters, of the load parameters, and of the slip \(s\). The latter, however, is also a function of these parameters and can be obtained from the torque-balance equation of the ABFC:

\[ M_1 = M_c + M_2. \tag{32} \]

Substituting into it the values of \(M_1\), \(M_2\), and \(M_c\), after some transformations one can obtain the following expression for the slip \(s\):

\[ s^3 + As^2 + Bs + C = 0; \tag{33} \]

here

\[ A=\frac{2r_n'' r_1' r_2''-2k_u r_2'' x_n''^{2}-r_n'' r_1'^{2}-r_n'' x_{k1}''^{2}} {k_u r_2'' x_n''^{2}+r_n'' r_1'^{2}+r_n'' x_{k1}''^{2}}; \]

\[ B=\frac{k_u r_2''\left(\frac{r_n'}{s_3}\right)^2+k_u r_2'' x_n''^{2}+r_n'' r_2''^{2}-2r_n'' r_1' r_2''} {k_u r_2'' x_n''^{2}+r_n'' r_1'^{2}+r_n'' x_{k1}''^{2}}; \]

\[ C=\frac{r_n'' r_2''^{2}} {k_u r_2'' x_n''^{2}+r_n'' r_1'^{2}+r_n'' x_{k1}''^{2}}, \]

where \(x_{k1}''=x_1'+x_2''\), \(r_n''=r_c'+r_2''+r_3''+r_n'\), \(k_u=\dfrac{U_1^2}{U_3''^{2}}\).

The number of real roots of equation (33) depends on the parameters of the asynchronous brushless frequency converter (ABFC) and its load and is equal to the number of common points of the curves \(M_1=f(s)\) and \((M_2+M_c)=f(s)\).

4. Magnetizing currents and m.m.f. of windings 1 and 2

The magnetizing currents of windings 1 and 2 largely determine their total currents, electromagnetic loads, the ABFC and, consequently, its characteristics, weight, and dimensions. In calculations of ordinary asynchronous motors the magnetizing current is determined through the magnetizing ampere-turns per pole pair; the latter, for specified magnetic loads, are determined from special curves of specific ampere-turns for the teeth and yoke. However, these curves cannot be used to determine the magnetizing currents in an ABFC, since the indicated curves are valid only for the case in which there is only one magnetic flux in the magnetic circuit of the electrical machine.

In investigating the magnetizing currents of windings 1 and 2 of an ABFC, we adopt the following assumptions:

1. The line voltage and the voltage of the synchronous generator are sinusoidal and symmetrical.

2. The phenomenon of hysteresis is neglected. Since the hysteresis loop for electrical grades of iron is narrow, this assumption cannot introduce any substantial error into the results of the investigation.

3. The number of pole pairs of winding 2 is greater than the number of pole pairs of winding 1 \((p_2>p_1)\). The latter inequality is fulfilled in an ABFC.

a) Expressions for the magnetizing currents and m.m.f. of windings 1 and 2 without allowance for higher harmonics in the curves of magnetic induction of fields \(\Phi_1\) and \(\Phi_2\)

Let us consider each of these components separately.

Magnetomotive force for the air gap.

The magnetomotive force for the magnetic circuit: air gap—teeth—yoke, can be represented as the sum of its three components

\[ f=f_\delta+f_z+f_a, \tag{34} \]

where \(f_\delta, f_z, f_a\) are the m.m.f. for the air gap, teeth, and yoke.

If the variation of the magnetic inductions of the fields \(\Phi_1\) and \(\Phi_2\) in the air gap is assumed sinusoidal, then for any point \(N\), arbitrarily chosen along the circumference of the rotor and located at a distance \(x\) from the initial point \(M\), from which we shall measure distances, at any instant the magnitude of the magnetic induction of the field \(\Phi_1\) can be expressed by the equation

\[ B_{\delta 1}=B_{\delta m1}\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right), \tag{35a} \]

and the magnetic induction of the field \(\Phi_2\) by the equation

\[ B_{\delta 2}=B_{\delta m2}\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right), \tag{35б} \]

where \(\omega_s=\omega_1 s\) is the angular velocity of rotation of the field \(\Phi_2\) relative to the rotor. The magnetic permeability of air is a constant quantity. Therefore the m.m.f. is

\[ f_\delta=F_{\delta 1}\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right) +F_{\delta 2}\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right). \tag{36} \]

Magnetomotive force in the teeth.

If the magnetization curve of the steel is expressed in the form of a hyperbolic sine

\[ H=\alpha \operatorname{sh}\beta B, \]

then the magnetic-field strength in the teeth \(H_z\), at any point of the rotor circumference \(N\) and at any instant of time, can be expressed by the equation

\[ \begin{aligned} H_z&=\alpha \operatorname{sh}\beta(B_{z1}+B_{z2})=\\ &=\alpha \operatorname{sh}\left[\beta B_{zm1}\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right] \operatorname{ch}\left[\beta B_{zm2}\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)\right]+\\ &\quad+\alpha \operatorname{ch}\left[\beta B_{zm1}\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right] \operatorname{sh}\left[\beta B_{zm2}\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)\right]. \end{aligned} \tag{37} \]

Expanding each of the factors of the first and second terms of this equation into a series and multiplying the resulting equation by \(L_z\), we obtain the expression for the m.m.f. in the teeth (per pair of poles):

\[ \begin{aligned} f_z={}&-2L_z\alpha J_0(j\beta B_{zm2})\,j \sum_{n=0}^{n=\infty}J_{2n+1}(j\beta B_{zm1}) \sin\left[(2n+1)\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right]-\\ &-2L_z\alpha J_0(j\beta B_{zm1})\,j \sum_{n=0}^{n=\infty}J_{2n+1}(j\beta B_{zm2}) \sin\left[(2n+1)\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)\right]-\\ &-4L_z\alpha \sum_{n=0}^{n=\infty}\sum_{b=1}^{b=\infty} jJ_{2n+1}(j\beta B_{zm2})J_{2b}(j\beta B_{zm1})\times\\ &\quad\times \cos\left[2b\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right] \sin(2n+1)\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)-\\ &-4L_z\alpha \sum_{n=1}^{n=\infty}\sum_{c=0}^{c=\infty} J_{2n}(j\beta B_{zm2})jJ_{2c+1}(j\beta B_{zm1})\times\\ &\quad\times \sin\left[(2c+1)\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right] \cos\left[2n\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)\right]. \end{aligned} \tag{38} \]

The coefficients \(J_n(j\beta B)\) are Bessel functions of the \(n\)-th order of a purely imaginary argument; for even \(n\) they will be real quantities, and for odd \(n\)—purely imaginary.

After some transformations, neglecting quantities of the second order, we obtain:

\[ \begin{aligned} f_z &= F_{z1}\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right) +F_{z2}\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)- \\ &\quad -F_{z1_3}\sin\left(3\omega_s t-\frac{\pi}{\tau_{1_3}}x\right) +F'_z\sin\left[(2\omega_2-\omega_s)t-\frac{\pi}{\tau'}x\right]- \\ &\quad -F''_z\sin\left[(2\omega_2+\omega_s)t-\frac{\pi}{\tau''}x\right] -F'''_z\sin\left[(\omega_2-2\omega_s)t-\frac{\pi}{\tau'''}x\right]- \\ &\quad -F^{\mathrm{IV}}_z\sin\left[(\omega_2+2\omega_s)t-\frac{\pi}{\tau^{\mathrm{IV}}}x\right]. \end{aligned} \tag{39} \]

where

\[ F_{z1}=-2L_z\alpha J_0(j\beta B_{zm2})\,jJ_1(j\beta B_{zm1}); \]

\[ F_{z2}=-2L_z\alpha J_0(j\beta B_{zm1})\,jJ_1(j\beta B_{zm2}); \]

\[ F_{z1_3}=2L_z\alpha J_0(j\beta B_{zm2})\,jJ_3(j\beta B_{zm1}); \]

\[ F'_z=F''_z=2L_z\alpha jJ_1(j\beta B_{zm1})\,J_2(j\beta B_{zm2}); \]

\[ F'''_z=F^{\mathrm{IV}}_z=2L_z\alpha J_1(j\beta B_{zm2})\,J_2(j\beta B_{zm1}) \]

— are the amplitudes of the fundamental and higher harmonics of the m.m.f. on the teeth;

\[ \tau_{1_3}=\frac{\tau_1}{3};\qquad \tau'=\frac{p_1}{2p_2-p_1}\tau_1;\qquad \tau''=\frac{p_1}{2p_2+p_1}\tau_1; \]

\[ \tau'''=\frac{p_1}{p_2-2p_1}\tau_1;\qquad \tau^{\mathrm{IV}}=\frac{p_1}{p_2+2p_1}\tau_1 \]

— are the pole pitches of the higher harmonics of the m.m.f., the numbers of pole pairs of which are respectively equal to:

\[ p_{1_3}=3p_1;\qquad p'=(2p_2-p_1)p_1;\qquad p''=(2p_2+p_1)p_1; \]

\[ p'''=(p_2-2p_1)p_1,\qquad p^{\mathrm{IV}}=(p_2+2p_1)p_1. \]

Magnetomotive force in the yoke.

If the magnetic inductions in the air gap \(B_{\delta1}\) and \(B_{\delta2}\) at any arbitrarily chosen point \(N\) on the circumference of the rotor, located at a distance \(x\) from the initial point \(M\), are expressed by equations (35), then the magnetic inductions of the fields \(\Phi_1\) and \(\Phi_2\) in any section of the yoke lying in the plane passing through point \(N\) and the axis of the machine, at any instant of time will be expressed by the equations:

\[ B_{a1}=B_{am1}\cos\left(\omega_s t-\frac{\pi}{\tau_1}x\right); \]

\[ B_{a2}=B_{am2}\cos\left(\omega_2 t-\frac{\pi}{\tau_2}x\right), \tag{40} \]

and the magnetic field intensity—by the equation

\[ \begin{aligned} H_a &= \alpha \operatorname{sh}\beta (B_{a1}+B_{a2})=\\ &= \alpha \operatorname{sh}\left[\beta B_{am1}\cos\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right] \operatorname{ch}\left[\beta B_{am2}\cos\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)\right]+\\ &\quad + \alpha \operatorname{ch}\left[\beta B_{am1}\cos\left(\omega_s t-\frac{\pi}{\tau_1}x\right)\right] \operatorname{sh}\left[\beta B_{am2}\cos\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)\right]. \end{aligned} \tag{41} \]

Expanding each of the factors of the first and second terms of this equation into a series and neglecting quantities of the second order, we obtain

\[ \begin{aligned} H_a &= -2\alpha J_0(j\beta B_{am2})jJ_1(j\beta B_{am1}) \cos\left(\omega_s t-\frac{\pi}{\tau_1}x\right)-\\ &\quad -2\alpha J_0(j\beta B_{am1})jJ_1(j\beta B_{am2}) \cos\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)+\\ &\quad +2\alpha J_0(j\beta B_{am2})jJ_3(j\beta B_{am1}) \cos\left(3\omega_s t-\frac{\pi}{\tau_{13}}x\right)+\\ &\quad +2\alpha jJ_1(j\beta B_{am1})J_2(j\beta B_{am2}) \cos\left[(2\omega_2-\omega_s)t-\frac{\pi}{\tau'}x\right]+\\ &\quad +2\alpha jJ_1(j\beta B_{am1})J_2(j\beta B_{am2}) \cos\left[(2\omega_2+\omega_s)t-\frac{\pi}{\tau''}x\right]+\\ &\quad +2\alpha jJ_1(j\beta B_{am2})J_2(j\beta B_{am1}) \cos\left[(\omega_2-2\omega_s)t-\frac{\pi}{\tau'''}x\right]+\\ &\quad +2\alpha jJ_1(j\beta B_{am2})J_2(j\beta B_{am1}) \cos\left[(\omega_2+2\omega_s)t-\frac{\pi}{\tau^{\mathrm{IV}}}x\right]. \end{aligned} \tag{42} \]

The magnetomotive force of the \(n\)-th harmonic over an infinitesimal section of the yoke \(dl\) is equal to

\[ df_{an}=H_{an}\,dl=H_{an}\frac{L_{an}}{\pi}\,dQ_n, \tag{43} \]

where \(L_{an}\) is the length of the magnetic line of force in the yoke of the \(n\)-th harmonic (for two poles);

\(dQ_n\) is an infinitesimal angle in electrical radians of the \(n\)-th harmonic, corresponding to the arc \(dl\).

The magnetomotive force of the \(n\)-th harmonic on the yoke at any point \(N\) arbitrarily taken on the circumference of the rotor at any instant of time is determined by the equation

\[ f_{an}=\int_{-Q_n}^{+Q_n} H_{an}\frac{L_{an}}{\pi}\,dQ_n, \tag{44} \]

where \(Q_n\) is the angle in electrical radians of the \(n\)-th harmonic m.m.f. between the initial point \(M\) and the current point.

Then the m.m.f. on the yoke at any point \(N\) arbitrarily taken on the circumference of the rotor at any instant of time will be equal to:

\[ f_a=\sum f_{an}=\sum \int_{-Q_n}^{+Q_n} H_{an}\frac{L_{an}}{\pi}\,dQ_n. \tag{45} \]

After transformations we shall have:

\[ \begin{aligned} f_a={}&F_{a1}\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right) +F_{a2}\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)+{}\\ &+F_{a1_3}\sin\left(3\omega_s t-\frac{\pi}{\tau_{1_3}}x\right) +F'_a\sin\left[(2\omega_2-\omega_s)t-\frac{\pi}{\tau'}x\right]+{}\\ &+F''_a\sin\left[(2\omega_2+\omega_s)t-\frac{\pi}{\tau''}x\right] +F'''_a\sin\left[(\omega_2-2\omega_s)t-\frac{\pi}{\tau'''}x\right]+{}\\ &+F^{\mathrm{IV}}_a\sin\left[(\omega_2+2\omega_s)t-\frac{\pi}{\tau^{\mathrm{IV}}}x\right], \end{aligned} \tag{46} \]

where

\[ F_{a1}=-\frac{4}{\pi}L_{a1}\alpha J_0(j\beta B_{am2})\,jJ_1(j\beta B_{am1}); \]

\[ F_{a2}=-\frac{4}{\pi}L_{a2}\alpha J_0(j\beta B_{am1})\,jJ_1(j\beta B_{am2}); \]

\[ F_{a1_3}=\frac{4}{\pi}\cdot\frac{p_1}{p_{1_3}}L_{a1}\alpha J_0(j\beta B_{am2})\,jJ_3(j\beta B_{am1}); \]

\[ F'_a=\frac{4}{\pi}\cdot\frac{p_1}{p'}L_{a1}\alpha jJ_1(j\beta B_{am1})\,J_2(j\beta B_{am2}); \]

\[ F''_a=\frac{4}{\pi}\cdot\frac{p_1}{p''}L_{a1}\alpha jJ_1(j\beta B_{am1})\,J_2(j\beta B_{am2}); \]

\[ F'''_a=\frac{4}{\pi}\cdot\frac{p_1}{p'''}L_{a1}\alpha jJ_1(j\beta B_{am2})\,J_2(j\beta B_{am1}); \]

\[ F^{\mathrm{IV}}_a=\frac{4}{\pi}\cdot\frac{p_1}{p^{\mathrm{IV}}}L_{a1}\alpha jJ_1(j\beta B_{am2})\,J_2(j\beta B_{am1}). \]

Substituting the values of the m.m.f. \(f_\delta\), \(f_z\), \(f_a\) according to (36), (39), and (46) into equation (34), we obtain the expression for the instantaneous values of the m.m.f. (per pair of poles) over the magnetic circuit: air gap—teeth—yoke, at any point of the rotor circumference \(N\) at any instant of time \(t\), in the form:

\[ \begin{aligned} f={}&F_1\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right) +F_2\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right)+{}\\ &+F_{1_3}\sin\left(3\omega_s t-\frac{\pi}{\tau_{1_3}}x\right) +F'\sin\left[(2\omega_2-\omega_s)t-\frac{\pi}{\tau'}x\right]+{}\\ &+F''\sin\left[(2\omega_2+\omega_s)t-\frac{\pi}{\tau''}x\right] +F'''\sin\left[(\omega_2-2\omega_s)t-\frac{\pi}{\tau'''}x\right]+{}\\ &+F^{\mathrm{IV}}\sin\left[(\omega_2+2\omega_s)t-\frac{\pi}{\tau^{\mathrm{IV}}}x\right], \end{aligned} \tag{47} \]

where

\[ F_1=F_{\delta1}+F_{z1}+F_{a1},\quad F_2=F_{\delta2}+F_{z2}+F_{a2},\quad F_{1_3}=F_{a1_3}-F_{z1_3}; \]

\[ F'=F'_a+F'_z,\quad F''=F''_a-F''_z,\quad F'''=F'''_a-F'''_z,\quad F^{\mathrm{IV}}=F^{\mathrm{IV}}_a-F^{\mathrm{IV}}_z. \]

The last expression for the m.m.f. \(f\) was derived, as indicated above, from the conditions that in the air gap of the converter there exist only two magnetic fluxes \(\Phi_1\) and \(\Phi_2\) with sinusoidal distribution of the magnetic inductions \(B_{\delta1}\) and \(B_{\delta2}\).

However, owing to the known properties of three-phase windings, in the m.m.f. curve \(f\) there will be absent those higher harmonics \(f_{1_3}, f', f'', f''', f^{IV}\) whose winding coefficients with respect to windings 1, 2, 3 of the ABFC will be equal to zero or close to zero; this will lead to the appearance in the air gap of the corresponding higher harmonics of magnetic induction [12], which will be in antiphase with the absent harmonics of the m.m.f.

If it is assumed that all higher harmonics will be absent from the m.m.f. curve \(f\), then the expression for the m.m.f. \(f\) will take the form:

\[ f=F_1\sin\left(\omega_s t-\frac{\pi}{\tau_1}x\right)+F_2\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right), \tag{48} \]

whence the expressions for the m.m.f. of winding 1, \(f_1\), in the stator axes, and the m.m.f. of winding 2, \(f_2\), in the rotor axes, may be written as follows:

\[ f_1=F_1\sin\left(\omega_1 t-\frac{\pi}{\tau_1}x\right), \]

\[ f_2=F_2\sin\left(\omega_2 t-\frac{\pi}{\tau_2}x\right). \tag{49} \]

The effective value of the magnetizing current of winding 1 is

\[ I_1=\frac{p_1F_1}{2.7w_1k_{w1}}. \tag{50} \]

The effective value of the magnetizing current of winding 2 is

\[ I_2=\frac{p_2F_2}{2.7w_2k_{w2}}. \tag{51} \]

b) Magnetizing currents and m.m.f. of windings 1 and 2 with allowance for higher harmonics of magnetic induction

Above we have already noted that the curves of variation of the magnetic inductions may contain higher harmonics. Let us consider them in more detail.

If, in the m.m.f. curve \(f\) according to expression (47), the harmonic

\[ f_{1_3}=(F_{a1_3}-F_{z1_3})\sin\left(3\omega_s t-\frac{\pi}{\tau_{1_3}}x\right) \tag{52} \]

takes place, while in reality, owing to the properties already indicated of three-phase windings, it is absent, then in the air gap there will appear a harmonic of magnetic induction

\[ B_{\delta1_3}=\pm B_{\delta m1_3}\sin\left(3\omega_s t-\frac{\pi}{\tau_{1_3}}x\right). \tag{53} \]

It is obvious that \(B_{\delta1_3}\) will be the greater, the greater \(F_{1_3}=F_{a1_3}-F_{z1_3}\) in expression (47). Calculations carried out by the author showed that, with a more or less rational distribution of the magnetic loadings between the yoke and the teeth (for economic considerations), the m.m.f. \(F_{1_3}\) in expression (47) is small. Consequently, in these cases the harmonic of magnetic induction \(B_{\delta1_3}\) will be small and may be neglected. If, in an ABFC, the ratio

\[ \frac{p_2}{p_1}=3, \]

then, in selecting the magnetic loadings on the yoke and the teeth, the condition \(F_{1_3}=F_{a1_3}-F_{z1_3}\approx 0\) must necessarily be satisfied.

Indeed, if in expression (47) \(F_{i_3}\ne 0\), then \(B_{\delta 1}\ne 0\). The harmonic \(B_{\delta 1}\) and the field of winding 2 in this case will have the same number of pole pairs, but different rotational speeds in space, which will lead to the appearance of vibrational forces.

If, in the m.m.f. curve \(f\) according to expression (47), the harmonics \(f'\), \(f''\), \(f'''\), \(f^{\mathrm{IV}}\) are present, but in reality are absent, then in the magnetic-induction curve there will appear higher harmonics of the magnetic induction \(B'_{\delta}\), \(B''_{\delta}\), \(B'''_{\delta}\), \(B^{\mathrm{IV}}_{\delta}\).

In order to find expressions for the m.m.f.’s \(F_{z1}\), \(F_{z2}\), \(F_{a1}\), \(F_{a2}\), taking into account the influence on them of the higher harmonics of the magnetic induction \(B'_{\delta}\), \(B''_{\delta}\), \(B'''_{\delta}\), \(B^{\mathrm{IV}}_{\delta}\), it is necessary to introduce these harmonics into equations (36), (37), and (41) and to carry out mathematical operations similar to those performed above. As a result we obtain an expression analogous to expression (47), in which, however, the m.m.f.’s \(F_{z1}\), \(F_{z2}\), \(F_{a1}\), \(F_{a2}\) will be functions of the magnetic inductions of the fundamental components and of the higher harmonics taken into account. The magnitude of the higher harmonics of the magnetic induction for given fundamental components is determined by solving the system of equations:

\[ \begin{aligned} F'&=0;\\ F''&=0;\\ F'''&=0;\\ F^{\mathrm{IV}}&=0. \end{aligned} \tag{54} \]

Usually not all of the higher harmonics \(f'\), \(f''\), \(f'''\), \(f^{\mathrm{IV}}\) are absent in the m.m.f. curve. Consequently, not all of the higher harmonics of the magnetic induction \(B'_{\delta}\), \(B''_{\delta}\), \(B'''_{\delta}\), \(B^{\mathrm{IV}}_{\delta}\) will occur, and among those that do, when calculating the m.m.f.’s \(F_{z1}\), \(F_{z2}\), \(F_{a1}\), \(F_{a2}\), it is sufficient to take into account one or two of the most strongly expressed. Below are given expressions for the m.m.f.’s \(F_{z1}\), \(F_{z2}\), \(F_{a1}\), \(F_{a2}\) with allowance for the higher harmonic magnetic induction:

\[ B'_{\delta}=-B'_{\delta m}\sin\left[(2\omega_2-\omega_s)t-\frac{\pi}{\tau'}x\right]: \]

\[ F_{z1}=2L_z\alpha\left\{-J_0(j\beta B'_{zm})J_0(j\beta B_{zm2})jJ_1(j\beta B_{zm1})-\right. \]

\[ \left.-jJ_1(j\beta B'_{zm})J_2(j\beta B_{zm2})\left[J_0(j\beta B_{zm1})-J_2(j\beta B_{zm1})\right]\right\}; \]

\[ F_{z2}=2L_z\alpha\left\{-J_0(j\beta B'_{zm})J_0(j\beta B_{zm1})jJ_1(j\beta B_{zm2})-\right. \]

\[ \left.-jJ_1(j\beta B'_{zm})jJ_1(j\beta B_{zm1})\left[jJ_3(j\beta B_{zm2})-jJ_1(j\beta B_{zm2})\right]\right\}; \]

\[ F_{a1}=\frac{4}{\pi}L_{a1}\alpha\left\{-J_0(j\beta B'_{am})J_0(j\beta B_{am2})jJ_1(j\beta B_{am1})-\right. \]

\[ \left.-jJ_1(j\beta B'_{am})J_2(j\beta B_{am2})\left[J_0(j\beta B_{am1})-J_2(j\beta B_{am1})\right]\right\}; \]

\[ F_{a2}=\frac{4}{\pi}L_{a2}\alpha\left\{-J_0(j\beta B'_{am})J_0(j\beta B_{am1})jJ_1(j\beta B_{am2})-\right. \]

\[ \left.-jJ_1(j\beta B'_{am})jJ_1(j\beta B_{am1})\left[jJ_3(j\beta B_{am2})-jJ_1(j\beta B_{am2})\right]\right\}. \tag{55} \]

The magnetic inductions in the teeth \(B'_{zm}\) and in the yoke \(B'_{am}\) are determined by the equation.

\[ \begin{aligned} &2L_z a\,[J_0(j\beta B'_{zm})jJ_1(j\beta B_{zm1})J_2(j\beta B_{zm2})+\\ &\quad +J_0(j\beta B_{zm1})J_0(j\beta B_{zm2})jJ_1(j\beta B'_{zm})]+\\ &\quad +\frac{4}{\pi}\frac{p_1}{p'}L_{a1}a\,[J_0(j\beta B'_{am})jJ_1(j\beta B_{am1})J_2(j\beta B_{am2})+\\ &\quad +J_0(j\beta B_{am1})J_0(j\beta B_{am2})jJ_1(j\beta B'_{am})]-1.66B'_{\delta m}=0. \end{aligned} \tag{56} \]

In deriving equations (55), (56), quantities of the second order were neglected; it was assumed that

\[ (B_{zm1}+B_{zm2})<22000\ \text{Gs},\quad \text{and}\quad (B_{am1}+B_{am2})<20000\ \text{Gs}. \]

The calculations performed by the author showed that the harmonics can exert a rather large influence on the e.m.f. of the teeth and yoke. A comparison of calculated and experimental data confirms the proposed method for calculating the e.m.f. and the magnetizing currents of the ABFC.

Conclusions

1. As follows from the operating principle of the ABFC, the latter makes it possible to obtain a “stiff” external characteristic by means of compounding, which cannot be obtained in the known brushless frequency converters.

2. The operating mode of the individual windings of the ABFC and the distribution of powers in it depend on the ratio of the numbers of pole pairs \(p_1\), \(p_2\), \(p_c\) and on the direction of rotation of the field \(\Phi_2\) relative to the rotor. Economically the most advantageous mode is that in which the direction of rotation of the field \(\Phi_2\) relative to the rotor and the rotation of the rotor itself coincide.

3. The obtained analytical expressions for currents, e.m.f., powers, torques, and slip \(s\) of the ABFC make it possible to calculate its operating characteristics without any graphical constructions.

4. The derived expressions for the e.m.f. and magnetizing currents, giving accuracy sufficient for practical calculations, may be recommended for calculating the magnetic circuit and selecting the electromagnetic loadings of the ABFC, which determine its weight and volume.

References

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