Synchronous Brushless Frequency Converter

V. S. Novokshonov, A. B. Tsukublin

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

At present, in the Soviet Union and abroad, asynchronous electric motors of increased frequency are widely used. The use of such electric motors makes it possible to significantly reduce the weight of hand-held power tools (electric saws, electric drills, etc.), to simplify the design and operation of high-speed machine tools in the woodworking and metalworking industries, and also to increase labor productivity [3, 4].

To supply asynchronous motors of increased frequency, either two-machine asynchronous frequency converters or an aggregate consisting of an asynchronous motor and a synchronous generator of increased frequency are usually used. The disadvantage of these converters is the presence in them of sliding contact, the operation of which in damp, dusty rooms, in the open air, and also in explosive atmospheres is extremely unreliable or even impermissible under safety regulations.

The present article gives a description and the principle of operation of a synchronous brushless frequency converter (SBFC), intended for supplying 2–3 electric drills of increased frequency.

Design and Principle of Operation

A synchronous brushless frequency converter is a synchronous machine with an externally closed magnetic flux. The stator core (2) of the SBFC practically does not differ in any way from the stator core of an ordinary three-phase alternating-current machine. Two three-phase windings are placed in the slots of the stator core: the winding of the synchronous motor (3), having the number of pole pairs \(p_1\), and the armature winding of the synchronous generator (4), having the number of pole pairs \(p_2\).

The rotor of the SBFC consists of a shaft and two identical magnetically separated parts, having beak-shaped poles of opposite polarity with pole pitch \(\tau_1\). The frame (1), bearing shields, and the magnetically separated parts of the rotor are made of steel with high magnetic permeability, for example, St 10. The shaft of the SBFC is made of nonmagnetic steel. The excitation winding consists of two coils located concentrically to the rotor axis in annular

Fig. 1.

Visible labels in the figure: 1, 2, 3, 4, 5, 6, 7, 8.

Fig. 2.

Visible labels in the figure: $\tau_1$, $e$, $B_\delta$, $\tau_1$.

$a$  $b$

in the projections of the bearing shields, and is supplied with direct current from semiconductor rectifiers.

The m.m.f. of the field winding creates the main magnetic flux, which closes not inside the machine, but along the external magnetic circuit formed by the shields, the frame, and the rotor of the converter.

Each rotor pole, along the length of the pole pitch \(\tau_1\), has clearly expressed slots. With the appropriate number and arrangement of these slots, the curve of induction distribution in the air gap will have, in addition to the fundamental harmonic with the number of pole pairs \(p_1\), a clearly expressed harmonic with the desired number of pole pairs \(p_2\).

In [2, 3] a number of magnetic systems are presented for obtaining the desired harmonics in the curve of magnetic induction in the air gap. Fig. 2 shows the developments of two magnetic systems of the rotor of the SBFC, the magnetic-induction curve, and the first and third harmonics extracted from it for a converter at 150 Hz.

When the primary winding is supplied by three-phase alternating current, the rotor of the converter rotates, like the rotor of a synchronous machine, synchronously with the field at the speed

\[ n=\frac{60 f_1}{p_1}, \]

where \(f_1\) is the frequency of the supply network.

The extracted harmonic with the number of pole pairs \(p_2\) then induces in the secondary (generator) winding an e.m.f. with frequency

\[ f_2=\frac{p_2 n}{60}=\frac{p_2}{p_1}\cdot f_1 . \]

Thus, the SBFC is an electric machine that consists of a synchronous motor and a synchronous generator with a common magnetic circuit and a common magnetic field, containing in the air gap two clearly expressed harmonics with the number of pole pairs \(p_1\) and \(p_2\). Starting of the converter is asynchronous, similar to the starting of an induction motor with a solid rotor.

The windings of the generator and the motor must be selected in such a way that there is no direct transformer coupling between them. The mathematical condition for the absence of transformer coupling between two windings can be written in the form

\[ k_{w12}=k_{y12}\cdot k_{з12}\cdot k_{c12}=0; \]

\[ k_{w21}=k_{y21}\cdot k_{з21}\cdot k_{c21}=0, \]

where \(k_{w12}\), \(k_{y12}\), \(k_{з12}\), \(k_{c12}\) are, respectively, the winding factor, the pitch factor, the section-distribution factor, and the coil-distribution factor of winding 1 for the field of winding 2, and \(k_{w21}\), \(k_{y21}\), \(k_{з21}\), \(k_{c21}\) are the winding factor, the pitch factor, the section-distribution factor, and the coil-distribution factor of winding 2 for the field of winding 1. If the winding factor of winding 2 for the field \(\Phi_1\) with the number of pole pairs \(p_1\) is equal to zero, then the winding factors of winding 2 are also equal to zero for all fields of higher spatial harmonics with \(p\ne p_2\), which are obtained when the curve of the fundamental field of the SBFC is expanded into a harmonic series. Consequently, the e.m.f.s induced by these higher harmonics in winding 2 are equal to zero; therefore, in analyzing the physical processes in the SBFC we do not consider them.

Basic Equations of the Synchronous Brushless Frequency Converter

Let us consider the physics of the processes occurring in the converter. In the no-load mode, the m.m.f. of the field windings creates the main magnetic flux of the machine. The distribution of the magnetic inductions of this flux in the air gap, with low saturation of the magnetic circuit and a small air gap, may be regarded as rectangular (Fig. 3). Such a nonsinusoidal field can be decomposed into a first harmonic with the number of pole pairs \(p_1\), a harmonic with the number of pole pairs \(p_2\), and higher harmonics, which will not be taken into account below for the reasons indicated earlier. These selected fluxes, with numbers of pole pairs \(p_1\) and \(p_2\), create the main e.m.f. of the motor \(E_1\) and of the generator \(E_2\), between which, as in reactive converters [3], there exists a rigid relationship

Fig. 3.

\[ \frac{E_{01}}{E_{02}} = \frac{k_{w1}\cdot w_1 \cdot f_1}{k_{w2}\cdot w_2 \cdot f_2}\cdot k_{\Phi}, \]

where \(k_{w1}, w_1\) are the winding factor and the number of turns of the motor winding,

\(k_{w2}, w_2\) are the winding factor and the number of turns of the generator winding,

\(f_1; f_2\) are the frequencies of the motor and generator.

\[ k_{\Phi}=\frac{\Phi_{01}}{\Phi_{02}}= \frac{B_{m01}\cdot \tau_1 \cdot e}{B_{m02}\cdot \tau_2 \cdot e} = \frac{B_{m01}\cdot p_1}{B_{m02}\cdot p_2}, \]

where \(B_{m01}\) is the amplitude of the first harmonic with the number of pole pairs \(p_1\),

\(B_{m02}\) is the amplitude of the selected harmonic with the number of pole pairs \(p_2\).

When the converter operates under load, an armature reaction is produced both as a motor and as a generator. But since two synchronous machines—a motor and a generator—are combined in one magnetic circuit, then

The complexity of considering the processes lies in the fact that the armature reaction of the motor affects not only the motor itself, but also the generator, while the armature reaction of the generator also affects the operation of the motor. When considering all the processes occurring in the converter, we shall use, as in the case of ordinary synchronous machines with salient poles, the two-reaction theory, assuming that the direct axis \((d—d)\) coincides with the axis of the rotor poles.

Fig. 4.

Fig. 5.

The fields \(\Phi_{ad2}\) and \(\Phi_{aq2}\), produced by the m.m.f. of the generator armature along the direct and quadrature axes, have in the air gap the form shown respectively in Figs. 4 and 5. The nonsinusoidal reaction field-

the generator-armature reaction along the direct axis \(\Phi_{ad2}\) can be decomposed into the field of the generator-armature reaction on the generator itself, \(\Phi_{ad22}\), with number of pole pairs \(p_2\), and into the field of the generator-armature reaction on the motor, \(\Phi_{ad12}\), with number of pole pairs \(p_1\). The field of the generator-armature reaction along the quadrature axis \(\Phi_{aq2}\) can correspondingly be decomposed into \(\Phi_{aq22}\) and \(\Phi_{aq12}\).

Fig. 6.

The fields along the direct and quadrature axes, \(\Phi_{ad1}\) and \(\Phi_{aq1}\), created by the m.m.f. of the motor armature, have in the air gap the form shown respectively in Figs. 6, 7.

Fig. 7.

These fields, just like the fields of the generator-armature reaction, can be decomposed into fields of the motor-armature reaction, on the motor itself

\(\Phi_{ad11}\) and \(\Phi_{aq11}\) with the number of pole pairs \(p_1\) and the armature-reaction fields of the motor on the generator \(\Phi_{ad21}\) and \(\Phi_{aq21}\) with the number of pole pairs \(p_2\). Each of the armature-reaction fluxes of the motor and generator will induce in the corresponding windings emfs which, according to the general rule, may be written in the following form:

\[ \begin{aligned} \dot E_{ad11} &= -j\dot I_{d1}\cdot x_{d11};& \dot E_{aq11} &= -j\dot I_{q1}x_{q11};\\ \dot E_{ad12} &= -j\dot I_{d2}\cdot x_{d12};& \dot E_{aq12} &= -j\dot I_{q2}\cdot x_{q12};\\ \dot E_{ad22} &= -j\dot I_{d2}\cdot x_{d22};& \dot E_{aq22} &= -j\dot I_{q2}\cdot x_{q22};\\ \dot E_{ad21} &= -j\dot I_{d1}\cdot x_{d21};& \dot E_{aq21} &= -j\dot I_{q1}\cdot x_{q21}, \end{aligned} \tag{1} \]

where \(x_{d11}, x_{d22}, x_{q11}, x_{q22}\) are the direct- and quadrature-axis synchronous self-induction reactances of the generator and motor, and \(x_{d12}, x_{d21}, x_{q12}, x_{q21}\) are the direct- and quadrature-axis synchronous mutual-induction reactances, determined by the formulas:

\[ x_{dik}=2m_k\frac{\mu_0}{K_{\mu d}K_{\delta}\delta}\cdot f_i \frac{w_k\cdot w_i\cdot K_{wk}\cdot K_{wi}}{P_iP_k}\cdot Dl\cdot k_{dik}, \]

\[ x_{qik}=2m_k\frac{\mu_0}{K_{\mu q}\cdot K_{\delta}\delta}\cdot f_i \frac{w_k\cdot w_i\cdot K_{wk}\cdot K_{wi}}{P_iP_k}\cdot Dl\cdot k_{qik}, \]

where \(i=1,2,\ k=1,2\).

The values \(k_{dik}\) and \(k_{qik}\) are determined as

\[ k_{dik}=\frac{B_{ad ikm}}{B_{ad im}};\qquad k_{qik}=\frac{B_{aq ikm}}{B_{aq im}}. \]

If the active resistances of the windings are neglected in view of their smallness—which is quite permissible for synchronous salient-pole machines—and if the physics of the processes occurring in the converter is taken into account, then the emf balance equations for both the motor and the generator can be written as follows (equations (2)):

\[ \begin{aligned} \dot U_1&=-\dot E_{01}+j\dot I_{d1}x_{d11}+j\dot I_{q1}x_{q11} +j\dot I_{d2}x_{d12}+j\dot I_{q2}x_{q12},\\ \dot U_2&=\dot E_{02}-j\dot I_{d2}x_{d22}-j\dot I_{q2}x_{q22} -j\dot I_{d1}x_{d21}-j\dot I_{q1}\cdot x_{q21}. \end{aligned} \tag{2} \]

Proceeding from the emf balance equation and from the physics of the processes, the vector diagram of the converter may be represented in the following form (Fig. 8).

Analysis of the vector diagram makes it possible to obtain certain equations necessary for calculating the converter:

\[ \begin{aligned} U_1\cos\Theta_1&=E_{01}+I_{d1}x_{d11}-I_2x_{d12}\sin(\varphi_2+\Theta_2),\\ U_1\sin\Theta_1&=I_{q1}\cdot x_{q11}-I_2x_{q12}\cos(\varphi_2+\Theta_2),\\ U_2\cos\Theta_2&=E_{02}+I_{d1}x_{d21}-I_2x_{d22}\sin(\varphi_2+\Theta_2),\\ U_2\sin\Theta_2&=I_2x_{q22}\cdot\cos(\varphi_2+\Theta_2)-I_{q1}x_{q21}. \end{aligned} \tag{3} \]

The electromagnetic power can be found, under the adopted assumptions of neglecting active resistances, in the following way:

\[ P_{\mathrm{em}2}=m_2U_2I_2\cos\varphi_2 =m_2U_2I_2\cos(\psi_2-\Theta_2)= \]

\[ =m_2U_2I_2\sin\psi_2\cdot\sin\Theta_2 +m_2U_2I_2\cos\psi_2\cdot\cos\Theta_2= \]

\[ =m_2U_2I_{d2}\cdot\sin\Theta_2 +m_2U_2I_{q2}\cos\Theta_2. \]

Substituting into this equation the values \(I_{d2}\) and \(I_{q2}\) from equations (3), we finally obtain the electromagnetic power of the generator

\[ P_{\mathrm{em}2} = m_2 U_2 \left( \frac{E_{02}x_{d11}}{x'_d} - \frac{E_{01}x_{d21}}{x'_d} \right)\sin\Theta_2 + m_2 \frac{U_2^2}{2}\left( \frac{x_{q11}}{x'_q} - \frac{x_{d11}}{x'_d} \right)\sin 2\Theta_2 + \]

\[ +\, m_2 U_1 U_2 \left( \sin\Theta_1\cos\Theta_2 \frac{x_{q21}}{x'_q} + \sin\Theta_2\cos\Theta_1 \frac{x_{d21}}{x'_d} \right), \]

Fig. 8.

where:

\[ x'_d = x_{d11}\cdot x_{d22} - x_{d21}\cdot x_{d12}, \]

\[ x'_q = x_{q11}\cdot x_{q22} - x_{q21}\cdot x_{q12}. \]

The electromagnetic power of the motor is written analogously

\[ P_{\mathrm{em}1} = m_1 U_1 \left( \frac{E_{01}x_{d22}}{x'_d} - \frac{E_{02}x_{d12}}{x'_d} \right)\sin\Theta_1 + m_1 \frac{U_1^2}{2}\cdot \left( \frac{x_{q22}}{x'_q} \right. \]

\[ \left. - \frac{x_{d22}}{x'_d} \right)\sin 2\Theta_1 + m_1 U_1\cdot U_2 \left( \sin\Theta_2\cos\Theta_1 \frac{x_{q12}}{x'_q} + \sin\Theta_1\cos\Theta_2 \frac{x_{d12}}{x'_d} \right). \]

On the basis of the equations presented, a methodology for calculating the SBFC converter is currently being developed.

References

1. Author’s certificate, No. 130966, 1960.

2. V. S. Novokshenov. Investigation of an asynchronous brushless frequency converter. Dissertation, Tomsk, 1961.

3. M. M. Tarashchansky. Synchronous-reluctance frequency converters. Ukrtekhizdat, 1962.

4. P. P. Barzilovich, G. M. Yanchuk et al. Frequency converters for supplying mine electric drills. Information handbook for the introduction of new technology, EMZ “Krasnyi Metallist,” Konotop, 1957.