PROCEEDINGS
OF THE TOMSK ORDER OF THE LABOR RED BANNER
POLYTECHNIC INSTITUTE named after S. M. KIROV

Volume 117
1963

WINDING FACTOR AND COMPATIBILITY CONDITIONS OF TWO DIFFERENT-POLE MACHINES IN A SINGLE MAGNETIC CORE

V. S. Novokshenov

(Presented by the scientific seminar of the Electromechanical Faculty)

In recent years, in the design of known and the creation of new frequency converters [1, 2, 3] and cascade asynchronous motors, the principle of combining two machines with different pole numbers in a single magnetic core has been used increasingly widely.

However, the theoretical questions of combined machines have not been covered sufficiently fully in the technical literature.

In the present work, on the basis of the expressions obtained for the winding factor of combined machines, an analysis is given of the compatibility conditions for two different-pole electrical machines in a single magnetic core.

Winding factor of combined machines

Usually the winding factor of alternating-current windings for the harmonic fields of these windings is expressed as the product of the coefficients of section distribution and winding-pitch shortening.

When calculating the emf induced in one winding by the field of another winding having a different number of poles, such an expression for the winding factor becomes insufficient, since in this case the emf of the individual coils usually does not coincide in phase; that is, when such coils are connected in series, the emf induced in them are added geometrically. This can be taken into account by introducing into the winding factor a third multiplier—the coefficient of distribution of the winding coils, by which is meant the ratio of the geometric sum of the emf of the coils to their algebraic sum.

Consequently, the winding factor of a winding having the number of pole pairs \(P_o\) for a field \(\Phi_n\), having the number of pole pairs \(P_n\), may be expressed as follows:

\[ K_{\mathrm{won}} = K_{\mathrm{yon}} \cdot K_{\mathrm{pon}} \cdot K_{\mathrm{zon}} . \tag{1} \]

where \(K_{\mathrm{yon}}, K_{\mathrm{pon}}, K_{\mathrm{zon}}\) are the coefficients of pitch shortening, section distribution, and distribution of the winding coils for the field \(\Phi_n\).

The coefficients of pitch shortening and section distribution are expressed, as is known, by the formulas:

\[ k_{y\mathrm{op}}=\sin \frac{\beta\pi\nu}{2}, \qquad k_{\rho\mathrm{op}}=\frac{\sin \dfrac{q\gamma\nu}{2}}{q\sin \dfrac{\gamma\nu}{2}} \tag{2} \]

where \(\nu\) is the ratio of the number of pole pairs of the field \(\Phi_p\) to the number of pole pairs of the winding;
\(\gamma\) is the angle of displacement between two adjacent slots, in electrical degrees of the winding.

Let us consider the distribution coefficient of the winding coils in more detail; first we shall consider the distribution coefficients of characteristic groups of coils, and then give the general expression for the distribution coefficient of the coils of alternating-current windings.

Figure 1 shows a group of \(q'_k\) coils displaced relative to one another by the angle \(\gamma'_k\) in electrical degrees of the winding, or by \(\gamma'_k\nu\) electrical degrees of the field. For simplicity of representation, the number of sections in a coil is taken equal to unity. The coils are connected in series in such a way that the beginning of each following coil is connected to the beginning of the preceding one. The magnetic fluxes created by these coils at any instant have one polarity, the same for all coils; therefore we call them coils of one polarity.

Fig. 1.

The e.m.f.s induced in the coils by the field \(\Phi_p\) may be represented as a system of \(q'_k\) vectors displaced relative to one another by the angle \(\gamma'_k\nu\); such a system of vectors is analogous to a system of \(q\) vectors of section e.m.f.s displaced relative to one another by the angle \(\gamma\nu\).

Therefore it is evident that the formula for the distribution coefficient of the coils will be analogous to the formula for the distribution coefficient of sections, namely

\[ k'_{\xi\mathrm{op}}= \frac{\sin \dfrac{q'_k\gamma'_k\nu}{2}} {q'_k\sin \dfrac{\gamma'_k\nu}{2}}, \tag{3} \]

as is not difficult to verify by repeating the geometrical constructions and mathematical operations used in deriving the well-known formula for the distribution coefficient of sections.

In Fig. 2, a, two coils of different polarity are shown. The second coil is displaced relative to the first by \(\gamma_k^{\prime\prime}\nu\) electrical degrees of the field \(\Phi_\nu\) and is reversed.

According to the vector diagram shown in Fig. 2, b, the e.m.f. induced in the pair of coils is equal to

\[ \overline{E}_{12}=\overline{E}_1+(-\overline{E}_2). \]

The distribution factor of such a pair of coils is

\[ \kappa_{\xi\mathrm{op}}^{\prime\prime} = \frac{E_{12}}{2E_1} = \frac{AB}{2AC} = \sin\frac{\gamma_k^{\prime\prime}\nu}{2}. \tag{4} \]

Fig. 2.

A group of coils of alternating polarity (Fig. 3) may be regarded as consisting of \(q_k'\) pairs of coils of different polarity, where \(q_k'\) is the number of series-connected coils of one polarity. Any two adjacent coils of one polarity are displaced with respect to one another by \(\gamma_k'\nu\) electrical degrees of the field.

The distribution factor of the pairs of coils may be represented by formula (3), and the distribution factor of each pair of coils of different polarity by formula (4). Then the distribution factor of coils of alternating polarity is obtained as the product of the factors \(\kappa_{\xi\mathrm{op}}'\) and \(\kappa_{\xi\mathrm{op}}^{\prime\prime}\):

\[ \kappa_{\xi\mathrm{op}} = \kappa_{\xi\mathrm{op}}' \cdot \kappa_{\xi\mathrm{op}}^{\prime\prime} = \frac{ \sin \dfrac{q_k'\gamma_k'\nu}{2} }{ q_k'\sin \dfrac{\gamma_k'\nu}{2} } \sin\frac{\gamma_k^{\prime\prime}\nu}{2}. \tag{5} \]

Each phase of an alternating-current winding (or each parallel branch of a phase, if the number of parallel branches \(a>1\)) consists of a group of series-connected coils of one polarity or of alternating polarity. Therefore expression (5) should be regarded as the formula for the distribution factor of coils of alternating-current windings; in this case, when only coils of one polarity are connected in series, the second factor in formula (5) should be taken equal to unity.

For each particular type of winding, the coil-distribution factor can be expressed as a function of the number of pole pairs of the winding \(p_0\), the number of pole pairs of the field \(p_{\mathrm{n}}\), and the number of parallel branches of the winding \(a\); to do this, the values \(\gamma'_{\kappa}, \gamma''_{\kappa}, q_{\kappa}\) corresponding to these windings must be substituted into formula (5).

From the point of view of calculating the coefficient \(\kappa_{\mathrm{bp}}\) for various types of ac windings, it is convenient to subdivide them into windings with the number of coils per phase \(q_{\kappa}=p_0\) and windings with the number of coils per phase \(q_{\kappa}=2p_0\).

Fig. 3.

Each of these types of windings, in turn, is divided into:

a) windings with a concentrated arrangement of coils in each parallel branch, when a row of adjacent coils of the given phase forms one parallel branch, the next row of coils another, etc.;

b) windings with a distributed arrangement of coils in each parallel branch, when the coils of each parallel branch are uniformly distributed over the entire circumference of the stator (rotor). For example, when \(a=2\), all odd-numbered coils of the phase belong to one parallel branch, and all even-numbered coils to the other.

The coil-distribution factors of the indicated types of windings, expressed in terms of \(p_0\), \(p_{\mathrm{n}}\), and \(a\), are given in Table 1.

Analysis of the compatibility conditions of two electric machines in one magnetic core

It is known [4] that, in order to combine two electric machines in one magnetic core, two conditions must be satisfied:

1. The superposition of the magnetic fields of the combined machines must not cause the appearance of forces of unilateral magnetic attraction and vibrations; according to R. Richter, this is fulfilled if the inequality is satisfied

\[ p_1 \pm p_2 \ne \pm 1, \tag{6} \]

where \(p_1\) and \(p_2\) are the numbers of pole pairs of the combined machines.

1. There must be no transformer coupling between the windings of the combined machines. The feasibility of the latter condition has been analyzed by a number of authors [4, 5]. However, in doing so, they used

Table 1

cumbersome and inconvenient method of complex analytical or graphical addition of the e.m.f. vectors of the sectional sides of the winding.

The condition for the absence of transformer coupling between two different-pole windings can be analyzed briefly and simply if one uses the winding-coefficient formulas given above. Indeed, the condition for the absence of transformer coupling between two windings (let us denote them by the indices 1 and 2) is satisfied if

\[ k_{\omega12}=k_{y12}\cdot k_{p12}\cdot k_{\xi12}=0, \]

\[ k_{\omega21}=k_{y21}\cdot k_{p21}\cdot k_{\xi21}=0, \tag{7} \]

where \(k_{\omega12}, k_{y12}, k_{p12}, k_{\xi12}\) are the winding coefficient and the coefficients of pitch shortening, section distribution, and coil distribution of winding 1 for the field of winding 2; \(k_{\omega21}, k_{y21}, k_{p21}, k_{\xi21}\) are the winding coefficient and the coefficients of pitch shortening, section distribution, and coil distribution of winding 2 for the field of winding 1.

Formulas (6) and (7) constitute a generalized mathematical expression of the compatibility conditions for two electrical machines in a single magnetic core.

In order for the winding coefficients (7) to be equal to zero, it is necessary and sufficient that, in each of them, either the winding pitch-shortening coefficient, or the winding coefficient of section distribution, or the winding coefficient of coil distribution be equal to zero.

The values of \(\beta\), \(p_{\mathrm{b}}\), \(p_{\mathrm{m}}\), and \(a\) for which the winding coefficients of coil distribution, section distribution, and pitch shortening are zero are given in Table 1.

REFERENCES

1. V. S. Novokshenov. Investigation of an Asynchronous Brushless Frequency Converter. Dissertation, 1951.

2. V. S. Novokshenov. Some Questions in the Theory of an Asynchronous Brushless Frequency Converter. Izvestiya TPI, vol. 94, 1958.

3. V. M. Pavlinich, N. S. Simunov. Properties and Efficiency of a Single-Machine Frequency Converter. Izvestiya vysshikh uchebnykh zavedenii, Elektromekhanika, No. 1, 1960.

4. A. D. Imas. Development and Investigation of High-Frequency Electric Drills. Report on Topic No. 16, DONUGI, Donetsk, 1948.

5. B. M. Milnich. On Winding Schemes of Combined Machines. Information-Technical Collection of TsBTI, NIIZ, No. 11, 1959.

Received by the editors
in June 1962.