ELECTROMAGNETIC LOADS AND EFFICIENCY OF COMBINED WINDINGS

V. M. Pavlinin, Yu. V. Baryshnikov, N. S. Skunov

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

In recent years, increasing attention has been paid to the synthesis of electrical machines. In a single magnetic core, two (or more) electrical machines with different numbers of poles and with mutually noninductive windings are combined.

In a number of cases, two mutually noninductive windings of such machines can be electrically combined. Combined windings are known in single-machine frequency converters [1, 2, 3], brushless synchronous machines [4], cascade AC exciters for brushless excitation of synchronous machines [5], asynchronous motors for two-current supply [6], and a number of other devices. According to all researchers, the combination of windings leads to an increase in the utilization factor of the machine.

However, when assessing the efficiency of combination, such essential details are often not taken into account as the reduction of winding factors, the forced ratio of the numbers of turns of the windings being combined, changes in the length of the end portions, and also the conditions for preserving the parameters and characteristics of the machines being designed. The limits at which replacing two windings by one becomes inadvisable are not determined.

Analysis of a large number of combined-winding circuits shows that the principle of their formation can be reduced mainly to two typical cases, presented in Fig. 1. In both cases,

Fig. 1. Principal winding diagrams

two windings are combined. One of them is the “primary” (index 1), for example, a three-phase winding of a motor or generator with specified values of voltage or emf. The other is the “secondary” (index 2), with a specified value of ampere-turns. The selection of the winding data of the primary winding is carried out in accordance with the required parameters, which is determined by the ratio of the linear loading to the induction in the air gap, \(A/B\delta\). The winding data of the “secondary” circuit in the combined winding are thereby obtained as constrained values.

When selecting the electromagnetic loadings and when evaluating the efficiency of a combined winding, it is also necessary to proceed from the fact that, when replacing two separate windings by one combined winding, the heating, losses, and efficiency of the unit must remain at the former level.

Selection of Current Density and Linear Loadings

From the condition of equality of the specific electrical losses per unit cylindrical surface of the stator bore of machines with separate and combined windings, it follows that

\[ A_1\Delta_1 + A_2\Delta_2 = A_c\Delta_{c\beta}. \tag{1} \]

Here

\[ A_1, A_2, \Delta_1 \text{ and } \Delta_2 \]

are the linear loadings and current densities of the separate windings, determined by the method set forth in [7];

\[ A_c \text{ and } \Delta_{c\beta} \]

are the effective linear loading and current density of the combined winding; the index “\(\beta\)” corresponds to the condition of maintaining the specific electrical losses when replacing the separate windings by one combined winding.

The effective linear loading is

\[ A_c=\frac{u_{\mathrm{pc}} I_c}{t}, \tag{2} \]

where

\[ u_{\mathrm{pc}} \]

is the number of effective conductors of the combined winding in a slot;

\[ t \]

is the tooth pitch;

\[ I_c \]

is the effective current in a conductor.

\[ I_c=\sqrt{I_{1c}^{\,2}+I_{2c}^{\,2}}. \tag{3} \]

Here \(I_{1c}\) and \(I_{2c}\) are the currents of the primary and secondary circuits flowing in one and the same conductor of the combined winding.

Substituting (3) into (2), we obtain

\[ A_c^{2}=A_{1c}^{2}+A_{2c}^{2}. \tag{4} \]

From (1), taking (4) into account, the effective current density is

\[ \Delta_{c\beta}= \frac{A_1\Delta_1 + A_2\Delta_2} {\sqrt{A_{1c}^{2}+A_{2c}^{2}}}. \tag{5} \]

Let us express \(A_{1c}\) and \(A_{2c}\) of the combined winding through \(A_1\) and \(A_2\) of the corresponding separate windings, proceeding from the following conditions.

1. To preserve the parameters and characteristics when replacing two separate windings by one combined winding, the primary circuit of the latter must be designed so that the equality is satisfied:

\[ \frac{A_1}{B_{\delta 1}}=\frac{A_{1c}}{B_{\delta 1c}}. \tag{6} \]

Hence, expanding the expressions \(A\) and \(B\delta\), we obtain that the number of turns of the primary circuit of the combined winding-

\[ w_{1c}=w_1\sqrt{\frac{k_{01}}{k_{01c}}}, \tag{7} \]

and, with equality of power,

\[ A_{1c}=A_1\sqrt{\frac{k_{01}}{k_{01c}}}. \tag{8} \]

Here and below \(k_0\) is the winding factor of the corresponding winding.

1. For secondary circuits, the number of turns of which in the combined winding cannot be chosen arbitrarily, it is necessary to ensure equality of the magnetomotive forces \(F_{2c}=F_2\). From this we obtain:

\[ A_{2c}=A_2\,\frac{k_{02}}{k_{02c}}. \tag{9} \]

In most cases it may be assumed approximately that \(\Delta_1\approx \Delta_2=\Delta\). Then from (5), taking (8) and (9) into account and denoting

\[ \frac{A_2}{A_1}=a,\qquad \frac{k_{01}}{k_{01c}}=\kappa_1 \quad \text{and} \quad \frac{k_{02}}{k_{02c}}=\kappa_2, \]

we obtain the ratio of the current density in the combined winding to the current density in the separate windings, for equality of specific losses:

\[ \frac{\Delta_{c\beta}}{\Delta} = \frac{1+a}{\sqrt{\kappa_1+\kappa_2^{\,2}a^2}}. \tag{10} \]

As a first approximation it may be assumed that the ratio of the winding factors is \(\kappa_1\approx 1\). Analysis of combined windings shows that this assumption gives a very small error when the ratio of the numbers of pole pairs of the machines being combined is equal to any even number or to an odd number greater than five.

In the secondary circuit the winding data are obtained as constrained values; therefore in a number of cases the winding factor \(k_{02c}\) may differ substantially from \(k_{02}\), and their ratio \(\kappa_2\) is not equal to unity.

In the idealized case, for \(\kappa_1=\kappa_2=a=1\), the current density is \(\Delta_{c\beta}=\sqrt{2}\Delta\). For \(a\ne 1\), but \(\kappa_1=\kappa_2=1\), we obtain

\[ \frac{\Delta_{ck}}{\Delta} = \frac{1+a}{\sqrt{1+a^2}}. \tag{11} \]

The dependence

\[ \frac{\Delta_{ck}}{\Delta}=f(a) \]

is given in Fig. 2.

Taking into account that in reality \(\kappa_2\ne 1\), from (10) and (11) we obtain

\[ \Delta_{c\beta}=\Delta_{ck}\beta . \tag{12} \]

where

\[ \beta = \frac{\Delta_{c\beta}}{\Delta_{ck}} = \sqrt{\frac{1+a^2}{1+\kappa_2^{\,2}a^2}}. \]

Families of curves \(\beta=f(\kappa_2)\) for different values of the ratio of linear loads \(a\) are given in Fig. 3.

The choice of current density according to (10), or according to the graphs of Figs. 2 and 3 in accordance with (12), ensures constancy of the specific losses per unit cylindrical surface of the stator bore. However, equality of copper losses is not maintained in this case, since any part of the combined winding, in comparison with one of the separate windings, is either lengthened,

Fig. 2. Relative current density as a function of the ratio of linear loads in the idealized case \((k_1=k_2=1)\), with equality of specific electrical losses

Fig. 3. Coefficient \(\beta\), taking into account the inequality of the winding coefficients

either is shortened. As shown below, in order to preserve the copper losses at the same power, the current density in the combined winding must be determined from the equation

\[ \frac{\Delta_{c\varepsilon}}{\Delta} = \frac{\xi_1+\xi_2 a}{\sqrt{k_1+k_2 a^2}}, \tag{13} \]

where

\[ \xi_1=\frac{l_{\mathrm{cp}1}}{l_{\mathrm{cpc}}} \quad \text{and} \quad \xi_2=\frac{l_{\mathrm{cp}2}}{l_{\mathrm{cpc}}} \]

are the ratios of the mean half-turn lengths of the corresponding separate windings and of the combined winding.

From (10) and (13) we find

\[ \Delta_{c\varepsilon}=\Delta_{c\beta}\,\varepsilon, \tag{14} \]

where

\[ \varepsilon=\frac{\Delta_{c\varepsilon}}{\Delta_{c\beta}} = \frac{\xi_1+\xi_2 a}{1+a}. \]

A family of curves \(\varepsilon=f(a)\) for various values of \(\xi_1\) and \(\xi_2\) is presented in Fig. 4. The values of \(\varepsilon\) were calculated under the assumption that one of the coefficients (\(\xi_1\) or \(\xi_2\)) is equal to unity. For example, if the pitch of the combined winding is close to the pitch of the primary separate winding, then \(\xi_1 \approx 1\). The other coefficient (\(\xi_2\)) may then differ considerably from unity in either direction, since the second of the windings being combined may have a smaller or larger number of poles and, accordingly, elongated or shortened end connections.

With an elongated end connection of the combined winding in comparison with one of the separate windings, the coefficient \(\varepsilon<1\).

In Fig. 5 are shown the critical values of the ratio of winding coefficients \(k_{gkr}\), at which \(\dfrac{\Delta_{ce}}{\Delta}=1\). The values \(k_{2kr}\) for various \(\xi_1\) and \(\xi_2\) are determined from (13).

\[ \begin{aligned} k_{2kr} &= \frac{1}{a}\sqrt{(1+\xi_2 a)^2-1}\quad &&\text{for } \xi_1=1;\\ k_{2kr} &= \frac{1}{a}\sqrt{(\xi_1+a)^2-1}\quad &&\text{for } \xi_2=1. \end{aligned} \tag{15} \]

Fig. 4. Coefficient \(\varepsilon\), taking into account the change in the lengths of the end parts when the total copper losses of the separate and combined windings are equal.
Solid line: \(\xi_1=1,\ \xi_2\ne 1\).
Dashed line: \(\xi_2=1,\ \xi_1\ne 1\).

Fig. 5. Critical values of the ratio of winding coefficients \(k_{2kr}\) as a function of the ratio of linear loads.
Solid line: \(\xi_1=1,\ \xi_2\ne 1\).
Dashed line: \(\xi_2=1,\ \xi_1\ne 1\).

It is obvious that for \(k_2>k_{2kr}\), replacing separate windings by one combined winding becomes inadvisable from the standpoint of copper consumption.

Thus, the relationships presented make it possible to select the optimal scheme of the combined winding and to determine the current density in it under the condition that the power and copper losses are maintained. For \(k_1\approx 1\), the current density can be found graphically from the curves in Figs. 2, 3, and 4 in accordance with the equation

\[ \frac{\Delta_{ce}}{\Delta}=\frac{\Delta_{ck}}{\Delta}\,\beta\cdot\varepsilon \tag{16} \]

Comparison of Copper Weight, Power, and Dimensions

Let us estimate the consumption of conductor material in a machine with separate windings and with combined windings, under the condition of equal copper losses.

Transforming the formula for determining the electrical losses \(p_m\) so that it includes the current density \(\Delta\) and the copper weight \(G\) and

Taking the specific weight of copper to be \(8.9\ \text{kg}/\text{dm}^3\), and the specific resistance at \(75^\circ\text{C}\) to be \(0.02135\ \dfrac{\Omega\cdot\text{mm}^2}{\text{m}}\), we obtain

\[ p_{\mathrm{m}} = 2.4 \Delta^2 G_{\mathrm{bm}} . \tag{17} \]

Here the dimension of \(\Delta\) is \(\text{A}/\text{mm}^2\); \(G\) is kg.

Then, assuming \(\Delta_1 \approx \Delta_2 = \Delta\), we write the ratio of the copper losses of the separate and combined windings:

\[ \frac{p_{\mathrm{m}1}+p_{\mathrm{m}2}}{p_{\mathrm{mc}}} = \left(\frac{\Delta}{\Delta_c}\right)^2 \frac{G}{G_c}, \tag{18} \]

where

\[ G = G_1 + G_2 \]

is the sum of the copper weights of the two separate windings.

Equation (18) makes it possible to estimate the reduction in copper consumption while maintaining the electrical losses, or the reduction in losses if the copper weight does not change when two windings are replaced by one combined winding. When the current density is chosen from the condition that the electrical losses remain unchanged, the ratio of copper weights is

\[ \frac{G}{G_c} = \left(\frac{\Delta_{ce}}{\Delta}\right)^2 . \tag{19} \]

As can be seen from Figs. 2, 3, 4, the quantity \(\dfrac{\Delta_{ce}}{\Delta}\) may be greater or less than unity, depending on the ratios of the linear loads, winding factors, and mean half-turn lengths. With rational design and choice of the combined-winding scheme, the copper consumption can be substantially reduced in comparison with the separate arrangement.

Let us consider the influence of electrical combination on the power of the unit.

Suppose the combined winding is placed in the same slots in which the two separate windings were located. Then the slot area \(S_{pc}=S_p\). Expressing \(S_p\) through the number of effective conductors per slot \(u_p\), their cross section \(Q\), and the copper slot fill factor \(k_z\), we obtain

\[ \frac{u_{pc} Q_{cm}}{k_{zc}} = \frac{u_{p1}Q_1 + u_{p2}Q_2}{k_z}. \tag{20} \]

Here, in accordance with (7),

\[ u_{pc} = u_{p1}\frac{\sqrt{\kappa_1}}{\alpha_1}, \tag{21} \]

where

\[ \alpha_1=\frac{a_1}{a_{1c}} \]

is the ratio of the numbers of parallel branches of the primary circuits of the separate and combined windings.

From condition (20), the maximum possible cross section of the effective conductor of the combined winding placed in the same slot is determined as

\[ Q_{cm} = \frac{k_{zc}}{k_z} \frac{\alpha_1}{\sqrt{\kappa_1}} \left( Q_1+\frac{u_{p2}}{u_{p1}}Q_2 \right). \tag{22} \]

From the condition of permissible heating at a given current \(I_c\) and the current density \(\Delta_{ce}\) selected according to (16), the required cross section of the effective conductor is

\[ Q_c=\frac{I_c}{\Delta_{ce}} . \tag{23} \]

For comparison with \(Q_{cm}\), let us express \(Q_c\) in terms of the cross sections \(Q_1\) and \(Q_2\) of the separate windings, assuming that the rated powers remain unchanged when two windings are replaced by one. It follows from the latter that

\[ I_{1c}=I_1\alpha_1,\quad \text{and}\quad I_{2c}=I_2\frac{k_2}{\sqrt{k_1}}\frac{u_p}{u_{p1}}\alpha_1 . \]

From (3) and (23), expressing the current as the product of its density and the cross section, we obtain

\[ Q_c=\frac{\Delta}{\Delta_{ce}}\frac{z_1}{\sqrt{k_1}} \sqrt{k_1Q_1^2+k_2^2Q_2^2\left(\frac{u_{p2}}{u_{p1}}\right)^2}. \tag{24} \]

The ratio of the cross section \(Q_{cm}\), the maximum possible under the condition of placement in the slot, to the cross section \(Q_c\) is

\[ k_Q=\frac{Q_{cm}}{Q_c}. \tag{25} \]

Substituting expressions (22) and (24) into (25) and expressing, in the latter,

\[ Q=\frac{At}{u_p\Delta}, \tag{26} \]

taking (11) and (16) into account, we obtain

\[ k_Q=\left(\frac{\Delta_{ck}}{\Delta}\right)^2 \beta^2 \varepsilon \frac{k_{zc}}{k_z}. \tag{27} \]

Similarly, from equation (19), taking (21), (24), and (26) into account, and expressing the ratio of weights through the numbers of effective conductors in the slot, their cross sections, and the mean half-turn lengths, we obtain the above relation (13).

Practical use of the maximum cross section \(Q_{cm}\) is possible at some new current density \(\Delta_{cm}\), at which the losses in the combined winding remain at the level of those in the separate windings. Equating (18) to unity, we obtain

\[ \left(\frac{\Delta_{cm}}{\Delta}\right)^2=\frac{G}{G_{cm}}, \]

where \(G_{cm}=G_c k_Q\) is the weight of copper at the maximum cross section. Taking (16), (19), and (27) into account,

\[ \frac{\Delta_{cm}}{\Delta}=\sqrt{\varepsilon\frac{k_z}{k_{zc}}}. \tag{28} \]

With the maximum cross section \(Q_{cm}\), the increase in current and power of the combined winding admissible under the heating conditions is determined by the following relation:

\[ k_p=\frac{I_{cm}}{I_c}=\frac{P_{cm}}{P_c} =\frac{Q_{cm}\Delta_{cm}}{Q_c\Delta_{ce}}, \]

which, after transformations, gives

\[ k_p=\frac{\Delta_{ck}}{\Delta}\,\beta \sqrt{\varepsilon\frac{k_{zc}}{k_z}} =\sqrt{k_Q}. \tag{29} \]

The coefficient \(k_p\) makes it possible to estimate the effectiveness of replacing two windings by one already at the initial design stage. However, in making this estimate it must be taken into account that increasing the power by means of the current may lead to an impermissible change in the linear loads, parameters, and characteristics.

It is of interest to compare the dimensions of machines of the same power with separate and combined windings.

For the cross section \(Q_c\), determined from (23), the required slot area is

\[ S_{pc}=\frac{u_{pc}Q_c}{k_{zc}}. \]

Hence, taking (20) into account, the coefficient of reduction of the slot area is

\[ k_s=\frac{S_{pc}}{S_p}=\frac{Q_c}{Q_{cm}}=\frac{1}{k_Q}. \tag{30} \]

Let us investigate the influence of \(k_s\) on the principal dimensions of the electromagnetic core. In doing so we shall assume that the slot has a rectangular shape and that the length of the stator core does not change.

Let us denote:

\(b_p,\ b_z\) — the width of the slot and of the tooth;

\(h_p\) — the height of the slot;

\(D,\ D_a\) — the inner and outer diameters of the stator.

To preserve the level of saturation, the parameters, and the characteristics, the geometrical dimensions must be changed in such a way that the following remain constant:

1) magnetic flux \(\Phi_c=\Phi\),

2) induction in the tooth \(B_{zc}=B_z\),

3) induction in the yoke \(B_{ac}=B_a\).

From the first condition it follows that the inductions in the air gap are

\[ \frac{B_{\delta c}}{B_\delta}=\frac{D}{D_c}. \tag{31} \]

From the first and second conditions, using the known relations and taking (31) into account, we obtain

\[ \frac{D_c}{D}=1-\frac{b_p-b_{pc}}{t}. \tag{32} \]

From the first and third conditions we find that the difference between the outer and inner stator diameters with a combined winding is

\[ D_{ac}-D_c=(D_a-D)-2(h_p-h_{pc}). \tag{33} \]

Substituting (32) into (33) and assuming that the slot dimensions in height and width change proportionally,

\[ \frac{h_{pc}}{h_p}=\frac{b_{pc}}{b_p}=\sqrt{k_s}, \]

after transformations we find the ratio

\[ \frac{D_{ac}}{D_a} = 1- \left(1-\sqrt{k_s}\right) \left( \frac{D}{D_a}\frac{b_p}{t} + \frac{2h_p}{D_a} \right). \tag{34} \]

The expression in parentheses contains the ratio of known quantities for a machine with separate windings. Approximately, for 4-pole asynchronous machines, which in most cases are basic when two machines are combined in one magnetic core, equation (32) may be represented in the form

\[ \frac{D_{ac}}{D_a}\approx 1-0.5\left(1-\sqrt{k_s}\right). \tag{35} \]

A similar analysis for the rotor under the same conditions and with an unchanged shaft diameter gives the ratio of the increments of slot width to height

\[ \frac{b_{\mathrm{pr}}-b_{\mathrm{prs}}}{h_{\mathrm{pr}}-h_{\mathrm{prs}}} = \frac{6.28}{z_p}, \tag{36} \]

where \(z_p\) is the number of rotor slots.

Let us evaluate the effectiveness of replacing two windings by one combined winding using as an example the single-machine frequency converter OPCh-200-5, manufactured by the Yaroslavl Electromechanical Plant [8]. In this converter, two asynchronous three-phase machines with separate windings and with a pole-number ratio of 1:3 are combined in one magnetic system.

The rotor windings of the OPCh-200-5—the motor three-phase short-circuited winding and the three-phase generator winding—can be replaced by one combined winding. The basic circuit arrangements for constructing a phase of the rotor windings are shown in Fig. 6.

Fig. 6. Principle of combining the rotor windings of the OPCh-200-5 converter (for one phase of the motor and generator); from top to bottom: separate windings, a combined winding with extended end connections, the same with shortened end connections.

Calculations of the OPCh-200-5 by the method presented showed (Table 1) that, when two windings are combined into one, the copper consumption in the rotor, for equal electrical losses, is reduced by a factor of 1.53 with extended end connections and by a factor of 1.88 with shortened ones. In the same rotor slots, its power could be increased respectively by factors of 1.37 and 1.24, or, with unchanged power, the rotor diameter could be reduced by 30 and 20 mm.

Conclusions

1. The proposed method makes it possible, at the initial design stage, to assess the advisability and effectiveness of replacing two windings by one combined winding.

2. Taking the presented relationships into account ensures the optimal choice of the combined-winding circuit and the choice of its electromagnetic loadings, a prescribed level of losses, parameters, and characteristics.

Table 1

Basic design data for the separate and combined rotor windings of the OPCh-200-5

REFERENCES

1. V. S. Novokshchenov. Some questions in the theory of an asynchronous brushless frequency converter. Izv. Tomsk Polytechnic Institute, vol. 94, 1958.

2. V. I. Popov. A contactless single-machine frequency multiplier with two pairs of combined windings. Proceedings of the Third All-Union Conference on Contactless Electrical Machines, issue 2, Riga, 1966.

3. Yu. V. Baryshnikov, V. M. Pavlinin. A single-machine frequency converter with one combined winding on the rotor. Proceedings of the Ural Polytechnic Institute named after S. M. Kirov, collection 138, 1964, Sverdlovsk.

4. Yu. V. Baryshnikov, A. T. Plastun. A contactless synchronous motor with combined windings. State and Prospects for the Development of Semiconductor Excitation of Synchronous Motors (conference proceedings). NTOEP, 1966, Sverdlovsk.

5. A. T. Plastun. Investigation and development of a system of brushless excitation of synchronous generators of small and medium power with a cascade asynchronous-synchronous exciter. Candidate dissertation, 1967, Sverdlovsk, UPI.

6. Yu. V. Baryshnikov, A. T. Plastun, N. S. Siyunov. Asynchronous short-circuited electric motor with combined winding on the stator. TsNTPO No. 17-67-1668/93, GOSINTI, Moscow, 1967.

7. V. M. Pavlinin, N. S. Siyunov. Optimal dimensions and electromagnetic loadings of a single-machine frequency converter. Izv. vuzov, Elektromekhanika, 1962, No. 12.

8. Asynchronous single-machine high-frequency converters OPCh-200-5. YaEMZ information sheet, 1966.