Vol. 228 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

ON SELECTING THE OPTIMAL DIMENSIONS OF THE POLE SYSTEM OF DC MACHINES WITH SERIES COMPOUND EXCITATION

Yu. N. Kroneberg

(Presented by the scientific and technical seminar of the Department of Electrical Equipment, AEMF)

Existing methods for calculating the machines under consideration (Fig. 1) make it possible, for given dimensions, to calculate the output characteristics reliably [1, 2, 3]. Practically no attention has been paid to the selection of optimal dimensions. This problem is considerably more complicated and, in its general formulation, has not been solved even for simpler electromagnetic excitation machines. Their design is based on a generalization of accumulated experience, which in many respects also extends to compound-excitation machines, making it possible, in particular, to choose satisfactorily the geometry of the air gap, as well as all dimensions of the yoke and armature. The main difficulties are associated with the selection of the dimensions

Fig. 1. Magnetic system. 1—armature, 2—yoke, 3—magnet, 4—pole core, 5—excitation-winding coil.

(materials are usually predetermined) most specific unit—the pole system. The criterion of its optimality, as a rule, is the minimum rated excitation power

\[ P_{\mathrm{ex}}=\frac{F_{\mathrm{ex}}^{2}}{w^{2}}\,R_t, \tag{1} \]

where

\[ \begin{aligned} F_{\mathrm{ex}}&\text{ is the rated m.m.f. of the excitation coil, A;}\\ R_t&\text{ is the winding resistance at operating temperature, ohm;}\\ w&\text{ is the number of turns in the coil.} \end{aligned} \]

Fig. 2. Equivalent circuit of the magnetic circuit.

At the same time, reliable self-excitation and a sufficient control range must be ensured, i.e., the initial e.m.f. (with the field winding de-energized) at no load must have a specified value or lie within prescribed limits. The imposed conditions make it possible to reduce the search for the optimum to an analysis of the dependence of the pole flux \(\Phi_p\) on the excitation m.m.f. at no load. To this end, using methods for calculating electrical machines, it is necessary to find the rated armature e.m.f. (the error of such a calculation is associated only with a negligible overestimate of the armature-reaction flux, mainly its longitudinal component), and then, knowing the leakage permeance of the poles, to determine the rated \(\Phi_{p\mathrm{n}}\) and initial \(\Phi_{p0}\) (or the range of its variation) fluxes.

The next stage consists in obtaining the analytical dependence of the rated excitation m.m.f. \(F_{\mathrm{ex}}\), which is substantially complicated by the nonlinearity of the equivalent circuit (Fig. 2) of the magnetic circuit (the nonlinear resistances are shaded) underlying the verified calculation methods [1, 2, 3].

The available data are sufficient to calculate and plot (Fig. 3) only the external, with respect to the pole, weber-ampere characteristic \(\Phi_j(F_\Sigma)\). To do this, after specifying a series of values of the armature flux \(\Phi_a\), it is necessary to calculate the corresponding drops of m.m.f. across the magnetic resistances of the air gap \(R_\delta\), the teeth \(R_z\), and the armature back \(R_a\); then, knowing the magnetic leakage resistance \(R_\sigma\), to compute the leakage fluxes \(\Phi_\sigma\) and, adding them to the fluxes \(\Phi_a\), obtain the fluxes

Fig. 3. Magnet diagram

of the yoke \(\Phi_j\) and of the m.m.f. drop in the magnetic resistance of the yoke \(R_j\); the sum of the m.m.f. drops in the air gap, teeth, back iron, and yoke will be equal to the m.m.f. \(F_\Sigma\). Suppose that the cross sections of the magnet \(m\) and the core \(s\) are known (in essence, this concerns only their widths \(b_m\) and \(b_s\), since the length \(l_p\) is predetermined by the armature dimensions). Then, knowing the calculated height of the magnet \(h_m\) and of the core \(h_s\), one can construct (Fig. 3) the characteristics of the branches of the unstabilized magnet (the switch in Fig. 2 in position 1), \(\Phi'_m(F'_m)\), and of the core \(\Phi_s(F_s)\). The sum of these fluxes at \(F_s = F'_m = F'_p\) gives the pole characteristic \(\Phi'_p(F'_p)\), while the addition of the m.m.f.’s \(F'_p\) and \(F_\Sigma\) at \(\Phi'_p = \Phi_j\) gives the dependence of the pole flux with an unstabilized magnet on the m.m.f. of the excitation winding, \(\Phi'_p(F_v)\).

Stabilization of the magnets (the switch in Fig. 2 is moved to position 2) in the machines under consideration is usually carried out [1] at no load with the excitation winding disconnected (if, of course, its opposing connection is impossible). The point at which the working Weber–ampere characteristic departs is the straight line \(\Phi_m(F_m)\), which is then used instead of the curve \(\Phi'_m(F'_m)\) to construct the required dependence \(\Phi_p(F_v)\); it is found graphically (Fig. 3) from the flux \(\Phi_{p0}\). The equation of this straight line has the form

\[ \Phi_m = \Phi_\rho + \gamma F_m, \tag{2} \]

where

\[ \Phi_\rho \]
— the residual flux of the stabilized magnet, Wb,

\[ \gamma = \frac{\rho}{h_p + 1.6 \cdot 10^6 \Delta_m \rho} \]
— the specific permeance of the magnet branch (per unit area), Wb/m\(^2\)A,

\(\rho\) — the recoil coefficient of the alloy, H/m,

\(\Delta_m\) — the butt-joint gap of the magnet, m.

The flux \(\Phi_\rho\) is (Fig. 3) a complex function of many variables. However, owing to the small demagnetization, which is explained not only by the presence of magnetically soft pole shoes and a small longitudinal mmf, but also by the effective shunting of the magnets by the cores, the difference between the flux \(\Phi_\rho\) and the residual flux \(\Phi_r\) becomes insignificant against the background of technological scatter. Therefore demagnetization is sometimes neglected even when calculating the characteristics [2]. On the basis of the foregoing, it is proposed, without calculating the stabilization mode, to adopt approximately (with subsequent verification)

\[ \Phi_m \simeq \eta m B_r + m \gamma F_m, \tag{2a} \]

where

\[ B_r \]

is the residual induction, \(\mathrm{Wb}/\mathrm{m}^2\),

\[ \eta=\frac{\Phi_\rho}{\Phi_r}=0.9 \div 1.0 \]

is the demagnetization coefficient.

When the excitation winding is switched off, then according to Figs. 2 and 3 the equations are valid

\[ F_{\Sigma 0}+F_{s0}=0, \tag{3} \]

\[ \Phi_{m0}+\Phi_{s0}=\Phi_{p0}, \tag{4} \]

in which, in addition to the flux \(\Phi_{p0}\) (for the time being we shall consider its fixed value to be specified), the mmf \(F_{\Sigma 0}(\Phi_{p0})\) is known and, consequently, \(F_{s0}=F_{m0}=-F_{\Sigma 0}\). From this it is already easy to find the core induction \(B_{s0}\) (it is always negative), since the dependence \(B_s(F_s)\), when the calculated core height \(h_s\) and butt-joint gap \(\Delta_s\) are specified, is known (Fig. 4). Then, according to equation (2a), we have

\[ \Phi_{m0}=\eta m B_r-m\gamma F_{\Sigma 0}=mB_{m0}, \tag{5} \]

where \(B_{m0}\) is the magnet induction with the excitation winding switched off. Taking into account that \(\Phi_{s0}=sB_{s0}\), from equations (4) and (5) we obtain the important expression

\[ s=\frac{\Phi_{p0}-m(\eta B_r-\gamma F_{\Sigma 0})}{B_{s0}},\ \mathrm{m}^2, \tag{6} \]

from which it follows that, for a constant initial flux \(\Phi_{p0}\), the cross sections of the magnet and the core may take many values, but must remain proportional. The optimization problem consists in selecting from this set such a combination as would ensure a minimum nominal excitation power. The existence of an extremum is obvious: with a small magnet cross section the core cross section is small and, consequently, the mmf drop is large and the excitation power is large; with an increase in the magnet cross section the total pole width increases and the coil cross section correspondingly decreases, i.e., its resistance increases, and the excitation power also increases.

When the machine at no load develops an emf corresponding to the nominal mode, then according to Figs. 2 and 3 the following equations are valid:

\[ F_{\Sigma n}+F_{sn}=F_{\mathrm{vn}}, \tag{7} \]

\[ \Phi_{mn}+\Phi_{sn}=\Phi_{pn}. \tag{8} \]

Substituting into equation (8) the flux \(\Phi_{mn}\), found from the equation

(2a), for \(F_m=F_{m\mathrm{n}}=F_{s\mathrm{n}}\) and taking into account that \(\Phi_{s\mathrm{n}}=sB_{s\mathrm{n}}\), we obtain the relation

\[ \eta mB_r + m\gamma F_{s\mathrm{n}} + sB_{s\mathrm{n}} = \Phi_{r\mathrm{n}}, \tag{9} \]

in which \(B_{s\mathrm{n}}\) is a known graphically specified (Fig. 4) nonlinear function of the m.m.f. \(F_{s\mathrm{n}}\). In order, with the aid of equation (9), to represent the m.m.f. \(F_{s\mathrm{n}}\) analytically, the graphical dependence must be approximated analytically. Since the approximating expression must correspond satisfactorily to the actual dependence only for the subsequently found optimal dimensions (i.e., ultimately only one point of the curve is needed, and in practice some assumed range), it is expedient to adopt the simplest linear approximation:

\[ B_s = B_x + \varkappa F_s, \tag{10} \]

where the coefficient \(\varkappa\) has the dimension of specific permeance, \(\mathrm{Wb}/\mathrm{m}^2\mathrm{A}\).

Fig. 4. Magnetization curve of the pole core

The most probable position of the operating point on the curve of Fig. 4 is in the region of the “knee” (as long as the core is unsaturated, it is advantageous to load it in order to reduce the pole width and increase the cross section of the excitation winding; however, passing beyond the “knee,” while having little effect on the winding cross section, leads to a sharp increase in the excitation m.m.f.; from this consideration the parameters \(B_x\) and \(\varkappa\) should be chosen).

Putting \(B_s=B_{s\mathrm{n}}\) and \(F_s=F_{s\mathrm{n}}\) in equation (10) and solving it simultaneously with equation (9), we find the nominal m.m.f. drop in the core:

\[ F_{s\mathrm{n}} = \frac{\Phi_{r\mathrm{n}}-\eta mB_r-sB_x}{m\gamma+s\varkappa}. \tag{11} \]

Substituting the value of \(s\) from equation (6) into equation (11), and then the result into equation (7), we obtain the desired analytical expression for the m.m.f. of the excitation winding as a function of the magnet cross section:

\[ F_{\mathrm{w}\mathrm{n}} = \frac{ B_{s0}(\eta B_r-\gamma F_{\Sigma\mathrm{n}}) - (\eta B_r-\gamma F_{\Sigma0})(B_x-\varkappa F_{\Sigma\mathrm{n}}) }{ \varkappa(\eta B_r-\gamma F_{\Sigma0})-\gamma B_{s0} } \cdot \frac{A_1-m}{A_2-m}, \tag{12} \]

where

\[ A_1 = \frac{ \Phi_{r\mathrm{n}}B_{s0}-\Phi_{r0}(B_x-\varkappa F_{\Sigma\mathrm{n}}) }{ B_{s0}(\eta B_r-\gamma F_{\Sigma\mathrm{n}}) - (\eta B_r-\gamma F_{\Sigma0})(B_x-\varkappa F_{\Sigma\mathrm{n}}) }, \ \mathrm{m}^2, \]

\[ A_2 = \frac{ \varkappa \Phi_{r0} }{ \varkappa(\eta B_r-\gamma F_{\Sigma0})-\gamma B_{s0} }, \ \mathrm{m}^2. \]

It remains now to express, through the magnet cross section, the active resistance of the excitation coil,

\[ R_t = 2\rho \alpha \varphi_t \frac{L}{g},\ \mathrm{ohm}, \tag{13} \]

where

\[ \begin{aligned} \rho_t &\text{ is the specific electrical resistivity, } \Omega\,\mathrm{mm}^2/\mathrm{m},\\ p &\text{ is the number of pole pairs,}\\ L &\text{ is the mean length of a turn, } \mathrm{m},\\ q &\text{ is the conductor cross section, } \mathrm{mm}^2. \end{aligned} \]

Assuming, as usual, a flat coil, according to Fig. 1 we have:

\[ L=2(l_p+b_m+b_s)+\frac{\pi}{2}(b_f-b_m-b_s), \tag{14} \]

\[ q=\frac{h_f k_3\cdot 10^6}{2w}(b_f-b_m-b_s), \tag{15} \]

where \(k_3\) is the ratio of the total copper cross section to the coil cross section.

Taking into account that \(b_m=\dfrac{m}{l_p}\) and \(b_s=\dfrac{s}{l_p}\), with the aid of equations (6), (13), (14), and (15), we obtain:

\[ R_t=2p(4-\pi)\cdot 10^{-6}\rho_t\,\frac{w^2}{h_f k_3}\cdot \frac{A_3-m}{A_4-m}, \tag{16} \]

where

\[ A_3= \frac{ \dfrac{1}{4-\pi}(4l_p+\pi b_f)l_p B_{s0}+\Phi_{p0} }{ \eta B_r-B_{s0}-\gamma F_{\Sigma0} }, \ \mathrm{m}^2; \]

\[ A_4=\frac{b_f l_p B_{s0}-\Phi_{p0}}{\eta B_r-B_{s0}-\gamma F_{\Sigma0}}, \ \mathrm{m}^2. \]

Finally, the nominal excitation power according to equations (1), (12), and (16) is

\[ P_{\mathrm{vn}}= A\left(\frac{A_1-m}{A_2-m}\right)^2 \frac{A_3-m}{A_4-m}, \ \mathrm{W}, \tag{17} \]

where

\[ A= 2p(4-\pi)\cdot 10^{-6}\frac{\rho_t}{h_f k_3}\cdot \frac{ \left[ B_{s0}(\eta B_r-\gamma F_{\Sigma n}) -(\eta B_r-\gamma F_{\Sigma0})(B_x-xF_{\Sigma n}) \right]^2 }{ \left[ x(\eta B_r-\gamma F_{\Sigma0})-\gamma B_{s0} \right]^2 }, \ \mathrm{W}. \]

When a fixed value \(\Phi_{p0}=\mathrm{const}\) is specified, the nominal excitation power becomes a function of only one variable—the cross section of the magnet. To find its optimal value in the sense indicated above, it is necessary, as usual, to differentiate equation (17) and set the derivative equal to zero. As a result we obtain a quadratic equation, whose solution gives the optimal value of the magnet cross section,

\[ m^*=\frac{-b+\sqrt{b^2-4ac}}{2a}, \ \mathrm{m}^2, \tag{18} \]

where

\[ a=2(A_1-A_2)-(A_3+A_4), \ \mathrm{m}, \]

\[ b=3A_1A_3+2A_2A_4-A_1A_4-A_2A_3, \ \mathrm{m}^2, \]

\[ c=2(A_1-A_2)A_3A_4-(A_3+A_4)A_1A_2, \ \mathrm{m}^3. \]

The optimal core cross section \(s^*\) is determined from the equation

(6) for \(m=m^*\), and from equations (14), (15), the mean turn length \(L^*\) and the conductor cross section \(q^*\) (the number of turns is assumed known).

In the general case, the dependence \(\Phi_p(F_{\mathrm{v}})\), calculated by means of existing methods [1, 2, 3] for the pole-system dimensions found above, will not pass through the points \((O,\Phi_{p0})\) and \((F_{\mathrm{vn}},\Phi_p)\), which is caused by the somewhat arbitrary choice of the magnet return line and by the possible error in approximating the pole-core magnetization curve (inaccuracy of the parameters \(\eta\), \(B_x\), and \(\chi\)). The magnitudes of the deviations are not difficult to find by carrying out a checking calculation. If they do not exceed \(5\div 15\%\), i.e., are at the level of the usual errors in the calculation of electrical machines, then no further refinement is required. Otherwise it is necessary to change the parameters \(\eta\), \(B_x\), and \(\chi\) and carry out a recalculation.

Sometimes a relatively rough estimate of the deviations of the initial \(\Delta\Phi_{p0}\) and nominal \(\Delta\Phi_{pn}\) fluxes may be quite sufficient; this can also be made without performing a checking calculation. To do this it is necessary to construct the demagnetization curve of the magnet with cross section \(m^*\) and, at m.m.f. \(F_{\Sigma 0}\), find the value of the flux that will differ from the flux \(\Phi^*_{m0}\), found from equation (5), by the amount \(\Delta\Phi_{m0}\). It is easy to show that the signs of the deviations \(\Delta\Phi_{m0}\) and \(\Delta\Phi_{p0}\) coincide and

\[ |\Delta\Phi_{p0}|<|\Delta\Phi_{m0}|. \tag{19} \]

It can also be shown that

\[ |\Delta\Phi_{pn}|<|\Delta\Phi_{m0}|+|s^*\Delta B_{s0}|, \tag{20} \]

where \(\Delta B_{sn}\) is found from the curve of Fig. 4 at m.m.f. \(F_{sn}\), calculated by means of equation (11).

The situation is considerably more complicated when the initial flux \(\Phi_{p0}\) may be chosen within a certain range. In this case, into equation (17), all coefficients of which will now be functions of the initial flux \(\Phi_{p0}\), one would have to substitute from equation (18) the value of the magnet cross section \(m^*\), which now is also a function of the flux \(\Phi_{p0}\), and differentiate the resulting expression for the nominal excitation power with respect to the argument \(\Phi_{p0}\), set the derivative equal to zero, and from this find the extremal value of the initial flux \(\Phi_{p0}\). However, the resulting equation is so cumbersome that solving it is clearly impractical (besides, it is unclear whether the extremal value lies in the specified region). It is much simpler to specify several values of the flux \(\Phi_{p0}\), to find for them, by equation (18), a series of magnet cross sections, and, by equation (17), to calculate a series of nominal excitation powers. Having constructed the dependences \(P_{\mathrm{vn}}(\Phi_{p0})\) and \(m^*(\Phi_{p0})\), it is then not difficult to find the optimal values of \(\Phi_{p0}\) and \(m^x\).

REFERENCES

1. Yu. B. Panferov. Calculation of the magnetic circuit of controlled direct-current machines with permanent magnets. Vestnik elektropromyshlennosti, No. 12, 1961.

2. L. M. Palastin. A graphoanalytical method for determining the no-load characteristics of controlled direct-current machines with permanent magnets. Elektrichestvo, No. 11, 1963.

3. Yu. N. Kroneberg. Analytical calculation of the no-load characteristics of direct-current machines with series compound excitation. In: Electronic and Electromechanical Devices. “Energiya,” 1969.