Optimal Geometry of Combined Contactless Synchronous Machines

OPTIMAL GEOMETRY OF COMBINED CONTACTLESS SYNCHRONOUS MACHINES

V. T. KARAVAEV

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

At present, much attention is being given to the problem of creating contactless electric machines, and their theory is being developed. The relevance of this problem is confirmed by the need to increase the reliability of electric machines and power systems as a whole.

In designing contactless electric machines, the choice of a rational geometry for the laminations of the stator and rotor cores is of great importance, since all the principal characteristics of the machine depend on the internal geometry of the magnetic core. An unsuccessful choice of lamination geometry—for example, making the machine too “iron-rich” or, conversely, too “copper-rich”—cannot be corrected by changing the winding data; therefore, determining the geometry of the magnetic core is one of the most critical aspects of design.

The distinctive feature of the contactless synchronous machine considered here is that both the exciter and the generator, while having the same overall dimensions, differ substantially in power. The exciter, having relatively low power, may have a magnetic-circuit geometry that is far from optimal.

The present work considers the possibility of obtaining an optimal geometry for the exciter while preserving the optimal geometry of the contactless synchronous generator.

Let us give several relations for individual sections of the magnetic circuit of two salient-pole synchronous electric machines combined in a single magnetic core and windings, taking into account the permissible total magnetic loadings.

The minimum value of the height of the rotor yoke \(h_{ap}\) is determined from the condition that the magnetic flux per pair of poles is equal to

\[ \Phi = \frac{\pi D l}{2p_2}\cdot B_\delta \cdot \alpha' = 2h_{ap}\cdot B_{ap}\cdot k_c. \]

Hence

\[ h_{ap}=\frac{\pi D\alpha'}{4p_2\gamma_{ap}\cdot k_c}, \tag{1} \]

where

\(D\) — rotor diameter;
\(l\) — calculated length of the machine;
\(\alpha'\) — calculated pole overlap;

\[ \gamma_{ap}=\frac{B_{ap}}{B_{\delta}} \]
—the ratio of the induction in the rotor yoke to the induction in the air gap;

\(K_c\)—the coefficient of filling of the stack with steel.

In view of the fact that the value of the induction in the rotor yoke \(B_{ap}\) is lower than the induction in the rotor tooth \(B_{zp}\), and has little effect on the magnitude of the inductive reactance, we take the height of the rotor yoke \(h_{ap}\) and the value of the induction in the rotor yoke \(B_{ap}\) as constant and determined by the total magnetic induction.

The minimum value of the tooth width \(b_{zp}\) is determined under the condition that the entire magnetic flux passes through the rotor tooth, and is found from the equality

\[ B_{\delta}\frac{\pi D}{z_p}=B_{zp}\cdot b_{zp}\cdot k_c, \]

whence

\[ b_{zp}=\frac{\pi D}{z_p\cdot \gamma_{zp}\cdot k_c}, \tag{2} \]

where

\(z_p\)—number of rotor slots,

\[ \gamma_{zp}=\frac{B_{zp}}{B_{\delta}} \]
—the ratio of the maximum induction in the rotor tooth to the induction of the air gap.

In the calculation we take \(K_c=1\).

To estimate the dependence of the dimensionless machine parameters on the geometry of the magnetic core, let us find an approximate dependence of the rotor slot area on the overall dimensions of the machine. The area of the rotor slots, taking into account (1) and (2), is

\[ S_{np}=b_{np}\cdot h_{np}\cdot z_p =\frac{\pi D_n^2}{2}\cdot k_D\left(1-\frac{1}{\gamma_{zp}}\right) \left(1-\frac{\lambda_c}{k_D}-\frac{\pi\alpha'}{2\rho_2\gamma_{ap}}\right), \tag{3} \]

where the width of the rotor slot is determined by the formula

\[ b_{np}=\frac{\pi D}{z_p}\left(1-\frac{1}{\gamma_{zp}}\right); \tag{4} \]

the height of the rotor slot is

\[ h_{np}=\frac{D-D_0-2h_{ap}}{2}; \]

\[ k_D=\frac{D}{D_n} \]
—the ratio of the rotor bore diameter to the outside diameter;

\[ \lambda_c=\frac{D_0}{D_n} \]
—the ratio of the inner diameter of the yoke to the outside diameter.

On the other hand, the rotor slot area can be determined by the formula

\[ S_{np}\cdot K_{zp}=2m_1\cdot W_1\cdot a_1\cdot a_2\cdot q_{np}, \tag{5} \]

where

\(K_{zp}\)—rotor slot fill factor;
\(m_1\)—number of phases of the exciter armature winding;
\(W_1\)—number of turns in the exciter phase;
\(q_{np}\)—cross section of the conductor of the rotor winding;

\(I_{\Sigma\mathrm{pr}}\)—total current in the conductor of the rotor winding;

\(j_{\mathrm{pr}}\)—total current density in the conductor of the rotor winding;

\(a_1\)—number of parallel branches of the armature winding of the exciter for one parallel branch with equal-pitch sections of the rotor winding;

\(a_2\)—number of parallel branches with equal-pitch sections of the rotor winding;

hence

\[ S_{\mathrm{pr}}= \frac{2m_1\cdot a_1\cdot a_2\cdot I_{\Sigma\mathrm{pr}}\cdot W_1} {k_{\mathrm{zp}}\cdot j_{\mathrm{pr}}}. \tag{6} \]

The value of the total current \(I_{\Sigma\mathrm{pr}}\) in the conductor is determined from the relations of the magnetomotive forces of the contactless synchronous generator

\[ \left. \begin{aligned} \beta_1\left(k_{\mu1}\frac{k_\delta\delta}{\mu_0}B_1 +0.45\,\frac{A_1D}{p_1}\,k_{01}\cdot k_{a1}\right) &= \frac{m\cdot\sqrt{2}}{\pi}\cdot \frac{a_1\cdot a_2\cdot W_1\cdot k_{01}}{p_1}\cdot I_{\mathrm{pr}}, \\ \beta_2\left(k_{\mu2}\frac{k_\delta\delta}{\mu_0}B_2 +0.45\,\frac{A_2D}{p_2}\,k_{02}\cdot k_{a2}\right) &= a_2\cdot W_{\mathrm{vg}}\cdot k_{\mathrm{obg}}\cdot k_{\mathrm{n}}\cdot I_{\mathrm{vg\text{-}pr}} . \end{aligned} \right\} \tag{7} \]

Carrying out simple transformations of relation (7) and taking into account that

\(W_{\mathrm{vg}}\)—number of turns of the generator excitation winding in one parallel branch with equal-pitch sections per pole, determined by the winding scheme;

\(p_1\)—number of pole pairs of the exciter;

\(\beta_1\) and \(\beta_2\)—coefficients taking into account the vector addition of the main m.m.f. and the armature reaction for the exciter and the generator;

\(A_1\) and \(A_2\)—linear loadings of the exciter and the generator;

\(B_1\) and \(B_2\)—magnetic flux densities of the exciter and generator in the air gap;

\[ A=\frac{2m_1a_1a_2W_1I_{\Sigma\mathrm{pr}}}{\pi D} \]

—the ratio of the total linear loading and the total current in the conductor of the rotor winding;

\(B_{\mathrm{zr}}\)—total magnetic flux density in the rotor tooth;

\(B=B_1+B_2\)—total magnetic flux density in the air gap;

\(k_{01}\), \(k_{02}\), and \(k_{\mathrm{obg}}\)—winding factors of the armature winding of the exciter, of the generator, and of the excitation winding of the generator;

\[ \xi_1=\frac{A_1}{A},\quad \xi_2=\frac{A_2}{A},\quad b_1=\frac{B_1}{B},\quad b_2=\frac{B_2}{B} \]

—quantities characterizing the relative distribution of linear and magnetic loadings of the contactless generator 2;

\(k_{\mathrm{n}}\)—coefficient of current nonuniformity in the generator excitation winding;

\(k_{a1}\), \(k_{a2}\)—coefficients of reduction of the armature reaction of the exciter and generator;

\(k_\delta\), \(\delta\)—air-gap coefficient and the air gap of the machine;

\(k_{\mu1}\), \(k_{\mu2}\)—saturation coefficient of the exciter and generator;

\[ I_{1\mathrm{pr}}=0.33\cdot k_{\mathrm{n}} I_{\mathrm{vgpr}} \]

—the ratio of alternating and direct current in the conductor of the rotor winding;

\[ I_{1\mathrm{pr}}=\frac{I_{\Sigma\mathrm{pr}}}{3.19} \]

—the ratio of alternating and total current in the conductor of the rotor winding,

we obtain

\[ I_{\mathrm{pr}}=\frac{k}{\gamma_{zp}}, \qquad k_{\mathrm{n}}\cdot I_{\mathrm{vgpr}}=\frac{k_1}{\gamma_{zp}}, \]

\[ I_{\Sigma \mathrm{pr}}=\frac{1}{\gamma_{zp}}\sqrt{k^2+k_1^2}, \tag{8} \]

where

\[ k=\frac{2\beta_1\cdot k_{\mu 1}\delta\cdot k_\delta\cdot b_1\cdot B_{zp}\cdot p_1} {\mu_0\cdot m_1\cdot a_1\cdot a_2\cdot W_1\cdot k_{01}\cdot \left(0.9-5.68\xi_1\cdot k_{a2}\right)}, \]

\[ k_1= \frac{\beta_2\cdot k_{\mu 2}\cdot \delta\cdot k_\delta\cdot b_2\cdot B_{zp}\cdot p_2} {a_2\cdot p_2\cdot W_{\mathrm{vg}}\cdot k_{0\mathrm{vg}}-0.475\beta_2\cdot k_{02}\cdot \xi_2\cdot m_1\cdot a_1\cdot a_2\cdot W_1\cdot k_{a2}}. \]

These are coefficients dependent on the electromagnetic loadings of the contactless generator.

Substituting the value of the total current (8) in the conductor of the rotor winding into formula (6), we obtain the dependence of the rotor-slot area \(S_{\mathrm{pr}}\) on the electromagnetic loadings of the combined electric machines:

\[ S_{\mathrm{pr}}=\frac{k_2}{\gamma_{zp}}\sqrt{k^2+k_1^2}; \tag{9} \]

where

\[ k_2=\frac{2m_1\cdot a_1\cdot a_2\cdot W_1}{k_{\mathrm{zp}}\cdot j_{\mathrm{pr}}}. \]

Solving equations (3) and (9) with respect to \(\lambda_c\), we obtain:

\[ \lambda_c = k_D\cdot \left( 1-\frac{\pi a'}{2p_2\gamma_{zp}} \right) - \frac{2k_2\sqrt{k^2+k_1^2}} {\pi D_{\mathrm{n}}\cdot k_D(\gamma_{zp}-1)}. \tag{10} \]

Equation (10), \(\lambda_c=f_1(\gamma_{zp})\), for various values of the coefficient \(k_D\) and with constant values of the coefficients \(k\), \(k_1\), and \(k_2\), is shown by curves 1, 2, 3 in Fig. 1.

The optimal operating mode of the exciter is possible with the minimum total resistance of the exciter armature winding, i.e., with the optimal geometry of the rotor sheet and with equality of the generator field-winding resistance to the critical resistance of the exciter armature winding.

The total resistance of the exciter armature winding is equal to:

\[ z=\sqrt{r_1^2+x_1^2} = \sqrt{ \frac{N_1}{S_{\mathrm{pr}}} - N_2 \left[ \frac{S_{\mathrm{pr}}-h_2\cdot b_{\mathrm{pr}}\cdot z_p} {9z_p l_{\mathrm{pr}}^2} + \frac{h_2}{b_{\mathrm{pr}}} + \Sigma\lambda \right]^2 }, \tag{11} \]

where

\[ r_1 \]
— active resistance of the exciter armature winding;

\[ x_1 \]
— leakage reactance of the exciter armature winding;

\[ \Sigma\lambda \]
— specific magnetic permeance, independent of the geometry of the rotor slot;

\[ h_2 \]
— height of the rotor-slot wedge and thickness of the slot insulation;

\[ N_1= \frac{ 16m_1^2\cdot \rho_1^2 l^2 \left(1+\frac{l_1}{l}\right)^2 W_1^4 } {k_{\mathrm{zp}}^2}; \]

\[ \rho \]
— specific resistance of the winding;

\[ l_1 \]
— length of the end portion of the rotor winding;

\[ N_2 = 16\pi^2 \mu_0^2 f_1^2 l^2 \frac{W_1^4}{p_1^2 q_1^2}; \]

\(\mu_0\)—magnetic permeability of vacuum;
\(f_1\)—output frequency of the exciter;
\(q_1\)—number of slots per pole and phase of the exciter.

Performing simple transformations of equation (1), taking into account equations (3) and (4), we obtain:

\[ z_1 = \sqrt{ \frac{N_1} { \left[ \frac{\pi D_{\mathrm{H}}^2}{2} \, k_D \left( 1-\frac{1}{\gamma_{zp}} \right) (0.48 k_D-\lambda_c) \right]^2 } + N_2 \left\{ \frac{ [0.5D_{\mathrm{H}}(0.48k_D-\lambda_c)+8.63]\,z_p } { 9\pi D_{\mathrm{H}} \left( 1-\frac{1}{\gamma_{zp}} \right) } +\Sigma\lambda \right\}^2 }. \tag{12} \]

Equation (12) was investigated on the “Promin’-M” electronic digital computer for the minimum total impedance of the exciter armature winding \(z_1\) as a function of the value \(\lambda_c\), for various values of \(\gamma_{zp}\), \(k_D\), and \(z_p\), for a machine constructed with the following data: \(D_{\mathrm{H}}=24.5\ \text{cm}\), \(D=15\ \text{cm}\), \(D_0=4.8\ \text{cm}\), \(\delta=0.04\ \text{cm}\), \(k_\delta=1.585\), \(\alpha'=0.72\), \(p_1=6\), \(z_p=54\), \(p_2=3\), \(m_1=3\), \(W=90\), \(W_1=54\), \(k_{01}=0.83\), \(k_{02}=0.682\), \(k_{\mathrm{avg}}=0.735\), \(q_{\mathrm{p\text{-}r}}=0.00152\ \text{cm}\), \(a_1=6\), \(a_2=4\), \(a_3=9\), \(\beta_1=0.75\), \(k_{a1}=0.8\), \(k_{a2}=0.8\), \(k_{\mathrm{H}}=0.4\), \(B_{zp}=16{,}200\ \text{Gs}\), \(\beta_2=0.75\), \(k_{\mu1}=1.49\), \(k_{\mu2}=1.2\), \(\xi=0.316\), \(b_1=0.15\). From this investigation the dependence

\[ \lambda_c=f_2(\gamma_{zp}), \]

shown in Fig. 1 by curves 4, 5, 6, was obtained; the points of intersection of these with curves 1, 2, 3 determine the optimum values of the coefficient \(\gamma_{zp}\), making it possible to determine the optimum value of \(\lambda_c\) and the slot geometry

Fig. 1

according to formulas (2), (3), and (4). Analysis of Fig. 1 shows that, as the number of rotor slots \(z_p\) decreases and \(k_D\), and consequently \(\lambda_c\), increases—that is, as the pole pitch of the combined electric machines increases—the optimum value of the coefficient \(\gamma_{zp}\) decreases; consequently, the principal magnetizing forces of the exciter and the generator will correspondingly decrease.

Matching the load resistance of the exciter (the resistance of the generator excitation winding) with the critical resistance of the exciter armature winding is possible both by increasing the number of parallel branches with equal-pitch sections of the combined rotor winding 1, and by somewhat increasing \(\lambda_c\) at the optimum value of \(\gamma_{zp}\), since in this case the inductive resistance decreases, while the active resistance increases.

REFERENCES

1. V. T. Karavaev. Contactless synchronous machine. USSR Author’s Certificate No. 264521, 1970.

2. V. T. Karavaev. Principal dimensions and optimum electromagnetic loadings of a contactless synchronous generator with combined magnetic and electric circuits. Proceedings of the Gorky Polytechnic Institute, vol. 25, issue 15, 1969.