Chapter 8: variable - reluctance machines and stepping motors  (Page 2/12)

$\left[\begin{array}{c}{\lambda }_1\\ {\lambda }_2\end{array}\right]=\left[\begin{array}{cc}{L}_{\text{11}}({\theta }_m)& L_{\text{12}}({\theta }_m)\\ L_{\text{12}}({\theta }_m)& L_{\text{22}}({\theta }_m)\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\end{array}\right]$ (8.1)

Here $L_{\text{11}}({\theta }_m)$ and $L_{\text{22}}({\theta }_m)$ are the self-inductances of phases 1 and 2, respectively, and $L_{\text{12}}({\theta }_m)$ is the mutual inductances. Note that, by symmetry

$L_{\text{22}}({\theta }_m)=L_{\text{11}}({\theta }_m-{\text{90}}^0)$ (8.2)

Figure 8.1 Basic two-phase VRMs: (a) singly-salient

and (b) doubly-salient.

Figure 8.2 Plots of inductance versus 0m for (a) the singly-salient

VRM of Fig. 8.1a and (b) the doubly-salient VRM of Fig. 8.1b.

• These inductances are periodic with a period of ${\text{180}}^0$ because rotation of the rotor through ${\text{180}}^o$ from any given angular position results in no change in the magnetic circuit of the machine.
• The electromagnetic torque of this system can be determined from the coenergy as

$T_{\text{mech}}=\frac{\partial W_{\text{fld}}^{\text{'}}(i_1,i_2,{\theta }_m)}{\partial {\theta }_m}$ (8.3)

where the partial derivative is taken while holding currents $i_1$ and $i_2$ constant. Here, the coenergy can be found,

$W_{\text{fld}}^{\text{'}}=\frac{1}{2}{L}_{\text{11}}({\theta }_m)i_1^2+L_{\text{12}}({\theta }_m)i_1{i}_2+\frac{1}{2}L{}_{\text{22}}({\theta }_m)i_2^2$ (8.4)

Thus, combining Eqs. 8.3 and 8.4 gives the torque as

$T_{\text{mech}}=\frac{1}{2}{i}_1^2\frac{{\text{dL}}_{\text{11}}({\theta }_m)}{{\mathrm{d\theta }}_m}+i_1{i}_2\frac{{\text{dL}}_{\text{12}}({\theta }_m)}{{\mathrm{d\theta }}_m}+\frac{1}{2}{i}_2^2\frac{{\text{dL}}_{\text{22}}({\theta }_m)}{{\mathrm{d\theta }}_m}$ (8.5)

• For the double-salient VRM of Fig. 8. lb, the mutual-inductance term ${\text{dL}}_{\text{12}}({\theta }_m)/{\mathrm{d\theta }}_m$ is zero and the torque expression of Eq. 8.5 simplifies to

$T_{\text{mech}}=\frac{1}{2}{i}_1^2\frac{{\text{dL}}_{\text{11}}({\theta }_m)}{{\mathrm{d\theta }}_m}+\frac{1}{2}{i}_2^2\frac{{\text{dL}}_{\text{22}}({\theta }_m)}{{\mathrm{d\theta }}_m}$ (8.6)

Substitution of Eq. 8.2 then gives

$T_{\text{mech}}=\frac{1}{2}{i}_1^2\frac{{\text{dL}}_{\text{11}}({\theta }_m)}{{\mathrm{d\theta }}_m}+\frac{1}{2}{i}_2^2\frac{{\text{dL}}_{\text{11}}({\theta }_m-{\text{90}}^0)}{{\mathrm{d\theta }}_m}$ (8.7)

• Equations 8.6 and 8.7 illustrate an important characteristic of VRMs in which mutual-inductance effects are negligible. In such machines the torque expression consists of a sum of terms, each of which is proportional to the square of an individual phase current. As a result, the torque depends only on the magnitude of the phase currents and not on their polarity. Thus the electronics which supply the phase currents to these machines can be unidirectional; i.e., bidirectional currents are not required.
• Since the phase currents are typically switched on and off by solid-state switches such as transistors or thyristors and since each switch need only handle currents in a single direction, this means that the motor drive requires only half the number of switches (as well as half the corresponding control electronics) that would be required in a corresponding bidirectional drive. The result is a drive system which is less complex and may be less expensive.
• The assumption of negligible mutual inductance is valid for the doubly-salient VRM of Fig.8.1b both due to symmetry of the machine geometry and due to the assumption of negligible iron reluctance. In practice, even in situations where symmetry might suggest that the mutual inductances are zero or can be ignored because they are independent of rotor position (e.g., the phases are coupled through leakage fluxes), significant nonlinear and mutual-inductance effects can arise due to saturation of the machine iron.
• At the design and analysis stage, the winding flux-current relationships and the motor torque can be determined by using numerical-analysis packages which can account for the nonlinearity of the machine magnetic material. Once a machine has been constructed, measurements can be made, both to validate the various assumptions and approximations which were made as well as to obtain an accurate measure of actual machine performance.
• The symbol Ps is used to indicate the number of stator poles and $P_r$ to indicate the number of rotor poles, and the corresponding machine is called a $P_s/P_r$ machine.
• Example 8.1 examines a 4/2 VRM.