ELECTROMECHANICAL MOTION DEVICES
Rotating Magnetic Field Based Analysis
3^rd Edition

Speed control is achieved by controlling the voltage $v_a$. This is generally accomplished by using a dc-to-dc converter. A two-quadrant chopper is shown in Figure 1. The chopper is used to control the applied voltage, where $$v_a = kv_s~~~~~~~~~~\text{(7)}$$ Here, $k$ is the duty cycle, which can be varied between zero and one.

Figure 1. Dc machine supplied by two quadrant dc-dc "chopper" converter

The acceleration characteristics with $T_L=0$ and $k = 0.6$ are shown below in Figure 2. The parameters of the machine are the following: $r_a=7~\Omega$, $L_{AA}=120$ mH, $k_v=1.41\times 10^{-2}$ V$\cdot$s/rad, and $J=1.06\times 10^{-6}$ kg$\cdot$m$^2$. In order to account for the no-load $i_a$ of 0.15 A at no-load speed $B_m$ was calculated to be $6.04\times 10^{-6}$ N$\cdot$m$\cdot$s.

Figure 2. Dc machine free acceleration.

Figure 2 above also shows the simulation of free acceleration of the permanent-magnet machine driven by a 20 kHz switching frequency of a two-quadrant chopper. This is superimposed on an average-value simulation response of the dc drive. They are essentially the same.

Example 3D. Torque Control. The parameters of a permanent-magnet dc machine are $V_a = 6$ V rated, $r_a = 7\,\Omega$, $k_v = 1.41\times 10^{-2}$ V$\cdot$s/rad, $L_{AA}=120$ mH, $J = 1.06\times 10^{-6}$ kg$\cdot$m$^2$, $B_m = 0$ N$\cdot$m$\cdot$s. We are to limit the torque $T_e^*$ to $0.423\times 10^{-2}$ N$\cdot$m or $I_a^* = 0.3$ A where the asterisk denotes commanded values. The control is shown symbolically in Fig. 3D-1.

Figure 3D-1. Torque Control Diagram

Since the current is controlled the electric dynamics are neglected only the mechanical dynamics are considered. The equations involved in Fig. 3D-1 are $$V_a = r_aI_a^*+\omega_rk_v.~~~~~~~~~~~~~~~~\text{(3D-1)}$$ $$T_e^* = J\frac{d\omega_r}{dt}+T_L~~~~~~~~~~~~~~~\text{(3D-2)}$$ where $T_e^* = k_vI_a^* = 0.423\times 10^{-2}$N$\cdot$m. The load line is $$T_L = K\omega_r^2~~~~~~~~~~\text{(3D-3)}$$ where $$K = 5.529\times 10^{-8}\,\text{N$\cdot$m$\cdot$s}^2~~~~~~~~~~\text{(3D-4)}$$ This intersects rated $V_a$ torque versus rotor speed plot at Operating Point 1 where $\omega_r = 276.6$ rad/s, as shown in Fig. 3D-2.

The dc machine is operating at Operating Point 1. The commanded torque is suddenly switched to $1/2$ the original value, which intersects the limiting torque $\left(I_a^*\right)$ at Operating Point 2 where $\omega_r = 195.6$ rad/s. The mechanical dynamics are $$T_e^* = J\frac{d\omega_r}{dt}+T_L~~~~~~~~~~\text{(3D-5)}$$ Assuming the torque control is functioning perfectly, the rotor slows and steady-state is reached at Operating Point 2. The voltage at Operating Point 2 is $$\begin{split} V_a & = r_aI_a^*+k_v\omega_r \\ & = (7)(0.15)+(1.41\times 10^{-2})(195.6)\\ & = 1.05+2.75 = 3.8\text{ V} \end{split}~~~~~~~~~~\text{(3D-6)}$$

The commanded torque $T_e^*$ is returned to the original value. The rotor speeds up and reaches steady-state at Operating Point 1. The trajectory from Operating Point 1 to Operating Point 2 and then back to 1 is shown in Fig. 3D-2.

Figure 3D-2. Drive operation during $\mathbf{T}^*_e$ switching