Question 10.3.5: Assuming that the motor is armature-controlled, draw the blo......
Assuming that the motor is armature-controlled, draw the block diagram of the speed-control system shown in Figure 10.3.4 and obtain its transfer functions.
The system inputs are the desired or required value ω_r of the load speed ω_L and the load torque T_L . The block diagram is shown in Figure 10.3.8 and was obtained by modifying the motor diagram in Figure 6.5.5 in Chapter 6.
The transfer functions can be obtained either by reducing the block diagram or by transforming the equations and eliminating all variables except the inputs and the output. The latter method is perhaps the easier one when the diagram has multiple inputs and loops containing comparators within them. Recalling that K_dK_{pot} = K_{tach}, we can write the following equation from the diagram
From the diagram we see that the motor torque is given by
T_m(s)=\frac{K_T}{L_a s + R_a}\left[V_m(s)-K_b \Omega_m(s)\right]Finally, we have \Omega_L (s) = \Omega_m(s)/N.
Using algebra to eliminate V_m(s), T_m(s), and \Omega_m(s) from these equations, we obtain the following transfer functions. To simplify the expressions, we have defined a new constant K_P = K_1K_aK_{tach}.
\frac{\Omega_L(s)}{\Omega_r(s)}=\frac{K_P K_T}{D(s)} (1)
\frac{\Omega_L(s)}{T_L(s)}=-\frac{\left(L_a s + R_a\right) / N}{D(s)} (2)
where the denominator is
D(s)=N L_a I_e s^2+N\left(R_a I_e+ c_e L_a\right) s+N R_a c_e+N K_T K_b+K_P K_T (3)
Using the parameter K_P , the block diagram can be simplified as shown in Figure 10.3.9. These transfer functions can be analyzed to determine whether or not the system is able to control the speed successfully and to determine an appropriate value for the gain K_1. Such analysis would consider the system’s stability, steady-state error, and transient response. We will illustrate this analysis in Sections 10.6, 10.7, and 10.8.
