Question 10.3.4: Derive the transfer functions of the system shown in Figure ......
Derive the transfer functions of the system shown in Figure 10.3.6 for the case where the field time constant L_f /R_f is small.
If T_L = 0, Figure 10.3.6 shows that the transfer function Ω_m(s)/V_m(s) is
\frac{\Omega_m(s)}{V_m(s)}=\frac{K_T}{\left(L_f s+R_f\right)\left(I_e s+c_e\right)}=\frac{K_T / R_f c_e}{\left(\tau_f s+1\right)\left(\tau_m s+1\right)} (1)
where τ_f = L_f /R_f , which is called the field time constant, and τ_m = I_e/c_e, which is the mechanical time constant. Quite often τ_f \ll τ_m, in which case equation (1) reduces to a first-order model (by setting τ_f = 0 in equation (1)):
\frac{\Omega_m(s)}{V_m(s)}=\frac{K_T / R_f c_e}{\tau_m s+1}=\frac{K_T / R_f}{I_e s+c_e}
The approximation τ_f \ll τ_m is equivalent to setting L_f = 0. If this approximation is valid, equations (1) and (2) of Example 10.3.3 reduce to the following first-order models:
\frac{\Omega_L(s)}{\Omega_r(s)}=\frac{K_P K_T}{N R_f I_e s+N c_e R_f+K_P K_T}
\frac{\Omega_L(s)}{T_L(s)}=-\frac{R_f / N}{N R_f I_e s+N c_e R_f+K_P K_T}
These correspond to the block diagram shown in Figure 10.3.7a. Moving the factor 1/N gives the diagram shown in part (b). Note that we may thus absorb the factor NK_T /R_f into the controller gain K_P. Part (c) shows the case where the control algorithm is some arbitrary transfer function G_c(s), where I = N^2 I_e and c = N^2c_e. We will use this simplified diagram quite often in discussing the use of various control algorithms in Sections 10.6 and 10.7.
