Question

Starting from the dynamic equations of the PM DC motor obtain the voltage-to-speed transfer function in s-domain. Draw the block diagram.

Answer 1

the armature current current controlled DC motor is intrinsically a velocity control scheme. In particular, consider the steady-state equations and the dependence of the speed ω upon the values of the inputs va and TL. Considering, for the sake of simplicity, the case F = 0,

it turns out:

ω ≈ 1 /KΦ Φ va − Ra/ (KΦ Φ)2 TL.

It can be seen that if the load torque TL is different from zero, a steady state error will be experienced on the system. Now, it can be seen that, the more KΦ Φ is high, the less the steady state error is large. Whence, it seems obvious to perform a control technique in which an additional speed feedback is performed. By means of a tachometer. Suppose the tachometer has a constant transfer function with a gain KT , then placing a gain KA in the direct branch. noticing that for the control scheme in Figure 16, it holds

va = KA (vr − KT ω),

then the following equation is obtained:

ω ≈ KA /KΦ Φ + KT KA vr − Ra /(KΦ Φ)(KΦ Φ + KT KA) TL

The transfer matrix of the system may be written as

ω(s) = [W1(s) W2(s)] [ va(s) TL(s) ],

where W1(s) := Ka KΦ Φ Km/ (1 + τa s)(1 + τm s) + Ka Km (KΦ Φ)2 ,

W2(s) := Km (1 + τa s) /(1 + τa s)(1 + τm s) + Ka Km (KΦ Φ)2

From the above equation, for sufficiently high values of KA, it turns out:

ω ≈ 1/ KT vr,

Block diagram: