Question

Consider the model of a small DC motor, where the following parameter values are assumed: R-10 L-10mH,J-0.01kgm0.0.05 2. ra a. Write down the transfer function of the system b. Choose a sample time for the system c. Find the pulse transfer function (use MATLAB 'c2d' command) d. Find the range of K for stability for the closed-loop sampled-data system

Answer 1

(a) The transfer function is between position and the voltage applied. The differential equations can be written as, Let the torque produced by motor be T and the Backemf and Torque constant are considered as of same values k. Then we can write,

di dt

In Laplace domain we can write,

$\\ (Js^2 + bs)\theta(s) = KI(s) \\ (Ls + R)I(s) = V(s) - K s \theta(s)$

Eliminating I(s) between the equations we get the transfer function as,

0 (s)

Substituting values we get,

θ(s) V(s) s(0.01s0.1) (0.01s +(0.05)s 0.0001s3 0.011s0.1025s 0.05 0.05

(b) Let us say that the sampling time is 0.1 seconds (10 Hz).

(c) The pulse transfer function will be,

0.01536220.01577z 0.0002003 23-1.358:2 0.3578: _ (1.67 10-5)

(d) Now the closed loop system will be stable for values of K such that the poles remain in left side of the s-plane (ie. the real part is negative). For that we can use the Root locus plot, which gives,

(MATLAB Code: rlocus(sys)) sys is the transfer function in discrete domain.

The plot is,

Root Locus Ğ侶凸 a N 2 0.5 -0.5 -1 1.5 2 -6 -4 2 Real Axis

Hence the range of K values should be greater than 106.