Question

1. Set up each of the following systems in a form suited for plotting the root locus. Give the appropriate loop transfer function F(s) and the gain K in terms of the original parameters. a. Closed-loop characteristic equation (s+a) +b(s+2c)=0 with b as the parameter; b. Closed-loop transfer function T($)= with a as the parameter; s? +as+7+a b(s +2c) c. Open-loop transfer function G(s) = with c as the parameter. (s + a) 1

Answer 1

Solution:

(a)

divide by

This is the standard form for root locus:

$\Rightarrow 1+b\frac{(s+2c)}{(s+a)^{3}}=0$

The standard transfer function of a closed-loop system is:

G 1+G

is the given characteristic equation.

The open Loop transfer function

$G=\frac{(s+2c)}{(s+a)^{3}}$

Gain = b

(b)

$T(s)=\frac{1}{s^{2}+as+7+a}$

divide by

$\Rightarrow T(s)=\frac{\frac{1}{s^{2}+as+7}}{1+\frac{a}{s^{2}+as+7}}$

Std.Form:

$ch.eqn : 1+\frac{a}{s^{2}+as+7}=0$

The open Loop transfer function

$G=\frac{1}{s^{2}+as+7}$

Gain = a

(c)

$G = \frac{b(s+2c)}{(s+a)^3}$

The closed-loop transfer function is:

$T = \frac{G}{1+G}$

$T = \frac{ \frac{b(s+2c)}{(s+a)^3}}{1+ \frac{b(s+2c)}{(s+a)^3}}$

$\Rightarrow T=\frac{b(s+2c)}{(s+a)^{3}+b(s+2c)}$

$\Rightarrow T = \frac{ b(s+2c)}{(s+a)^3+ bs+2bc}$

Characteristic equation:

divide by $(s+a)^3+bs$

$1+c\frac{2b}{(s+a)^3+ bs} = 0$

Gain = c

Note: Hope you like the solution, kindly upvote.