# Control Systems Root Locus - Control Systems

## What is Root Locus?

We can find the path of the closed loop poles in the root locus diagram. So that, we can find the nature of the control system. Here we can use an open loop transfer function to know more about the stability of the closed loop control system.

## What are the basics of Root Locus?

The Root locus is the locus of the roots of the characteristic equation by varying system gain K from zero to infinity.

However the characteristic equation of the closed loop control system is $1+G\left(s\right)H\left(s\right)=0$

We can represent G(s)H(s) as $G\left(s\right)H\left(s\right)=K\frac{N\left(s\right)}{D\left(s\right)}$

Where,

• K represents the multiplying factor
• N(s) represents the numerator term having (factored)nthorder polynomial of ‘s’.
• D(s) represents the denominator term having (factored)mthorder polynomial of ‘s’.

Now substitute,G(s)H(s)value in the characteristic equation. $1+k\frac{N\left(s\right)}{D\left(s\right)}=0$ $⇒D\left(s\right)+KN\left(s\right)=0$

Case 1 − K = 0

If $K=0$ then $D\left(s\right)=0$

Which means that the closed loop poles are equal to open loop poles when K is zero?

Case 2 − K = ∞

Now Re-write the above characteristic equation as $K\left(\frac{1}{K}+\frac{N\left(s\right)}{D\left(s\right)}\right)=0⇒\frac{1}{K}+\frac{N\left(s\right)}{D\left(s\right)}=0$ $\frac{1}{\mathrm{\infty }}+\frac{N\left(s\right)}{D\left(s\right)}=0⇒\frac{N\left(s\right)}{D\left(s\right)}=0⇒N\left(s\right)=0$

IfK=, thenN(s)=0. So that the closed loop poles are equal to the open loop zeros when K is infinity.

We can finalize that the root locus branches start at open loop poles and end at open loop zeros.

## What are Angle Condition and Magnitude Condition?

If the points on the root locus branches supports the angle condition. Here the angle condition is used to know whether the point exist on root locus branch or not. By using magnitude condition we can find that the value of K for the points on the root locus branches or not. So, use the magnitude condition for the points to satisfy the angle condition.

Characteristic equation of closed loop control system is $1+G\left(s\right)H\left(s\right)=0$ $⇒G\left(s\right)H\left(s\right)=-1+j0$

The phase angle of $G\left(s\right)H\left(s\right)$is

 $\mathrm{\angle }G\left(s\right)H\left(s\right)={\mathrm{tan}}^{-1}\left(\frac{0}{-1}\right)=\left(2n+1\right)\pi$1+G(s)H(s)=0

G(s)H(s)=1+j0

The angle condition is the point at which the angle of the open loop transfer function is an odd multiple of1800

Magnitude of $G\left(s\right)H\left(s\right)$is -$|G\left(s\right)H\left(s\right)|=\sqrt{\left(-1{\right)}^{2}+{0}^{2}}=1$

If the magnitude of the open loop transfer function is one then magnitude condition satisfied the angle condition.