K=520 A=104 B=-64 C=-6656

H=1000V/m

Max input X=3.5mm

Current rating of amplifier=0.8 A continuous and 6A for max 10 ms impulse

First we plot the rot locus of the plant transfer function to have a deep sight into the system behavior, the Matlab code used is given below which plots the root locus of the plant transfer function.

Matlab code for s plane root locus:

*% Given data or variables*

*A = 104;*

*B = -64;*

*C = -6656;*

*K = 520;*

*num=K;*

*den=[1 A B C];*

*%open loop transfer function*

*open_loopTF=tf(num,den)*

*figure;*

*rlocus(open_loopTF)*

*xlabel(**'real axis'**)*

*ylabel(**'imaginary axis'**)*

*title(**'root locus for open loop transfer function'**)*

*% closed loop transfer function*

*Figure;*

*H=1000;*

*CL=feedback(open_loopTF,C)*

*rlocus(CL)*

*xlabel(**'real axis'**)*

*ylabel(**'imaginary axis'**)*

*title(**' root locus for closed loop transfer function'**)*

Both open loop and closed loop systems are unstable. The open loop has one root on the left hand plane. Also the closed loop has one pole on the left hand plane and by varying ** k** we don’t get a system stable.

The system cannot be stabilized by a pure gain compensator. The root locus of the plant function is given in the figure which clearly shows that the pole lies in the right plane and the system is unstable. From the root locus we observe that by increasing the gain the system can be marginally stable. This may be further clarified with the help of Bode plot. From Bode plot the phase of the system response is beyond the stability limits. So it clearly indicates that we have to include a phase lead to get the stability. Only gain has no effect over the phase.

**Figure 1: root locus of open loop transfer function**

**Figure 2. root locus with feedback**

**Figure 3. Bode plot ot the system**

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