Compute the open loop DC gain if the closed loop transfer function is $\frac{2s+6}{2s^2+10s+14}$ with unity feedback factor.

Concept:

For unity negetive feedback

OLTF = CLTF/(1 - CLTF)

OLTF → Open Loop Transfer Function

CLTF → Closed Loop Transfer Function

To determine the DC gain of the open-loop transfer function we will substitute s =  0 in the expression and calculate the value.

Calculate:

GIven;

$\mathrm{CLTF}=\frac{2s+6}{2s^2+10s+14}$

Then

$\mathrm{OLTF}=\frac{\frac{2s+6}{2s^2+10s+14}}{1-\frac{2s+6}{2s^2+10s+14}}=\frac{2s+6}{2s^2+8s+8}$

Put s = 0

DC gain = 6/8 = 0.75 

Additional Information DC gain is the ratio of the magnitude of the response to the steady-state step to the magnitude of the step input.

The final value theorem demonstrates that DC gain is the value of the transfer function assessed at 0 for stable transfer functions.

DC gain is nothing but the error coefficient

For type 0 system : $K_p=\underset{s\to 0}{lim}G(s)$

For type 1 system : $K_v=\underset{s\to 0}{lim}sG(s)$

For type 2 system : $K_a=\underset{s\to 0}{lim}{s}^{2}G(s)$