Solution 11 :

(a)

We first note the subtle point that the question talks of H(s) and H₁(s), thus implying the assumptions that the transfer function and the inverse function have Laplace transforms. Then we get the relation :

First, we note that the total transfer function is unity. This can also be noted from the fact that h (t) = u(t) and the laplace transform of u(t) is U(s) =1. Then we use the convolution property of the Laplace transform to write, 1 = U(s) = H(s) H₁(s) . Hence, H₁(s) = 1/H(s)

(b)

Since H₁(s) = 1/H(s) we see that the poles and zeroes will get interchanged. Or, we can deal with the given figure explicitly to note that H(s) is of the form where k is some arbitrary nonzero constant. Then we get H₁(s) to be The pole - zero plot of this function is given at the right.