GATE 2022 EE Q.11: solution and answer

Official answer: B or DSource: Official answer key, GATE 2022 EE (IIT Kharagpur)

1. Stability. The denominator $s^2+Cs+D$ has $C>0$ and $D>0$. For a second-order polynomial, positive coefficients put both roots in the left half-plane, so $H(s)$ is stable and its frequency response $H(j\omega )$ is meaningful.

2. Low frequency. $H(j0)=\frac{B}{D}$, which is finite and non-zero because $B,D>0$.

3. High frequency. The numerator has degree 1 and the denominator degree 2, so $|H(j\omega )|\to 0$ as $\omega \to \infty$ (for large enough $\omega$ it falls off as $A/\omega$).

4. High-pass is impossible. A high-pass filter must pass high frequencies: $|H(j\omega )|$ has to approach a non-zero value as $\omega \to \infty$. Here $|H(j\omega )|\to 0$ for every choice of positive $A,B,C,D$, so this system cannot be a high-pass filter. That is option (B).

5. Low-pass is possible. Example: $A=0.01$, $B=1$, $C=1.4$, $D=1$ gives $|H|=1.00$ at $\omega =0$, $0.71$ at $\omega =1$ and $0.010$ at $\omega =10$.

6. Band-pass is possible. Example: $A=1$, $B=0.01$, $C=0.2$, $D=1$ gives $|H|=0.010$ at $\omega =0$, $5.0$ at $\omega =1$ and $0.010$ at $\omega =100$.

7. About (D). An ideal integrator is $K/s$, with a pole at $s=0$. $H(s)$ has no pole at the origin, because $D>0$ keeps the constant term of the denominator non-zero, so it is never an ideal integrator. It can behave like one — magnitude falling at 20 dB per decade with phase near $-90^{\circ }$ — over a limited band of frequencies, but not at all frequencies.

8. The answer is (B). The official key accepts both (B) and (D).