GATE 2024 EC Q.12: solution and answer

Official answer: ASource: Official answer key, GATE 2024 EC (IISc Bengaluru)

1. Definitions. A decade is a frequency ratio of $10$ and an octave is a frequency ratio of $2$. On a Bode plot the horizontal axis is ${\log }_{10}\omega$, so a slope in dB per decade becomes dB per octave once you know how many octaves fit in one decade.

2. Octaves in one decade. $$\text{number of octaves}={\log }_{2}10=\frac{1}{{\log }_{10}2}=\frac{1}{0.301}\approx 3.32$$

3. Convert the slope. $$\frac{40\text{dB}}{\text{decade}}=\frac{40\text{dB}}{3.32\text{octaves}}\approx 12.04\text{dB/octave}\approx 12\text{dB/octave}$$ Put the other way round, one octave is ${\log }_{10}2\approx 0.301$ decade, so the change over one octave is $40\times 0.301\approx 12.04\text{dB}$.

4. Check with poles. A slope of $40\text{dB/decade}$ is what 2 poles (or 2 zeros) give well above their corner frequency. Each one adds $20{\log }_{10}2\approx 6.02\text{dB}$ per octave, so 2 of them add $2\times 6.02\approx 12\text{dB/octave}$.

5. Only option (A), $12\text{dB/octave}$, matches.