Question

Select the correct statement from the following:

• A. The frequency response of a pure capacity process is unbounded
• B. The phase lag of a pure time delay system decrease with increasing frequency
• C. The amplitude ratio of a pure capacity process is inversely proportional to the frequency
• D. The amplitude ratio of a pure time delay system increases with frequency

Hint:

For a pure capacity process: - \(\text{AR} = \frac{K}{\omega}\) (Inversely proportional to \(\omega\)) - \(\phi = -90^\circ\) (Constant phase lag across all frequencies)

Answer 1

The Correct Option is C

Concept: To determine the frequency response behavior of dynamic operations, we analyze their core transfer functions by setting \(s = j\omega\). From the resulting complex values, we calculate the Amplitude Ratio (AR) and Phase Angle (\(\phi\)).

Step 1: Analyzing a pure capacity process.
A pure capacity (or pure integrator) process has a standard transfer function defined as: \[ G(s) = \frac{K}{s} \] Substituting \(s = j\omega\): \[ G(j\omega) = \frac{K}{j\omega} = -j\frac{K}{\omega} \] Calculating the Amplitude Ratio (AR), which is the absolute magnitude of the complex expression: \[ \text{AR} = |G(j\omega)| = \sqrt{0^2 + \left(-\frac{K}{\omega}\right)^2} = \frac{K}{\omega} \] From this derivative, it is clear that \(\text{AR} \propto \frac{1}{\omega}\). Thus, the amplitude ratio of a pure capacity process is explicitly inversely proportional to the frequency, making Statement (C) absolutely correct.

Step 2: Checking alternative statements for confirmation.

• Pure time delay process: Represented as \( G(s) = e^{-\tau_d s} \). Substituting \(s = j\omega\) gives \( G(j\omega) = e^{-j\omega\tau_d} \).

• The magnitude is \(\text{AR} = |e^{-j\omega\tau_d}| = 1\) (constant for all frequencies, disproving statement D).

• The phase angle is \(\phi = -\omega\tau_d\). As frequency \(\omega\) increases, the phase lag becomes increasingly negative (it increases in magnitude, disproving statement B).