Question

The frequency at which maximum AR occurred is called

• A. Corner frequency
• B. Cross over frequency
• C. Resonant frequency
• D. Cyclic frequency

Hint:

Resonance only occurs in second or higher-order systems with a damping ratio $\zeta \lt 0.707$.
First-order systems can never exhibit resonance because their AR decreases monotonically with frequency.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks for the terminology used to describe the specific frequency where the Amplitude Ratio (AR) of a system reaches its maximum peak value.
This behavior is associated with resonance, which is a key concept in frequency response analysis of dynamic systems.

Step 2: Key Formula or Approach:
For a second-order system, the Amplitude Ratio (AR) is given by: \[ AR = \frac{1}{\sqrt{(1 - \eta^2)^2 + (2\zeta\eta)^2}} \] where $\eta = \omega/\omega_n$ is the normalized frequency, and $\zeta$ is the damping ratio.
If $\zeta \lt 1/\sqrt{2} \approx 0.707$, the AR curve exhibits a peak at a specific frequency called the resonant frequency ($\omega_r$).
To find the resonant frequency, we differentiate the AR expression with respect to $\omega$ and set it to zero:
\[ \frac{d(AR)}{d\omega} = 0 \quad \implies \quad \omega_r = \omega_n \sqrt{1 - 2\zeta^2} \]

Step 3: Detailed Explanation:
The physical significance of this peak is that if the system is excited by a sinusoidal input of frequency $\omega_r$, the output will oscillate with maximum amplitude.
This phenomenon is called resonance, and the frequency $\omega_r$ is called the resonant frequency.
Let us review the other frequencies listed in the options to show why they are incorrect:
1. Corner frequency: The frequency where the low-frequency and high-frequency asymptotes of a Bode plot intersect ($\omega = 1/\tau$).
2. Cross over frequency: The frequency where the open-loop phase angle is $-180^\circ$ (phase crossover) or where the open-loop gain is 1 (gain crossover).
3. Cyclic frequency: Standard frequency measured in Hertz ($f = \omega / 2\pi$), unrelated to the peak of the AR curve.
Therefore, the peak of the AR curve uniquely defines the resonant frequency.

Step 4: Final Answer
The correct designation is the resonant frequency, corresponding to option (C).