Question

Bode diagram is generated from output response of the system subjected to _____________________ input

• A. Impulse
• B. Step
• C. Ramp
• D. Sinusoidal

Hint:

Think of a Bode analyzer, an instrument used to experimentally generate Bode plots. It works by injecting sine waves of varying frequencies into the system and measuring the magnitude and phase of the output signal at each frequency.

Answer 1

The Correct Option is D

Step 1: Understand the purpose of a Bode diagram.
A Bode diagram is a graphical representation of the frequency response of a linear time-invariant (LTI) system. It consists of two plots:
Magnitude plot: Shows how the magnitude of the system's transfer function (typically in decibels) varies with the frequency of the input signal (on a logarithmic scale).
Phase plot: Shows how the phase shift of the system's transfer function (in degrees or radians) varies with the frequency of the input signal (on a logarithmic scale).
The Bode diagram provides valuable information about the system's behavior at different frequencies, including its gain, phase margin, bandwidth, and stability. 
Step 2: Consider the input signal required to determine the frequency response.
The frequency response of a system describes how the system responds to sinusoidal inputs of different frequencies. By applying a sinusoidal input to a stable linear system, the steady-state output will also be a sinusoid of the same frequency but with a different magnitude and phase shift. The Bode diagram essentially plots this magnitude ratio (output amplitude / input amplitude) and phase shift as a function of the input frequency. Let the input be a sinusoidal signal \(u(t) = A \sin(\omega t)\). For a linear system with a transfer function \(G(s)\), the steady-state output \(y_{ss}(t)\) will be:
$$y_{ss}(t) = |G(j\omega)| A \sin(\omega t + \angle G(j\omega))$$ where \(|G(j\omega)|\) is the magnitude and \(\angle G(j\omega)\) is the phase of the frequency response at frequency \(\omega\), obtained by substituting \(s = j\omega\) into the transfer function \(G(s)\). The Bode magnitude plot is related to \(20 \log_{10} |G(j\omega)|\) versus \(\log_{10} \omega\), and the Bode phase plot is \(\angle G(j\omega)\) versus \(\log_{10} \omega\). 
Step 3: Evaluate the other input signals and their relation to frequency response.
Impulse input: The impulse response of a system \(h(t)\) is the output when the input is a Dirac delta function \(\delta(t)\). The Fourier transform of the impulse response, \(H(j\omega) = \mathcal{F}\{h(t)\} = G(j\omega)\), gives the frequency response. While the impulse response contains all frequency information, practically, Bode diagrams are constructed by analyzing the system's steady-state response to sinusoidal inputs at various frequencies. Step input: The step response of a system is its output when the input is a unit step function \(u(t) = U(t)\). The step response provides information about the transient behavior (rise time, settling time, overshoot) and low-frequency characteristics, but it doesn't directly reveal the system's response across a wide range of frequencies in the same straightforward manner as sinusoidal inputs. Ramp input: The ramp response is the output for an input \(u(t) = t U(t)\). Similar to the step response, it primarily provides information about the system's steady-state error for a ramp input (velocity lag constant) and doesn't directly give the frequency response. 
Step 4: Conclude the type of input used to generate a Bode diagram.
The Bode diagram is fundamentally a plot of the system's frequency response, which is determined by examining the system's steady-state output when subjected to sinusoidal inputs over a range of frequencies.