Which value of s represents a damped sinusoid?

Which value of s represents a damped sinusoid?

• A. s = σ
• B. s = jω
• C. s = σ + jω
• D. s = 0

Answer: (C)

A damped sinusoid can be represented in terms of a complex frequency variable, which takes into account both the oscillatory and decaying nature of the signal. The variable s encapsulates these characteristics by incorporating both a real part and an imaginary part. In this context, s = σ + jω effectively represents a damped sinusoidal signal. Here, σ (the real part) represents the damping factor. When σ is positive, it indicates that the amplitude of the sinusoidal waveform is decreasing over time, which is characteristic of damping. The term jω (the imaginary part) represents the oscillation frequency, where ω is the angular frequency of the sinusoid. Thus, the combination σ + jω allows for the representation of a waveform that not only oscillates but also gradually decreases in amplitude—qualifying it as a damped sinusoid. This formulation arises commonly in the analysis of differential equations and control systems, where damping plays a crucial role in the behavior of dynamic systems. In contrast, having solely σ denotes no oscillation, solely jω indicates oscillation without decay, and a value of zero does not represent any signal characteristics relevant to damping.

A damped sinusoid can be represented in terms of a complex frequency variable, which takes into account both the oscillatory and decaying nature of the signal. The variable s encapsulates these characteristics by incorporating both a real part and an imaginary part.

In this context, s = σ + jω effectively represents a damped sinusoidal signal. Here, σ (the real part) represents the damping factor. When σ is positive, it indicates that the amplitude of the sinusoidal waveform is decreasing over time, which is characteristic of damping. The term jω (the imaginary part) represents the oscillation frequency, where ω is the angular frequency of the sinusoid.

Thus, the combination σ + jω allows for the representation of a waveform that not only oscillates but also gradually decreases in amplitude—qualifying it as a damped sinusoid. This formulation arises commonly in the analysis of differential equations and control systems, where damping plays a crucial role in the behavior of dynamic systems.

In contrast, having solely σ denotes no oscillation, solely jω indicates oscillation without decay, and a value of zero does not represent any signal characteristics relevant to damping.