First-order systems under sinusoidal excitation: When a sinusoidal input passes through a linear first-order system, how does the output amplitude and frequency change?

Introduction / Context: The frequency response of linear time-invariant (LTI) systems is foundational for control and signal processing. A first-order process behaves like a low-pass filter. Understanding what such a system does to a sinusoidal input—without solving differential equations every time—helps predict attenuation and phase lag across frequencies.

Given Data / Assumptions:

• System is linear and time-invariant, first order with transfer function form K / (1 + sτ) or similar.
• Input is sinusoidal at some angular frequency ω.
• Steady-state sinusoidal response is considered.

Concept / Approach: For LTI systems, a sinusoidal input at frequency ω produces a sinusoidal output at the same frequency, scaled in magnitude by |G(jω)| and shifted in phase by ∠G(jω). For a first-order low-pass, |G(jω)| = K / sqrt(1 + (ωτ)^2), which is ≤ K, showing attenuation that increases with ω. The frequency does not change; only amplitude is reduced and phase lags.

Step-by-Step Solution:

Verification / Alternative check: Bode plots of a first-order system show a 0 dB plateau at low ω and a −20 dB/decade roll-off past the corner at 1/τ, confirming attenuation at higher frequencies.

Why Other Options Are Wrong:

Common Pitfalls: Confusing transient behaviour with steady-state sinusoidal response; only amplitude and phase change at the same frequency in steady state.

Final Answer: Gets attenuated (magnitude decreases)