Magnitude at the critical (cutoff) frequency of first-order filters Is the output voltage equal to 63.3% of the maximum at the critical frequencies, or is a different percentage correct for the −3 dB point?

Introduction / Context:
Critical (cutoff) frequencies for first-order low-pass or high-pass filters are commonly defined at the −3 dB point. Remembering the exact amplitude ratio prevents errors when reading Bode plots or specifying bandwidths.

Given Data / Assumptions:

• First-order filter (RC or RL), linear, sinusoidal steady state.
• Critical frequency fc = 1 / (2 * pi * τ) where τ is the time constant.
• At fc, the magnitude response is down by 3 dB relative to the passband.

Concept / Approach:

At −3 dB, the ratio of output to maximum magnitude is 1 / sqrt(2) ≈ 0.707 (70.7%). The number 63.3% is associated with time-domain exponential settling (1 − e^(−1)) at one time constant, not with frequency-domain cutoff.

Step-by-Step Solution:

Verification / Alternative check:

Time-domain vs frequency-domain: at t = τ after a step input to a first-order low-pass, the output reaches 63.2% (often rounded 63.3%) of its final value. This figure is unrelated to the −3 dB amplitude in the steady-state sinusoidal frequency response.

Why Other Options Are Wrong:

• “True” variants conflate time-constant settling with −3 dB amplitude.
• The RC vs RL distinction is irrelevant; both first-order filters share the same normalized magnitude at cutoff.
• Current vs voltage does not change the 1 / sqrt(2) relationship for the transfer variable.

Common Pitfalls:

Memorizing 63% without context. Always separate time-constant step responses (63.2% at t = τ) from frequency response cutoffs (70.7% at −3 dB).

Final Answer:

False