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A filter's transfer function H(jω) = V_out/V_in is a ratio of phasors that characterizes frequency response. The magnitude |H(jω)| and phase ∠H(jω) show which frequencies are passed or attenuated. Passive filters (built with R, L, C) have transfer functions that are ratios of polynomials in jω, leading to characteristic rolloff rates.
From your work with phasor-domain KVL, you know how to write voltage divider expressions with complex impedances: V̅_out = V̅_in · Z₂ / (Z₁ + Z₂). The transfer function H(jω) = V̅_out / V̅_in is exactly that ratio — but instead of thinking of it as a number for one particular frequency, you treat it as a function of ω and ask how the circuit responds across all frequencies. This is the conceptual shift from phasor analysis (one frequency at a time) to filter analysis (all frequencies simultaneously).
Consider a simple RC low-pass filter: a resistor R in series with the input, a capacitor C to ground, with the output taken across the capacitor. The capacitor's impedance is Z_C = 1/(jωC). Writing the voltage divider: H(jω) = Z_C / (R + Z_C) = 1 / (1 + jωRC). Now compute the magnitude: |H(jω)| = 1 / √(1 + (ωRC)²). At low frequencies (ω → 0), the denominator approaches 1, so |H| ≈ 1 — the signal passes through unchanged. At high frequencies (ω → ∞), the denominator grows without bound, so |H| → 0 — the signal is blocked. The transition happens around the cutoff frequency ω_c = 1/RC, where |H| = 1/√2 ≈ 0.707, which corresponds to a −3 dB attenuation. The capacitor acts like an open circuit at low ω (blocks DC... but wait, at DC, Z_C → ∞, so the output equals the input!) and a short circuit at high ω (shunting the signal to ground).
The phase is ∠H(jω) = −arctan(ωRC). At low frequencies the phase shift is near zero; at the cutoff frequency it is −45°; at very high frequencies it approaches −90°. Phase shift matters because it represents a time delay — a sinusoid at the output lags behind the input. For audio applications this is often acceptable; for control systems the accumulated phase shift can cause instability. Understanding both magnitude and phase is essential for using filters in larger systems.
For passive filters, the rolloff rate beyond the cutoff frequency is determined by circuit order. A first-order RC or RL filter rolls off at −20 dB/decade (the magnitude halves every time frequency doubles). A second-order RLC filter rolls off at −40 dB/decade and can exhibit resonance: the denominator polynomial has complex roots, causing the magnitude to peak near the resonant frequency ω₀ = 1/√(LC) before falling. By combining filter stages or using higher-order RLC networks, you can build steeper rolloffs. The transfer function's polynomial structure — specifically the locations of its poles and zeros in the complex plane — fully predicts these behaviors, connecting passive filter analysis directly to the poles-and-zeros framework you'll use for more general system analysis.