Frequency Response Basics

A circuit that contains a capacitor or an inductor treats different frequencies differently. Topic 8 gave us a single impedance $Z$; if we sweep the frequency we can watch $|Z|$ and the phase angle walk through a whole curve. That curve — the frequency response — is the behaviour pattern of every filter, amplifier stage, audio crossover, and signal-conditioning block in an AC system.

The transfer function H(jω)

Pick any two nodes in a linear circuit — call them input and output — and the ratio of their phasor voltages is the transfer function:

$H(j\omega)$ is a complex number, so it carries two pieces of information per frequency:

• Magnitude $|H(j\omega)|$: how big the output is compared to the input. 1.0 = same size; 0.1 = tenth; 0 = fully blocked.
• Phase $\angle H(j\omega)$: how much the output sinusoid is shifted relative to the input. Positive = leads; negative = lags.

Plot $|H|$ in dB against frequency on a log axis and you get a Bode plot. The straight-line slopes on a Bode plot are the visual shorthand every filter designer works in.

Two reflexes to build up front:

• 0 dB means the output is equal to the input (ratio = 1). Pass-band gain in a simple passive filter caps at 0 dB — you can attenuate, not amplify, without an active element.
• −3 dB means the output is $1/\sqrt{2} \approx 0.707$ of the input — half the power. The frequency where this happens is the cutoff (or corner) frequency, the universally-agreed edge of a filter's pass-band.

RC low-pass — the simplest filter there is

Two components, one node for the output:

The maths uses the complex voltage divider from Topic 8. With $V_\text{out}$ taken across the capacitor,

Magnitude:

At very low $\omega$, $\omega RC \ll 1$ and $|H| \to 1$ (unity pass). At very high $\omega$, $|H| \to 1/(\omega RC)$ — falling by a factor of 10 every time frequency rises 10×, which is exactly −20 dB/decade. The two asymptotes meet at the cutoff:

At $f_c$, $\omega RC = 1$, so $|H| = 1/\sqrt{2}$ = −3 dB. The canonical filter marker.

RC high-pass — swap the two components

Put the capacitor in series and the resistor to ground and everything inverts:

At low frequency the cap is an open, so nothing reaches the output. At high frequency the cap is a short and the full input lands on $R$. The cutoff is still at $f_c = 1/(2\pi RC)$; above it, the signal passes freely. The Bode plot is mirror-reflected: a rising asymptote at +20 dB/decade below the corner, flattening to 0 dB above.

RL filters — same idea, inductor instead of capacitor

The inductor behaves opposite to the capacitor, so:

• RL low-pass — inductor in series, R across the output. L looks like a short at DC (passes) and a big impedance at high frequency (blocks). Cutoff: $f_c = R/(2\pi L)$. Same −20 dB/decade roll-off.
• RL high-pass — R in series, L to ground. At DC the inductor is a short (no output); at high frequency its impedance is huge (all the signal lands on it). Cutoff: same $f_c = R/(2\pi L)$.

RC filters dominate because capacitors are cheaper and smaller than inductors, but RL filters are common in power-supply input stages and RF matching networks where the big inductor is already there for another reason.

Band-pass — the series RLC

Stack L and C in series, take the output across R, and you get a band-pass: signals away from $f_0$ are dropped across the LC branch; at $f_0$ the reactances cancel and the full input reaches the output. Topic 10 pulls on this thread at length — resonance is the peak of this curve.

Voltage ratios and dB in one sentence each

• Ratio → dB: $|H|_\text{dB} = 20 \log_{10}|H|$. A 10× ratio is 20 dB. A 100× ratio is 40 dB. Doubling is +6 dB; halving is −6 dB.
• −3 dB ≈ 0.707× voltage = 0.5× power. The convention on the border of the pass-band.
• −20 dB/decade = one factor-of-10 drop per decade of frequency. The universal first-order slope.
• −40 dB/decade comes from second-order filters (two energy-storage elements). Topic 10.

Common applications

• Audio tone controls. Tweeters live behind an RC high-pass; woofers live behind an RL low-pass. Pick the cutoff to split the spectrum cleanly around 2 – 4 kHz for a typical speaker crossover.
• Anti-aliasing filters. Every ADC needs an RC (or higher-order) low-pass ahead of it so frequencies above half the sample rate can't alias into the baseband.
• Coupling caps between amplifier stages. A series capacitor plus the downstream input impedance is an RC high-pass: blocks DC bias, passes the audio or video signal through.
• Power-supply ripple filters. An LC low-pass on the output of a rectifier shaves the 100 / 120 Hz ripple down to millivolt levels before feeding the downstream regulator.