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Every real electrical signal occupies a range of frequencies. Speech sits between roughly 300 Hz and 3.4 kHz; the audible bandwidth of music is 20 Hz to 20 kHz; AM radio broadcasts occupy 9 kHz channels around carriers between 540 kHz and 1.6 MHz; FM broadcasts occupy 200 kHz channels around carriers between 88 MHz and 108 MHz; Wi-Fi runs at 2.4 GHz and 5 GHz. Almost every electronic system needs to select a particular band of frequencies and reject the rest — to separate a wanted signal from interference, hum, hiss, or adjacent channels. The device that does this is a filter. The simplest filters consist of just a resistor and a capacitor connected in two arrangements: the low-pass filter, which passes signals below a cutoff frequency and attenuates higher frequencies; and the high-pass filter, which does the opposite. More elaborate filters combine these blocks with op-amps to give sharper roll-offs and to add band-pass and band-stop responses. This lesson develops the mathematics of first-order RC filters, derives the −3 dB cutoff frequency, introduces Bode plots in dB and log frequency, explains the universal −20 dB-per-decade roll-off, and surveys the active filters that extend these ideas to higher orders.
Spec mapping (AQA 7408 §3.13.3 — Analogue signal processing, option E, continued): This lesson covers passive RC low-pass and high-pass filters; the −3 dB cutoff frequency f_c = 1/(2πRC); the asymptotic gain and phase responses (Bode plots); the −20 dB/decade roll-off characteristic of a first-order filter; active filters using op-amps for sharper roll-offs and unity-gain pass-band response; band-pass and band-stop filter concepts; and the use of filters in audio, radio, and instrumentation signal chains. (Refer to the official AQA specification document for exact wording.)
Synoptic links: RC filters connect directly to the time-constant picture from Year 12 capacitor charging/discharging (§3.7.1) — the same RC product τ that sets the half-life of a discharging capacitor sets the cutoff frequency of a filter through f_c = 1/(2πRC). Active filters build on the op-amp topologies of Lesson 2 — placing a capacitor in the inverting feedback path produces an integrator (effectively a low-pass with unbounded DC gain) and the more elaborate Sallen–Key topology produces second-order active filters. Filters are also essential to the digital-signal lesson (Lesson 1) — every ADC needs an anti-aliasing low-pass filter and every DAC needs a reconstruction filter; the choice of filter order sets how close the sampling rate must sit above the Nyquist limit.
The canonical low-pass filter is a single resistor in series with a single capacitor, with the output taken across the capacitor:
V_in ──/\/\──┬─── V_out
R │
C
│
GND
At low frequency the capacitor's impedance Z_C = 1/(jωC) is very large; almost no current flows; V_out ≈ V_in (the filter passes the signal unchanged). At high frequency Z_C → 0; the capacitor short-circuits the output to ground; V_out → 0 (the filter blocks the signal). The cross-over happens at a characteristic frequency at which |Z_C| = R.
By the voltage-divider rule with complex impedances:
V_out / V_in = Z_C / (R + Z_C) = (1/jωC) / (R + 1/jωC) = 1 / (1 + jωRC)
The transfer function of the filter is:
H(ω) = 1 / (1 + jωRC)
Take the magnitude:
|H(ω)| = 1 / √(1 + (ωRC)²)
And the phase:
φ(ω) = −arctan(ωRC)
The cutoff frequency is defined as the frequency at which |H| has fallen by a factor of √2 from its DC value — that is, the gain has dropped by 3 dB. This corresponds to ωRC = 1, or equivalently:
f_c = 1 / (2πRC)
At f = f_c:
For f ≪ f_c: |H| ≈ 1, φ ≈ 0° — the filter is "transparent". For f ≫ f_c: |H| ≈ 1/(ωRC), φ ≈ −90°. The gain falls in inverse proportion to frequency.
In the high-frequency limit |H| ≈ 1/(ωRC). Expressed in decibels:
|H|_dB = 20 log₁₀ |H| = −20 log₁₀(ωRC)
So when ω increases by a factor of 10 (one decade), |H|_dB falls by 20 dB. This is the signature first-order roll-off: gain falls at exactly −20 dB per decade above cutoff. A second-order filter (two cascaded poles) gives −40 dB/decade; a fourth-order filter −80 dB/decade. The order of the filter sets how steeply it rejects out-of-band signals.
Question: Design a passive RC low-pass filter with a cutoff frequency of 1.0 kHz. Choose a sensible R and C.
Solution: f_c = 1/(2πRC) = 1000 Hz, so RC = 1/(2π × 1000) = 1.59 × 10⁻⁴ s. Choose R = 1.59 kΩ and C = 100 nF (RC = 1.59 × 10³ × 100 × 10⁻⁹ = 1.59 × 10⁻⁴ s ✓).
Equally valid: R = 16 kΩ, C = 10 nF; or R = 160 Ω, C = 1 μF. The choice within the constraint is governed by:
A typical sensible choice is R ≈ 1–10 kΩ with C chosen to match.
Swap R and C: take the output across the resistor instead of the capacitor.
V_in ──┤├──┬─── V_out
C │
R
│
GND
At low frequency Z_C is huge; almost no current flows; V_out ≈ 0 (the filter blocks DC and low frequencies). At high frequency Z_C → 0; the capacitor passes the signal to the resistor; V_out → V_in (the filter passes the signal).
By the voltage-divider rule:
V_out / V_in = R / (R + 1/(jωC)) = jωRC / (1 + jωRC)
H(ω) = jωRC / (1 + jωRC)
Magnitude: |H| = ωRC / √(1 + (ωRC)²)
Phase: φ = 90° − arctan(ωRC)
Same cutoff frequency formula:
f_c = 1 / (2πRC)
At f = f_c: |H| = 1/√2 = −3 dB, φ = +45° (output leads input by 45° — high-pass filters introduce phase advance rather than phase lag).
For f ≪ f_c: |H| ≈ ωRC, φ ≈ +90° — the gain falls into the stop-band at +20 dB/decade as you approach DC. For f ≫ f_c: |H| ≈ 1, φ ≈ 0° — the filter is transparent.
The most common use of a high-pass filter is AC coupling: blocking the DC component of a signal while passing the AC. An audio amplifier between stages often has a coupling capacitor that removes the DC bias of the previous stage but lets the audio signal through. A typical 1 μF coupling capacitor into a 10 kΩ input impedance has f_c = 1/(2π × 10⁴ × 10⁻⁶) = 16 Hz — comfortably below the audio band.
Question: An audio signal is coupled through a 0.10 μF capacitor into the 100 kΩ input impedance of the next stage. (a) Calculate the cutoff frequency. (b) Will the lowest audible frequency (20 Hz) be attenuated significantly?
Solution:
(a) f_c = 1/(2π × 10⁵ × 10⁻⁷) = 1/(2π × 10⁻²) = 15.9 Hz.
(b) At 20 Hz, ωRC = 20/15.9 = 1.26. |H| = 1.26/√(1 + 1.26²) = 1.26/1.61 = 0.78 — about 2 dB of attenuation. This is barely audible but a fastidious designer would drop f_c to 5 Hz by increasing C to 0.33 μF.
A Bode plot displays the frequency response of a filter (or any linear system) on log–log axes:
The advantage of these axes is that asymptotic behaviour becomes straight lines that are trivial to sketch.
|H|_dB
0 ━━━━━━━━━━┓
╲
╲ slope −20 dB/decade
╲
−20 ╲
╲
−40 ╲
╲
└─────────────────→ log f
f_c 10f_c 100f_c
Key features:
The mirror image: a straight line rising at +20 dB/decade up to f_c, then a flat pass-band at 0 dB above f_c.
Question: A low-pass filter has f_c = 1 kHz. (a) State the gain in dB at 10 kHz. (b) State the gain in dB at 100 kHz. (c) State the gain in dB at f = 0.1 kHz.
Solution:
(a) 10 kHz is one decade above f_c. The asymptotic stop-band slope is −20 dB/decade, so |H| = −20 dB (within rounding; the exact value is 20 log₁₀(1/√101) = −20.04 dB).
(b) 100 kHz is two decades above f_c, so |H| = −40 dB.
(c) 0.1 kHz is one decade below f_c — comfortably in the pass-band, where |H| ≈ 1 = 0 dB. (Exact: 20 log₁₀(1/√1.01) = −0.04 dB.)
graph LR
A["Input signal<br/>(broadband)"] --> B["Low-pass filter<br/>f_c = 1 kHz"]
B --> C["Wanted signal<br/>(0 - 1 kHz)"]
B --> D["Rejected:<br/>noise > 1 kHz"]
style C fill:#27ae60,color:#fff
style D fill:#e74c3c,color:#fff
Passive RC filters have two limitations: (i) the maximum roll-off of a single RC pair is −20 dB/decade, which is gentle; (ii) the filter's input impedance and output impedance vary with frequency, making cascading awkward. Active filters use op-amps to provide buffering, gain, and sharper roll-offs.
Take the inverting amplifier from Lesson 2 and replace the feedback resistor with a parallel combination of R_f and a capacitor C_f:
V_out / V_in = −(R_f / R_in) × 1/(1 + jωR_f C_f)
This is an inverting amplifier whose gain rolls off as a first-order low-pass with f_c = 1/(2πR_f C_f). It buffers the input (high input impedance) and the output (low output impedance), can have gain greater than 1, and is easily cascadable.
The Sallen–Key topology uses one op-amp configured as a non-inverting amplifier with two RC stages around it to give a second-order response (−40 dB/decade roll-off) from a single op-amp. AQA does not require derivation of the Sallen–Key transfer function but you should recognise the topology by name and know it as the standard second-order active filter.
A first-order filter rolls off at −20 dB/decade. To get from a 20 kHz cutoff to a 100× attenuation (−40 dB) requires the frequency to be 100× higher — at 2 MHz. A second-order filter rolls off at −40 dB/decade and reaches −40 dB only one decade above cutoff (200 kHz instead of 2 MHz). For an anti-aliasing filter feeding an ADC sampling at 48 kHz, a second-order or higher filter is essentially mandatory to avoid wasting most of the sample rate as guard band.
A band-pass filter passes signals within a band f_low to f_high and rejects all others. It can be built by cascading a high-pass (cutoff at f_low) with a low-pass (cutoff at f_high). In radio receivers, every tuned channel is selected by a band-pass filter centred on the carrier frequency.
A band-stop filter (or notch filter) rejects signals within a narrow band and passes everything else. The classic application is rejection of 50 Hz mains hum from an audio or biomedical signal: a narrow notch at exactly 50 Hz removes the mains pickup without significantly affecting wanted content above 100 Hz or below 30 Hz.
Question: An AM radio receiver needs to select a 9 kHz-wide audio channel centred at 1 MHz. Sketch the response of a band-pass filter that would do this, and state the two corner frequencies.
Solution: f_low = 1.000 − 0.0045 = 995.5 kHz; f_high = 1.000 + 0.0045 = 1004.5 kHz. The pass-band is the 9 kHz between these. The filter response is flat between f_low and f_high and rolls off above and below. A second-order band-pass is the textbook minimum; commercial AM receivers use ceramic filters of order 4 or higher to achieve sharper skirts.
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