Frequency response and filters

Because the impedances of capacitors and inductors depend on frequency, a circuit containing them treats signals of different frequencies differently. Plotting how a circuit's output amplitude and phase vary with frequency, its frequency response, tells an engineer what the circuit will do to any signal, since any signal can be built from sinusoids. Circuits designed to pass some frequencies and reject others are filters, and they are everywhere: in audio crossovers, radio tuners, the anti-aliasing stage before every analog-to-digital converter, the power supplies of electronic equipment and the harmonic filters of industrial power systems.

This lesson defines the transfer function, introduces decibels and Bode plots, analyses first-order low-pass and high-pass filters, studies series and parallel resonance and band-pass behavior, and finishes with active filters built from op amps.

The transfer function

For a circuit with a sinusoidal input at angular frequency \(\omega\), the transfer function is the ratio of the output phasor to the input phasor:

Its magnitude is the gain at that frequency and its angle is the phase shift. Because the transfer function is computed with impedances, it is found with the same tools as in Lesson 3, typically a voltage divider. The frequency response is the pair of plots of magnitude and phase against frequency.

Decibels

Gains vary over many orders of magnitude, so they are expressed in decibels:

A gain of 10 is +20 dB, 100 is +40 dB, 0.1 is −20 dB, 1 is 0 dB, \(1/\sqrt2\) is −3.01 dB and 2 is +6.02 dB. The factor 20 (rather than 10) applies because power is proportional to the square of voltage, so 20 log of a voltage ratio equals 10 log of the corresponding power ratio. Decibels have a further advantage: the gains of cascaded stages multiply, so their decibel values add.

First-order low-pass filter

In a series RC circuit with the output taken across the capacitor, the voltage divider gives

At low frequency the capacitor is nearly an open circuit and the output equals the input; at high frequency it is nearly a short and the output falls toward zero. The cutoff (corner, or half-power) frequency \(\omega_c\), or \(f_c = 1/(2\pi RC)\), is where \(|H| = 1/\sqrt2\) (−3 dB) and the phase is −45°. The magnitude and phase are

Well above cutoff, \(|H| \approx \omega_c/\omega\): each tenfold increase in frequency (a decade) reduces the output tenfold, a slope of −20 dB per decade (equivalently −6 dB per octave, a doubling of frequency). An RL circuit with the output across the resistor gives the same response with \(\omega_c = R/L\).

Bode plots

A Bode plot draws the magnitude in decibels and the phase in degrees against frequency on a logarithmic axis. On these axes, the response of each factor of a transfer function is approximately a set of straight lines, and the plot of a product of factors is the sum of their individual plots. The asymptotic rules for the common factors are:

• A constant gain K: a horizontal line at \(20\log_{10}K\) dB, with zero phase (or 180° if K is negative).
• A simple pole \(1/(1 + j\omega/\omega_p)\): 0 dB below \(\omega_p\), then −20 dB/decade; the true curve is 3 dB below the corner. The phase goes from 0° to −90°, passing −45° at \(\omega_p\), and is approximated by a straight line from \(0.1\omega_p\) to \(10\omega_p\).
• A simple zero \((1 + j\omega/\omega_z)\): the mirror image, +20 dB/decade above \(\omega_z\) and phase rising to +90°.
• A factor \(j\omega\) (a zero at the origin): a straight line of +20 dB/decade passing through 0 dB at \(\omega = 1\), with a constant +90° phase.

High-pass filter

Swapping the resistor and capacitor, so the output is taken across the resistor, gives a high-pass filter:

It blocks DC and passes high frequencies, with a slope of +20 dB/decade below cutoff and a phase lead from +90° at low frequency to 0° at high frequency. A series capacitor feeding an amplifier's input resistance forms such a filter; it is used to block a DC offset while passing an AC signal (AC coupling), and its cutoff must be well below the lowest signal frequency of interest.

Resonance and band-pass filters

A circuit containing both inductance and capacitance can resonate: at one frequency the inductive and capacitive reactances are equal and cancel. For a series RLC circuit, \(\mathbf{Z} = R + j(\omega L - 1/\omega C)\), which is purely resistive and smallest at

At resonance the current is largest (\(V/R\)), and the voltages across L and C, equal and opposite, can be many times the source voltage. Taking the output across R gives a band-pass filter, whose gain is 1 at \(\omega_0\) and falls on either side. Its selectivity is described by the bandwidth \(B\), the width of the band between the two half-power frequencies, and the quality factor \(Q\):

A high-Q circuit is sharply selective; a low-Q circuit passes a broad band. For \(Q\) well above 1, the half-power frequencies lie approximately at \(\omega_0 \pm B/2\). A parallel RLC circuit behaves as the dual: its impedance is largest (purely \(R\)) at the same \(\omega_0\), with \(B = 1/(RC)\) and \(Q = \omega_0 RC\).

Active filters

Passive RC filters have two drawbacks: their gain cannot exceed 1, and loading the output changes the response. Adding an op amp (ELE-202) solves both. An inverting amplifier with a capacitor across its feedback resistor is a first-order active low-pass filter:

with a passband gain of \(-R_f/R_i\). The op amp's low output impedance means the next stage does not disturb the response, so filters can be cascaded freely, their decibel gains adding. Second-order active sections (such as the Sallen-Key circuit) produce −40 dB/decade, and cascades of such sections designed to standard responses give sharper filters: Butterworth (maximally flat passband), Chebyshev (steeper roll-off with passband ripple) and Bessel (best preserved waveform shape).

Filters are classified by what they pass: low-pass, high-pass, band-pass and band-stop (notch). A notch filter at 60 Hz, for example, removes power-line interference from biomedical signals such as electrocardiograms. A key application in every digital system is the anti-aliasing filter: before a signal is sampled at rate \(f_s\), components above \(f_s/2\) must be removed, or they fold back and appear as false lower frequencies (ELE-351).

Lesson 10 shows how transfer functions and their poles connect frequency response to the time-domain behavior of Lessons 1 and 2, and Lesson 12 designs and verifies a complete filter.