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Frequency response, RC filters, Bode plots and resonance

The frequency response function, RC low-pass and high-pass filters, decibels and Bode plots, higher-order filters, resonance and band-pass.

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A circuit with capacitors and inductors responds differently to different frequencies: it lets some components of a signal through and attenuates others. This is the principle of filters, found in every electronic device: they separate signal from noise, remove 50 Hz hum, select a radio station, protect a converter from aliasing. In this lesson we cover the frequency response, Bode plots, first-order filters and resonance.

Frequency response

With the phasor method, the ratio of a circuit’s output phasor to its input phasor is a function of angular frequency:

    \[ H(j\omega) = \frac{\bar V_{out}}{\bar V_{in}} \]

It is the Laplace transfer function evaluated at s = j\omega (Mathematical Methods lesson). The magnitude |H| says how much each frequency is amplified or attenuated, the argument by how much it is phase-shifted.

The RC low-pass filter

With the output taken across the capacitor, by the impedance divider:

    \[ H(j\omega) = \frac{1/(j\omega C)}{R + 1/(j\omega C)} = \frac{1}{1 + j\omega RC} = \frac{1}{1 + j\,\omega/\omega_0},\qquad \omega_0 = \frac{1}{RC} \]

    \[ |H| = \frac{1}{\sqrt{1 + (\omega/\omega_0)^2}},\qquad \arg H = -\arctan\frac{\omega}{\omega_0} \]

  • at low frequency (\omega \ll \omega_0) the capacitor is almost an open circuit: |H| \approx 1, the signal passes;
  • at high frequency the capacitor is almost a short circuit: |H| \approx \omega_0/\omega, the signal is attenuated;
  • at the cut-off frequency \omega_0, |H| = 1/\sqrt2 and the phase shift is −45°.

Example. With R = 1.6 kΩ and C = 100 nF, f_0 = \frac{1}{2\pi RC} \approx 1 kHz.

Bode plots

The magnitude is plotted in decibels, 20\log_{10}|H|, together with the phase, both against frequency on a logarithmic scale. Logarithmic scales turn products into sums, and the plots of complex circuits are built by adding those of their parts.

Bode magnitude and phase plots of an RC low-pass filter
The magnitude is flat at 0 dB up to the cut-off frequency, where it is −3 dB, then falls by 20 dB per decade. The phase goes from 0° to −90°, with −45° at the cut-off frequency. The dashed lines are the asymptotes.

For the first-order low-pass:

  • below \omega_0 the magnitude is about 0 dB;
  • above \omega_0 it falls by 20 dB per decade (at ten times the frequency the signal is ten times smaller);
  • at \omega_0 the error with respect to the asymptotes is largest, −3 dB (half the power).

Decibels are also handy because chains of stages add up: a +40 dB amplifier followed by a filter attenuating 20 dB gives +20 dB overall.

The high-pass

Swapping R and C (output across the resistor):

    \[ H(j\omega) = \frac{j\omega RC}{1 + j\omega RC} \]

It blocks DC and low frequencies and passes high ones, with a slope of +20 dB/decade below cut-off. It is the coupling capacitor between two amplifier stages: it passes the audio signal and blocks the DC bias voltage. It must be sized so the cut-off lies below the lowest frequency to be passed (20 Hz for audio).

Higher-order filters

A first-order filter separates two nearby frequencies poorly: 20 dB per decade is not much. Cascading stages, or using inductors and capacitors together, gives filters of order n with a slope of 20n dB/decade. The classic families shape the response with different trade-offs:

  • Butterworth: the flattest possible passband response;
  • Chebyshev: a steeper transition, with ripple in the passband;
  • Bessel: nearly linear phase, so the waveform is preserved (suited to pulses).

With operational amplifiers, active filters without inductors are built, and that is how almost all filters below a few megahertz are made.

Resonance

In the series RLC circuit the impedance is R + j(\omega L - 1/(\omega C)): at the resonant frequency \omega_r = 1/\sqrt{LC} the reactances cancel, the impedance is smallest and the current largest (Physics 2 lesson on AC). Taking the output across the resistor gives a band-pass, centred on \omega_r, with bandwidth \Delta\omega = R/L and quality factor Q = \omega_r/\Delta\omega. The parallel LC circuit does the opposite: at resonance its impedance is largest, and it is the tuning circuit of oscillators and radio receivers.

Filters and signals

By superposition, a filter treats each Fourier component of a signal separately. A 1 kHz square wave through a low-pass with a 1 kHz cut-off loses most of its high harmonics (3, 5, 7 kHz…) and becomes almost a sine wave: the RC circuit that “rounds” the edges in the lesson on transients is the same phenomenon seen in frequency.

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