Analogue Filters
Overview
The page covers the design and analysis of analogue filters. It includes a discussion on passive and active filters, low-pass and high-pass filters, and first and second-order filters. There are also dedicated child pages that cover Sallen-Key and voltage-controlled voltage-source (VCVS) filters.
Child Pages
📄️ Filter Tunings
Butterworth, Chebyshev, Bessel, elliptic, transfer functions, polynomials, equations, graphs and more information on analogue filter tunings.
📄️ Sallen-Key Filters
Schematics, equations, worked examples, calculators and more info on low-pass/high-pass Sallen-Key filters.
Passive vs. Active
Even in the pass-band, passive filters almost always increase the impedance of the signal, post filter. For a trace on a circuit board, this actually makes the post-filter trace more susceptible to picking up external noise. For this reason, when using a passive filter to filter out induced noise of a sensitive trace, always place a passive filter as close as possible to the receiving end of the signal (e.g. as close as possible to an ADC pin on a microcontroller).
Low-pass filters have an additional advantage when used on the analogue outputs from microcontrollers. If the DAC does not work properly for some reason (assuming you are using a DAC), you can sometimes implement the desired behaviour by using PWM instead. With the cut-off frequency set correctly, the PWM signal will be filtered so that a DC voltage proportional to the duty cycle remains, which is what you wished to implement with the DAC in the first place.
Active filters are electronic waveform filters which require their own power source (such as any filter powered with an op-amp), as opposed to passive filters (such as RC filters) which do not require an external power source. Active filters allow higher roll-of and better transfer characteristics than passive filters, but require more componentry and consume power.
Filter Topologies vs. Tunings
- Filter types describe the purpose of the filter. Filter types include low-pass, high-pass, band-pass, notch (band reject), and all-pass.
- Filter topologies define the what components go where. Filter topologies include
- Filter tunings define the values of the components in a particular topology. Filter tunings include Butterworth, Chebyshev and Bessel.
Filter Parameters
Cutoff Frequency (fc)
The cutoff frequency (a.k.a. corner frequency or break frequency) is the frequency which marks the transition from a pass band to a stop band. It marks the frequency at which the energy (whether it be voltage, current or both) stops being passed through and begins being blocked. For any real filter, there is a transition from the passband to the stopband, as so the cutoff frequency is usually defined at the "-3dB" point --- the point at which the signal degrades to -3dB (half power) of the nominal passband value.
The symbol is usually used to represent the cutoff frequency. Sometimes you may see instead.
Gain Factor (K)
At frequencies , the circuit multiplies the input signal by gain factor .
Component Spread
Component spread is a measure of ratio between the highest and lowest valued components required to construct a filter. Low component spread is a good property for a filter to have, as it aids manufacturability.
Filter Optimizations
Filter optimizations maximize a certain characteristic of a filter topology, such as maximum pass-band flatness or steepest roll-off. Butterworth, Chebyshev and Bessel are examples of filter optimizations.
1st Order Filters
First-Order Low-Pass RC Filter
Schematic
The below schematic shows a first-order low-pass RC filter consisting of a single series resistor and then a single capacitor to ground.
The low-pass RC filter lets through low frequencies but dampens high frequencies.
How To Choose R And C
The cut-off frequency is determined by both the value of the resistor and the value of the capacitor, and is equal to:
where:
is the resistance, in Ohms
is the capacitance, in Farads
is the cut-off frequency, in Hertz
As usual, the choice of and is a design decision which involves trade-offs. In terms of choosing :
- A resistance which is too small could draw too much current, either presenting too much load to the input, or overheating. It also could mean that the capacitor has to be very large and/or expensive to get the desired cut-off frequency.
- A resistance which is too large increases the output impedance of the filter, resulting in distortions if too much load is applied to . It also increases the noise susceptibility of the circuit.
Typically, a resistance between and is used. Then the capacitance is chosen to give the desired cut-off frequency.
Frequency Response
To plot the frequency response, we first need to find the transfer function for the low-pass RC filter. We can use the voltage divider rule to write as a function of :
where:
is the magnitude of the input signal at frequency , in Volts
is the impedance of the capacitor at frequency , in Ohms
is the resistance of the resistor, in Ohms
Recall that we can write in the Laplace domain as . Also, is just our transfer function :
Multiply top and bottom by to clean things up:
Now replace with so we can then take the magnitude:
Now take the magnitude (for more info on why and how to do this, see the What Are Transfer Functions, Poles, And Zeroes? page):
And let's find the phase response:
This shows us that an RC filter "delays" signals as they pass through. The higher the frequency, the greater the delay.
The following plot shows the frequency response (also known as a bode plot) of a low-pass filter, with values and . Magnitude is plotted in blue and phase in green.
![](/assets/images/rc-low-pass-filter-frequency-response-0e32ead6c72c13241e9bedbe6a45e5ec.png)
The cut-off frequency (also called the break frequency or turnover frequency1), is not the frequency at which all higher frequencies are stopped (remember, this is an ideal filter, but in real-life they always let through some fraction of the higher-frequencies). Instead, it is the frequency at where: