ANALOGUE FILTERS
Analogue Filters
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1. Overview
1.1. Passive vs. Active
Even in the passband, passive filters almost always increase the impedance of the signal, post filter. For a trace on a circuit board, this actually makes the postfilter 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).
Lowpass 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 cutoff 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 opamp), as opposed to passive filters (such as RC filters) which do not require an external power source. Active filters allow higher rollof and better transfer characteristics than passive filters, but require more componentry and consume power.
1.2. Filter Topologies vs. Tunings

Filter topologies define the what components go where.

Filter tunings define the values of the components in a particular topology. Filter tunings include Butterworth, Chebyshev and Bessel.
1.3. Filter Parameters
1.3.1. 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 \(f_c\) is usually used to represent the cutoff frequency. Sometimes you may see \(f_{3dB}\) instead.
1.3.2. Gain Factor (K)
At frequencies \(f << f_c\), the circuit multiplies the input signal by gain factor K
.
1.3.3. 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.
1.3.4. Filter Optimizations
Filter optimizations maximize a certain characteristic of a filter topology, such as maximum passband flatness or steepest rolloff. Butterworth, Chebyshev and Bessel are examples of filter optimizations.
2. 1st Order Filters
2.1. FirstOrder LowPass RC Filter
2.1.1. Schematic
Figure 1 shows a firstorder lowpass RC filter consisting of a single series resistor and then a single capacitor to ground.
The lowpass RC filter lets through low frequencies but dampens high frequencies.
2.1.2. How To Choose R And C
The cutoff frequency is determined by both the value of the resistor and the value of the capacitor, and is equal to:
where:
\(R\) is the resistance, in Ohms
\(C\) is the capacitance, in Farads
\(f_c\) is the cutoff frequency, in Hertz
As usual, the choice of \(R\) and \(C\) is a design decision which involves tradeoffs. In terms of choosing \(R\):

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 cutoff frequency.

A resistance which is too large increases the output impedance of the filter, resulting in distortions if too much load is applied to \(V_{out}\). It also increases the noise susceptibility of the circuit.
Typically, a resistance between \(1k\Omega\) and \(100k\Omega\) is used. Then the capacitance is chosen to give the desired cutoff frequency.
2.1.3. Frequency Response
To plot the frequency response, we first need to find the transfer function for the lowpass RC filter. We can use the voltage divider rule to write \(v_{out}\) as a function of \(v_{in}\):
where:
\(v_{in}\) is the magnitude of the input signal at frequency \(f\), in Volts
\(X_C\) is the impedance of the capacitor at frequency \(f\), in Ohms
\(R\) is the resistance of the resistor, in Ohms
Recall that we can write \(X_C\) in the Laplace domain as \(X_C = \frac{1}{sC}\). Also, \(\frac{v_{out}}{v_{in}}\) is just our transfer function \(H(s)\):
Multiply top and bottom by \(\frac{1}{sC}\) to clean things up:
Now replace \(s\) with \(j\omega\) 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 lowpass filter, with values \(R = 1k\Omega\) and \(C = 1\mu F\). Magnitude is plotted in blue and phase in green.
The cutoff frequency (also called the break frequency or turnover frequency[1]), \(f_c\) is not the frequency at which all higher frequencies are stopped (remember, this is an ideal filter, but in reallife they always let through some fraction of the higherfrequencies). Instead, it is the frequency at where:
The choice of resistor and capacitor above gives a cutoff frequency of \(f_c = 159Hz\).
Lowpass RC filters are typically used for applications up to 100kHz, above 100kHz RLC filters are used[2].
2.1.5. Typical Uses
The lowpass RC filter is one (if not) the most commonly used filters on circuit board designs. Its popularity results from it’s simplicity (two passive components), low cost (one resistor, one capacitor), small size, and it’s myriad of uses.
Due to the presence of the resistor, it is a lossy filter, and therefore not suited for highpower applications (use a lowpass LC filter instead).
The lowpass RC filter can be used to provide filtering on analogue inputs to a microcontroller before being sampled by the ADC. One example could be to filter the output of an analogue temperature sensor. Note that is normally advantageous to place the filter as close as possible to the microcontroller, rather than close to the sensor producing the voltage. This is because the series resistor of the RC filter increases the source impedance of the analogue signal, making the PCB track less immune to noise once it passes through the resistor.
Another way to reduce the reduction in noise immunity due to the resistor in the RC lowpass filter is to make the capacitor as large as practically possible (for a particular cutoff frequency). Both the resistance and the capacitance influence the cutoff frequency. If you increase the capacitance by 10x, and reduce the resistance by 10x, you get the same cutoff frequency, but far better noise immunity since the source impedance is not altered as much.
Another consideration is the effect of the increase in source impedance (due to the resistor in the RC filter) when connecting the output to something like a microcontroller ADC). The input impedance of an nonbuffered ADC pin on a microcontroller is usually somewhere between \(20500k\Omega\) (note that this is usually variable, and can change with sampling rate). This will form a resistor divider with the RC filter resistance, increasing the ADC measurement error. As a general rule, you want the RC filter resistance to be much lower than the ADC input impedance.
A RC filter resistance which is at least 50x lower than the ADC input impedance is acceptable in most cases. For a standard ADC input impedance of \(50k\Omega\), this means that the resistor in the RC filter should be no more than \(1k\Omega\).
2.1.6. Transient Response
The equation for the voltage across the capacitor is:
where:
\(V_c\) = voltage across the capacitor, Volts
\(V_s\) = supply voltage, Volts
\(t\) = time since supply was turned on, Seconds
\(R\) = resistance, Ohms
\(C\) = capacitance, Farads
This equation can be rearranged to find the time \(t\), and which the capacitor is at a certain voltage:
This form of the equation can be useful to calculate the delay (aka the time \(t\)), that the RC circuit will provide before something happens.
2.2. Building A VDAC From An ADC And Lowpass RC Filter
Lowpass RC filters can also be used to create a VDAC (voltagebased digitaltoanalogue converter) from a PWM signal. This is useful since many microcontrollers have one (or more) PWM peripherals, but rarely a builtin VDAC. A simple RC filter placed on the output pin of the PWM signal can convert it into a VDAC, in where the duty cycle determines the analogue voltage output.
2.3. LowPass LC Filter
The basic lowpass LC filter consists of a single inductor and capacitor.
Unlike the lowpass RC filter, the lowpass LC filter is theoretically lossless. This means that it does not dissipate energy as heat. However, realworld inductors have coil resistance and magnetic losses, as well as ESR in the capacitor. Also, the presence of the inductance usually makes the LC filter larger and more expensive than the RC filter.
This makes an LC lowpass filter suitable for higherpower applications. You will see LC lowpass filters being used on the output of buck converters (they are essentially part of the buck converter), to filter the output of an Hbridge, and to filter audio signals before they reach the speakers.
The cutoff frequency of a lowpass LC filter is given by the following equation:
The characteristic impedance is:
which you will notice is also present in the cutoff frequency equation.
2.3.1. Parasitic elements
The main parasitic element to consider with a lowpass LC filter is the parasitic coil resistance of the inductor, \(R_L\). Larger valued inductors typically have a larger coil resistance (due to more windings). This dampens the output signal.
This is equivalent to a lowpass RLC filter.
2.4. Lowpass RLC Filter
The quality factor is equal to:
As you increase the series resistance, the quality factor decreases.
The damping factor is equal to:
2.5. LowPass Pi And t Filters
Lowpass Pi (π) and tfilters are one step better than the lowpass LC or RC filter.
A 1storder lowpass πfilter has two capacitors and one inductor. The first capacitor absorbs the most AC by shunting it to ground (assuming the input has a finite source impedance). The inductor then blocks remaining AC, allowing only DC to pass through to the second capacitor. The second capacitor then shunts any remaining AC signal back through ground.
The equations for a 1st order filter are:
where:
\(C\) = total capacitance in \(F\)
\(L\) = total inductance, in \(H\)
\(z_o\) = characteristic impedance, in \(\Omega\)
\(f_c\) = 3dB cutoff frequency, in \(Hz\)
Mentioned is total capacitance or total inductance, as in the case of the πfilter each capacitor is C/2, and in the case of the tfilter, each inductor is L/2. 
The typical value to use for the characteristic impedance is \( z_o = 50 \Omega \). Use this if you are unsure on what to set it to. This value is only important if your are matching two RF circuits.
A tfilter is usually better at suppressing highfrequencies than a πfilter, as parasitic coupling between input and output due to PCB layout tends to turn the π filter into a notch filter. However, πfilters are more common because they are cheaper (capacitors are cheaper than inductors).
Both π and t filters may use feedthrough capacitors instead of standard caps for better performance (feedthrough capacitors have lower parasitic series inductance).
2.6. Prepackaged Pi And T Filters
π and t filters can come in prepackaged components which take all the hassle out of designing the filter correctly and reduce the BOM count of your design. They are commonly in EIAxxxx chip packages.
One such example is the TDK Corporation MEM Series.
3. 2ndOrder Passive Filters
This chaining is also called cascading. The benefit of doing this is that a secondorder filter has a rolloff of 40dB/decade, twice that of a firstorder filter.
3.1. SecondOrder LowPass RC
The corner frequency \(f_c\) is equal to:
Is is important to remember that for a secondorder filter, the gain at the corner frequency is no longer 3dB. Instead it is 6dB. In general, the gain can be described for \(n\) stages with:
The reduce the effects of each stages dynamic impedance effecting it’s neighbours, its recommended that the following stages resistance should be around 10x the previous stage, and the capacitance 1/10th of the previous stage.
3.2. Passive RC Networks With Voltage Gain > 1
It might seem hard to believe, but you can build RC networks which increase the input voltage at specific frequencies. See Herman Epstein  Synthesis Of Passive RC Networks With Gains Greater Than Unity (cached copy, 20210123) for a detailed analysis.
4. Filter Optimizations
Filter optimizations are specific tunings of filters to maximise a particular characteristic of it’s response. Filter optimization directly specifies what the filter coefficients must be.

Butterworth Optimized for the flattest response through the passband, at the expense of having a low transition between the pass and stopband.

Chebyshev: Designed to have a steep transition between the pass and stopband, at the expense of gain ripple in either the pass or stopband (type 1 or type 2). Also called Chevyshev, Tschebychev, Tschebyscheff or Tchevysheff, depending on exactly how you translate the original Russian name. There are two types of Chebyshev filters:

Type 1: Type 1 Chebyshev filters (a.k.a. just a Chebyshev filter) have ripple in the passband, but no ripple in the stopband.

Type 2: Type 2 Chebyshev filters (a.k.a. an inverse Chebyshev filter) have ripple in the stopband, but no ripple in the passband.

Bessel: Optimized for linear phase response up to (or down to for highpass filters) the cutoff frequency \(f_c\), at the expense of a slower transition to the stopband. This is useful to minimizing the signal distortion (a linear phase response in the frequency domain is a constant time delay in the time domain).

Elliptic: Designed to have the fastest transition from the passband to the stopband, at the expense of ripple in both of these bands (Chebyshev optimization only produces ripple in one of the bands but is not as fast in the transition). Also called Cauer filters or Rational Chebyshev filters.
The graphs below show the differences in response (bode plots, gain and phase) for these various filter optimizations:
Sometimes the differences can been visualized better by display the gain as V/V:
The linear phase delay of the Bessel filter is best visualized in the below plot where the phase in plotted on a linear scale rather than a logarithmic:
4.1. Chebyshev Optimization
Chebyshev filters with even order numbers (e.g. 2nd order, 4th order, …) generate ripples above the 0dB line, filters with odd order numbers (e.g. 3rd order, 5th order, …) generate ripples below the 0dB line.
Because Chebyshev filters have ripple in the passband, their cutoff frequency is usually defined in a completely different way to all other filter optimizations. Rather than specifying \(f_c\) as the 3dB point, the \(f_c\) for Chebyshev filters is defined at the point at which the gain leaves the allowed ripple region (i.e. > 0.5dB for a 0.5dB Chebyshev filter, > 3dB for a 3dB Chebyshev filter).
4.2. Bessel Optimization
Commonly used in analoguecrossover circuitry.
4.3. Filter Coefficient Tables

\(n\) is the filter order

\(i\) is the partial filter order

\(a_i\) and \(b_i\) are the filter coefficients

\(k_i\) is the ratio between the corner frequency of the partial filter \(f_{ci}\) and the corner frequency of the overall filter \(f_c\). In equation form:
\[\begin{align} k_i = \frac{f_{ci}}{f_c} \end{align}\] 
\(Q_i\) is the quality factor of the partial filter
4.3.1. Butterworth Coefficients
\(n\)  \(i\)  \(a_i\)  \(b_i\)  \(k_i\)  \(Q_i\) 

1 
1 
1.0000 
0.0000 
1.000 
n/a 
2 
1 
1.4142 
1.0000 
1.000 
0.71 
3 
1 
1.0000 
0.0000 
1.000 
n/a 
2 
1.0000 
1.0000 
1.272 
1.00 
4.3.2. Chebyshev Coefficients For 3dB Passband Ripple
\(n\)  \(i\)  \(a_i\)  \(b_i\)  \(k_i\)  \(Q_i\) 

1 
1 
1.0000 
0.0000 
1.000 
n/a 
2 
1 
1.0650 
1.9305 
1.000 
1.30 
3 
1 
2.7994 
0.0000 
0.357 
n/a 
2 
0.4300 
1.2036 
1.378 
2.55 
4.3.3. Bessel Coefficients
\(n\)  \(i\)  \(a_i\)  \(b_i\)  \(k_i\)  \(Q_i\) 

1 
1 
1.0000 
0.0000 
1.000 
n/a 
2 
1 
1.3617 
0.6180 
1.000 
0.58 
3 
1 
0.7560 
0.0000 
1.323 
n/a 
2 
0.9996 
0.4772 
1.414 
0.69 
4.4. 2nd Order Filter Topologies
A filter topology is an actual circuit configuration which can realize a number of different filter designs. This is different from the configurations such as Butterworth, Chebyshev and Bessel which define the component tuning

SallenKey (a.k.a. KRC filters)

TowThomas

MultipleFeedback Filters (a.k.a. infinitegain filters)

StateVariable Filters: As known as KHN filters after the inventors W. J. Kerwin, L. P. Huelsman and R. W. Newcomb, first reported in 1967[3].
5. SallenKey Filters
The SallenKey filter is one of the most popular active 2ndorder analogue filters. It can be configured as a lowpass, highpass, bandpass or bandstop filter. Also called a Sallen and Key filter. It was first introduced in 1955 by R.P. Sallen and E.L. Key of MIT’s Lincoln Labs, whose last names give this filter it’s name.
It has low component spread (low ratios of highest to lowest capacitor and resistor values). It also has a high input impedance and low output impedance, allowing for multiple filters to be chained together without intermediary buffers. However, one issue with the SallenKey filter is the strong dependence on the opamp having low output impedance. An opamp’s output impedance rises with frequency, and thus the filtering ability begins to suffer around the 50500kHz range.
It is closely related to a voltagecontrolled voltage source (VCVS) filter, however the VCVS filter also includes gain by connected a resistor divider from the output to the inverting terminal of the opamp.
5.1. LowPass SallenKey Filter
The schematic for a unitygain lowpass SallenKey filter is shown below:
It looks like 2 cascaded RC filters, except with the other terminal of the 1st capacitor connected to the opamp’s output rather than ground! What does this mean?
TODO: Add more info here.
Take note of labelling of the resistors and capacitors if reading other material on SallenKey filters, there is no one popular convention as the resistor and capacitor orders are switched frequently. 
A SallenKey filter has a gain which begins to increase again after a certain frequency in the stop band.
We will simulate the response of a SallenKey filter designed with a cutoff frequency of 1kHz. Below is the KiCad schematic used for the simulation:
The KiCad schematic for this simulation can be <a href="lowpasssallenkey/lowpasssallenkey.sch" download>downloaded here</a>. The simulated gain (magnitude) and phase response is shown in Figure 10.
The transfer function:
The resistance of the resistors \(R1\) and \(R2\) are related to the capacitances and filter coefficients by the following equation:
You use the \(\) sign when calculating \(R1\) and the \(+\) sign for calculating \(R2\).
To obtain real values under the square root, \(C1\) must obey the follow condition:
These equations give you enough info to calculate all the resistances and capacitors for a SallenKey filter. See the design example below to show how you would go about it.
5.1.1. Design Example: 2ndOrder LowPass UnityGain 3dBChebyshev SallenKey Filter
The task is to design a 2ndorder unitygain SallenKey filter optimized with Chebyshev 3dB ripple coefficients (this will give us a sharp transition from the passband to the stopband) and a corner frequency must be \(f_c = 1kHz\).

Look up the Chebyshev filter coefficients. From the table we get:
\[\begin{align} a_1 = 1.0650 \\ b_1 = 1.9305 \end{align}\] 
Choose a capacitance for \(C2\). This is rather arbitrary, but a good recommended starting range is something between \(1100nF\). Lets pick:
\[\begin{align} C2 = 10nF \end{align}\] 
Calculate the capacitance of \(C1\) from \(Eq. \ref{eqn:c1geq}\):
\[\begin{align} C1 &\geq C2 \frac{4b_1}{a_1^2} \\ &\geq 10nF \frac{4\cdot1.9305}{1.0650^2} \\ &\geq 68.1nF \end{align}\]Pick the next largest E12 value:
\[\begin{align} C1 = 82nF \end{align}\] 
Calculate \(R1\) and \(R2\) using \(Eq. \ref{eqn:r1r2eq}\):
\[\begin{align} R1 &= \frac{a_1 C1  \sqrt{(a_1 C1)^2  4 b_1 C1C2}}{4\pi f_c C1 C2} \\ &= \frac{1.0650 \cdot 82nF  \sqrt{1.0650^2 \cdot 82nF^2  4 \cdot 1.9305 \cdot 10nF \cdot 82nF}}{4\pi \cdot 1kHz \cdot 10nF \cdot 82nF} \\ &= 4.98k\Omega \end{align}\]\[\begin{align} R2 &= \frac{a_1 C2 + \sqrt{a_1^2 C2^2  4 b_1 C1C2}}{4\pi f_c C1 C2} \\ &= \frac{1.0650 \cdot 82nF + \sqrt{1.0650^2 \cdot 82nF^2  4 \cdot 1.9305 \cdot 10nF \cdot 82nF}}{4\pi \cdot 1kHz \cdot 10nF \cdot 82nF} \\ &= 12.0k\Omega \end{align}\]Pick the closest E96 values:
\[\begin{align} R1 = 4.99k\Omega \\ R2 = 12.1k\Omega \end{align}\] 
Build the circuit! It should look like this:
Figure 11. Schematic of the design example (2ndorder 3dB Chebyshev SallenKey lowpass filter with a cutoff frequency of 1kHz) above. 
And just good measure this was simulated, to make sure the response is as expected.
5.2. Dependence On OpAmp Output Impedance
A SallenKey filter is strongly dependent on the opamp having a low output impedance. A opamp’s output impedance increases with increasing frequency, thus the performance of the SallenKey begins to suffer around the 50500kHz range.
This can be seen in the following bode plot for a 2ndorder lowpass SallenKey filter, with a cutoff frequency \(f_c\) of 1kHz:
6. VoltageControlled VoltageSource (VCVS) Filters
Voltagecontrolled voltagesource (VCVS) filters are an extension of the SallenKey filter (in that sense, the SallenKey filter can be thought of as a simplification of the VCVS filter in where the voltage gain of the opamp is set to one) in where standard resistor divider feedback is added between the opamp’s output and the inverting input, allowing the gain of the filter to be something other than \(1\).
They are called VCVS filters because the opamp is used as a voltage amplifier.
7. Design Tools
7.1. OKAWA Filter Design and Analysis
Great site with webbased calculators and design tools for active and passive filters. Very detailed site with many configuration options and the site even outputs graphs of your designed filter response.
7.2. PSoC Microcontrollers And The PSoC Creator IDE
The PSoC microcontroller features an inbuilt and versatile digital filter block, and the IDE has a graphicallyedited method of configuring the DFB to do exactly what you want. The IDE even shows you graphs of the expected response (magnitude, phase, step plots e.t.c).
8. External Resources

The New Jersey Institute of Technology EE 494 Laboratory IV Part B lab manual is a great practical resource for learning how to design active filters.

The Design With Operational Amplifiers And Analog Integrated Circuits by Sergio Franco, Fourth Edition is a great book to purchase if you are interesting in further reading and getting right into the weeds of analogue filter design!

Op Amps For Everyone by Ron Mancini (SLOD006B) has some detailed sections on opamp filter circuits.

SLOA024B: Analysis of the SallenKey Architecture  Application Report, by Texas Instruments can be used for further reading on the SallenKey and VCVS amplifiers (<a href="sloa024banalysisofthesallenkeyarchitectureapplicationreporttexasinstruments.pdf" download>cached local copy</a>).

Texas Instruments Filter Designer is a free online tool to design filters.

The Analog Devices Filter Wizard is an alternative to the Texas Instruments version.
References

[1] Wikipedia. Lowpass filter. Retrieved 20210325, from https://en.wikipedia.org/wiki/Lowpass_filter.

[2] Electronic Tutorials. Passive Low Pass Filter. Retrieved 20210325, from https://www.electronicstutorials.ws/filter/filter_2.html.

[3] Franco, Sergio. Design With Operational Amplifiers And Analog Integrated Circuits. Fourth Edition. McGrawHill Education. Copyright 2015.
Authors
This work is licensed under a Creative Commons Attribution 4.0 International License .
Related Content:
Tags
 electronics
 circuit design
 filters
 passive filters
 active filters
 RC
 lowpass
 highpass
 LC
 bode plot
 frequency response
 SallenKey
 voltagecontrolled voltage source
 VCVS
 cutoff frequency
 corner frequency
 break frequency