e or responsible for its content.
Analog Filter Design Demystified www.maxim-ic.com/an1795 Page 1 of 10


ANALOG FILTER CIRCUITS
Application Note 1795: Dec 03, 2002

Analog Filter Design Demystified

This article shows the reader how to design analog filters. It starts by covering the
fundamentals of filters, it then goes on to introduce the basic types like Butterworth,
Chebyshev, and Bessel, and then guides the reader through the design process for
lowpass and highpass filters. Includes the derivation of the equations and the circuit
implementation.

It's a jungle out there.
A small tribe, in the dense wilderness, is much sought after by head hunters from the
surrounding plains. The tribe knows it is threatened, because its numberskilled off by the
accelerating advance of modern technologyare dwindling at an alarming rate. This is the
tribe of the Analog Engineers.
The guru of Analog Engineers is the Analog Filter Designer, who sits on the throne of his
kingdom and imparts wisdom while reminiscing of better days. You never get to see him even
with an appointment, and you call him "Sir."
The countless pages of equations found in most books on filter design can frighten small dogs
and children. This article unravels the mystery of filter design, enabling you to design
continuous-time analog filters quickly and with a minimum of mathematics. The throne will
soon be vacant.
The Theory of Analog Electronics
Analog electronics has two distinct sides: the theory taught by academic institutions (equations
of stability, phase-shift calculations, etc.), and the practical side familiar to most engineers
(avoid oscillation by tweaking the gain with a capacitor, etc.). Unfortunately, filter design is
based firmly on long-established equations and tables of theoretical results. Filter design from
theoretical equations can prove arduous. Consequently, this discussion employs a minimum of
matheither in translating the theoretical tables into practical component values, or in deriving
the response of a general-purpose filter.
The Fundamentals
Simple RC lowpass filters have the transfer function
.
Cascading such filters complicates the response by giving rise to quadratic equations in the
denominator of the transfer function. Thus, the denominator of the transfer function for any www.maxim-ic.com/an1795 Page 2 of 10

second-order lowpass filter is as
2
+ bs + c. Substituting values for a, b and c determines the filter
response over frequency. Anyone who remembers high school math will note that the above
expression equals zero for certain values of "s" given by the equation

At the values of "s" for which this quadratic equation equals zero, the transfer function has
theoretically infinite gain. These values, which establish the performance of each type of filter
over frequency, are known as the poles of the quadratic equation. Poles usually occur as pairs,
in the form of a complex number (a + jb) and its complex conjugate (a - jb). The term jb is
sometimes zero.
The thought of a transfer function with infinite gain may frighten nervous readers, but in
practice it isn't a problem. The pole's real part "a" indicates how the filter responds to transients,
and its imaginary part "jb" indicates the response over frequency. As long as this real part is
negative, the system is stable. The following text explains how to transfer the tables of poles
found in many text books into component values suitable for circuit design.
Filter Types
The most common filter responses are the Butterworth, Chebyshev, and Bessel types. Many
other types are available, but 90% of all applications can be solved with one of these three.
Butterworth ensures a flat response in the pass band and an adequate rate of rolloff. A good "all
rounder," the Butterworth filter is simple to understand and suitable for applications such as
audio processing. The Chebyshev gives a much steeper rolloff, but passband ripple makes it
unsuitable for audio systems. It is superior for applications in which the passband includes only
one frequency of interest (e.g., the derivation of a sinewave from a square wave, by filtering out
the harmonics).
The Bessel filter gives a constant propagation delay across the input frequency spectrum.
Therefore, applying a square wave (consisting of a fundamental and many harmonics) to the
input of a Bessel filter yields an output square wave with no overshoot (all the frequencies are
delayed by the same amount). Other filters delay the harmonics by different amounts, resulting
in an overshoot on the output waveform. One other popular filter, the elliptical type, is a much
more complicated beast that will not be discussed in this text. Similar to the Chebyshev
response, it has ripple in the passband and severe rolloff at the expense of ripple in the stop
band.
Standard Filter Blocks
The generic filter structure (Figure 1a) lets you realize a highpass or lowpass implementation
by substituting capacitors or resistors in place of components G1-G4. Considering the effect of
these components on the op-amp feedback network, one can easily derive a lowpass filter by
making G2/G4 into capacitors and G1/G3 into resistors. (Vice versa yields the highpass
implementation.) www.maxim-ic.com/an1795 Page 3 of 10


Figure 1. By substituting for G1-G4 in the generic filter block (a), you can implement a
lowpass filter (b) or a highpass filter (c).
The transfer function for the lowpass filter (Figure 1b) is

This equation is simpler with conductances. Replace the capacitors with a conductance of sC,
and the resistors with a conductance of G. If this looks complicated, you can "normalize" the
equation. Set the resistors equal to 1 or the capacitors equal to 1F, and change the
surrounding components to fit the response. Thus, with all resistor values equal to 1 , the
lowpass transfer function is
www.maxim-ic.com/an1795 Page 4 of 10

This transfer function describes the response of a generic, second-order lowpass filter. We now
take the theoretical tables of poles that describe the three main filter responses, and translate
them into real component values.
The Design Process
To determine the filter type required, you should use the above descriptions to select the
passband performance needed. The simplest way to determine filter order is to design a second-
order filter stage, and then cascade multiple versions of it as required. Check to see if the result
gives the desired stopband rejection, and then proceed with correct pole locations as shown in
the tables in the appendix. Once pole locations are established, the component values can soon
be calculated.
First, transform each pole location into a quadratic expression similar to that in the denominator
of our generic second-order filter. If a quadratic equation has poles of (a ± jb), then it has roots
of (s - a - jb) and (s - a + jb). When these roots are multiplied together, the resulting quadratic
expression is s
2
- 2as + a
2
+ b
2
.
In the pole tables a is always negative, so for convenience we declare s
2
+ 2as + a
2
+ b
2
, and use
the magnitude of a regardless of its sign. To put this into practice, consider a fourth-order
Butterworth filter. The poles and the quadratic expression corresponding to each pole location
are as follows:
Poles (a ± jb)
Quadratic expression
-0.9239 ± j0.3827
s
2
+ 1.8478s + 1
-0.3827 ± j0.9239
s
2
+ 0.7654s + 1
You can design a fourth-order Butterworth lowpass filter with this information. Simply
substitute values from the above quadratic expressions into the denominator of Equation 1.
Thus, C2C4 = 1 and 2C4 = 1.8478 in the first filter, implying that C4 = 0.9239F and C2 =
1.08F. For the second filter, C2C4 = 1 and 2C4 = 0.7654, implying that C4 = 0.3827F and C2 =
2.61F. All resistors in both filters equal 1 . Cascading these two second-order filters yields a
fourth-order Butterworth response with rolloff frequency of 1rad/s, but the component values
are impossible to find. If the frequency or component values above are not suitable, read on.
It so happens that if you maintain the ratio of the reactances to the resistors, the circuit response
remains unchanged. You might therefore choose 1k resistors. To ensure that the reactances
increase in the same proportion as the resistances, divide the capacitor values by 1000.
We still have the perfect Butterworth response, but unfortunately the rolloff frequency is
1rad/s. To change the circuit's frequency response, we must maintain the ratio of reactances to
resistances but simply at a different frequency. For a rolloff of 1kHz rather than 1rad/s, the
capacitor value must be further reduced by a factor of 2 x 1000. Thus, the capacitor's
reactance does not reach the original (normalized) value until the higher frequency. The
resulting fourth-order Butterworth lowpass