Electronic filters are analog circuits which perform signal processing functions, specifically
to remove unwanted frequency components from the signal, to enhance wanted ones, or both.
Electronic filters can be: passive or active
analog or digital high-pass, low-pass, band-pass, band-stop,
or all-pass. discrete-time or continuous-time
linear or non-linear infinite impulse response or finite impulse
response The most common types of electronic filters
are linear filters, regardless of other aspects of their design. See the article on linear
filters for details on their design and analysis.
History
The oldest forms of electronic filters are passive analog linear filters, constructed
using only resistors and capacitors or resistors and inductors. These are known as RC and RL
single-pole filters respectively. More complex multipole LC filters have also existed for
many years, and their operation is well understood. Hybrid filters are also possible, typically
involving a combination of analog amplifiers with mechanical resonators or delay lines.
Other devices such as CCD delay lines have also been used as discrete-time filters. With
the availability of digital signal processing, active digital filters have become common.
Classification by technology Passive filters
Passive implementations of linear filters are based on combinations of resistors, inductors
and capacitors. These types are collectively known as passive filters, because they do
not depend upon an external power supply and/or they do not contain active components such
as transistors. Inductors block high-frequency signals and
conduct low-frequency signals, while capacitors do the reverse. A filter in which the signal
passes through an inductor, or in which a capacitor provides a path to ground, presents
less attenuation to low-frequency signals than high-frequency signals and is therefore
a low-pass filter. If the signal passes through a capacitor, or has a path to ground through
an inductor, then the filter presents less attenuation to high-frequency signals than
low-frequency signals and therefore is a high-pass filter. Resistors on their own have no frequency-selective
properties, but are added to inductors and capacitors to determine the time-constants
of the circuit, and therefore the frequencies to which it responds.
The inductors and capacitors are the reactive elements of the filter. The number of elements
determines the order of the filter. In this context, an LC tuned circuit being used in
a band-pass or band-stop filter is considered a single element even though it consists of
two components. At high frequencies, sometimes the inductors
consist of single loops or strips of sheet metal, and the capacitors consist of adjacent
strips of metal. These inductive or capacitive pieces of metal are called stubs.
Single element types
The simplest passive filters, RC and RL filters, include only one reactive element, except
hybrid LC filter which is characterized by inductance and capacitance integrated in one
element. L filter
An L filter consists of two reactive elements, one in series and one in parallel.
T and π filters
Three-element filters can have a 'T' or 'π' topology and in either geometries, a low-pass,
high-pass, band-pass, or band-stop characteristic is possible. The components can be chosen
symmetric or not, depending on the required frequency characteristics. The high-pass T
filter in the illustration, has a very low impedance at high frequencies, and a very
high impedance at low frequencies. That means that it can be inserted in a transmission
line, resulting in the high frequencies being passed and low frequencies being reflected.
Likewise, for the illustrated low-pass π filter, the circuit can be connected to a
transmission line, transmitting low frequencies and reflecting high frequencies. Using m-derived
filter sections with correct termination impedances, the input impedance can be reasonably constant
in the pass band. Multiple element types
Multiple element filters are usually constructed as a ladder network. These can be seen as
a continuation of the L,T and π designs of filters. More elements are needed when it
is desired to improve some parameter of the filter such as stop-band rejection or slope
of transition from pass-band to stop-band. Active filters
Active filters are implemented using a combination of passive and active components, and require
an outside power source. Operational amplifiers are frequently used in active filter designs.
These can have high Q factor, and can achieve resonance without the use of inductors. However,
their upper frequency limit is limited by the bandwidth of the amplifiers.
Other filter technologies Digital filters
Digital signal processing allows the inexpensive construction of a wide variety of filters.
The signal is sampled and an analog-to-digital converter turns the signal into a stream of
numbers. A computer program running on a CPU or a specialized DSP calculates an output
number stream. This output can be converted to a signal by passing it through a digital-to-analog
converter. There are problems with noise introduced by the conversions, but these can be controlled
and limited for many useful filters. Due to the sampling involved, the input signal must
be of limited frequency content or aliasing will occur.
Quartz filters and piezoelectrics
In the late 1930s, engineers realized that small mechanical systems made of rigid materials
such as quartz would acoustically resonate at radio frequencies, i.e. from audible frequencies
up to several hundred megahertz. Some early resonators were made of steel, but quartz
quickly became favored. The biggest advantage of quartz is that it is piezoelectric. This
means that quartz resonators can directly convert their own mechanical motion into electrical
signals. Quartz also has a very low coefficient of thermal expansion which means that quartz
resonators can produce stable frequencies over a wide temperature range. Quartz crystal
filters have much higher quality factors than LCR filters. When higher stabilities are required,
the crystals and their driving circuits may be mounted in a "crystal oven" to control
the temperature. For very narrow band filters, sometimes several crystals are operated in
series. Engineers realized that a large number of
crystals could be collapsed into a single component, by mounting comb-shaped evaporations
of metal on a quartz crystal. In this scheme, a "tapped delay line" reinforces the desired
frequencies as the sound waves flow across the surface of the quartz crystal. The tapped
delay line has become a general scheme of making high-Q filters in many different ways.
SAW filters SAW filters are electromechanical devices
commonly used in radio frequency applications. Electrical signals are converted to a mechanical
wave in a device constructed of a piezoelectric crystal or ceramic; this wave is delayed as
it propagates across the device, before being converted back to an electrical signal by
further electrodes. The delayed outputs are recombined to produce a direct analog implementation
of a finite impulse response filter. This hybrid filtering technique is also found in
an analog sampled filter. SAW filters are limited to frequencies up to 3 GHz. The filters
were developed by Professor Edward George Sydney Paige and others.
BAW filters BAW filters are electromechanical devices.
BAW filters can implement ladder or lattice filters. BAW filters typically operate at
frequencies from around 2 to around 16 GHz, and may be smaller or thinner than equivalent
SAW filters. Two main variants of BAW filters are making their way into devices: thin film
bulk acoustic resonator or FBAR and solid mounted bulk acoustic resonators.
Garnet filters
Another method of filtering, at microwave frequencies from 800 MHz to about 5 GHz,
is to use a synthetic single crystal yttrium iron garnet sphere made of a chemical combination
of yttrium and iron. The garnet sits on a strip of metal driven by a transistor, and
a small loop antenna touches the top of the sphere. An electromagnet changes the frequency
that the garnet will pass. The advantage of this method is that the garnet can be tuned
over a very wide frequency by varying the strength of the magnetic field.
Atomic filters For even higher frequencies and greater precision,
the vibrations of atoms must be used. Atomic clocks use caesium masers as ultra-high Q
filters to stabilize their primary oscillators. Another method, used at high, fixed frequencies
with very weak radio signals, is to use a ruby maser tapped delay line.
The transfer function see also Filter for further analysis
The transfer function of a filter is the ratio of the output signal to that of the input
signal as a function of the complex frequency :
with . The transfer function of all linear time-invariant
filters, when constructed of discrete components, will be the ratio of two polynomials in , i.e.
a rational function of . The order of the transfer function will be the highest power
of encountered in either the numerator or the denominator.
Classification by topology Electronic filters can be classified by the
technology used to implement them. Filters using passive filter and active filter technology
can be further classified by the particular electronic filter topology used to implement
them. Any given filter transfer function may be
implemented in any electronic filter topology. Some common circuit topologies are:
Cauer topology – passive Sallen Key topology – active
Multiple Feedback topology – active State Variable Topology – active
Biquadratic topology biquad filter – active Classification by design methodology
Historically, linear analog filter design has evolved through three major approaches.
The oldest designs are simple circuits where the main design criterion was the Q factor
of the circuit. This reflected the radio receiver application of filtering as Q was a measure
of the frequency selectivity of a tuning circuit. From the 1920s filters began to be designed
from the image point of view, mostly being driven by the requirements of telecommunications.
After World War II the dominant methodology was network synthesis. The higher mathematics
used originally required extensive tables of polynomial coefficient values to be published
but modern computer resources have made that unnecessary.
Direct circuit analysis Low order filters can be designed by directly
applying basic circuit laws such as Kirchhoff's laws to obtain the transfer function. This
kind of analysis is usually only carried out for simple filters of 1st or 2nd order.
Image impedance analysis
This approach analyses the filter sections from the point of view of the filter being
in an infinite chain of identical sections. It has the advantages of simplicity of approach
and the ability to easily extend to higher orders. It has the disadvantage that accuracy
of predicted responses relies on filter terminations in the image impedance, which is usually not
the case.
Network synthesis
The network synthesis approach starts with a required transfer function and then expresses
that as a polynomial equation of the input impedance of the filter. The actual element
values of the filter are obtained by continued-fraction or partial-fraction expansions of this polynomial.
Unlike the image method, there is no need for impedance matching networks at the terminations
as the effects of the terminating resistors are included in the analysis from the start.
Here is an image comparing Butterworth, Chebyshev, and elliptic filters. The filters in this
illustration are all fifth-order low-pass filters. The particular implementation –
analog or digital, passive or active – makes no difference; their output would be
the same.
As is clear from the image, elliptic filters are sharper than all the others, but they
show ripples on the whole bandwidth. See also
Notes, references and external links