
ELECTRONIC DESIGN NOTES #9  ANALOG FILTERS

The first thing we do after we
capture/collect an analog or a digital signal is to filter it. Filtering can be analog or digital: between the two,
digital filters are way more efficient. However, digital filters do not work very well if we do not use an analog
filter module first. In other words, analog filters are mandatory no matter what!
Analog filters are built with electronic hardware components, while digital filters are best implemented in firmware.
However, there are also hardware digital filters built with PLG (Programmable Logic Gates) when the speed factor
is imperative.
The most basic roots of Digital Signal Processing (DSP) technology is
using filtersof course, that is digital
filters. DSP, however, is a late child, therefore it works as an improved version of the good old analog filtering
techniques. If you intend to start working with DSP, you need to study analog filters very,
very well first.
The simplified structure employed to present analog filters on this page is:
1. Types of Filters 2. LowPass Filters 3. HighPass Filters 4. BandPass Filters 5. BandReject Filters
NOTE The basic notions highlighted on this page are related to
a few electronic design topics presented in the first part, Hardware Design, of
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1. TYPES OF FILTERS
There are 2 main types of filters:
1. passive filters, made of passive components: resistors, capacitors, and inductors;
2. active filters, employing Operational Amplifiers.
Passive filters are built with:
A. ResistorCapacitor: these are
RC filters, and they are the most used
ones since they are cheaper to build; B. InductorCapacitor: they are noted as
LC filters, and they have
(somewhat) better performances. However, the problems are the fact that inductors are expensive,
they are difficult to tune up to exact/precise values, and they
require shielding of their electromagnetic field.
Both passive and active filters may be serialized (cascaded): in this way we obtain 1, 2, 3 ... n order
filters. Of course, the higher the order, the better is the analog filtering employed. Commonly,
we build a 3 up to 9 order analog
filter, and then we use a digital firmware filter (a DSP filter) of 500, 1000, or even
of a 2000 order.
Again, although digital filtersthey are in fact firmware and software routinesare way more efficient, they do
not work very well if you do not have a first, basic, analog hardware filter.
Now, depending on their functionality, both passive and active filters can be:
1. LowPass 2. HighPass 3. BandPass 4. BandReject


Active Filters
are a bit more complex since they are built using one of the following Schematics:
1. Butterworth 2. Chebichev 3. Inverse Chebichev 4. Eliptic Integral (or Zolatarev, or complete Chebichev) 5. Legendre 6. Bessel

ACTIVE FILTERS COMPARISON CHART 
Filter Response 
Specifications 
Comments 

Fig 5: Butterworth
The best amplitude flat response in passband. 
Most popular, generalpurpose filters. 

Fig 6: Chebyshev
Built for equal amplitude ripples in passband. 
The transition slope attenuation is steeper than the
Butterworth one. 

Fig 7: Inverse Chebyshev
Built for equal amplitude ripples in stopband. 
No passband ripples. 

Fig 8: Elliptic Integral (or
Zolatarev, or complete Chebychev)
Equal amplitude ripples in both the passband and the stopband. 
Some of the best analog filters. 

Fig 9: Legendre
Similar to Butterworth with no ripple in passband, and steeper transition. 
Good filters,
though not very flat in passband. 

Fig 10: Bessel
Almost linear in passband but very poor transient slope. 
Excellent for pulse generator circuits since they minimizes
ringing and overshooting.
Particularly good when combined with firmware digital filters. 

In order to help designing active filters employing Operational Amplifiers, Microchip offers their FilterLab
application software for free download. That is a nice useful program, easy to learn and master. 
2. LOWPASS FILTERS
LowPass filters will stop all frequencies greater than the cutoff frequency.

LOWPASS FILTERS

Graphic Representation 
Description 

Fig 11: LowPass Attenuation curve
In Fig 11 you can see that everything is fine and perfect until we reach the HP_{p}
(HalfPower point) corresponding to f_{c} (the cutoff frequency).
That is when our filter starts working, because its purpose is to cut all frequencies greater than f_{c}. 

Fig 12: LowPass, first order,
simple RC circuit
This circuit is going to give us the above attenuation curve. A few formulas are needed when working with RC
filters:
A = Xo /√(R^{2} + Xc^{2})
The above formula becomes:
A = 1 /√[1 + (2*PI*f*R*C)^{2}]
Note that A = 0.707 in HP_{p}. This allows us to calculate:
f_{c} = 1/(2*PI*R*C) 

Fig 13: LowPass, first order,
simple LC circuit
Using inductors and capacitors we obtain the same output attenuation curve pictured in Fig 11. The
formulas used to calculate the filter are a bit different.
First of all, because we deal with AC signals, we have a Characteristic Equivalent Resistance
Re = √(L/C)
In this case the cutoff frequency is:
fc = 1/[2*PI*√(L*C)] 

Fig 14: LowPass, first order,
"T" LC circuit
The "T" LC filter is a common circuit, and we would like to point out that
2C needs 2 times
the value of C in the previous case. The formulas used to calculate the circuit are the same as above.
Note that at high frequencies L behaves like a capacitor, while C behaves like a resistor, due the
reactance formulas presented in the previous Design Notes. 

Fig 15: LowPass, first order,
"PI" LC circuit
The "PI" LC filter is another common filter circuit. In order to simplify things
2L has
double the value in previous circuit. 

Fig 16: LowPass, first order,
active filter
This is the simplest possible active LowPass filter. Note that the OA is used only to amplify the
output of a simple RC filter. 

Fig 17: LowPass, second order, RC
circuit
Better filtering results are obtained if we cascade 2 or more filterscommonly, up to 7..9 

Fig 18: LowPass, second order, LC
circuit
Same as the above. 

Fig 19: LowPass, second order, active filter circuit
This is a simple, second order Butterworth filter. Again, for best results it is recommend
using some professional design software, as is FilterLab. 

3. HIGHPASS FILTERS
HighPass filters stop al frequencies smaller than the cutoff frequency. 
HIGHPASS FILTERS 
Graphic Representation 
Description 

Fig 20: HighPass filter
attenuation curve
The graph on left tells us that the HighPass filters work to stop all frequencies up to the cutoff f_{c}.
The cutoff frequency appears when the attenuation reaches the HalfPower point (0.707*Vrms).


FIG 21: HighPass, first order,
simple RC filter
Two formulas are used to calculate this HighPass simple RC circuit:
A = 1 /√[1 + 1/(2*PI*R*C)^{2}]
f_{c} = 1/(2*PI*R*C)
Above the cutoff A is almost 1 and A [db] appx = 0 [db]. Below the cutoff A is (2*PI*R*C)
and
A [db] appx = 20log(2*PI*R*C) 

Fig 22: HighPass, first order,
simple LC filter
First we determine the Characteristic Equivalent Resistance: Re = √(L/C)
then the cutoff frequency: f_{c} = 1/[2*PI*√(L*C)]
Re must have the same impedance as the source one; this allows us to calculate:
L = Re/2*PI*fc
C = 1/2*PI*fc*Re 

Fig 23: HighPass, first order,
"T" LC filter
Again it is improper to name this "T" circuit "a first order one", because it is in
fact a second order in disguise. In order to facilitate calculations, the inductance is
selected as L/2 of
the previous circuit. 

Fig 24: HighPass, first order,
"PI" LC filter
Same considerations as the above. This time C is half the value it had previously. 

Fig 25: HighPass, first order
active filter
In this case the OA doesn't do too much in terms of amplification; however, first order active
filters are almost never used. Things start being a bit more interesting beginning with the second order
active filters up. 

Fig 26: HighPass, second order,
simple RC filter
We can improve filtering efficiency by using higher order filters. 

Fig 27: HighPass. second order,
simple LC filter
Same considerations as above.


Fig 28: HighPass, second order,
active filter
In this particular case we can calculate:
f_{c} = 1/2*PI*√(C_{2}*C_{3}*R_{1}*R_{2})
A_{v} = C_{2}/C_{1} 

4. BANDPASS FILTERS
Logically, a HighPass filter in series with a LowPass one results in a BandPass filter. The following
table presents a few common instances.


5. BANDREJECT FILTERS
There are many good Schematics used to build BandReject filters;
following are presented two of
the most basic.


Last word: active filters are way more powerful, except they are
fairly difficult to tune up. In addition,
active filters may introduce some unwanted noise.
Passive filters need to be cascaded for some significant results: you need a minimum of 5..7 order filter.
However, you can be certain your passive filter will not introduce any
noiseexcepting the unshielded LC filters, naturally.

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