COROLLARY THEOREMS - ELECTRONINC DESIGN NOTES: Analog Filters

The first thing we do after we capture/collect an analog or 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 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). 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) theory is using filters--of course, that is digital filters. DSP, however, is a late child, therefore it works as an improved version of the old analog filtering techniques. If you intend to start working with DSP, you need to study analog filters very well.

The structure employed to present analog filters is:
1. Types of filters
2. Low-Pass filters
3. High-Pass filters
4. Band-Pass filters
5. Band-Reject filters

NOTE
The basic notions highlighted in this page are related to electronic design topics presented in the first part Hardware Design of Learn Hardware Firmware and Software Design.

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. resistor-capacitor: these are RC filters, and they are the most used ones since they are easier and cheaper to build.
B. inductor-capacitor: they are noted as LC filters, and they have better performances. The problems are: inductors are expensive, very difficult to tune to exact values, and they may require shielding of their electromagnetic field.

Both the passive and the 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 filtering. It is common to build 3 to 9 order analog filters, and then to use digital firmware filters (DSP filters) of 500, 1000 or even 2000 order.

Again, although digital filters--they are in fact firmware and software routines--are way more efficient, they do not work very well if you do not have a first, basic, analog hardware filter.

Depending on their functionality, both passive and active filters can be:
1. Low-Pass
2. High-Pass
3. Band-Pass
4. Band-Reject

FILTERS CLASSIFICATION BASED ON FUNCTIONALITY
Frequency response	Filter Type
	Fig1: Low-Pass Filter Avp = attenuation in passband Avs = attenuation in stopband HPp = Half-Power point (0.707 of voltage) fc = cutoff frequency
	Fig 2: High-Pass Filter Avp = attenuation in passband Avs = attenuation in stopband HPp = Half-Power point (0.707 of voltage) fc = cutoff frequency
	Fig 3: Band-Pass Filter Avp = attenuation in passband Avs = attenuation in stopband HPp = Half-Power point (0.707 of voltage) fcl = cutoff frequency low fch = cutoff frequency high
	Fig 4: Band-Reject Filter Avp = attenuation in passband Avs = attenuation in stopband freject = frequency (band) reject

Active Filters are a bit more complex, and they are constructed as one of the following type:
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, general-purpose 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, but 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 you design Active Filters using Operational Amplifiers, Microchip offers their FilterLab application software for free download. It is a nice, useful program, easy to learn and master.

LOW-PASS FILTERS

Low-Pass filters will stop all frequencies greater than the cutoff frequency.

LOW-PASS FILTERS
Graphic Representation	Description
	Fig 11: Low-Pass Attenuation curve In Fig 11 you can see that everything is fine and perfect until we reach the HP_p (Half-Power point) corresponding to f_c (cutoff frequency). That is when our filter starts working, because its purpose is to cut all frequencies greater than f_c
	Fig 12: Low-Pass, first order, simple RC circuit This circuit is going to give us the above attenuation curve. Few formulas are needed when working with RC filters: A = Xo /√(R² + Xc²) The above formula becomes: *A = 1 /√[1 + (2PIfRC)²]* Note that A = 0.707 in HP_p. This allows us to calculate: *f_c = 1/(2PIRC)**
	Fig 13: Low-Pass, 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√(LC)]*
	Fig 14: Low-Pass, first order, "T" LC circuit The "T" LC filter is a common circuit, and we would like to point out that C 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 the C behaves like a resistor, due the reactance formulas presented in previous Design Notes.
	Fig 15: Low-Pass, first order, "PI" LC circuit The "PI" LC filter is another common filter circuit. In order to simplify things L has double the value in previous circuit.
	Fig 16: Low-Pass, first order, active filter This is the simplest possible active Low-Pass filter. Note that the OA is used only to amplify the output of a simple RC filter.
	Fig 17: Low-Pass, second order, RC circuit Better filtering results are obtained if we cascade 2 or more filters--normally up to 7..9.
	Fig 18: Low-Pass, second order, LC circuit Same as the above.
	Fig 19: Low-Pass, second order, Active filter circuit This is a simple, second order Butterworth filter. Again, for best results it is recommend the use of specialized design software, as is FilterLab.

HIGH-PASS FILTERS

High-Pass filters stop al frequencies smaller than the cutoff frequency.

HIGH-PASS FILTERS
Graphic Representation	Description
	Fig 20: High-Pass filter attenuation curve The graph on left tells us the High-Pass filters work to stop all frequencies up to the cutoff f_c. The cutoff frequency is when the attenuation reaches the Half-Power point (0.707*Vrms).
	FIG 21: High-Pass, first order, simple RC filter Two formulas are used to calculate this High-Pass simple RC circuit: A = 1 /√[1 + 1/(2PIRC)²] f_c = 1/(2PIRC) Above the cutoff A is almost 1 and A [db] appx = 0 [db]. Below the cutoff A is (2**PIRC)* and A [db] appx = 20log(2**PIRC)*
	Fig 22: High-Pass, first order, simple LC filter First we determine the Characteristic Equivalent Resistance: Re = √(L/C) then the cutoff frequency: f_c = 1/[2**PI√(LC)]* Re must be the same impedance as the source one: this allows us to calculate: L = Re/2PIfc C = 1/2**PIfcRe*
	Fig 23: High-Pass, 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 chosen as L/2 of the previous circuit.
	Fig 24: High-Pass, first order, "PI" LC filter Same considerations as the above. This time C is half the value it had previously.
	Fig 25: High-Pass, first order Active filter In this case the OA doesn't do too much in terms of amplification; however, the first order active filters are almost never used. Things start being a bit more interesting beginning with the second order filters.
	Fig 26: High-Pass, second order, simple RC filter We can improve filtering efficiency by using higher order filters.
	Fig 27: High-Pass. second order, simple LC filter Same considerations as the above.
	Fig 28: High-Pass, second order, Active filter In this particular case we can calculate: f_c = 1/2PI√(C₂C₃R₁*R₂) A_v = C₂/C₁

BAND-PASS FILTERS

Logically, a High-Pass filter in series with a Low-Pass one results in a Band-Pass filter. The following table presents few simple particular cases:

BAND-PASS FILTERS
Graphic Representation	Description
	Fig 29: Band-Pass Attenuation Note that we have 2 cutoff frequencies in this case: f_cl = low-cutoff frequency f_ch = high-cutoff frequency
	Fig 30: Simple RC Band-Pass Filter f_ch = 1/(2PIR₂C₂) f_cl = 1/(2PIR₁C₁) with R2 >10*R1
	Fig 31: Simple LC Band-Pass Filter f_ch = 1/[2PI√(L₂C₂)] f_cl = 1/[2PI√(L₁C₁)]
	Fig 32: Second order active Band-Pass Filter Use FilterLab to build your active filters, in order to save your time. That program has a Wizard for building filters; even more, you can adjust the value of each component and analyze the generated Baude diagram for changes.

BAND-REJECT FILTERS

There are many good schematics used to build Band-Reject filters; here are presented briefly only two of them.

BAND-REJECT FILTERS
Graphic Representation	Description
	Fig 33: Band-Reject Attenuation Note that we have a frequency reject value here, marked as fr
	Fig 34: Simple Band-Reject Filter using RC components Considering this particular case, we have: C₁ = C₂ C₃ = 2C₁ R₁ = R₂ R₃ = R₁/2 fr = 1/2PIR₂C₂
	Fig 35: Simple Band-Reject Filter using OA (also known as "Twin T" circuit) The gain is 1, and the reject frequency is: *fr = 1/2PIR₂C₂** with the same considerations as presented above

Last word: active filters are way more powerful, but they are difficult to tune. 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 noise--excepting the LC filters.

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