1. FILTERS

2. DEFINITION
Filters - electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both.
It can be done by altering the amplitude and/or phase characteristics of a signal with respect to frequency.
Ideally, a filter will not add new frequencies to the input signal, nor will it change the component frequencies of that signal, but it will change the relative amplitudes of the various frequency components and/or their phase relationships.

3. TYPES OF FILTERS Passive filters
- made up of a combination of passive components such as
resistors (R), inductors (L) and capacitors (C).
- they do not depend upon an external power supply and/or they do not contain active (amplifying) components such as transistors and operational amplifiers.
- not restricted by the bandwidth limitations of op-amps.
- it can be used at a very high frequencies and it can handle large value of current or voltage levels than active devices.
- simplest passive filters; RC and RL filters

4. Active filters
- a combination of passive and active (amplifying) components, and require an external power source.
- Operational amplifiers are frequently used in active filter designs, besides the passives components such as resistors and capacitors, but there is no inductors used .
- The active filters are easier to design.
- At high frequencies, it is limited by the bandwidth of the operational amplifiers used.

5. Differentiate between passive filter and active filter.
made up of a combination of passive components such as resistors (R), inductors (L) and capacitors (C).
made up of op-amps, resistors and capacitors and except inductors
NO external power source require
require an external power source
are not restricted by the bandwidth limitations of op-amps
At high frequencies, it is limited by the bandwidth of the operational amplifiers used.

6. Operation of passive filter
Passive filters are based on combinations of resistors (R), inductors (L) and capacitors (C).
Inductors block high-frequency signals and conduct low-frequency signals, while capacitors do the reverse.
If the signal passes through an inductor, attenuate high-frequency signals than low-frequency signals and is a low-pass filter.
If the signal passes through a capacitor, attenuate low frequency signals than high-frequency signals and 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.

7. The types of passive filter
Low-pass filters that allow only low frequency signals to pass through
High-pass filters that allow only high frequency signals to pass through
Band-pass filters that allow signals falling within a certain frequency range to pass through.

8. Types of passive filters - “Ideal”
lowpass
highpass
bandpass

9. Realistic Filters:
lowpass
highpass
bandpass

10. The types of passive filter
Low-pass filter
- A simple passive Low Pass Filter or LPF, series a single Resistor with a single Capacitor (RC circuit)
- In this type of filter arrangement the input signal (Vin) is applied to the series combination (both the Resistor and Capacitor together) but the output signal (Vout) is taken across the capacitor only.

11. Low Pass Filter Circuit
The reactance of a capacitor varies inversely with frequency, while the value of the
resistor remains constant as the frequency changes.
At low frequencies the capacitive reactance, (Xc) of the capacitor will be very large
compared to the resistive value of the resistor, R and as a result the voltage across the
capacitor, Vc will also be large while the voltage drop across the resistor, Vr will be much
lower.
At high frequencies the reverse is true with Vc being small and Vr being large.

12. The circuit gain, Av which is given as Vout/Vin (magnitude) and is calculated as:@

13. Frequency Response Curve
Cut-off frequency
Bode plot : Frequency Response curve
Output of the filter nearly flat for low frequencies and all of the input signal is passed directly to the output, resulting in a gain of nearly 1, called unity, until it reaches its Cut-off Frequency point ( ƒc ) because the reactance of the capacitor is high at low frequencies and blocks any current flow through the capacitor.
After this cut-off frequency point the response of the circuit decreases giving a slope of -20dB/ Decade "roll-off" as signals above this frequency become greatly attenuated, until at very high frequencies the reactance of the capacitor becomes so low that it gives the effect of a short circuit condition on the output terminals resulting in zero output.

14. Cut-off Frequency and Phase Shift

15. Example No1
A Low Pass Filter circuit consisting of a resistor of 4k7Ω in series with a capacitor of 47nF is connected across a 10v sinusoidal supply. Calculate the output voltage (Vout) at a frequency of 100Hz and again at frequency of 10,000Hz or 10kHz.
At a frequency of 100Hz.
At a frequency of 10KHz.

16. High-pass filter
A High Pass Filter or HPF, is opposite to the Low Pass filter circuit, as now the two components have been interchanged with the output signal (Vout) being taken from across the resistor.
High pass filter circuit only passes signals above the selected cut-off point, ƒc eliminating any low frequency signals from the waveform.
High Pass Filter Circuit

17. In this circuit arrangement, the reactance of the capacitor is very high at low frequencies so
the capacitor acts like an open circuit and blocks any input signals at Vin until the cut-off
frequency point (ƒc) is reached.
Above this cut-off frequency point the reactance of the capacitor has reduced sufficiently
as to now act more like a short circuit allowing all of the input signal to pass directly to the
output.
Frequency Response Curve
Bode plot : Frequency Response curve
a High Pass filter is the exact opposite to that of a low pass filter.
The signal is attenuated or damped at low frequencies with the output increasing at
+20dB/Decade until the frequency reaches the cut-off point (ƒc) where again R = Xc.

18. Cut-off Frequency and Phase Shift
The circuit gain, Av which is given as Vout/Vin (magnitude) and is calculated as:
at high frequency: Xc , Vout = Vin
at low frequency : Xc ∞ , Vout = 0

19. Example No1.
Calculate the cut-off or "breakpoint" frequency (ƒc) for a simple high pass filter consisting of an 82pF capacitor connected in series with a 240kΩ resistor.

20. Band-pass filter
The cut-off frequency or ƒc point in a simple RC passive filter can be accurately controlled using just a single resistor in series with a non-polarized capacitor, and depending upon which way around they are connected either a low pass or a high pass filter is obtained.
One simple use for these types of filters is in audio amplifier applications or circuits such as in loudspeaker crossover filters or pre-amplifier tone controls.
By connecting or "cascading" together a single Low Pass Filter circuit with a High Pass Filter circuit, we can produce another type of passive RC filter that passes a selected range or "band" of frequencies that can be either narrow or wide while attenuating all those outside of this range.
This new type of passive filter arrangement produces a frequency selective filter known commonly as a Band Pass Filter or BPF.

21. Band Pass Filter Circuit
a Band Pass Filters passes signals within a certain "band" or "spread" of frequencies
without distorting the input signal or introducing extra noise.
- This band of frequencies can be any width and is commonly known as the filters Bandwidth. - Bandwidth is defined as the frequency range between two specified frequency cut-off
points (ƒc), that are 3dB below the maximum centre or resonant peak while attenuating or
weakening the others outside of these two points.
- "bandwidth” is the difference between the lower cut-off frequency ( ƒcLOWER ) and the higher cut-off frequency ( ƒcHIGHER ) points. BW = ƒH - ƒL.

22. Bode plot : Frequency Response curve
- the signal is attenuated at low frequencies with the output increasing at a slope of
+20dB/Decade until the frequency reaches the "lower cut-off" point ƒL.
At this frequency the output voltage is again 1/√2 = 70.7% of the input signal value or -3dB
(20 log (Vout/Vin)) of the input.
The output continues at maximum gain until it reaches the "upper cut-off" point ƒH where the
output decreases at a rate of -20dB/Decade attenuating any high frequency signals.
The point of maximum output gain is generally the geometric mean of the two -3dB value
between the lower and upper cut-off points and is called the "Centre Frequency" or "Resonant
Peak" value ƒr.

23. Completed Band Pass Filter Circuit
For example.
The High Pass Filter Stage.
The value of the capacitor C1 required to give a cut-off frequency ƒL of 1kHz with a resistor value of 10kΩ is calculated as:
Then, the values of R1 and C1 required for the high pass stage to give a cut-off frequency of 1.0kHz are,
R1 = 10kΩs and C1 = 15nF.
The Low Pass Filter Stage.
The value of the capacitor C2 required to give a cut-off frequency ƒH of 30kHz with a resistor value of 10kΩ is calculated as:
Then, the values of R2 and C2 required for the low pass stage to give a cut-off frequency of 30kHz are,
R = 10kΩ´s and C = 510pF. However, the nearest preferred value of the calculated capacitor value of 510pF is 560pF so this is used instead.

24. Applications of each passive filter.
Low pass filter
- Electronic low-pass filters - drive subwoofers and other types of loudspeakers, to block high
pitches that they can't efficiently broadcast.
- Radio transmitters use low-pass filters to block harmonic emissions which might cause
interference with other communications.
- The tone knob found on many electric guitars is a low-pass filter used to reduce the amount
of treble in the sound.
High Pass filters
- As an audio crossover to direct high frequencies to a tweeter while attenuating bass signals
which could interfere with, or damage, the speaker.
- High-pass filters are also used for AC coupling at the inputs of many audio amplifiers, for
preventing the amplification of DC currents which may harm the amplifier, rob the amplifier
of headroom, and generate waste heat at the loudspeakers voice coil.
Digital image processing to perform image modifications, enhancements and noise reduction.
Band Pass filter
- Audio crossovers - a class of electronic filter used in audio applications. (Most individual
loudspeaker drivers are incapable of covering the entire audio spectrum from low
frequencies to high frequencies with acceptable relative volume and lack of distortion so
most hi-fi speaker systems use a combination of multiple loudspeakers or drivers, each
catering to a different frequency band.) 

25. Cut-off frequency, Frequency pass-band and Frequency bandwidth
Low Pass filters
- known as "Cut-off", "Corner" or "Breakpoint" frequency
- defined as the frequency at which higher frequencies are blocked and lower
frequencies are passed.
- the frequency point where the capacitive reactance and resistance are equal, R = Xc
Formula:
-all the frequencies below the cut-off frequency, ƒc point that are allow to passes through the filter
-area that unaltered or no attenuation happen
- known as the filters Pass band zone
- the filter operating frequency range
High pass filters
- defined as the frequency at which higher frequencies
are passed and lower frequencies are blocked.
-all the frequencies above the cut-off frequency, ƒc point that are allow to passes through the filter

26. Cut-off frequency
Frequency pass band
Frequency bandwidth.
Band Pass filters
- known as "Cut-off", "Corner" or "Breakpoint" frequency
-have two cut-off frequencies (representing lower and upper limits).
Formula:
ƒL is the lower -3dB cut-off frequency point;
ƒH is the upper -3dB cut-off frequency point;
-area that unaltered or no attenuation happen between "lower cut-off" point ƒL and "upper cut-off" point ƒH
- known as the filters Pass band zone
- the filter operating frequency range between the "lower cut-off" point ƒL and "upper cut-off" point ƒH

27. TYPES OF FILTERS Active filters
- a combination of passive and active (amplifying) components, and require an external power source.
- Operational amplifiers are frequently used in active filter designs, besides the passives components such as resistors and capacitors, but there is no inductors used .
- The active filters are easier to design.
- At high frequencies, it is limited by the bandwidth of the operational amplifiers used.
A pole is nothing more than an RC circuit
n-pole filter  contains n-RC circuit.

28. The types of active filter
Active Low Pass Filter
- operation and frequency response is exactly the same as Passive filter
uses an op-amp for amplification and gain control.
- the simplest form of a low pass active filter is to connect an inverting or
non-inverting amplifier to the basic RC low pass filter circuit
First-order Low-pass filter 28

29. input of a non-inverting operational amplifier.
- consists a passive RC filter stage providing a low frequency path to the
input of a non-inverting operational amplifier.
- frequency response of the circuit : same as that for the passive RC filter,
except the amplitude of the output is increased by the pass band gain,
Af of the amplifier.
- non-inverting amplifier circuit gain:
Filter Gain:
- Therefore, the gain of an active low pass filter as a function of frequency
will be:
Voltage Gain:
Where:
  Af = the gain of the filter, (1 + R2/R1)
  ƒ = the frequency of the input signal in Hertz, (Hz)
  ƒc = the cut-off frequency in Hertz, (Hz)

30. Thus, the operation of a low pass active filter can be verified from the frequency gain equation above as:
                
3. At very high frequencies, ƒ > ƒc,
                                  
2. At the cut-off frequency, ƒ = ƒc,
1. At very low frequencies, ƒ < ƒc,

31. Frequency response curve
Cut-off frequency of Low-pass filter:

32. Second-order (Sallen-Key) Low-pass filter
- a first-order low pass active filter can be converted into a second-order
low pass filter simply by using an additional RC network in the input path.
- the frequency response of the second-order low pass filter is identical to
that of the first-order type except that the stop band roll-off will be twice
the first-order filters at 40dB/decade

33. Cut-off frequency of second order Low-pass filter:

34. Active High Pass Filter
- operation same as passive high pass filter circuit uses an op-amp for
amplification and gain control.
the simplest form of a high pass active filter is to connect an inverting or
non-inverting amplifier to the basic RC high pass passive filter circuit
First-order High-pass filter

35. passes high frequency signals.
- a first-order Active High Pass Filter, attenuates low frequencies and
passes high frequency signals.
- consists of a passive filter section followed by a non-inverting
operational amplifier.
- the frequency response of the circuit is the same as that of the passive
filter, except that the amplitude of the signal is increased by the gain of
the amplifier
- non-inverting amplifier circuit gain:
Filter Gain:
- Therefore, the gain of an active high pass filter as a function of
frequency will be:
Voltage Gain:
Where:
  Af = the gain of the filter, (1 + R2/R1)
  ƒ = the frequency of the input signal in Hertz, (Hz)
  ƒc = the cut-off frequency in Hertz, (Hz)

36. Just like the low pass filter, the operation of a high pass active filter can be verified from the frequency gain equation above as:
1. At very low frequencies, ƒ < ƒc,
2. At the cut-off frequency, ƒ = ƒc,
3. At very high frequencies, ƒ > ƒc,

37. Frequency response curve
Cut-off frequency of High-pass filter:

38. Second-order (Sallen-Key) High-pass filter
- A first-order high pass active filter can be converted into a second-order high
pass filter simply by using an additional RC network in the input path.
- The frequency response of the second-order high pass filter is identical to that
of the first-order type except that the stop band roll-off will be twice the first
order filters at 40dB/decade

39. Frequency response curve
Cut-off frequency of High-pass filter:

40. Band-Pass Filter
- Active Band Pass Filter is slightly different in that it is a frequency
selective filter circuit used in electronic systems to separate a signal at
one particular frequency, or a range of signals that lie within a certain
"band" of frequencies from signals at all other frequencies.
- This band or range of frequencies is set between two cut-off or corner
frequency points labelled the "lower frequency" (ƒL) and the "higher
frequency" (ƒH) while attenuating any signals outside of these two points.
- simple Active Band Pass Filter can be easily made by cascading together
a single Low Pass Filter with a single High Pass Filter as shown. 40

41. - The cut-off or corner frequency of the low pass filter (LPF) is higher than the cut-
off frequency of the high pass filter (HPF) and the difference between the
frequencies at the -3dB point will determine the "bandwidth" of the band pass
filter while attenuating any signals outside of these points.
- One way of making a very simple Active Band Pass Filter is to connect the
basic passive high and low pass filters we look at previously to an amplifying op-
amp circuit as shown.

42. - this cascading together of the individual low and high pass passive
Active Band Pass Filter
- this cascading together of the individual low and high pass passive
filters produces a wide pass band.
- the first stage of the filter will be the high pass stage that uses the
capacitor to block any DC biasing from the source.

43. - the higher corner point (ƒH) as well as the lower corner frequency cut-off point
(ƒL) are calculated the same as before in the standard first-order low and high
pass filter circuits.
- a reasonable separation is required between the two cut-off points to prevent
any interaction between the low pass and high pass stages.
- the amplifier provides isolation between the two stages and defines the overall
voltage gain of the circuit.
- the bandwidth of the filter is therefore the difference between these upper and
lower at -3dB points.
For example, if the -3dB cut-off points are at 200Hz and 600Hz then the bandwidth of the filter would be given as: Bandwidth (BW) = = 400Hz.

44. High-pass section: fcH
Low-pass section: fcL
Center frequency: fR
Where:
ƒr is the resonant or Centre Frequency
ƒL is the lower -3dB cut-off frequency point
ƒH is the upper -3db cut-off frequency point