UNIT-III
OP-A MP APPLICATIONS-II
PART-A
OP-AMP COMPARATOR
We know that when the op-amp is used in open-loop configuration any input signal, which even
slightly exceeds zero, drives the output into saturation because of very high open-loop voltage
gain of op-amp. It means that the application of a small differential input signal of appropriate
polarity causes the output to switch to its either saturation. Thus op-amp comparator is a
circuit with two inputs and one output. The two inputs can be compared with each other i.e.
one of them can be considered a reference voltage, Vref.
Figure shows an op-amp comparator circuit. A fixed reference voltage Vref is applied to the
inverting input terminal and sinusoidal signal Vin is applied to the non-inverting input terminal.
When Vin exceeds Vref the output voltage goes to positive saturation because the voltage at the
inverting input is smaller than at the non-inverting input. On the other hand, when Vin is less
than Vref the output voltage goes to negative saturation. Thus output voltage Vout changes from
one saturation level to another whenever Vin = Vref ,as illustrated in figure. In short, the
comparator is a type of an analog-to-digital converter (ADC). At any given time the output
voltage waveform shows whether Vin is greater or less than Vref. The comparator is sometimes
referred to as a volt-level detector because for a desired value of Vref, the voltage level of the
input voltage Vin can be detected.
Diodes D1 and D2 are provided in the circuit to protect the op-amp against damage due to
excessive input voltage. Because of these diodes, the differential input voltage Vd is clamped to
either + 0.7 V or -0.7 V, hence the diodes are called clamp diodes. There are some op-amps
with built-in input protection. Such op-amps need not to be provided with protection diodes.
The resistance R1 in series with Vin is used to limit the current through protection diodes D1 and
D2 while resistance R is connected between the inverting input terminal and Vref to reduce the
offset problem.
When the reference voltage Vref is negative with respect to ground, with a sinusoidal signal
applied to the non-inverting input terminal, the output voltage will be as illustrated in
figure. Obviously, the amplitude of Vin must be large enough to pass through Vref for switching
action to take place. Since the sinusoidal input signal is applied to the non-inverting terminal,
this circuit is called the non-inverting op-amp comparator.
Similarly an inverting op-amp comparator can be had by applying the sinusoidal input to the
inverting input terminal to the op-amp.
Figure shows the circuit for an inverting comparator in which the sinusoidal input signal Vin is
applied to the inverting input terminal while the reference voltage Vref is applied to the
non-inverting input terminal. In this circuit Vref is obtained by the use of a potentiometer
forming a potential divider arrangement with dc supply voltage + Vcc and – VEE. As the wiper
connected to non-inverting input terminal is moved toward + Vcc, Vref becomes more positive,
while if it is moved toward – VEE, Vref becomes more negative.
The input and output waveforms are shown in figures.Comparators are used in circuits such
as discriminators, voltage level detectors, oscillators, digital interfacing, Schmitt trigger etc.
Zero Crossing Detector :
Zero-crossing detector is an applied form of comparator. Either of the op-amp
basic comparator circuits discussed can be employed as the zero-crossing detector
provided the reference voltage Vref is made zero. Zero-crossing detector using inverting
op-amp comparator is depicted in figure. The output voltage waveform shown in figure
indicates when and in what direction an input signal Vin crosses zero volt.
SCHMITT TRIGGER USING OP-AMP
In some applications the input signal may be low frequency one (i.e. input may be a slowly
changing waveform). In such a case output voltage VOUT may not switch quickly from
one saturation state to the other. Because of the noise at the input terminals of the op-amp,
there may be fluctuation in output voltage between two saturation states (+Vsat and
–Vsat voltages). Thus zero crossings may be detected for noise voltages as well as input
signal Vin. Both of these problems can be overcome, if we use regenerative or positive
feeding causing the output voltage Vout to change faster and eliminating the false output
transitions that may be caused due to noise at the input of the op-amp. This can be achieved
using a Schmitt Trigger Circuit.
A Schmitt trigger circuit is a fast-operating voltage-level detector. When the input voltage
arrives at the upper or lower trigger levels, the output changes rapidly. The circuit operates
with almost any type of input waveform, and it gives a pulse-type output.
.
The circuit of an op-amp Schmitt trigger circuit is shown in figure. The input voltage Vin is
applied to the inverting input terminal and the feedback voltage goes to the non-inverting
terminal. This means the circuit uses positive voltage feedback instead of negative feedback,
that is, in this circuit feedback voltage aids the input voltage rather than opposing it. For
instance, assume the inverting input voltage to be slightly positive. This will produce a negative
output voltage. The voltage divider feeds back a negative voltage to the non-inverting input,
which results in a larger negative voltage. This feeds back more negative voltage until the circuit
is driven into negative saturation. If the input voltage is slightly negative instead of positive, the
circuit would be driven into the positive saturation. This is the reason the circuit is also referred
to as regenerative comparator.
When the circuit is positively saturated, a positive voltage is fed back to the non-inverting input.
This positive input holds the output in the high state. Similarly, when the output voltage is
negatively saturated, a negative voltage is fed back to the non-inverting input, holding the
output in the low state. In either case, the positive feedback reinforces the existing output
state.
The feedback fraction (β) = R2/R1 + R2
When the output is positively saturated, the reference voltage applied to the non-inverting
input is
Vref = + βVsat
When the output is negatively saturated, the reference voltage is
Vref = - βVsat
The output voltage will remain in a given state until the input voltage exceeds the reference
voltage for that state. For instance, if the output is positively saturated, the reference voltage is
+ βVsat. The input voltage Vin must be increased slightly above + βVsat to switch the output
voltage from positive to negative, as shown in figure. Once the output is in the negative state, it
will remain there indefinitely until the input voltage becomes more negative than – βVsat. Then
the output switches from negative to positive. This can be explained from the input-output
characteristics of the Schmitt trigger shown in figure.
Characteristics of the Schmitt trigger:
Assume that input voltage Vin is greater than the +βVsat, and output voltage VOUT is at its
negative extreme (point 1). The voltage across R2 in the figure is a negative quantity.
As a result, Vin must be reduced to this negative voltage level (point 2 on the characteristics)
before the output switches positively (point 3). If the input voltage is made more negative than
the –βVsat, the output remains at +VOUT (points 3 to 4). For the output to go negative once
again, Vin must be increased to the +βVsat level (point 5 on the characteristics).
In figure, the trip points are defined as the two input voltages where the output changes states.
The upper trip point (abbreviated UTP) has a value
UTP = β Vsat
and the lower trip point has a value
LTP = – β Vsat
The difference between the trip points is the hysteresis H and is given as
H = + β Vsat – (-β Vsat) = 2 β Vsat
The hysteresis is caused due to positive feedback. If there were no positive feedback, β would
equal zero and the hysteresis would disappear, because the trip points would both equal zero.
Hysteresis is desirable in a Schmitt trigger because it prevents noise form causing false
triggering.
To design a Schmitt trigger, potential divider current I2 is once again selected to be very much
larger than the op-amp input bias current. Then the resistor R2 is calculated from equation
R2 = UTP/I2
And R1 is determined from
R1 = (VOUT – UTP) / I2
ASTABLE OR FREE RUNNING MULTI VIBRATOR USING 741 OP-AMP
 The simple op-amp square wave generator is shown in the figure is also called a free
running oscillator the principle of generation of square wave output is to force an
op-amp to operate in the saturation region.
 In figure the feedback fraction (β) = R3/R2 + R3 of the output is fed back to the noninverting input terminal. Thus the reference voltage VRef is βVout and may take value as
+βVout or -βVout.
 The output is also fed back to the inverting input terminal after integrating by means of
a low pass RfC combination.
 Whenever input at the inverting input terminal just exceeds VRef switching takes place
resulting in a square wave output.
 In Astable multi-vibrator both the states are quasi stable.
 Consider an instant of time when the output is at +Vout .
 The comparator now starts charging towards +Vout through resistance Rf as shown in the
figure.
 The voltage at the non-inverting input terminal is held at +βVout by R2 and R3
combination, this condition continues as the charge on C rises until it has just exceeds
+βVout, the reference voltage.
 When the voltage at the inverting input terminal becomes just grater than this
reference voltage the output is driven to –Vout . At this instant the voltage on the
capacitor is +βVout. It begins to discharge through Rf i.e charges towards –Vout.When the
output voltage switches to –Vout the capacitor charges more and more negatively until
its voltage just exceeds –βVout. The output switches back to +Vout. The cycle repeats
itself.
 The frequency is determined by, the time taken by the capacitor to charge from –βVout
to +βVsat and vice versa.
The voltage across the capacitor as a function of time is given by
VC(t) =Vf + (Vi-Vf) e-t/RfC
Where Vf = Final Value = +Vout
Vi = Initial Value= - βVout
VC(t) =Vout + (-βVout-Vout) e-t/RfC
Therefore
VC(t) =Vout - Vout (1+ β) e-t/RfC
At t=T/2 Voltage across the capacitor reaches +βVout and switching takes place.
Therefore
VC (T/2) = βVout= Vout - Vout (1+ β) e-T/2RfC
Vout (1+ β) e-T/2RfC = Vout - βVout
Vout (1+ β) e-T/2RfC = Vout (1- β)
e-T/2RfC = (1- β) / (1+ β)
-T/2RfC = ln{(1- β) / (1+ β)}
T/2RfC = ln{(1+ β) / (1- β)}
T = 2RfC * ln{(1+ β) / (1- β)}
The output wave form is symmetrical
If R2=R3 then β=0.5
and
T = 2RfC ln(3)
and for R2=1.16R3 it can be seen that
T = 2RfC
Or
f0=1/2RfC
The output swings from +Vout to -Vout
So
Vo peak to peak = 2Vout
MONOSTABLE OR ONE SHOT MULTI VIBRATOR USING 741 OP-AMP
A mono-stable multi-vibrator (MMV) has one stable state and one quasi-stable state. The
circuit remains in its stable state till an external triggering pulse causes a transition to the quasistable state. The circuit comes back to its stable state after a time period T. Thus it generates a
single output pulse in response to an input pulse and is referred to as a one-shot or single shot.
Mono-stable multi-vibrator circuit illustrated in figure is obtained by modifying the Astable
multi-vibrator circuit by connecting a diode D1 across capacitor C so as to clamp Vc at Vd during
positive excursion.
Under steady-state condition, this circuit will remain in its stable state with the output
VOUT = + VOUT or + Vz and the capacitor C is clamped at the voltage VD (on-voltage of diode VD =
0.7 V). The voltage VD must be less than βVOUT for Vin < 0. The circuit can be switched to the
other state by applying a negative pulse with amplitude greater than βVOUT – VD to the noninverting input terminal.
When a trigger pulse with amplitude greater than βVOUT – VD is applied, Vin goes positive
causing a transition in the state of the circuit to -Vout. The capacitor C now charges
exponentially with a time constant τ = RfC toward — VOUT (diode Dl being reverse-biased).
When capacitor voltage Vc becomes more negative than –βVOUT, Vin becomes negative and,
therefore, output swings back to +VOUT (steady- state output). The capacitor now charges
towards +VOUT till Vc attain VD and capacitor C becomes clamped at VD. The trigger pulse,
capacitor voltage waveform and output voltage waveform are shown in figures respectively.
The width of the trigger pulse T must be much smaller than the duration of the output pulse
generated i.e. TP« T. For reliable operation the circuit should not be triggered again before T.
The pulse width T of mono-stable multi-vibrator is calculated as follows.
The general solution for a single time constant low passes RC circuit with Vi and Vf as initial and
final values is
VC(t) =Vf + (Vi-Vf) e-t/RfC
Where Vf =– VOUT and Vi= VD
Vc = – VOUT + (VOUT + VD) e-t/τ
At instant
So
t = T,
Vc = – β VOUT and τ = RfC
- β VOUT =- VOUT + (VOUT + VD) e-T/τ or
T = RfC loge (1 + VD/VOUT)/ 1- β
Usually VD << VOUT and if R2 = R3 so that if β = R3/(R2+R3) = ½
then,
T = RfC loge 2
T= 0.693 Rf C
TRIANGUALR WAVE FORM GENERATOR
 We know that the integrator output waveform will be triangular if the input to it is a
square-wave. It means that a triangular-wave generator can be formed by simply
cascading an integrator and a square-wave generator, as illustrated in figure.
 This circuit needs a dual op-amp, two capacitors, and at least five resistors. The
rectangular-wave output of the square-wave generator drives the integrator which
produces a triangular output waveform. The rectangular-wave swings between
+Vsat and -Vsat .
 The triangular-waveform has the same period and frequency as the square-waveform.
The amplitude of the triangular wave form decrease with an increase in its frequency
and vice versa.
 The input of integrator A2 is a square wave and its output is a triangular waveform, the
output of integrator will be triangular wave only when R4 C2 > T/ 2 where T is the period
of square wave.
 As a general rule, R4C2 should be equal to T. It may also be necessary to shunt the
capacitor C2 with resistance R5 = 10 R4 and connect an offset volt compensating network
at the non-inverting input terminal of op-amp A2 so as to obtain a stable triangular
wave. Since the frequency of the triangular-wave generator like any other oscillator, is
limited by the op-amp slew-rate, a high slew rate op-amp, like LM 301, should be used
for the generation of relatively higher frequency waveforms.
PART-B
ACTIVE FILTERS
A filter is a frequency selective circuit that passes a specified band of frequencies and blocks or
attenuates signals of frequencies outside this band. Filter may be classified on a number of
ways.
1. Analog or digital
2. Passive or active
3. Audio or radio frequency
 Analog filters are designed to process only signals while digital filters process analog
signals using digital technique. Depending on the type of elements used in their
consideration, filters may be classified as passive or active.
 Elements used in passive filters are resistors, capacitors and inductors. Active filters, on
the other hand, employ transistors or Op-Amps, in addition to the resistor and
capacitors. Depending upon the elements the frequency range is decided.
 RC filters are used for audio or low frequency operation. LC filters are employed at RF or
high frequencies.
The most commonly used filters are:
1.
2.
3.
4.
5.
Low pass filters
High pass filter
Band pass filter
Band reject filter.
All pass filter
Fig shows the frequency response characteristics of the five types of filters. The ideal response
is shown by dashed line. While the solid lines indicates the practical filter response.
Fig
 A low pass filter has a constant gain from 0 Hz to a high cutoff frequency fH. Therefore,
the bandwidth is fH. At fH the gain is down by 3db. After that the gain decreases as
frequency increases. The frequency range 0 to fH Hz is called pass band and beyond fH is
called stop band.
 Similarly, a high pass filter has a constant gain from very high frequency to a low cutoff
frequency fL. Below fL the gain decreases as frequency decreases. At fL the gain is down
by 3db. The frequency range fL Hz to ∞ (infinity) is called pass band and below fL is
called stop band.
 When the filter circuit passes signals that are above one cut-off frequency and below a
second cut-off frequency, it is called a band-pass filter, Thus a band-pass filter has a
pass band between two cut-off frequencies fLand fH where fH > fL . The bandwidth of the
band-pass filter is, therefore, equal to fH – fL where fL and fH are lower and higher cut-off
frequencies respectively.
 The band-stop or band-reject filter performs exactly opposite to the band-pass i.e. it
has a band-stop between two cut-off frequencies fH and fL and
two pass-bands: 0 < f < fH and f > fL. The ideal response of a band-stop filter is illustrated
in fig. This is also called a band-elimination or notch filter.
 The ideal response of an all-pass filter is shown in fig. This filter passes all frequencies
equally well, i.e., output and input voltages are equal in amplitude for all frequencies.
The important feature of this filter is that it provides predictable phase shift for
frequencies of different input signals.
 The filter discussed above has ideal characteristics and a sharp cut-off but unfortunately, ideal filter response is not practical because linear networks cannot produce the
discontinuities. However, it is possible to obtain a practical response that approximates
the ideal response by using special design techniques, as well as precision component
values and high-speed op-amps.
ACTIVE AND PASSIVE FILTERS – A COMPARISON
 The simplest approach to building a filter is with passive components (resistors,
capacitors, and inductors). In the R-F range it works quite well but with the lower
frequencies, inductors create problems. AF inductors are physically larger and
heavier, and therefore expensive. For lower frequencies the inductance is to be
increased which needs more turns of wire. It adds to the series resistance which
degrades the inductor’s performance.
 Input and output impedances of passive filters are both a problem, especially
below RF. The input impedance is low, that loads the source, and it varies with the
frequency. The output impedance is usually relatively high, which restricts the load
impedance that the passive filter can drive. There is no isolation between the load
impedance and the passive filter. Thus the load will have to be considered as a
component of the filter and will have to be taken into consideration while
determining filter response or design. Any change in load impedance may
significantly alter one or more of the filter response characteristics.
 An active filter uses an amplifier with R-C networks to overcome these problems of
passive filters. Originally built with vacuum tubes and then transistors, active filters
now normally are centered on op-amps. By enclosing a capacitor in a feedback loop,
the inductor (with all its low frequency problems) can be eliminated. By proper
configuration input impedance can be increased. The load is driven from the output
of the op-amp, giving very low output impedance. Not only does this improve load
drive capability, but the load is now isolated from the frequency determining
network. Thus variation in load will have no effect on the characteristics of the
active filter.
 The amplifier allows us to specify and easily adjust pass-band gain, pass-band
ripple, cutoff frequency, and initial roll-off. Because of high input impedance of the
op-amp, large value resistors can be used and therefore size and cost of the
capacitors used are reduced. By selecting a quad op-amp IC, steep roll-off can be
built in very little space and at very little cost.
 Active filters also have their limitations. High frequency response is limited by the
gain bandwidth (GBW) and slew-rate of the op-amps. High frequency op-amps are
expensive, making passive filters a more economical choice for RF applications.
Active filter needs a power supply. For op-amps this may be two supplies. Variations
in the power supplies output voltage may affect, to some extent, the signal output
from the active filter. In multi-stage applications, the common power supply
provides a bus for high frequency signals. Feedback along the power supply lines
may cause oscillations unless decoupling techniques are rigorously applied. Active
devices, and therefore active filters, are much more susceptible to RF interference
and ionization than are passive R-L-C filters. Practical considerations limit the Q of
the band-pass and notch filters to less than 50. For circuits requiring very selective
(narrow) filtering, a crystal filter, because of its high Q value, will prove to be the
best.
 Although active filters are most widely employed in the field of
communications and signal processing, they are used in one form or another in
almost all sophisticated electronic systems. Radio, TV, telephone, RADAR, spacesatellites, and biomedical equipment are but a few systems that make use of active
filters.
FIRST ORDER LOW PASS FILTER
Fig. shows a first order low pass Butter-worth filter that uses an RC network for filtering, OpAmp is used in non-inverting configuration, R1 and Rf decides the gain of the filter
.
According to voltage divider rule, the voltage at the non-inverting terminal is:
Thus the low pass filter has a nearly constant gain Af from 0 Hz to high cut off frequency fH. At
fH the gain is 0.707 Af and after fH it decreases at a constant rate with an increases in frequency.
fH is called cutoff frequency because the gain of filter at this frequency is reduced by 3dB from
0Hz.
Filter Design:
A low pass filter can be designed using the following steps:
1. Choose a value of high cutoff frequency fH.
2. Select a value of C less than or equal to 1 µF.
3. Calculate the value of R using
.
4. Finally, select values of R1 and RF to set the desired gain using
Example - 1
Design a low pass filter at a cutoff frequency of 1 kH z with a pass band gain of 2.
Solution:
Given fH = 1 kHz. Let C = 0.01 µF.
Therefore, R can be obtained as
A 20 kΩ potentiometer can be used to set the resistance R.
Since the pass band gain is 2, R1 and RF must be equal. Let R1 = R2 = 10 kΩ.
.
SECOND ORDER LOW-PASS BUTTERWORTH FILTER
A stop-band response having a 40-dB/decade at the cut-off frequency is obtained with the
second-order low-pass filter. A first order low-pass filter can be converted into a second-order
low-pass filter by using an additional RC network as shown in fig.
Fig.
Fig.
The gain of the second order filter is set by R1 and RF, while the high cut-off frequency fH is
determined by R2, C2, R3 and C3 as follows:
Furthermore, for a second-order low pass Butterworth response, the voltage gain magnitude is
given by
where,
Except for having the different cut off frequency, the frequency response of the second order
low pass filter is identical to that of the first order type as shown in fig.
Filter Design:
The design steps of the second order filter are identical to those of the first order filter as given
bellow:
1. Choose a value of high cutoff frequency fH.
2. To simplify the design calculations, set R2 = R3 = R and C2 = C3 = C. Then choose a value
of C less than 1 µF.
3. Calculate the value of R using
.
4. Finally, because of the equal resistor (R2 = R3) and capacitor (C2 = C3) values, the pass
band voltage gain AF has to be equal to 1.586. This gain is necessary to guarantee
Butterworth response. Therefore, RF = 0.586 R1. Hence choose a value of R1= 100 kΩ
and calculate the value of RF.
First Order High Pass Butterworth filter:
Fig. shows the circuit of first order high pass filter. This is formed by interchanging R and C in
low pass filter.
The lower cut off frequency is fL. This is the frequency at which the magnitude of the gain is
0.707 times its pass band value. All frequencies higher than fL are pass band frequencies with
the highest frequency determined by the closed loop bandwidth of the OP-AMP.
The magnitude of the gain of the filter
is
HIGHER ORDER FILTERS
 From the discussion made so far on the filters, it may be concluded that in the stopband the gain of the filter changes at the rate of 20 db/decade for first-order filters and
40 db/decade for second-order filters. This means that as the order of the filter is
increased, the actual stop-band response of the filter approaches its ideal stop-band
characteristics. In general, a third-order filter produces 60 db/decade, a fourth-order
filter produces 80 db/decade and so on.
 Higher-order filters, such as third, fourth, fifth, and so on, are built simply by using the
first and second-order filters.
 The simplest way to build a third-order low-pass filter is by cascading a first order filter
with a second-order. Similarly a fourth-order low-pass filter can be formed by cascading
two second-order low-pass filters. Although there is no limit to the order of the filter
that can be formed, as the order of the filter increases, so does its size. Also the
accuracy declines, in that the difference between the actual stop-band response and
the theoretical stop-band response increases with an increase in the order of the
filter.
A third-order low-pass Butterworth filter is illustrated in figure.
 The voltage-gain of the first section is optional, it can be set, whatever is required. The
voltage-gain of the second section, however affects the flatness of the overall response.
If closed-loop gain is kept 1.586, then the overall gain will be down 6 db (3 db for each
section) at the cut-off frequency. By increasing the voltage gain of the second section
slightly, cumulative loss of voltage gain is offset. By using an advanced mathematical
derivation, it can be proved that an Af , of 2 is the critical value required for a maximally
flat response.
In this case Rf = R1
When Af = 2, the cut-off frequency is given as
fH = 1 / 2∏RC
Where R and C are the resistance and capacitance of each section. At cut-off frequency, the
overall voltage gain is down 3 db. Above the cut-off frequency, the voltage gain drops at a rate
of 60 db per decade equivalent to 18 db per octave.
 A fourth-order low-pass Butterworth filter is illustrated in figure. It is formed by
cascading two second-order low-pass filters. If Af , of 1.586 is used for both sections, the
voltage gain will be down 6 db at the cut-off frequency. By using different gain for each
section, we can strike a compromise that produces a maximally flat response. An
advanced derivation shows that we need to use Af = 1.152 for the first section and
Af = 2.235 for the second section.
Also, the overall filter gain is equal to the product of the individual voltage gains of the filter
sections. Hence, the overall gain of a fourth-order filter is 1.152 x 2.235 = 2.575.
 In all our Butterworth designs, the cut-off frequency is given as 1 / 2∏RC
As with the first- and second-order filters, the third- and fourth-order high-pass filters are
formed by simply interchanging the positions of the frequency determining resistors and
capacitors in the corresponding low-pass filters. The high-order filters can be designed by
following the procedures outlined for the first- and second-order filters.
 Generally, the minimum-order filter required depends on the application specifications.
Although a high-order filter than necessary provides a better stop-band response, the
high-order filter is more complex, occupies more space and is more expensive.
 It is worth mentioning here that in all filters, the same resistance and capacitance values
are used in the bypass or R-C networks, a definite convenience in selection of
components and ease of construction. This fixes the overall gain of the high-order
filters. Furthermore, the 3-db cut-off frequency is always the same and is equal to
1/2∏RC
BAND PASS FILTER
 A band-pass filter is a circuit which is designed to pass signals only in a certain band of
frequencies while attenuating all signals outside this band. The parameters of
importance in a band-pass filter are the high and low cut-off frequencies (fH and fL), the
bandwidth (BW), the centre frequency fc, centre-frequency gain, and the selectivity or
Q.
 There are basically two types of band-pass filters viz wide band-pass and narrow
band-pass filters. Unfortunately, there is no set dividing line between the two.
However, a band-pass filter is defined as a wide band-pass if its figure of merit or quality
factor Q < 10 while the band-pass filters with Q > 10 are called the narrow band-pass
filters. Thus Q is a measure of selectivity, meaning the higher the value of Q the more
selective is the filter, or the narrower is the bandwidth (BW). The relationship between
Q, 3-db bandwidth, and the centre frequency fc is given by an equation
Q=
𝒇𝒄
𝑩𝑾
For a wide band-pass filter the centre frequency can be defined as fc =√𝐟𝐋𝐟𝐇
BW= fH - fL
Where fH and fL are respectively the high and low cut-off frequencies in Hz.In a narrow bandpass filter, the output voltage peaks at the centre frequency fc.
WIDE BANDPASS FILTER
A wide band-pass filter can be formed by simply cascading high-pass and low-pass sections and
is generally the choice for simplicity of design and performance though such a circuit can be
realized by a number of possible circuits. To form a ± 20 db/ decade band-pass filter, a firstorder high-pass and first-order low-pass sections are cascaded; for a ± 40 db/decade band-pass
filters, second-order high- pass filter and a second-order low-pass filter are connected in series,
and so on. It means that, the order of the band-pass filter is governed by the order of the highpass and low-pass filters it consists of.
A ± 20 db/decade wide band-pass filter composed of a first-order high-pass filter and a firstorder low-pass filter, is illustrated in fig, Its frequency response is illustrated in fig.
A narrow bandpass filter employing multiple feedback is depicted in figure. This filter employs
only one op-amp, as shown in the figure. In comparison to all the filters discussed so far, this
filter has some unique features that are given below.
1. It has two feedback paths, and this is the reason that it is called a multiple-feedback filter.
2. The op-amp is used in the inverting mode.
The frequency response of a narrow bandpass filter is shown in fig(b).
Generally, the narrow bandpass filter is designed for specific values of centre frequency fc and
Q or fc and BW. The circuit components are determined from the following relationships. For
simplification of design calculations each of C1 and C2 may be taken equal to C.
R1 = Q/2∏ fc CAf
R2 =Q/2∏ fc C(2Q2-Af)
and R3 = Q / ∏ fc C
where Af, is the gain at centre frequency and is given as
Af = R3 / 2R1
The gain Af however must satisfy the condition Af < 2 Q2.
The centre frequency fc of the multiple feedback filter can be changed to a new frequency fc‘
without changing, the gain or bandwidth. This is achieved simply by changing R 2 to R’2 so that
R’2 = R2 [fc/f’c]2
Band Stop Filter:
The band-pass filter passes one set of frequencies while rejecting all others. The band-stop filter
does just the opposite. It rejects a band of frequencies, while passing all others. This is also
called a band-reject or band-elimination filter. Like band-pass filters, band-stop filters
may also be classified as (i) wide-band and (ii) narrow band reject filters.
The narrow band reject filter is also called a notch filter. Because of its higher Q, which exceeds
10, the bandwidth of the narrow band reject filter is much smaller than that of a wide band
reject filter.
Wide Band-Stop (or Reject) Filter:
A wide band-stop filter using a low-pass filter, a high-pass filter and a summing amplifier
is shown in figure. For a proper band reject response, the low cut-off frequency fL of high-pass
filter must be larger than the high cut-off frequency fH of the low-pass filter. In addition, the
pass-band gain of both the high-pass and low-pass sections must be equal.
This is also called a notch filter. It is commonly used for attenuation of a single frequency such
as 60 Hz power line frequency hum. The most widely used notch filter is the twin-T network
illustrated in fig. (a). This is a passive filter composed of two T-shaped networks. One T-network
is made up of two resistors and a capacitor, while the other is made of two capacitors and a
resistor.One drawback of above notch filter (passive twin-T network) is that it has relatively low
figure of merit Q. However, Q of the network can be increased significantly if it is used with the
voltage follower, as illustrated in fig. (a). Here the output of the voltage follower is supplied
back to the junction of R/2 and 2 C. The frequency response of the active notch filter is shown
in fig (b).
Notch filters are most commonly used in communications and biomedical instruments for
eliminating the undesired frequencies.
A mathematical analysis of this circuit shows that it acts as a lead-lag circuit with a phase angle,
shown in fig. (b). Again, there is a frequency fc at which the phase shift is equal to 0°. In fig. (c),
the voltage gain is equal to 1 at low and high frequencies. In between, there is a frequency fc at
which voltage gain drops to zero. Thus such a filter notches out, or blocks frequencies near fc.
The frequency at which maximum attenuation occurs is called the notch-out frequency given by
fn = Fc = 2∏RC
Notice that two upper capacitors are C while the capacitor in the centre of the network is 2C.
Similarly, the two lower resistors are R but the resistor in the centre of the network is 1/2R.
This relationship must always be maintained.
All pass filters:
An all-pass filter is that which passes all frequency components of the input signal without
attenuation but provides predictable phase shifts for different frequencies of the input signals.
The all-pass filters are also called delay equalizers or phase correctors. An all-pass filter with the
output lagging behind the input is illustrated in figure.
The output voltage Vout of the filter circuit shown in fig. (a) can be obtained by using the
superposition theorem
Vout = -Vin +[ -jXC/R-jXC]2Vin
Substituting -jXC = [1/j2∏fc] in the above equation, we have
Vout = Vin [-1 +( 2/ j2∏Rfc)]
or Vout / Vin = 1- j2∏Rfc/1+ j2∏Rfc
where f is the frequency of the input signal in Hz.
From equations given above it is obvious that the amplitude of Vout / Vin is unity, that
is |Vout | = |Vin| throughout the useful frequency range and the phase shift between the
input and output voltages is a function of frequency.
By interchanging the positions of R and C in the circuit shown in fig. (a), the output can be
made leading the input.
These filters are most commonly used in communications. For instance, when signals are
transmitted over transmission lines (such as telephone wires) from one point to another
point, they undergo change in phase. To compensate for such phase changes, all-pass filters
are employed.