Linear and Non-Linear

Applications of IC Op-Amps:
The Non-Inverting Op-Amp
• The non-inverting amplifier with input signal applied to the non-inverting
input and the output voltage feedback to the inverting input, i.e. in voltage-
series mode is shown below
• The op-amp provides an internal gain A. The external resistors R1 and Rf
form the feedback voltage divider circuit with an attenuation factor of β.
• Since the feedback voltage is at the inverting input, it opposes the input
voltage at the non-inverting input terminal, and hence, the feedback is
negative or degenerative.
The Closed Loop Voltage Gain
V0  A(V1  V2 )
v1  vi
R1v0
v2  v f 
R1  RF
 R1vo 
vo  A  vi  
 R1  RF 
A  R1  RF  vi
vo 
R1  RF  AR1
 vo A  R1  RF 
AF  
vi R1  RF  AR1
Since A is very large
AR1  R1  RF   R1  RF  AR1   AR1
vo RF
AF   1 
vi R1
Q For a non-inverting amplifier shown in Fig. below, determine (a) Av (b)
Vo (c) IL and (d) Io
The Inverting Op-Amp
• The inverting amplifier is shown below, and its alternate circuit arrangement
is also shown, with the circuit redrawn in a different way to illustrate how the
voltage shunt feedback is achieved.
• The input signal drives the inverting input of the op-amp through resistor R1.
• The op-amp has an open-loop gain of A, so that the output signal is much
larger than the error voltage.
• Because of the phase inversion, the output signal is 180° out-of-phase with
the input signal.
 iin  i f  ib

iin  i f
vi  vi 2 vi 2  v0

R1 Rf
vo
vi1  vi 2 
A
vo
vi 2  
A
 vin  vo / A   v0 / A   vo

R1 RF
vo ARF
AF  
vin R1  RF  AR1
Since A is very large
AR1 R1  RF R1  RF  AR1  AR1
vo RF
AF   
vi R1
Virtual Ground

• As the name indicates it is virtual, not real ground. For some purposes
we can consider it as equivalent to ground.

• In op-amps the term virtual ground means that the voltage at that
particular node is almost equal to ground voltage (0V).

• It is not physically connected to ground. This concept is very useful in

analysis of op-amp circuits and it will make a lot of calculations very
simple.
The Differential Amplifier

• The differential amplifier, also called difference amplifier, can be

constructed using a single op-amp with constant or variable gain in
closed-loop configuration.
  R f   R3  Rf
Vo  1    Vi1  Vi 2
 R1  R3  R2  R1

If R3  R f , R2  R1
Rf Rf
Vo  Vi1  Vi 2
R1 R1
If R3  R f  R2  R1  R

Vo  Vi1  Vi 2
The Adder or The Summing Amplifier

• The output of the adder shown is the linear addition of a number of input
signals. Since a virtual ground exists at the inverting input of op-amp at the
node a
V1 V2 Vn Vo
I    
R1 R2 Rn Rf

IF R1  R2   Rn  R
Rf
Vo   V1  V2   Vn 
R
The Ideal Integrator
• A circuit in which the output voltage waveform is the time integral of the
input voltage waveform is called integrator or integrating amplifier.

• The expression for the output

voltage vo(t) can be obtained by
writing Kirchhoff ’s current
equation at node a as given by
i1 = if + iB
• Since iB is negligibly small,
i1 = i f

vi (t )  va (t ) d
 C f  va (t )  vo (t ) 
R1 dt
 • However, Vb(t) = Va(t ) = 0 because the gain of the op-amp Av is
very large (Concept of virtual ground). Therefore
vi (t ) d
 C f  vo (t ) 
R1 dt
• Integrating both sides with respect to time, we get the output voltage as
vi (t ) d
0 R1 dt  0 C f dt  vo (t )  dt  C f vo (t )  vo (0)
t t

1 t
vo (t )  
R1C f  v (t )dt  v (0)
0 i o

• where vo(0) is the integration constant and is proportional to the value of the
output voltage vo(t) at t = 0. The output equation indicates that the output
voltage is directly proportional to the negative integral of the input voltage and
inversely proportional to the time constant R1Cf .
• In frequency (s) domain
1 Vo ( s ) 1
Vo ( s )   Vi ( s ) 
sR1C f Vi ( s ) sR1C f

• Letting s = jω in steady-state, we get

1
Vo ( j )   Vi ( j )
j R1C f

• Hence, the magnitude of the transfer function of the integrator is

Vo ( j ) j 1
| A |  
Vi ( j )   R1C f   R1C f
• At ω = 0, the gain of the integrator is infinite. Also the capacitor acts
as an open circuit and hence there is no negative feedback.

• Thus, the op-amp operates in open loop and hence the gain becomes
infinite (or the op-amp saturates).

• In practice, the output will never become infinite. As the frequency

increases, the gain of the integrator decreases.
Summing Integrator

1 t  v1 (t ) v2 (t ) v3 (t ) 
vo (t )  
Cf 0  R1  R2  R3 dt  vo (0)
• The integrator errors are the deviations from the ideal behaviour found
in a practical integrator circuit.
• The major sources of error are the offset and drift of the op-amp.
• The op-amp’s input offset voltage and bias current cause continuous
charging of feedback capacitor even in the absence of an input signal.
As a result, the op-amp output can drift into the positive or negative
saturation.
• Due to the above limitations, an ideal integrator is not used in practice.
• A few additional components are used along with the ideal integrator
circuit to minimise the effect of the error voltage. Such an integrator is
called practical integrator.
The Practical Integrator
• The practical integrator circuit (lossy integrator) is shown in figure below.
• Here, the feedback capacitor is shunted by a resistor Rf so that the gain of
the integrator at low frequency is limited to avoid any saturation problem.
• Since the parallel combination of resistor Rf and capacitor Cf dissipates
power, this circuit is called a lossy integrator.
• The resistor Rf provides the dc stabilisation, by limiting the low frequency
gain to –Rf/R1.
• The resistor Rcomp is given by Rcomp = R1||Rf and when Rf >> R1, Rcomp = R1.
 Vi ( s ) Vo ( s )
 sC f Vo ( s )  0
R1 Rf
1
Vo ( s )   Vi ( s)
sR1C f  R1 / R f

Vo ( s ) 1

Vi ( s ) sR1C f  R1 / R f

Vo 1 R f / R1
| A |  
Vi  2 R12C 2f  R12 / R 2f 1   R f C f 
2
• When Rf is very large, the lossy integrator will approximately become an
ideal integrator.
• At low frequencies, assuming the low level frequency to be fa, the gain is
approximately equal to Rf/R1.
• At -3 dB level the gain is 0.707 (Rf /R1).

1   R f C f 
2
 2

• Solving for f = fa, we get

1
fa 
2 R f C f
The Ideal Differentiator
• The ideal differentiator is obtained by interchanging the position of the
resistor and capacitor in the ideal integrator circuit, or it may be
constructed from a basic inverting amplifier, if the input resistor R1 is
replaced by a capacitor C1.
• The ideal differentiator circuit is shown below
 • The expression for the output voltage vo(t) can be obtained by
writing Kirchhoff ’s current equation at node ‘a’ as given by
ic = if + iB
• Since iB is negligibly small,
ic = if
d va  vo
C1  vi  va   d
C1  vi  
vo
dt Rf dt Rf

d
vo (t )   R f C1  vi (t ) 
dt
• Thus, the output vo is equal to the RfC1 times the negative instantaneous
rate of change of the input voltage vi with time.
• A differentiator performs the reverse of the integrator’s function. The upper
cut-off frequency is given by

1
fa 
2 R f C1
Summing Differentiator
• The circuit diagram of a summing differentiator is shown below, which is
derived from the simple differentiator.
• The output voltage for the summing differentiator can be written as

 dv1 (t ) dv2 (t ) 
vo (t )   R f C1  C2 
 d t dt 

For C1 = C2 , we get

 dv1 (t ) dv2 (t ) 
vo (t )   R f C1   
 d t dt 
Limitation of Ideal Differentiator
• When compared to integrator circuits, the differentiator circuits are
more susceptible to noise.
• The input noise fluctuations of small amplitudes will have large
derivatives.
• When differentiated, these noise fluctuations will generate large noise
signals at the output, which will introduce a poor signal to noise ratio.
• This problem may be minimised by placing a resistor in series with
the input capacitor.
• This modified circuit differentiates only low frequency signals with a
constant high frequency gain.
• In a differentiator circuit, the limitations due to noise, stability and
input impedance can pose problems.
• In order to minimise noise and aid in stability, a small capacitor may
be placed in parallel with Rf , which will reduce the high frequency
gain.
• In order to place a lower limit on the input impedance, a resistor may
be connected in series with the differentiating capacitor.
• The addition of either component will limit the upper range of
differentiation.
• As the frequency increases, the gain of the differentiator increases
due to the reduction of input impedance XC1.
• These limitations are overcome using a practical differentiator circuit.
The Practical Differentiator
• A practical differentiator circuit is shown below.
• This eliminates the limitations of noise and stability.
 Vo ( s ) Z f sR f C1
 
Vi ( s ) Z i 1  sR f C f  1  sR1C1 
Letting RfCf = R1C1, we get

Vo ( s ) sR f C1 sR f C1
 
1  sR1C1 
2 2
Vi ( s )  f 
1  j 
 fb 

1
fb 
2 R1C1
The Logarithmic Amplifier
The Antilogarithmic Amplifier
Voltage to Current Converter with Floating Load

I L  Vi / R1
Voltage to Current Converter with Grounded Load

I1  I 2  I L
Vi  Va  / R  Vo  Va  / R  I L
Vi  Vo  2Va  I L R
Va  Vi  Vo  I L R  / 2

Vo  2Va  Vi  Vo  I L R
Vi  I L R
I L  Vi / R
Comparators
• An op-amp comparator compares an input voltage signal with a known
voltage, called the reference voltage.
• In its simplest form, the comparator consists of an op-amp operated in
open-loop, and when fed with two analog inputs, it produces one of
the two saturation voltages ±Vsat (+VCC or -VEE ) at the output of the
op-amp.
Comparators

• The diodes D1 and D2 are connected to protect the op-amp from excessive
input voltages
Q Draw the transfer characteristics of the comparator circuit shown below, when
(a) op-amp is ideal and (b) open-loop gain of op-amp is 100000. Assume VZ1
= VZ2 = 5.5 V.
Schmitt Trigger
• The basic comparator is used in open-loop mode. Since the open loop
gain of the op-amp is very large, false triggering at the output can occur
even due to a few tenths of millivolts peak of the input or less.
• When the input changes slowly as compared to the output, noise is
coupled from the output of the comparator back to the input.
• The comparator circuit designed with a positive feedback to avoid such
an unwanted triggering is called the Schmitt Trigger or the Regenerative
Comparator.
• The positive feedback makes the gain very large and the transfer curve of
the comparator becomes closer to the ideal curve
Vref 
R2
R1  R2
Vsat  Vref   VUT

Vref 
R2
R1  R2
 Vsat  Vref   VLT
2 R2Vsat
VH  VUT  VLT 
R1  R2
Refer to the circuit shown below, R1 = 56 k Ω, R 2= 150 Ω, vi = 1Vpp sine
wave of frequency 50 Hz, Vref = 0V and op-amp 741 is used with supply
voltages of = ±15 V and the saturation voltages are ±13.5 V.
Determine the threshold voltages VUT and VLT and draw the input and
output waveforms of Schmitt trigger. Also, plot the hysteresis voltage curve.
 R2
VUT  VLT  Vsat
R1  R2
150
VUT   (13.5)  36mV
56150
VLT  VUT  36mV
Square-Wave Generator
• when vo = +Vsat, C charges from –βVsat to +βVsat and switches vo to –Vsat
and
• when vo = –Vsat, C charges from + βVsat to –βVsat and switches vo to +Vsat.
The frequency or time period of the free running multi-vibrator is determined
by the charging and discharging time of the capacitor between the voltage
levels -βVsat and +βVsat and vice versa.

 T1
 R2
Vsat  Vsat 1  (1   )e 
RC
 R1  R2
 
T1
(1   )  (1   )e RC

1  R1  2 R2
T1  RC ln  RC ln
1  R1
 1 
T  2T1  2 RC ln
1 
 R1  2 R2 
T  2 RC ln  
 R1 

1 1
fo  
T  1  
2 RC ln  
 1  
R
Considering R1  R2 , we have    0.5, T  2 RC ln 3and
2R
1 1
fo  
2 RC ln 3 2.2 RC
Q. For the circuit shown, assuming that R1 = 116 kΩ, R2 = 100 k Ω, and ±
Vsat = ± 14 V, find
(i) the time constant to produce 1 kHz output
(ii) the resistance R and
(iii) the maximum value of differential input voltage.
Triangular-Wave Generator
Triangular-Wave Generator
• Triangular-Wave Generator consists of two op-amps and several
passive components.
• The op-amp A1 forms a non-inverting comparator with hysteresis,
which is a Schmitt Trigger. The op-amp A2 forms an integrator
which integrates the output obtained from the Schmitt trigger.
• The op-amp A1 is a two level comparator whose outputs are
determined by ±Vsat.
• The square-wave output from A1 is applied to the (–) input
terminal of the op-amp A2.
• The output of A2 is a triangular wave and it is fed back as an input
to the comparator A1 through a voltage divider network formed
by R2 and R3.
• When the comparator output is at + Vsat, the effective voltage at the point P is

 Vsat   Vramp    0
R2
Vramp 
R2  R3  
 R2
Vramp   Vsat 
R3
• Similarly, at t = T2 , when the output of A1 switches from –Vsat to +Vsat ,
 R2 R2
Vramp   Vsat   Vsat
R3 R3
• Thus the peak-to-peak amplitude of the triangular wave is

vo ( pp )  Vramp   Vramp   2 R Vsat

R2
3
• The time taken for the output of A2 to switch from –Vramp to +Vramp is
half of the time period, i.e. T/2. From the basic integrator output
equation,
1 T /2 Vsat  T 
vo     Vsat dt   
R1C1 0 R1C1  2 
R1C1vo ( pp )
T 2
Vsat

• Substituting the value of vo(pp)

4R1C1 R2 1 R3
T f0  
R3 T 4 R1C1 R2
Assume that for the circuit shown below R1 = 100 kΩ, R2 = 10 kΩ, R3 = 20 kΩ,
C1 = 0.01 μF and ± Vsat = ± 14 V for the op-amps. Determine the (a) period, (b)
frequency, (c) peak value of square wave and (d) peak value of triangular wave.
Saw-Tooth Wave Generator
Precision Rectifier

• The signal processing applications with very low voltage,

current and power levels require rectifier circuits.
• The ordinary diodes cannot rectify voltages below the cut-in
voltage of the diode.
• A circuit which can act as an ideal diode or precision signal-
processing rectifier circuit for rectifying voltages which are
below the level of cut-in voltage of the diode can be designed
by placing the diode in the feedback loop of an op-amp.
Precision Diode
Full Wave Rectifier
 Clipper
• A clipper is an electronic circuit that produces an output by removing a part
of the input above or below a reference value. That means, the output of a
clipper will be same as that of the input for other than the clipped part.
• Due to this, the peak to peak amplitude of the output of a clipper will be
always less than that of the input.
• The main advantage of clippers is that they eliminate the unwanted noise
present in the amplitude of an ac signal.
• Clippers can be classified into the following two types based on the clipping
portion of the input.
• Positive Clipper
• Negative Clipper
Positive Clipper
• A positive clipper is a clipper that clips only the positive portion(s) of the
input signal.
Negative Clipper
• A negative clipper is a clipper that clips only the negative portion(s) of the
input signal.
• It can be obtained just by reversing the diode and taking the reverse polarity
of the reference voltage, in the circuit of a positive clipper.
Positive Clamper
Negative Clamper
Instrumentation Amplifier
• The instrumentation amplifier is a dedicated differential amplifier with
extremely high input impedance

• Its gain can be precisely set by a single resistor

• It has high common mode rejection ratio (CMRR) and slew rate

• It has accurate and high voltage gain

• It has low power consumption, thermal and time drift

• It is used as signal conditioner of low level signal in large amount of noise

Working
• It consist of two stages
• First stage offers high
input impedance also
allow to set gain
• Second stage is a
differential amplifier
which provide high
CMRR

R3  2 R1 
vo  1    v2  v1 
R2  Rg 
Fig. Instrumentation amplifier
The Buffer Amplifier (Stage 1)
• The analysis can be done using superposition theorem
• Provides high input impedance
• Isolates input circuit from loading
• Provides unity gain for common mode signals
 R1  R1
vo1  v  v  1 
' ''
 v1  v2
01 01  R  Rg
 g 

 R1  R1
vo 2  v02  v02  1 
' ''
 v2  v1
 R  Rg
 g 

 2 R1 
vo 2  vo1  1 
 
  v2  v1 
 R g 
Fig. Stage 1 of instrumentation amplifier
The Difference Amplifier (Stage 2)
• The analysis can be done using superposition theorem
• Provides high CMRR
 R3   R3  R3
vo  v  v   1   
' ''
 v02  v01
R2   R2  R3 
0 0
 R2
R3 R3
vo  v02  v01
R2 R2
R3
vo   v02  v01 
R2
R3  2 R1 
vo   1    v2  v1 
R2  Rg 
Fig. Stage 2 of instrumentation amplifier
Conclusion
• Advantages of Instrumentation amplifier
• The gain of the instrumentation amplifier can be varied by varying single resistor
• It improves the signal to noise ratio of the input electrical signal from the transducer
• High CMRR and slew rate
• High input impedance
• Low output impedance
• Applications of Instrumentation amplifier
• Isolation as well as providing gain in the measurement of temperature, pressure,
resistance etc
• In Data acquisition from low output transducers such as strain gauges,
Thermocouples, Wheatstone bridge measurements etc
• In Medical instrumentation, Navigation, Radar instrumentation etc
• In Audio applications involving low amplitude audio signals in noisy environments to
improve the signal to noise ratio
Mono-stable Multi-vibrator

• Mono-stable multi-vibrator, also called a one-shot multi-vibrator has a

stable state and a quasi-stable state.

• Single output pulse of adjustable time duration in response to a

triggering signal can be generated using the mono-stable multi-vibrator.

• The time duration for the output pulse is achieved by connecting the
required external components to the op-amp.
• For a low-pass RC circuit, the general solution is
vo  V fin  Vini  V fin  e  t / RC
Vfin = – Vsat and Vini = VD (diode forward voltage)

vc  Vsat  VD  Vsat  e t / RC

• At the end of time t = T, vc = –βVsat. Thus,

 Vsat  Vsat  VD  Vsat  e T / RC

 1  VD / Vsat  R2
T  RC ln  , 
 1   R1  R2
• When Vsat >> VD (0.7 V ) and R1 = R2 with β = 0.5,
T = 0.693 RC