EE101: Op Amp circuits (Part 1)
M. B. Patil
mbpatil@ee.iitb.ac.in
www.ee.iitb.ac.in/~sequel
Department of Electrical Engineering
Indian Institute of Technology Bombay
M. B. Patil, IIT Bombay
Op Amps: introduction
* The Operational Amplifier (Op Amp) is a versatile building block that can be
used for realizing several electronic circuits.
M. B. Patil, IIT Bombay
Op Amps: introduction
* The Operational Amplifier (Op Amp) is a versatile building block that can be
used for realizing several electronic circuits.
* The use of Op Amps frees the user from cumbersome details such as transistor
biasing and coupling capacitors.
M. B. Patil, IIT Bombay
Op Amps: introduction
* The Operational Amplifier (Op Amp) is a versatile building block that can be
used for realizing several electronic circuits.
* The use of Op Amps frees the user from cumbersome details such as transistor
biasing and coupling capacitors.
* The characteristics of an Op Amp are nearly ideal → Op Amp circuits can be
expected to perform as per theoretical design in most cases.
M. B. Patil, IIT Bombay
Op Amps: introduction
* The Operational Amplifier (Op Amp) is a versatile building block that can be
used for realizing several electronic circuits.
* The use of Op Amps frees the user from cumbersome details such as transistor
biasing and coupling capacitors.
* The characteristics of an Op Amp are nearly ideal → Op Amp circuits can be
expected to perform as per theoretical design in most cases.
* Amplifiers built with Op Amps work with DC input voltages as well → useful in
sensor applications (e.g., temperature, pressure)
M. B. Patil, IIT Bombay
Op Amps: introduction
* The Operational Amplifier (Op Amp) is a versatile building block that can be
used for realizing several electronic circuits.
* The use of Op Amps frees the user from cumbersome details such as transistor
biasing and coupling capacitors.
* The characteristics of an Op Amp are nearly ideal → Op Amp circuits can be
expected to perform as per theoretical design in most cases.
* Amplifiers built with Op Amps work with DC input voltages as well → useful in
sensor applications (e.g., temperature, pressure)
* The user can generally carry out circuit design without a thorough knowledge of
the intricate details (next slide) of an Op Amp. This makes the design process
simple.
M. B. Patil, IIT Bombay
Op Amps: introduction
* The Operational Amplifier (Op Amp) is a versatile building block that can be
used for realizing several electronic circuits.
* The use of Op Amps frees the user from cumbersome details such as transistor
biasing and coupling capacitors.
* The characteristics of an Op Amp are nearly ideal → Op Amp circuits can be
expected to perform as per theoretical design in most cases.
* Amplifiers built with Op Amps work with DC input voltages as well → useful in
sensor applications (e.g., temperature, pressure)
* The user can generally carry out circuit design without a thorough knowledge of
the intricate details (next slide) of an Op Amp. This makes the design process
simple.
* However, as Einstein has said, we should “make everything as simple as possible,
but not simpler.” → need to know where the ideal world ends, and the real one
begins.
M. B. Patil, IIT Bombay
Op Amp 741
Actual circuit
VCC
Q8
Q12
Q13
Q14
Q9
Q15
R6
+
Q2
Q1
Symbol
−
R7
Q19
R5
CC
Q3
Q18
OUT
OUT
R10
Q4
VCC
Q21
−VEE
Q20
Q23
Q7
Q16
Q10
Q5
Q6
Q17
R9
Q11
R4
R8
R1
R3
R2
Q22
Q24
−VEE
offset adjust
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
OUT
Ro
Vi
Ri
AV Vi
Vo
Vi
AV Vi
Vo
−VEE
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
OUT
Ro
Vi
Ri
AV Vi
Vo
Vi
AV Vi
Vo
−VEE
* The external resistances (∼ a few kΩ) are generally much larger than Ro and
much smaller than Ri → we can assume Ri → ∞, Ro → 0 without significantly
affecting the analysis.
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
OUT
Ro
Vi
Ri
AV Vi
Vo
Vi
AV Vi
Vo
−VEE
* The external resistances (∼ a few kΩ) are generally much larger than Ro and
much smaller than Ri → we can assume Ri → ∞, Ro → 0 without significantly
affecting the analysis.
* VCC and −VEE (∼ ±5 V to ±15 V ) must be supplied; an Op Amp will not work
without them!
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
OUT
Ro
Vi
Ri
AV Vi
Vo
Vi
AV Vi
Vo
−VEE
* The external resistances (∼ a few kΩ) are generally much larger than Ro and
much smaller than Ri → we can assume Ri → ∞, Ro → 0 without significantly
affecting the analysis.
* VCC and −VEE (∼ ±5 V to ±15 V ) must be supplied; an Op Amp will not work
without them!
In Op Amp circuits, the supply voltages are often not shown explicitly.
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
Ro
Vi
OUT
Ri
AV Vi
Vo
Vi
AV Vi
Vo
−VEE
* The external resistances (∼ a few kΩ) are generally much larger than Ro and
much smaller than Ri → we can assume Ri → ∞, Ro → 0 without significantly
affecting the analysis.
* VCC and −VEE (∼ ±5 V to ±15 V ) must be supplied; an Op Amp will not work
without them!
In Op Amp circuits, the supply voltages are often not shown explicitly.
*
Parameter
Ideal Op Amp
741
AV
Ri
Ro
∞
∞
0
105 (100 dB)
2 MΩ
75 Ω
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
Ro
Vi
OUT
Ri
Vi
Vo
AV Vi
Vo
AV Vi
−VEE
saturation
linear
saturation
Vsat
10
10
0
−5
slope = AV
−Vsat
−0.1
0
Vi (mV)
0.1
0
−5
−10
−0.2
saturation
5
Vo (V)
Vo (V)
5
0.2
linear
saturation
−10
−5
0
Vi (V)
5
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
Ro
Vi
OUT
Ri
Vi
Vo
AV Vi
Vo
AV Vi
−VEE
saturation
linear
saturation
Vsat
10
10
saturation
5
Vo (V)
Vo (V)
5
0
−5
−5
slope = AV
−Vsat
−10
−0.2
−0.1
0
Vi (mV)
0.1
0
0.2
linear
saturation
−10
−5
0
Vi (V)
5
* The output voltage Vo is limited to ±Vsat , where Vsat ∼ 1.5 V less than VCC .
M. B. Patil, IIT Bombay
Op Amp: equivalent circuit
VCC
Ro
Vi
OUT
Ri
Vi
Vo
AV Vi
Vo
AV Vi
−VEE
saturation
linear
saturation
Vsat
10
10
saturation
5
Vo (V)
Vo (V)
5
0
−5
−5
slope = AV
−Vsat
−10
−0.2
−0.1
0
Vi (mV)
0.1
0
0.2
linear
saturation
−10
−5
0
Vi (V)
5
* The output voltage Vo is limited to ±Vsat , where Vsat ∼ 1.5 V less than VCC .
* For −Vsat < Vo < Vsat , Vi = V+ − V− = Vo /AV , which is very small
→ V+ and V− are virtually the same.
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
−VEE
Vi
Ri
AV Vi
5
Vo
Vo (V)
OUT
Ro
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
Ro
−VEE
Vi
Ri
AV Vi
5
Vo
Vo (V)
OUT
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
* Broadly, Op Amp circuits can be divided into two categories:
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
Ro
−VEE
Vi
Ri
AV Vi
5
Vo
Vo (V)
OUT
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
* Broadly, Op Amp circuits can be divided into two categories:
- Op Amp operating in the linear region
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
Ro
−VEE
Vi
Ri
AV Vi
5
Vo
Vo (V)
OUT
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
* Broadly, Op Amp circuits can be divided into two categories:
- Op Amp operating in the linear region
- Op Amp operating in the saturation region
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
Ro
−VEE
Vi
Ri
AV Vi
5
Vo
Vo (V)
OUT
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
* Broadly, Op Amp circuits can be divided into two categories:
- Op Amp operating in the linear region
- Op Amp operating in the saturation region
* Whether an Op Amp in a given circuit will operate in linear or saturation region
depends on
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
Ro
Vi
Ri
AV Vi
−VEE
5
Vo
Vo (V)
OUT
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
* Broadly, Op Amp circuits can be divided into two categories:
- Op Amp operating in the linear region
- Op Amp operating in the saturation region
* Whether an Op Amp in a given circuit will operate in linear or saturation region
depends on
- input voltage magnitude
M. B. Patil, IIT Bombay
Op Amp circuits
Vsat
10
VCC
Ro
−VEE
Vi
Ri
AV Vi
5
Vo
Vo (V)
OUT
saturation
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
* Broadly, Op Amp circuits can be divided into two categories:
- Op Amp operating in the linear region
- Op Amp operating in the saturation region
* Whether an Op Amp in a given circuit will operate in linear or saturation region
depends on
- input voltage magnitude
- type of feedback (negative or positive)
(We will take a qualitative look at feedback later.)
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
VCC
−VEE
Ro
Vi
Ri
AV Vi
saturation
5
Vo
Vo (V)
OUT
Vsat
10
iin
0
−5
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
VCC
Ro
Vi
Ri
AV Vi
saturation
5
Vo
Vo (V)
OUT
Vsat
10
iin
0
−5
−VEE
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
In the linear region,
* V+ − V− = Vo /AV , which is very small
→ V+ ≈ V−
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
VCC
Ro
Vi
Ri
AV Vi
saturation
5
Vo
Vo (V)
OUT
Vsat
10
iin
0
−5
−VEE
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
In the linear region,
* V+ − V− = Vo /AV , which is very small
→ V+ ≈ V−
* Since Ri is typically much larger than other resistances in the circuit,
we can assume Ri → ∞ .
→ iin ≈ 0
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
VCC
Ro
Vi
Ri
AV Vi
saturation
5
Vo
Vo (V)
OUT
Vsat
10
iin
0
−5
−VEE
linear
saturation
−Vsat
−10
−5
0
Vi (V)
5
In the linear region,
* V+ − V− = Vo /AV , which is very small
→ V+ ≈ V−
* Since Ri is typically much larger than other resistances in the circuit,
we can assume Ri → ∞ .
→ iin ≈ 0
These two “golden rules” enable us to understand several Op Amp circuits.
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
R2
Vi
R1
ii
Vo
RL
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Since ii (current entering the Op Amp) is zero, i1 goes through R2 .
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Since ii (current entering the Op Amp) is zero, i1 goes through R2 .
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Since ii (current entering the Op Amp) is zero, i1 goes through R2 .
„ «
„ «
Vi
R2
R2 = −
Vi .
→ Vo = V− − i1 R2 = 0 −
R1
R1
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Since ii (current entering the Op Amp) is zero, i1 goes through R2 .
„ «
„ «
Vi
R2
R2 = −
Vi .
→ Vo = V− − i1 R2 = 0 −
R1
R1
The circuit is called an “inverting amplifier.”
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Since ii (current entering the Op Amp) is zero, i1 goes through R2 .
„ «
„ «
Vi
R2
R2 = −
Vi .
→ Vo = V− − i1 R2 = 0 −
R1
R1
The circuit is called an “inverting amplifier.”
Where does the current go?
Op Amp circuits (linear region)
R2
i1
Vi
R1
ii
R2
i1
Vi
Vo
R1
ii
RL
Vo
RL
Since V+ ≈ V− , V− ≈ 0 V → i1 = (Vi − 0)/R = Vi /R .
(The non-inverting input is at real ground here, and the inverting input is at
virtual ground.)
Since ii (current entering the Op Amp) is zero, i1 goes through R2 .
„ «
„ «
Vi
R2
R2 = −
Vi .
→ Vo = V− − i1 R2 = 0 −
R1
R1
The circuit is called an “inverting amplifier.”
Where does the current go?
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
5
f = 1 kHz
1k
Vi
R2
R1
Vo
RL
Vi , Vo (Volts)
10 k
Vm = 0.5 V
Vo
0
Vi
−5
0
0.5
1
t (msec)
1.5
2
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
5
f = 1 kHz
1k
Vi
R2
R1
Vo
RL
Vi , Vo (Volts)
10 k
Vm = 0.5 V
Vo
0
Vi
−5
0
0.5
1
t (msec)
1.5
2
* The gain of the inverting amplifier is −R2 /R1 . It is called the “closed-loop gain”
(to distinguish it from the “open-loop gain” of the Op Amp which is ∼ 105 ).
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
5
f = 1 kHz
1k
Vi
R2
R1
Vo
RL
Vi , Vo (Volts)
10 k
Vm = 0.5 V
Vo
0
Vi
−5
0
0.5
1
t (msec)
1.5
2
* The gain of the inverting amplifier is −R2 /R1 . It is called the “closed-loop gain”
(to distinguish it from the “open-loop gain” of the Op Amp which is ∼ 105 ).
* The gain can be adjusted simply by changing R1 or R2 !
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
5
f = 1 kHz
1k
Vi
R2
R1
Vo
RL
Vi , Vo (Volts)
10 k
Vm = 0.5 V
Vo
0
Vi
−5
0
0.5
1
t (msec)
1.5
2
* The gain of the inverting amplifier is −R2 /R1 . It is called the “closed-loop gain”
(to distinguish it from the “open-loop gain” of the Op Amp which is ∼ 105 ).
* The gain can be adjusted simply by changing R1 or R2 !
* For the common-emitter amplifier, on the other hand, the gain −gm (RC k RL )
depends on how the BJT is biased (since gm depends on IC ).
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
5
10 k
f = 1 kHz
1k
Vi
R2
R1
Vo
RL
Vo
Vi , Vo (Volts)
Vm = 0.5 V
0
Vi
−5
0
0.5
1
t (msec)
1.5
2
* The gain of the inverting amplifier is −R2 /R1 . It is called the “closed-loop gain”
(to distinguish it from the “open-loop gain” of the Op Amp which is ∼ 105 ).
* The gain can be adjusted simply by changing R1 or R2 !
* For the common-emitter amplifier, on the other hand, the gain −gm (RC k RL )
depends on how the BJT is biased (since gm depends on IC ).
(SEQUEL file: ee101 inv amp 1.sqproj)
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
1k
R2
R1
Vo
RL
Vi , Vo (Volts)
f = 1 kHz
Vi
15
10 k
Vm = 2 V
Vo
0
−15
0
Vi
0.5
1
t (msec)
1.5
2
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
1k
R2
R1
Vo
RL
Vi , Vo (Volts)
f = 1 kHz
Vi
15
10 k
Vm = 2 V
Vo
0
−15
0
Vi
0.5
1
t (msec)
1.5
2
* The output voltage is limited to ±Vsat .
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
1k
R2
R1
Vo
RL
Vi , Vo (Volts)
f = 1 kHz
Vi
15
10 k
Vm = 2 V
Vo
0
−15
0
Vi
0.5
1
t (msec)
1.5
2
* The output voltage is limited to ±Vsat .
* Vsat is ∼ 1.5 V less than the supply voltage VCC .
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
10
10 k
f = 25 kHz
R2
1k
Vi
Vo (expected)
R1
Vo
Vi , Vo (Volts)
Vm = 1 V
Vo
0
Vi
RL
−10
0
20
40
t (µsec)
60
80
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
10
10 k
f = 25 kHz
R2
1k
Vi
Vo (expected)
R1
Vo
Vi , Vo (Volts)
Vm = 1 V
Vo
0
Vi
RL
−10
0
20
40
t (µsec)
60
80
* If the signal frequency is too high, a practical Op Amp cannot keep up with the
input due to its “slew rate” limitation.
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
10
10 k
f = 25 kHz
R2
1k
Vi
Vo (expected)
R1
Vo
Vi , Vo (Volts)
Vm = 1 V
Vo
0
Vi
RL
−10
0
20
40
t (µsec)
60
80
* If the signal frequency is too high, a practical Op Amp cannot keep up with the
input due to its “slew rate” limitation.
* The slew rate of an Op Amp is the maximum rate at which the Op Amp output
can rise (or fall).
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
10
10 k
f = 25 kHz
R2
1k
Vi
Vo (expected)
R1
Vo
Vi , Vo (Volts)
Vm = 1 V
Vo
0
Vi
RL
−10
0
20
40
t (µsec)
60
80
* If the signal frequency is too high, a practical Op Amp cannot keep up with the
input due to its “slew rate” limitation.
* The slew rate of an Op Amp is the maximum rate at which the Op Amp output
can rise (or fall).
* For the 741, the slew rate is 0.5 V /µsec.
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
10
10 k
f = 25 kHz
R2
1k
Vi
Vo (expected)
R1
Vo
Vi , Vo (Volts)
Vm = 1 V
Vo
0
Vi
RL
−10
0
20
40
t (µsec)
60
80
* If the signal frequency is too high, a practical Op Amp cannot keep up with the
input due to its “slew rate” limitation.
* The slew rate of an Op Amp is the maximum rate at which the Op Amp output
can rise (or fall).
* For the 741, the slew rate is 0.5 V /µsec.
(SEQUEL file: ee101 inv amp 2.sqproj)
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
R2
Vi
R2
Vi
R1
R1
Vo
Vo
RL
RL
Circuit 1
Circuit 2
What if the + (non-inverting) and − (inverting) inputs of the Op Amp are
interchanged?
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
R2
Vi
R2
Vi
R1
R1
Vo
Vo
RL
RL
Circuit 1
Circuit 2
What if the + (non-inverting) and − (inverting) inputs of the Op Amp are
interchanged?
R2
Our previous analysis would once again give us Vo = −
Vi .
R1
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
R2
Vi
R2
Vi
R1
R1
Vo
Vo
RL
RL
Circuit 1
Circuit 2
What if the + (non-inverting) and − (inverting) inputs of the Op Amp are
interchanged?
R2
Our previous analysis would once again give us Vo = −
Vi .
R1
However, from Circuit 1 to Circuit 2, the nature of the feedback changes from
negative to positive.
→ Our assumption that the Op Amp is working in the linear region does not hold for
R2
Circuit 2, and Vo = −
Vi does not apply any more.
R1
M. B. Patil, IIT Bombay
Op Amp circuits: inverting amplifier
R2
Vi
R2
Vi
R1
R1
Vo
Vo
RL
RL
Circuit 1
Circuit 2
What if the + (non-inverting) and − (inverting) inputs of the Op Amp are
interchanged?
R2
Our previous analysis would once again give us Vo = −
Vi .
R1
However, from Circuit 1 to Circuit 2, the nature of the feedback changes from
negative to positive.
→ Our assumption that the Op Amp is working in the linear region does not hold for
R2
Circuit 2, and Vo = −
Vi does not apply any more.
R1
(Circuit 2 is also useful, and we will discuss it later.)
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
R2
i1
R1
Vi
Vo
RL
* V+ ≈ V− = Vi
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
R2
i1
R1
Vi
Vo
RL
* V+ ≈ V− = Vi
→ i1 = (0 − Vi )/R1 = −Vi /R1 .
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
R2
i1
R1
Vi
Vo
RL
* V+ ≈ V− = Vi
→ i1 = (0 − Vi )/R1 = −Vi /R1 .
«
„
«
„
Vi
R2
* Vo = V+ − i1 R2 = Vi − −
R2 = Vi 1 +
.
R1
R1
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
R2
i1
R1
Vi
Vo
RL
* V+ ≈ V− = Vi
→ i1 = (0 − Vi )/R1 = −Vi /R1 .
«
„
«
„
Vi
R2
* Vo = V+ − i1 R2 = Vi − −
R2 = Vi 1 +
.
R1
R1
* This circuit is known as the “non-inverting amplifier.”
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
R2
i1
R1
Vi
Vo
RL
* V+ ≈ V− = Vi
→ i1 = (0 − Vi )/R1 = −Vi /R1 .
«
„
«
„
Vi
R2
* Vo = V+ − i1 R2 = Vi − −
R2 = Vi 1 +
.
R1
R1
* This circuit is known as the “non-inverting amplifier.”
* Again, interchanging + and − changes the nature of the feedback from negative
to positive, and the circuit operation becomes completely different.
M. B. Patil, IIT Bombay
Inverting or non-inverting?
R2
Vs
Vs
R1
Vo = −
RL
R2
i1
Ro
R1
Vi
R2
Vs
R1
Ri
AV Vi
Vo
RL
Vo
RL
Inverting amplifier
R2
R2
Ro
R1
R1
Vs
Vo = 1 +
RL
R2
R1
!
Vi
Vs
Ri
AV Vi
Vs
Non−inverting amplifier
* If the sign of the output voltage is not a concern, which configuration should be
preferred?
M. B. Patil, IIT Bombay
Inverting or non-inverting?
R2
Vs
Vs
R1
Vo = −
RL
R2
i1
Ro
R1
Vi
R2
Vs
R1
Ri
AV Vi
Vo
RL
Vo
RL
Inverting amplifier
R2
R2
Ro
R1
R1
Vs
Vo = 1 +
RL
R2
R1
!
Vi
Vs
Ri
AV Vi
Vs
Non−inverting amplifier
* If the sign of the output voltage is not a concern, which configuration should be
preferred?
* For the inverting amplifier, since V− ≈ 0 V , i1 = Vs /R1 → Rin = Vs /i1 = R1 .
M. B. Patil, IIT Bombay
Inverting or non-inverting?
R2
Vs
Vs
R1
Vo = −
RL
R2
i1
Ro
R1
Vi
R2
Vs
R1
Ri
AV Vi
Vo
RL
Vo
RL
Inverting amplifier
R2
R2
Ro
R1
R1
Vs
Vo = 1 +
RL
R2
R1
!
Vi
Vs
Ri
AV Vi
Vs
Non−inverting amplifier
* If the sign of the output voltage is not a concern, which configuration should be
preferred?
* For the inverting amplifier, since V− ≈ 0 V , i1 = Vs /R1 → Rin = Vs /i1 = R1 .
* For the non-inverting amplifier, Rin ∼ Ri of the Op Amp, which is a few MΩ.
→ Non-inverting amplifier is better if a large Rin is required.
M. B. Patil, IIT Bombay
Non-inverting amplifier
R2
R1
Vo
Vo
Vi
RL
Vi
RL
Consider R1 → ∞ , R2 → 0 .
M. B. Patil, IIT Bombay
Non-inverting amplifier
R2
R1
Vo
Vo
Vi
RL
Vi
RL
Consider R1 → ∞ , R2 → 0 .
Vo
R2
→1+
→ 1 , i.e., Vo = Vi .
Vi
R1
M. B. Patil, IIT Bombay
Non-inverting amplifier
R2
R1
Vo
Vo
Vi
RL
Vi
RL
Consider R1 → ∞ , R2 → 0 .
Vo
R2
→1+
→ 1 , i.e., Vo = Vi .
Vi
R1
This circuit is known as unity-gain amplifier/voltage follower/buffer.
M. B. Patil, IIT Bombay
Non-inverting amplifier
R2
R1
Vo
Vo
Vi
RL
Vi
RL
Consider R1 → ∞ , R2 → 0 .
Vo
R2
→1+
→ 1 , i.e., Vo = Vi .
Vi
R1
This circuit is known as unity-gain amplifier/voltage follower/buffer.
What has been achieved?
M. B. Patil, IIT Bombay
Loading effects
Ro
Rs
Vs
Vi
Ri
AV Vi
Vo
RL
Consider an amplifier of gain AV . We would like to have Vo = AV Vs .
M. B. Patil, IIT Bombay
Loading effects
Ro
Rs
Vs
Vi
Ri
AV Vi
Vo
RL
Consider an amplifier of gain AV . We would like to have Vo = AV Vs .
However, the actual output voltage is,
Vo =
RL
RL
Ri
× AV Vi = AV ×
×
Vs .
Ro + RL
Ro + RL
Ri + Rs
M. B. Patil, IIT Bombay
Loading effects
Ro
Rs
Vs
Vi
Ri
AV Vi
Vo
RL
Consider an amplifier of gain AV . We would like to have Vo = AV Vs .
However, the actual output voltage is,
Vo =
RL
RL
Ri
× AV Vi = AV ×
×
Vs .
Ro + RL
Ro + RL
Ri + Rs
To obtain the desired Vo , we need Ri → ∞ and Ro → 0 .
M. B. Patil, IIT Bombay
Loading effects
Ro
Rs
Vs
Vi
Ri
AV Vi
Vo
RL
Consider an amplifier of gain AV . We would like to have Vo = AV Vs .
However, the actual output voltage is,
Vo =
RL
RL
Ri
× AV Vi = AV ×
×
Vs .
Ro + RL
Ro + RL
Ri + Rs
To obtain the desired Vo , we need Ri → ∞ and Ro → 0 .
The buffer (voltage follower) provides this feature (next slide).
M. B. Patil, IIT Bombay
Op Amp buffer
A
Vs
Vi
Vo
RL
Ro
RL
Ri
AV Vi
Vo
B
Vs
Op Amp
R′
* The current drawn from the source (Vs ) is small (since Ri of the Op Amp is
large) → the buffer has a large input resistance.
M. B. Patil, IIT Bombay
Op Amp buffer
A
Vs
Vi
Vo
RL
Ro
RL
Ri
AV Vi
Vo
B
Vs
Op Amp
R′
* The current drawn from the source (Vs ) is small (since Ri of the Op Amp is
large) → the buffer has a large input resistance.
* As we have seen earlier, AV is large → Vi ≈ 0 V → VA = VB = Vs .
M. B. Patil, IIT Bombay
Op Amp buffer
A
Vs
Vi
Vo
RL
Ro
RL
Ri
AV Vi
Vo
B
Vs
Op Amp
R′
* The current drawn from the source (Vs ) is small (since Ri of the Op Amp is
large) → the buffer has a large input resistance.
* As we have seen earlier, AV is large → Vi ≈ 0 V → VA = VB = Vs .
* The resistance seen by RL is R 0 ≈ Ro , which is small → the buffer has a small
output resistance. (To find R 0 , deactivate the input voltage source (Vs )
→ AV Vi = 0 V .)
M. B. Patil, IIT Bombay
Op Amp buffer
Vo1
Vs
i1
Vo
Vo2
Rs
Ro
buffer
Vi
i2
buffer
RL
load
Ri
AV Vi
source
amplifier
M. B. Patil, IIT Bombay
Op Amp buffer
Vo1
Vs
i1
Vo
Vo2
Rs
Ro
buffer
Vi
i2
buffer
RL
load
Ri
AV Vi
source
amplifier
Since the buffer has a large input resistance, i1 ≈ 0 A,
and V+ (on the source side) = Vs → Vo1 = Vs .
M. B. Patil, IIT Bombay
Op Amp buffer
Vo1
Vs
i1
Vo
Vo2
Rs
Ro
buffer
Vi
i2
buffer
RL
load
Ri
AV Vi
source
amplifier
Since the buffer has a large input resistance, i1 ≈ 0 A,
and V+ (on the source side) = Vs → Vo1 = Vs .
Similarly, i2 ≈ 0 A, and Vo2 = AV Vs .
M. B. Patil, IIT Bombay
Op Amp buffer
Vo1
Vs
i1
Vo
Vo2
Rs
Ro
buffer
Vi
i2
buffer
RL
load
Ri
AV Vi
source
amplifier
Since the buffer has a large input resistance, i1 ≈ 0 A,
and V+ (on the source side) = Vs → Vo1 = Vs .
Similarly, i2 ≈ 0 A, and Vo2 = AV Vs .
Finally, Vo = Vo2 = AV Vs , as desired, irresepective of RS and RL .
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
Vi3
Vi2
Vi1
R3 i3
Rf
R2 i2
R1 i1
i
if
ii
Vo
RL
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
Vi3
Vi2
Vi1
R3 i3
Rf
R2 i2
R1 i1
i
if
ii
Vo
RL
V− ≈ V+ = 0 V → i1 = Vi1 /R1 , i1 = Vi2 /R2 , i1 = Vi3 /R3 .
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
Vi3
Vi2
Vi1
R3 i3
Rf
R2 i2
R1 i1
i
if
ii
Vo
RL
V− ≈ V+ = 0 V → i1 = Vi1 /R1 , i1 = Vi2 /R2 , i1 = Vi3 /R3 .
„
«
Vi1
Vi2
Vi3
i = i1 + i2 + i3 =
+
+
.
R1
R2
R3
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
Vi3
Vi2
Vi1
R3 i3
Rf
R2 i2
R1 i1
i
if
ii
Vo
RL
V− ≈ V+ = 0 V → i1 = Vi1 /R1 , i1 = Vi2 /R2 , i1 = Vi3 /R3 .
„
«
Vi1
Vi2
Vi3
i = i1 + i2 + i3 =
+
+
.
R1
R2
R3
Because of the large input resistance of the Op Amp, ii ≈ 0 → if = i, which gives,
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
Vi3
Vi2
Vi1
R3 i3
Rf
R2 i2
R1 i1
i
if
ii
Vo
RL
V− ≈ V+ = 0 V → i1 = Vi1 /R1 , i1 = Vi2 /R2 , i1 = Vi3 /R3 .
„
«
Vi1
Vi2
Vi3
i = i1 + i2 + i3 =
+
+
.
R1
R2
R3
Because of the large input resistance of the Op Amp, ii ≈ 0 → if = i, which gives,
«
„
«
„
Vi1
Vi2
Vi3
Rf
Rf
Rf
V o = V − − if Rf = 0 −
+
+
Rf = −
Vi1 +
Vi2 +
Vi3 ,
R1
R2
R3
R1
R2
R3
i.e., Vo is a weighted sum of Vi1 , Vi2 , Vi3 .
M. B. Patil, IIT Bombay
Op Amp circuits (linear region)
Vi3
Vi2
Vi1
R3 i3
Rf
R2 i2
R1 i1
i
if
ii
Vo
RL
V− ≈ V+ = 0 V → i1 = Vi1 /R1 , i1 = Vi2 /R2 , i1 = Vi3 /R3 .
„
«
Vi1
Vi2
Vi3
i = i1 + i2 + i3 =
+
+
.
R1
R2
R3
Because of the large input resistance of the Op Amp, ii ≈ 0 → if = i, which gives,
«
„
«
„
Vi1
Vi2
Vi3
Rf
Rf
Rf
V o = V − − if Rf = 0 −
+
+
Rf = −
Vi1 +
Vi2 +
Vi3 ,
R1
R2
R3
R1
R2
R3
i.e., Vo is a weighted sum of Vi1 , Vi2 , Vi3 .
If R1 = R2 = R3 = R , the circuit acts as a summer, giving
Vo = −K (Vi1 + Vi2 + Vi3 ) with K = Rf /R .
M. B. Patil, IIT Bombay
Summer example
1.2
Vi1
0.6
Vi3
Vi2
Vi1
R3
Vi3
i3
Rf
R 2 i2
R 1 i1
i
0
if
Vi2
ii
Vo
RL
R1 = R2 = R3 = 1 kΩ
−0.6
Vo
−1
−2
Rf = 2 kΩ
→ Vo = −2 (Vi1 + Vi2 + Vi3 )
SEQUEL file: ee101_summer.sqproj
−3
0
1
2
t (msec)
3
4
M. B. Patil, IIT Bombay
Summer example
1.2
Vi1
0.6
Vi3
Vi2
Vi1
R3
Vi3
i3
Rf
R 2 i2
R 1 i1
i
0
if
Vi2
ii
Vo
RL
R1 = R2 = R3 = 1 kΩ
−0.6
Vo
−1
−2
Rf = 2 kΩ
→ Vo = −2 (Vi1 + Vi2 + Vi3 )
SEQUEL file: ee101_summer.sqproj
−3
0
1
2
t (msec)
3
4
* Note that the summer also works with DC inputs. This is true about the
inverting and non-inverting amplifiers as well.
M. B. Patil, IIT Bombay
Summer example
1.2
Vi1
0.6
Vi3
Vi2
Vi1
R3
Vi3
i3
Rf
R 2 i2
R 1 i1
i
0
if
Vi2
ii
Vo
RL
R1 = R2 = R3 = 1 kΩ
−0.6
Vo
−1
−2
Rf = 2 kΩ
→ Vo = −2 (Vi1 + Vi2 + Vi3 )
SEQUEL file: ee101_summer.sqproj
−3
0
1
2
t (msec)
3
4
* Note that the summer also works with DC inputs. This is true about the
inverting and non-inverting amplifiers as well.
* Op Amps make life simpler! Think of adding voltages in any other way.
M. B. Patil, IIT Bombay