Chapter 4:
Operational Amplifiers (Part 2)
Minkyu Je
Operational Amplifiers
EE201 Circuit Theory
1
Outline
¢
Introduction
¢
Op-Amp Models
¢
Fundamental Op-Amp Circuits
Operational Amplifiers
EE201 Circuit Theory
2
Inverting Amplifier
¢
Nonideal op-amp model
Op-amp
v1 - vS v1 v1 - vo
+
+
=0
R1
Ri
R2
v o - v 1 v o - Av e
+
=0
R2
Ro
v e = -v 1
Operational Amplifiers
EE201 Circuit Theory
3
Inverting Amplifier
¢
Nonideal op-amp model
é1
1
1
+
+
ê
ê R1 Ri R2
ê æ 1
Aö
ç
÷÷
ê çR
ë è 2 Ro ø
v1 - vS v1 v1 - vo
+
+
=0
R1
Ri
R2
v o - v 1 v o - Av e
+
=0
R2
Ro
v e = -v 1
£
vo
=
vS
æ 1 öù
÷÷ ú
- çç
év S ù
v
é
ù
R
è 2ø ú 1 =ê ú
R
1
1 ú êëv o úû ê 01 ú
+
ë û
R2 Ro úû
- (R2 R1 )
éæ 1
1
1 öæ 1
1 ö
÷÷
÷÷çç
1 - êçç
+
+
+
ëè R1 Ri R2 øè R2 Ro ø
æ 1 öæ 1
A öù
÷÷ ú
çç
÷÷çç
è R2 øè R2 Ro ø û
Employing typical values for the circuit parameters (e.g., A = 105, Ri = 108 W,
Ro = 10 W, R1 = 1 kW, and R2 = 5 kW),
vo
= -4.9996994 » -5.0000 .
vS
£
For the ideal op-amp model,
æv ö
R
lim çç o ÷÷ = - 2 = -5 .
A ® ¥è v S ø
R1
Operational Amplifiers
EE201 Circuit Theory
4
Inverting Amplifier
¢
Ideal op-amp model
i+ = i- = 0
v+ = v£
£
Since v+ = 0, v- = 0.
£
The gain is a simple resistor ratio.
0 - vS 0 - vo
+
=0
R1
R2
£
The gain is essentially independent of
op-amp parameters.
For the ideal op-amp model,
£
Despite the op-amp parameters Ao, Ri,
and Ro that are sensitive to fabrication
tolerance, temperature, and supply
voltage, the gain is stable.
vo
R
=- 2.
vS
R1
Operational Amplifiers
EE201 Circuit Theory
5
Noninverting Amplifier
¢
Ideal op-amp model
£
Employing the ideal op-amp
model conditions,
v - = v + = v in and i - = 0 .
£
The KCL equation at the negative
terminal of the op-amp is
v in v o - v in
or
=
RI
RF
æ 1
1 ö vo
÷÷ =
v in çç
+
.
R
R
R
è I
F ø
F
£
Thus,
vo
R
= 1+ F .
v in
RI
Operational Amplifiers
EE201 Circuit Theory
6
Gain Error
¢
Gain error in an amplifier is defined as
é actual gain - ideal gain ù
GE = ê
ú ´ 100%.
ideal
gain
ë
û
£
For a standard noninverting configuration
with finite gain Ao,
v S = v in + v 1, v o = Aov in ,
and v 1 =
£
Solving these equations,
vS =
£
R1
vo º b vo .
R1 + R2
é 1 + Ao b ù
1
vo + b vo = vo ê
ú.
Ao
A
ë
û
o
Thus, the actual gain is
vo
Ao
=
.
v S 1 + Ao b
Operational Amplifiers
EE201 Circuit Theory
7
Gain Error
¢
Gain error in an amplifier is defined as
é actual gain - ideal gain ù
GE = ê
ú ´ 100%.
ideal
gain
ë
û
£
Recalling that the ideal gain is (R1 + R2)/R1
= 1/b, the gain error is
1ù
é Ao
ê 1+ A b b ú
- 100%
o
GE = ê
.
ú ´ 100% =
1
1
+
A
b
ê
ú
o
êë
úû
b
Operational Amplifiers
EE201 Circuit Theory
8
Difference Amplifier
¢
Ideal op-amp model
£
For the ideal op-amp model,
v - = v + = v in and i - = i + = 0.
£
At the inverting terminal,
v1 - v - vo - v +
= 0.
R1
R2
£
At the noninverting terminal,
v2 - v - v =
.
R3
R4
£
Solving these two equations for vo results in the expression
vo =
£
R2 æ
R ö R4
R
çç 1 + 1 ÷÷
v 2 - 2 v1 .
R1 è
R2 ø R3 + R4
R1
Note that if R4 = R2 and R3 = R1, the expression reduces to v o =
Operational Amplifiers
EE201 Circuit Theory
R2
(v 2 - v 1) .
R1
9
Summing Amplifier
¢
Ideal op-amp model
£
Applying KCL at node a, I = I1 + I2 + I3 , where
I1 =
£
V1 - Va
V - Va
V - Va
V - Vo
, I2 = 2
, I3 = 3
, and I = a
.
R1
R2
R3
Rf
Solving these equations together for vo,
æR
ö
R
R
Vo = -çç f V1 + f V2 + f V3 ÷÷ .
R2
R3 ø
è R1
Operational Amplifiers
EE201 Circuit Theory
10
Precision Diff. Voltage-Gain Amp.
¢
It is used to provide a singeended input for an analog-todigital converter.
£
Since we are interested in an
expression for vo in terms of
v1 and v2, we eliminate va
terms from the following two
node equations:
v1 - vo v1 - v a v1 - v 2
+
+
=0
R2
R1
RG
v 2 - v a v 2 - v1 v 2
+
+
= 0.
R1
RG
R2
£
Therefore,
æ
R
2R2 ö
÷÷ .
v o = (v 1 - v 2 )çç 1 + 2 +
R
R
è
1
G ø
Operational Amplifiers
EE201 Circuit Theory
11
Electronic Ammeter
¢
The unknown current, I, through RI produces a voltage, VI . VI is amplified by
the op-amp to produce a voltage, Vo, which is proportional to I. The output
voltage is measured with a simple voltmeter. We want to find the value of R2
such that 10 V appears at Vo for each milliamp of unknown current.
æ
R ö
VI = IRI and Vo = VI çç 1 + 2 ÷÷
R1 ø
è
æ
Vo
R ö 10 V
= RI çç 1 + 2 ÷÷ =
, resulting in R2 = 9 kW .
I
R
1
mA
è
1ø
Operational Amplifiers
EE201 Circuit Theory
12
Two-Stage Amplifier
¢
The circuit consists of two noninverting amplifier.
£
Output of the first amplifier:
12k ö
æ
Va = 20m ´ ç 1 +
÷ = 100 mV
3
k
è
ø
£
Output of the second amplifier:
10k ö
æ
Vo = 100m ´ ç 1 +
÷ = 350 mV
4k ø
è
£ Io
through 10-kW resistor:
Io =
Vo - Vb 350m - 100m
=
= 25 μA
10k
10k
Operational Amplifiers
EE201 Circuit Theory
13
Output Voltage Range
¢
The two op-amp circuits shown in the figure produce an output given by Vo =
8V1 - 4V2 where 1 V £ V1 £ 2 V and 2 V £ V2 £ 3 V. Determine (a) the range of Vo
and (b) if both of the circuits will produce the full range of Vo given that the dc
supplies are ±10 V.
Circuit 1
£
Circuit 2
Range of Vo :
Vo,min = 8V1,min - 4V2,max = 8 ´ 1 - 4 ´ 3 = -4 V.
Vo,max = 8V1,max - 4V2,min = 8 ´ 2 - 4 ´ 2 = 8 V.
Operational Amplifiers
EE201 Circuit Theory
14
Output Voltage Range
¢
The two op-amp circuits shown in the figure produce an output given by Vo =
8V1 - 4V2 where 1 V £ V1 £ 2 V and 2 V £ V2 £ 3 V. Determine (a) the range of Vo
and (b) if both of the circuits will produce the full range of Vo given that the dc
supplies are ±10 V.
£
Vx is given by Vx = 2V1 - V2 .
£
Range of Vx:
Vx,min = 2V1,min - V2,max = 2 ´ 1 - 3 = -1 V.
Vx,max = 2V1,max - V2,min = 2 ´ 2 - 2 = 2 V.
£
Circuit 1
Operational Amplifiers
Since both the max and min values are
within the supply range, the first op-amp
will not saturate, and this circuit will
produce the full range of Vo.
EE201 Circuit Theory
15
Output Voltage Range
¢
The two op-amp circuits shown in the figure produce an output given by Vo =
8V1 - 4V2 where 1 V £ V1 £ 2 V and 2 V £ V2 £ 3 V. Determine (a) the range of Vo
and (b) if both of the circuits will produce the full range of Vo given that the dc
supplies are ±10 V.
£
Vy is given by Vy = -8V1.
£
Range of Vy:
Vy,min = -8V1,max
= -8 ´ 2 = -16 V.
Vy,max = -8V1,min
= -8 ´ 1 = -8 V.
£
Circuit 2
Since the range of Vy is
outside the power supply
limits, this circuit will
saturate and fail to produce
the full range of Vo.
Operational Amplifiers
EE201 Circuit Theory
16
Instrumentation Amplifier
¢
An instrumentation amplifier is an amplifier of low level signals used in
process control or measurement applications, and commercially available in
single-package units.
£
The 2nd-stage amplifier is a
difference amplifier, so
R
Vo = 2 (Vo 2 - Vo1) .
R1
£
Since the inverting inputs of A1
and A2 draw no current, I flows
through the resistors as if they
were in series. Hence,
Vo1 - Vo2 = I (2R3 + R4 ) and
I=
£
Va - Vb V1 - V2
=
.
R4
R4
As a result, Vo =
Operational Amplifiers
2R3 ö
R2 æ
çç 1 +
÷(V - V1) .
R1 è
R4 ÷ø 2
EE201 Circuit Theory
17
Negative Feedback
¢
We can note one common characteristic of all circuits studied in this chapter.
£
The output is connected to the inverting input of the op-amp through a
resistive network.
£
This connection where a portion of the output voltage is fed back to the
inverting input is referred to as negative feedback.
£
Feeding back the output voltage to the negative input terminal maintains
this voltage difference near zero to allow linear operation of the op-amp.
£
The negative feedback is necessary for the proper operation of nearly all opamp circuits.
£
Our analysis of op-amp circuit is based on the assumption that the voltage
difference at the input terminals is nearly zero.
£
Almost all op-amp circuits utilize negative feedback.
Operational Amplifiers
EE201 Circuit Theory
18
Negative Feedback
¢
The positive feedback is utilized in oscillator circuits, Schmitt triggers, and
comparators.
£
The circuit shown below looks similar to the inverting amplifier.
£
However, there is one very important difference: R2 is connected to the
positive input terminal instead of the negative one, resulting in positive
feedback.
£
As a result of positive feedback, the
output value of this op-amp circuit
has two possible values, VCC or VEE .
£
Analysis of this circuit using the
ideal op-amp model does not
predict this result.
£
Note that the ideal op-amp model
may only be utilized when negative
feedback is present in the op-amp
circuits.
Operational Amplifiers
EE201 Circuit Theory
19
Summary
¢
Fundamental op-amp circuits
£
Inverting amplifier
£
Noninverting amplifier
£
Gain error
£
Difference amplifier
£
Summing amplifier
£
Precision differential voltage-gain amplifier
£
Electronic ammeter
£
Two-stage amplifier
£
Output voltage
£
Instrumentation amplifier
£
Negative feedback
Operational Amplifiers
EE201 Circuit Theory
20
End of Slides
Operational Amplifiers
EE201 Circuit Theory
21