OPERATIONAL AMPLIFIERS
An amplifier is a signal conditioning device
which alters an analog signal in a particular
manner which can be described by the
relationship:
E out (t ) = h[ E in(t )] (6.43)
where Eo is the output voltage, Ei is the input
voltage, and h is some mathematical
relationship. (a transfer function!) For many
amplifiers, h = constant and the amplifier only
alters the gain of the signal. Giving:
Eout (t )
Avo Ein (t )
where the open loop gain of the amplifier is Avo.
E o(t ) = Avo [ E i 2(t ) - E i1(t )]
The Ideal Op Amp
Figure 1. Equivalent circuit for an ideal operational amplifier.
1. The voltage gain is infinite: Avo = ∞
2. The input resistance is infinite: rin = ∞
3. The output resistance is zero: ro = 0
4. The bandwidth is infinite: BW = ∞
5. There is zero input offset voltage: EO = 0 if
Ein = 0
AXIOMS
1. The differential input voltage is zero.
2. There is no current flow into either input
terminal
3. With the loop closed, the (-) input will be
driven to the potential of the (+), or
reference, input.
Some Basic Amplifying Circuits
1. the inverting configuration
2. the noninverting configuration
3. the differential configuration (combination of
1. and 2.)
4. the summing Inverter configuration
5. the Integrator configuration
6. the Differentiator configuration
7. the Voltage Follower configuration
Standard Op-Amp Schematic Symbol
If we operate the amplifier in a closed loop
feedback mode we can analyze the resulting
circuit based on some simplifying assumptions.
Consider the circuit shown in Figure 6.24a1,
6.21a2,3, 6.20a4. We will assume that the open
loop gain is infinite, that the input impedance is
infinite, and that the output impedance is zero.
For infinite input impedance, the current flowing
into the amplifier becomes zero.
E o - E i2 E i2
=
R1
R2
R 2 + R1
=
Eo
E i2
R2
E i1 - E i 2 = E o/A = 0
R 2 + R1
=
Eo
Ei
R2
if =
The closed loop gain is given by
+
G = E o = R 2 R1 (6.46)
Ei
R2
Remarkably, the closed loop gain is dependent
only on the two resistances and not on the
amplifier gain.
Now consider the circuit shown in
Figure 6.21 (e)
d
[ - ]
[ E o - E A ] = - i R2 = - E i E A
dt
R2
E B = R 1 i I 1 = 0 (note R 1 can be 0)
I f =C
E A - E B = E o /A = 0, so E A = 0
d
Ei
C
Eo = dt
R2
1
=
Eo
E i dt
R2C
(6.52)
Note the error in Eq. (6.52) in the text in the First
Edition.
By combining the elements shown in Figure
6.241, 6.212,3, 6.204, we can perform many
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mathematical operations, hence the term
operational amplifier. Only a rudimentary
knowledge of electronics is required to design
operational amplifier circuits. The performance
of the circuit depends only on the circuit
elements and is independent of the gain of the
operational amplifier, as long as it remains
constant. Thus, operational amplifiers can be
made quite inexpensively.
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Summary of Inverting Amplifier Characteristics
Rf
1. Gain = R , unlimited in range (Rf may be 0
in
for 0 gain, may in infinite).
2. Input Impedance = Rin.
3. If = Iin, regardless of Rf (Is=0!).
4. Summing point is a virtual ground at the
same potential as the (+) input.
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Inverting Amplifier
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Summary of Non-Inverting Op Amp
1. Gain =
Rin
Rin
Rf
lower limit of unity gain where
Rin = ∞, or Rf = 0.
2. Input Impedance = ∞
3. If = Iin, regardless of Rf
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Non-Inverting Op Amp
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Summary of Differential Op Amp
1. Differential-Mode Gain (Ein1 ≠ Ein2)
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Eo
Rf
Ein1 Ein 2
Rin
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2. Common-Mode Gain (Ein1 = Ein2)
a. When
Rf
Rf
Rin
Rin
Rf
Rf
R f Rin
R f Rin
Rin
Rin
Rin Rin
Rin R f
, 0
b.When
,
c. In terms of resistor match (worst case)
Rf
4
Rin R f
where δ = fractional unbalance of
resistors (i.e. 1%=0.01) and Rin and Rf
are the nominal resistance values.
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