en0216_opamp
TRANSCRIPT
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Operational Amplifiers
(revision)+VCC (15V)
-VCC (-15V)
output
non-inverting
input
inverting input
+
-
IMPORTANT: Basic assumptions are
• very high voltage gain (2 x 105)
ideal is infinite
• very high input impedance (2 Mohms)
ideal is infinite
•Also an ideal op.amp has infinite bandwidth
( roll-off starts at 10 Hz)
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“741” Op. Amp.
top view
1
2
3
4 5
6
7
8offset null
inverting
input
non-inverting
input
-VCC
(-15V)
+VCC
(+15V)
output
offset null
-
+
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Consequences of basic
assumptions are:
1. Vout = A(V+ - V-)
or V+ - V- = Vout/A
which means that if A is large
then V+ ≈ V-
The voltage difference at the
inputs tends to be very small
If one of the op. amp. inputs is
earthed then the other input is a
VIRTUAL EARTH.
2. The consequence is that the
inputs draw no (little) current
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Op. Amp stability with
feedback
Amplifier
Ao
Feedback
block β
XiXia
-
Xo
Xf
Xf = βXo
Xia = Xi - Xf (negative feedback)
Xo = AoXia
∴Xia = Xi - βXo
∴X0 = AoXi -A0βXo
βo
o
i
o
A
A
X
X
+=
1
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What does the feedback equation
approximate to if Ao β is much
greater than 1?
If Aoβ >> 1 then
β
1≅
i
o
X
X
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Definitions
• Open-loop gain: The voltage gain
of the op amp without feedback.
• Feedback factor (β): The fraction
of the output voltage fed back to the
input by a (negative) feedback
network.
• Loop gain (Aoβ): The product of
the open-loop gain and the feedback
factor.
• Closed-loop gain (Afb): The
voltage gain of an op amp with
feedback
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Ideal case:
ββ
1
1≅
+==∴
∞→
o
o
in
out
fb
o
A
A
V
VA
,A
Loop-gain crossing frequency (fCL):
The frequency at which the closed loop-gain is
equal to the open-loop gain.
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Oscillations
An op amp is usually used with
feedback. Negative feedback
gives several advantages:
• increased input impedance
• reduced output impedance
• reduced distortion
• increased stability
(Positive feedback may give
oscillations).
With negative feedback an op
amp circuit may generate an
oscillatory signal at the
frequency at which it has a
phase angle of -180o.
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The circuit will then generate an
a.c. output with no input (it
draws dc power from the power
supply).
An op amp negative feedback
circuit will oscillate if there
exists a frequency at which the
magnitude of the loop gain is
greater than unity at a phase-
shift of -180o.
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Sinusoidal oscillator
using positive feedback
Use a positive feedback loop
containing a frequency
selective network.
The loop is designed to have a
unity gain at a single frequency
determined by the selective
network.
To determine if a circuit will
oscillate:
(a) Find the magnitude of the
loop-gain at which the phase
angle is -180o
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Or
(b) Find the value of the phase angle at
which the magnitude of the loop-gain
is unity.
Phase-margin: The difference
between the phase shift of a signal
through a system and the phase shift
that will cause the system to oscillate.
(usually an extra -180o).
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Positive feedback
Ao
β
+Vin Vout
Note that in practice no input is required
for oscillations)
βo
o
fbA
AA
−=
1
Loop-gainNote - sign
At a specific frequency, ωo ,the loop-gain is
unity and Afb is infinite.
At this frequency the circuit will have a finite
output for no input and is by definition an
oscillator.
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Wien Bridge Oscillator
Without amplitude stabilisation)
R1
R2
-
+
C R
RC
Vo
Vin
Zs
Zp
Op amp is in non-inverting configuration.
Closed-loop gain is (1 + R2/R1)
Feedback network transfer function is :
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( )( )
( )
( )
−+
+
=∴
++
+
=∴
+=
RCRCj
R
R
jL
RCsRCs
R
R
sL
ZZ
Z
sV
sV
sp
p
o
i
ωω
ω1
3
1
13
1
1
2
1
2
Hence loop-gain is a real number (phase is zero)
at ω = 1/RC.
To obtain sustained oscillations at this frequency
set the magnitude of L(jω) to unity by selecting
R2 = 2R1
In practice start oscillations by choosing R2/R1
to be slightly greater than 2.
Note that these oscillations have no amplitude
control.
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For oscillations:
L(jωo) = Ao (jωo) β (jωo) = 1
At ωo the phase of the loop-gain is zero, and its
magnitude is 1.
This criterion should be satisfied at one frequency,
ωo, only.
ωo is determined solely by the phase characteristic
of the feedback loop.
If Aoβ becomes less than unity oscillations cease
and if Aoβ becomes greater than unity oscillations
will grow in amplitude.
A non-linear circuit is required for gain control.
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Amplitude control
-
+
R2 R
1a b
Vo
R2 is adjusted until oscillations just start to grow.
As oscillations grow the diodes conduct causing
the effective resistance between a and b to decrease.
Equilibrium will be reached at the output amplitude
that causes the loop-gain to be exactly unity.
The output amplitude can be varied by adjusting
R2.
Note that the output at a has lower distortion than at
b.
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To ensure that oscillations start design
the circuit so that Aoβ is slightly greater
than 1.
Turn power on and oscillations grow.
At the desired output level the non-linear
network comes into action and causes the
loop-gain to be reduced to exactly 1.
This gives sustained oscillations at the
required amplitude.
If the loop-gain is reduced below 1 the
amplitude of the output will diminish -
which is detected by the non-linear
network which will cause the
loop-gain to increase to exactly 1.
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RC Phase-shift oscillator
-
+
C CC
R R R
R f
R1
Inverting configuration gives 180o phase shift.
The three R-C sections give 180o phase shift at
a particular frequency.
Hence oscillations occur at this frequency
if the loop-gain is exactly 1.
Feedback ratio
−+
−
=
C
R
Cj
C
RR
R
ωωω
β2
3322
3
3
615
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For 180o phase-shift,
RC
C
R
C
6
1
61 2
33
=⇒
=
ω
ωω
giving
29
1−=β
(the minus sign confirms that the cascade
inverts the feedback at the oscillation frequency)
For unity loop-gain
291
=R
Rf
In practice Rf is made adjustable to allow for
small differences in component values, and to allow
for the loading caused by R1.