Physics 116A Notes
David E. Pellett
Draft v.0.95
• Notes Copyright 2004 David E. Pellett unless stated otherwise.
• References:
– Text for course:
Fundamentals of Electrical Engineering, second edition, by Leonard
S. Bobrow, published by Oxford University Press (1996)
– Others as noted
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Op-Amp Basics: Outline
• Operational amplifier and negative feedback
– Introduction and historical development: long distance telephone repeaters, analog computers, commodity electronic components...
– Motivation of golden rules: i+ = i− = 0, v+ = v− .
∗ Inverting amplifier with finite gain op amp
∗ Same thing using golden rules
– Current-to-voltage converter
– Summing amplifier
– Non-inverting amplifier
– i and v across a capacitor
– Integrator
– Differentiator
2
Negative Feedback Amplifier Development
1927: Repeater amplifiers needed every few miles for long distance telephone
transmission on coaxial cable (“pipe”)
• Frequency division multiplexing
(See text, Sec. 10.6 for details)
– Frequency shifted channels 4kHz wide for multiple conversations
– 3 to 4 channels per line max possible in 1927
– Harold Black, EE at Western Electric (NYC) wanted 1000 ch or more
– Required linear (low distortion) amplifier, stable gain (unavailable)
• H. Black sketched negative feedback amplifier idea on newspaper on ferry
boat to NYC office, witnessed for patent application
• Difficult to build: stability problems–tended to “sing” (oscillate)
• Development and theoretical understanding worked out with
– Harry Nyquist, Physics Ph.D., stability criterion
– Hendrik W. Bode, Physics Ph.D., phase shift and gain as function of
frequency
– Patent awarded in 1937
3
Negative Feedback Amplifier Development
First commercial IC Op Amp, mid 1960’s: Fairchild µA709, designed by
Robert J. Widlar. Followed by µA741.
“The technology of communications advances when new physical phenomena
are discovered and applied.”
– Pierce and Noll, Signals: the science of telecommunications
4
Op Amp: LM741 Schematic
LM741 Op Amp circuit diagram:
Diagram Copyright 2000 National Semiconductor Corporation
This is pretty complicated, but we can model it (at low frequencies) with a voltage-controlled voltage source and two resistors.
5
Op Amp: Simplified Model
A simple low frequency model uses a controlled source and two resistors:
The factor Av ≡ vo/vi for the dependent source is called the voltage gain.
For LM741, Ri = 2 MΩ, Ro = 75 Ω and Av = 200000 (DC).
For LF411, Ri = 1012 Ω
Ideal case with finite voltage gain: Ri → ∞, Ro = 0
since Ri → ∞ (i.e., open circuit), i+ = i− = 0.
(This is one of the two op-amp “golden rules.”)
6
Negative Feedback: Inverting Amplifier
Now use this model to make an inverting amplifier:
Note: all points connected to ground are wired together using wires which
are not shown.
[Draw figure on the blackboard now!]
R1 and R2 make a voltage divider between the Op Amp input and output
connected to the inverting input. It provides negative feedback.
Find vi and the voltage gain with feedback, AvF ≡ vo/vin.
Use nodal analysis at node 1 where v1 = −vi.
7
Inverting Amplifier – Finite-Gain Op Amp,
Find vi using nodal analysis at node 1:
i1 = i2
v1 − vo
vin − v1
=
R1
R2
vin + vi
vi + vo
=−
R1
R2
R2 (vin + vi) = −R1 (vi + vo) = −R1 vi(1 + Av )
vi(R1 (1 + Av ) + R2 ) = −R2 vin
vi = −
vin
1
1+ R
(1 + Av )
R2
In the limit Av → ∞,
vi = 0.
8
Inverting Amplifier, Gain → ∞
This means node 1 (the inverting input) is a “virtual ground”
(held at ground by the negative feedback). This requires the noninverting input to be grounded, as is the case for the inverting
amplifier.
More generally, the negative feedback insures that
v+ = v−
in the limit that the Op Amp gain, Av → ∞.
This is the other “golden rule.” It is true for any circuit involving
an ideal Op Amp with negative feedback.
9
Inverting Amplifier – Finite-Gain Op Amp,
Find AvF :
vo = Av vi = −
AvF ≡ vo/vin = −
Av vin
1
1+ R
(1 + Av )
R2
1
.
1
1/Av + R
(1/A
+
1)
v
R2
Again, in the limit Av → ∞,
AvF = −R2 /R1 .
You can get this result directly using the ideal op amp golden rules:
v+ = v−
and
i+ = i− = 0.
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