1. Lectures Week 8 The operational amplifier (“op amp”) Feedback
Comparator circuits
Ideal op amp
Unity-gain voltage follower circuit
Reading
Ch. 14, Secs , lightly,
, lightly
EE 42/100 Fall 2005
Week 8, Prof. White

2. The Operational Amplifier
The operational amplifier (“op amp”) is a basic building block used in analog circuits.
Its behavior is modeled using a dependent source.
When combined with resistors, capacitors, and inductors, it can perform various useful functions:
amplification/scaling of an input signal
sign changing (inversion) of an input signal
addition of multiple input signals
subtraction of one input signal from another
integration (over time) of an input signal
differentiation (with respect to time) of an input signal
analog filtering
nonlinear functions like exponential, log, sqrt, etc.
EE 42/100 Fall 2005
Week 8, Prof. White

3. Op Amp Circuit Symbol and Terminals
V +
positive power supply
non-inverting input + –
output
inverting input
V –
negative power supply
The output voltage can range from V – to V + (“rails”)
The positive and negative power supply voltages do not have to be equal in magnitude (example: 0V and +3V DC supplies)
EE 42/100 Fall 2005
Week 8, Prof. White

4. Op Amp Terminal Voltages and Currents
All voltages are referenced to a common node.
Current reference directions are into the op amp.
V +
ic+ ip + – io + vp – in + vo – + vn –
ic-
V – –
Vcc + +
Vcc –
common node
(external to the op amp)
EE 42/100 Fall 2005
Week 8, Prof. White

5. Op-Amp Voltage Transfer Characteristic
The op-amp is basically a
differentiating amplifier:
Vcc
slope = A >>1
vid = vp–vn
-Vcc
Regions of operation:
“negative saturation”
“linear”
“positive saturation”
 In the linear region, vo = A (vp – vn) = A vid
where A is the open-loop gain
 Typically, Vcc  20 V and A > 104
linear range: -2 mV  vid = (vp – vn)  2 mV
Thus, for an op-amp to operate in the linear region,
vp  vn
(i.e., there is a “virtual short” between the input terminals.)
EE 42/100 Fall 2005
Week 8, Prof. White

6. Achieving a “Virtual Short”
Recall the voltage transfer characteristic of an op-amp:
Q: How does a circuit maintain a virtual short at the input of an op-amp, to ensure operation in the linear region?
A: By using negative feedback. A signal is fed back from the output to the inverting input terminal, effecting a stable circuit connection. Operation in the linear region enforces the virtual short circuit.
Plotted using different scales
for vo and vp–vn
Plotted using similar scales
for vo and vp–vn vo vo
Vcc
Vcc
~10 V
slope = A >>1
slope = A >>1
~10 V
vp–vn
vp–vn
-Vcc
-Vcc
~1 mV
~10 V
EE 42/100 Fall 2005
Week 8, Prof. White

7. Negative vs. Positive Feedback
Familiar examples of negative feedback:
Thermostat controlling room temperature
Driver controlling direction of automobile
Pupil diameter adjustment to light intensity
Familiar examples of positive feedback:
Microphone “squawk” in sound system
Mechanical bi-stability in light switches
Fundamentally
pushes toward
stability
Fundamentally
pushes toward
instability or
bi-stability
EE 42/100 Fall 2005
Week 8, Prof. White

8. Op Amp Operation without Negative Feedback (Comparator Circuits for Analog-to-Digital Signal Conversion)
1. Simple comparator with 1 Volt threshold:
 V– is set to 0 Volts (logic “0”)
 V+ is set to 2 Volts (logic “1”)
 A = 100 V0
VIN 1 2
If VIN > 1.01 V,
V0 = 2V = Logic “1” +  V0
VIN 1V
If VIN < 0.99 V,
V0 = 0V = Logic “0”
2. Simple inverter with 1 Volt threshold:
 V– is set to 0 Volts (logic “0”)
 V+ is set to 2 Volts (logic “1”)
 A = 100
VIN 1 2 V0
If VIN > 1.01 V,
V0 = 0V = Logic “0”
If VIN < 0.99 V,
V0 = 2V = Logic “1” +  V0
VIN 1V
EE 42/100 Fall 2005
Week 8, Prof. White

9. Op Amp Circuits with Negative Feedback
Q: How do we know whether an op-amp is operating in the linear region?
A: We don’t, a priori.
Assume that the op-amp is operating in the linear region and solve for vo in the op-amp circuit.
If the calculated value of vo is within the range from -Vcc to +Vcc, then the assumption of linear operation might be valid. We also need stability – usually assumed for negative feedback.
If the calculated value of vo is greater than Vcc, then the assumption of linear operation was invalid, and the op-amp output voltage is saturated at Vcc.
If the calculated value of vo is less than -Vcc, then the assumption of linear operation was invalid, and the op-amp output voltage is saturated at -Vcc.
EE 42/100 Fall 2005
Week 8, Prof. White

10. Op Amp Circuit Model (Linear Region)
Ri is the equivalent resistance “seen” at the input terminals, typically very large (>1MW, as large as 1012 W for FET input op-amps), so that the input current is usually very small:
ip = –in  0 ip vp Ro io Ri vo + –
A(vp–vn) in vn
Note that significant output current (io) can flow when ip and in are negligible!
EE 42/100 Fall 2005
Week 8, Prof. White

11. ip = –in= 0 vp = vn ip + – io + vp – in + vo – + vn – Ideal Op-Amp
Assumptions:
Ri is large (105 W)
A is large (104)
Ro is small (<100 W)
Simplified circuit symbol:
power-supply terminals and
dc power supplies not shown
ip = –in= 0
vp = vn ip + – io + vp – in + vo –
Note: The resistances used in an op-amp circuit must be much larger than Ro and much smaller than Ri in order for the ideal op-amp equations to be accurate. + vn –
EE 42/100 Fall 2005
Week 8, Prof. White

12. Unity-Gain Voltage-Follower Circuit+ 
VIN vn
V0(V)
VIN(V) 1 2
IIN vp
vp = vn  V0 = VIN
( valid as long as V –  V0  V + )
Note that the analysis of this simple (but important) circuit required only one of the ideal op-amp rules.
Q: Why is this circuit important (i.e., what is it good for)?
A: A “weak” source can drive a “heavy” load; in other words, the source VIN only needs to supply a little power (since IIN = 0), whereas the output can drive a power-hungry load (with the op-amp providing the power).
EE 42/100 Fall 2005
Week 8, Prof. White

13. What’s Inside an Op-Amp?
EE 42/100 Fall 2005
Week 8, Prof. White

14. Lecture Week 8 (continued)
Op-Amp circuits continued: examples:
Inverting amplifier circuit
Summing amplifier circuit
Non-inverting amplifier circuit
Differential amplifier circuit
Current-to-voltage converter circuit
Reading
(Note: amplifiers are discussed
in great detail in Ch. 11)
EE 42/100 Fall 2005
Week 8, Prof. White

15. Review: Negative Feedback
Negative feedback is used to “linearize” a high-gain differential amplifier.
With feedback
Without feedback V0 + 
VIN V0 +  V+ V
V0(V) V0 5 5 4 3 2 1
VIN 1 2 3 4 5 5
EE 42/100 Fall 2005
Week 8, Prof. White

16. Gain vs. Frequency of the Basic Op-Amp without and
with Feedback (Hambley, Sec. 14.5)
Facts:
The open-loop gain of an op--amp (no feedback from output to inverting input)
is constant from DC to a frequency fBOL, after which the open-loop gain drops at a rate of
-20dB/decade of frequency (Fig – next slide).
As the negative feedback is made stronger, the gain decreases but the bandwidth
of the amplifier increases. (Bandwidth of an amplifier is the frequency range over which
it amplifies.)
One can show (Hambley) that the product of the dc gain and the bandwidth
is a constant that is independent of the amount of negative feedback. This is called the
gain-bandwidth product for the op-amp. Example: The op-amp you’ll use in the lab
(National Semiconductor LMC6482) is rather like that shown in Hambley Fig ,
with a DC open-loop gain of 100 dB (voltage amplification of 105) out to about only
40 Hz! With negative feedback, however, the amplifier has an increasing bandwidth
but with decreasing gain. The unity gain (0 dB) bandwidth is 4 MHz; one can get
40 dB of gain (Vout/Vin = 100) from DC to about 40 kHz.
EE 42/100 Fall 2005
Week 8, Prof. White

17. Op-Amp Frequency Response with and without Negative Feedback
EE 42/100 Fall 2005
Week 8, Prof. White

18. Application of Voltage Follower: Sample and Hold Circuit
EE 42/100 Fall 2005
Week 8, Prof. White

19. Inverting Amplifier CircuitRf is Rs in – + + vn – + vo – vs + – + vp –
in = 0  is = -if
vp = 0  vn = 0
EE 42/100 Fall 2005
Week 8, Prof. White

20. Analysis using Realistic Op-Amp Model
In the analysis on the previous slide, the op-amp was assumed to be ideal, i.e.
Ri = ; A = ; Ro = 0
In reality, an op-amp has finite Ri, finite A, non-zero Ro, and usually is loaded at its output terminals with a load resistance RL.
EE 42/100 Fall 2005
Week 8, Prof. White

21. Summing Amplifier CircuitRa ia if Rf Rb in va + – ib – + + vn – + vo – vb + – ic + vp –
superposition ! Rc vc + –
in = 0  ia + ib + ic = -if
vp = 0  vn = 0
EE 42/100 Fall 2005
Week 8, Prof. White

22. Application: Digital-to-Analog Conversion
A DAC can be used to convert the digital representation
of an audio signal into an analog voltage that is then
used to drive speakers -- so that you can hear it!
Binary
Analog
number
output
(volts)
0 1 .5
“Weighted-adder D/A converter”
0 1 0 1
0 1 1
0 1 0
1.5 2
4-Bit D/A 8V - + V0 5K
80K
40K
20K
10K S1
+ - S3 S2 S4
0 1
1 0 1
2.5 3
3.5 4
1 0
0 1
1 0 1 1
4.5 5
5.5 6
6.5 7
7.5
S1 closed if LSB =1
S " if next bit = 1
S " if " " = 1
S " if MSB = 1
(Transistors are used as electronic switches)
MSB
LSB
EE 42/100 Fall 2005
Week 8, Prof. White

23. Characteristic of 4-Bit DAC8 7 6 5
Analog Output (V) 4 3 2 1 2 4 6 8 10 12 14 16
0000
0001
1111
1000
0100
Digital Input
EE 42/100 Fall 2005
Week 8, Prof. White

24. Noninverting Amplifier CircuitRf Rs in – + + ip vn + vo – + – Rg vg + – vp –
ip = 0  vp = vg  vn = vg
in = 0  Rs & Rf form a voltage divider
EE 42/100 Fall 2005
Week 8, Prof. White

25. Differential Amplifier CircuitRa Rb in + vn – va + – Rc – + ip + vo – + vp – vb + – Rd
in = 0 
ip = 0 
EE 42/100 Fall 2005
Week 8, Prof. White

26. Differential Amplifier (cont’d)
More usual version of differential amplifier (Horowitz
& Hill, “Art of Electronics”, 2nd Ed., p (part of their
“smorgasbord” of op-amp circuits):
Let Ra = Rc = R1
and Rb = Rd = R2
Then v0 = (R2 / R1)(vb – va)
To get good “common-mode rejection” (next slide)
you need well-matched resistors (Ra and Rc,
Rb and Rd), such as 100kW 0.01% resistors
EE 42/100 Fall 2005
Week 8, Prof. White

27. Differential Amplifier – Another Perspective
Redefine the inputs in terms of two other voltages:
1. differential mode input vdm  vb – va
2. common mode input vcm  (va + vb)/2
so that
va = vcm – (vdm/2) and vb = vcm + (vdm/2)
Then it can be shown that
“common mode gain”
“differential mode gain”
EE 42/100 Fall 2005
Week 8, Prof. White

28. Differential Amplifier (cont’d)
An ideal differential amplifier amplifies only the differential mode portion of the input voltage, and eliminates the common mode portion.
provides immunity to noise (common to both inputs)
If the resistors are not perfectly matched, the common mode rejection ratio (CMRR) is finite:
EE 42/100 Fall 2005
Week 8, Prof. White

29. Op-Amp Current-to-Voltage Converter
EE 42/100 Fall 2005
Week 8, Prof. White

30. Summary Voltage transfer characteristic of op-amp:
A feedback path between an op-amp’s output and its inverting input can force the op-amp to operate in its linear region, where vo = A (vp – vn)
An ideal op amp has infinite input resistance Ri, infinite open-loop gain A, and zero output resistance Ro. As a result, the input voltages and currents are constrained:
vp = vn and ip = -in = 0 vo
Vcc
~10 V
slope = A >>1
vp–vn
-Vcc
~1 mV
EE 42/100 Fall 2005
Week 8, Prof. White

31. EE 42/100 Fall 2005
Week 8, Prof. White

32. EE 42/100 Fall 2005
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33. EE 42/100 Fall 2005
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34. Differentiator (from Horowitz & Hill)
EE 42/100 Fall 2005
Week 8, Prof. White

35. Op-Amp Imperfections
Discussed in Hambley, pp (of interest if you need to
use an op-amp in a practical application!)
Linear range of operation: Input and output impedances not
ideal values; gain and bandwidth limitations (including gain-bandwidth product);
Nonlinear limitations: Output voltage swing (limited by “rails” and more); output current limits; slew-rate limited (how fast can the output voltage change: 0.5V/ms for the 741 op-amp, to 6000V/ms for high
slew-rate op-amp); full-power bandwidth
DC imperfections: Offset current (input currents don’t sum
exactly to zero); offset voltage; There are pins (denoted offsets) for inputs to help control this.
EE 42/100 Fall 2005
Week 8, Prof. White