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Lectures Week 8

- The operational amplifier (op amp)
- Feedback
- Comparator circuits
- Ideal op amp
- Unity-gain voltage follower circuit
- Reading
- Ch. 14, Secs. 14.1-14.3, 14.5-14.7 lightly,
- 14.8-14.9, 14.10 lightly

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.

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)

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)

Op-Amp Voltage Transfer Characteristic

vo

The op-amp is basically a differentiating

amplifier

Vcc

slope A gtgt1

vid vpvn

-Vcc

positive saturation

linear

negative saturation

Regions of operation

- ? In the linear region, vo A (vp vn) A vid
- where A is the open-loop gain
- ? Typically, Vcc ? 20 V and A gt 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.)

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 vpvn

Plotted using similar scales for vo and vpvn

vo

vo

Vcc

Vcc

slope A gtgt1

slope A gtgt1

10 V

10 V

vpvn

vpvn

-Vcc

-Vcc

1 mV

10 V

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

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

- 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

Op Amp Circuits with Negative Feedback

- Q How do we know whether an op-amp is operating

in the linear region? - A We dont, 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.

Op Amp Circuit Model (Linear Region)

Ri is the equivalent resistance seen at the

input terminals, typically very large (gt1MW, 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(vpvn)

in

vn

Note that significant output current (io) can

flow when ip and in are negligible!

Ideal Op-Amp

- Assumptions
- Ri is large (?105 W)
- A is large (?104)
- Ro is small (lt100 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

Unity-Gain Voltage-Follower Circuit

vn

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).

Whats Inside an Op-Amp?

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)

Review Negative Feedback

Negative feedback is used to linearize a

high-gain differential amplifier.

With feedback

Without feedback

V0(V)

V0

5

5

4

3

2

1

VIN

1

2

3

4

5

?5

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. 14.20 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 youll use in the lab - (National Semiconductor LMC6482) is rather like

that shown in Hambley Fig. 14.22, - 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.

Op-Amp Frequency Response with and without

Negative Feedback

Application of Voltage Follower Sample

and Hold Circuit

Inverting Amplifier Circuit

if

Rf

Rs

in

is

vn

vo

vs

vp

in 0 ? is -if vp 0 ? vn 0

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.

Summing Amplifier Circuit

Ra

if

ia

Rf

in

Rb

va

ib

vn

vo

vb

ic

vp

superposition !

Rc

vc

in 0 ? ia ib ic -if vp 0 ? vn 0

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!

Analog

Binary

output

number

(volts)

0

0

0

0

0

0

0

0 1

.5

Weighted-adder D/A converter

S1 closed if LSB 1 S2 " if next bit

1 S3 " if " " 1 S4 "

if MSB 1

(Transistors are used as electronic switches)

LSB

MSB

Characteristic of 4-Bit DAC

8

7

6

5

Analog Output (V)

4

3

2

1

0

0

2

4

6

8

10

12

14

16

0000

1111

0001

1000

0100

Digital Input

Noninverting Amplifier Circuit

Rf

Rs

in

ip

vo

vn

Rg

vg

vp

ip 0 ? vp vg ? vn vg in 0 ? Rs Rf

form a voltage divider

Differential Amplifier Circuit

Ra

Rb

in

vn

va

Rc

ip

vo

vp

vb

Rd

in 0 ? ip 0 ?

Differential Amplifier (contd)

More usual version of differential amplifier

(Horowitz Hill, Art of Electronics, 2nd

Ed., p. 184-5 (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

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

Differential Amplifier (contd)

- 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

Op-Amp Current-to-Voltage Converter

Summary

- Voltage transfer characteristic of op-amp
- A feedback path between an op-amps 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

slope A gtgt1

10 V

vpvn

-Vcc

1 mV

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Differentiator (from Horowitz Hill)

Op-Amp Imperfections

Discussed in Hambley, pp. 651-665 (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

dont sum exactly to zero) offset voltage

There are pins (denoted offsets) for inputs to

help control this.