EE 201 op amps – 1

Operational amplifiers (Op amps)

View it as an ideal amp. Take the properties to the extreme: Ri → ∞, Ro = 0, A → ∞. ?!?!?!?!

Ri–

+vi

+– Avi

Ro

v+

v––

+vo

–

+vi

+– Avi

v+

v––

+vo

A → ∞

Consequences: No voltage dividers at input or output. (That’s good.) No current flows into the input. (That’s good.) The gain is infinite. (Is that good?)

EE 201 op amps – 2

How do we handle this infinite gain business?

vo = Avi

In order to keep vo finite, then as A → ∞ , vi → 0. In other words, we must force the difference signal at the input to go to zero. How do we do that? With feedback, of course.

6G =�

�+ $�6L

Recall that the difference signal in a feedback arrangement must become very small if the gain is very big.

→ 0, as A → ∞

One input is connected to the source voltage in some fashion. The other input is connected to the feedback network. If the feedback is working properly, then vi = v+ – v_ = 0.

The condition of v+ = v_ is called a virtual short at the input. This should be the case, if negative feedback is working in the circuit.

EE 201 op amps – 3

–+

–vo+

v–

v+

v+ → non-inverting input v_ → inverting input

Ideal op amp

v+ ≈ v_ → virtual short (assuming a proper negative feedback configuration.)

i+ = i_ = 0 → due to infinite input resistance

No voltage divider effects at output. This means that we can connect anything to the output without worrying about loading effects.

When using an op amp in a circuit:

EE 201 op amps – 4

Non-inverting amplifier

–+

+–

+vo–

R1

R2

vS

Write a node equation at the inverting terminal.

YR � Y�5�

=Y�5�

+ L�

v– = v+ = vs (virtual short due to feedback) i– = 0 (infinite input resistance)

YR � Y65�

=Y65�

* =YRY6

=5� + 5�5�

= �+5�5�

(Worth memorizing.)

EE 201 op amps – 5

Inverting amplifier

–++

–

R1

R2

–vo+

vS

Write a node equation at the inverting terminal.

v– = v+ = 0 (virtual ground!) i– = 0 (infinite input resistance)

YV � Y�5�

=Y� � YR

5�+ L�

YV5�

=�YR5�

* =YRY6

= �5�5�

Note the negative sign!

(Also worth memorizing.)

EE 201 op amps – 6

Summing amp (weighted summer)

–++

–R1

RF

–vo+

vS1+–

vS2

R2

R3

+–

vS3

At the inverting terminal: v– = v+ = 0 (virtual ground). Write a node equation there. (Or use superposition.)

YR = ��5)5�

Y6� +5)5�

Y6� +5)5�

Y6��

Y6�5�

+Y6�5�

+Y6�5�

=�YR5)

Use another inverter if you don’t like the negative sign.

The “virtually grounded” inverting terminal becomes a summing node.

EE 201 op amps – 7

Difference ampWe would like to amplify only the difference between va and vb. Anything this applied in common to both, will not be amplified.

–+va

vbR1

R2

R3 R4

At the inverting input:

YE � Y�5�

=Y� � YR

5�

YR =

��+

5�5�

�Y� � 5�

5�YE

At the non-inverting input:YD � Y+

5�=

Y+

5�Y+ =

5�5� + 5�

YD

v+ = v–

YR =

��+

5�5�

� �5�

5� + 5�YD

�� 5�

5�YE

5�5�

=5�5�

if then YR =5�5�

(YD � YE)Difference only! (Check it.)

EE 201 op amps – 8

Unity gain buffer

–+vs vo

Non-inverting amp with R2 = 0 and R1 → ∞ . So G = 1, meaning vo = vs. What good is that?

Connecting two circuits. If Ro1> Ri2, then much of the voltage is lost in the voltage divider, vi2 << v1.

+–

v1Ro1

Ri2 +–

v1Ro1 –

+

Ri2

High input resistance of op amp makes v+ = v1. Zero output resistance of op amp makes vi2 = vo. Since vo = v+, then vi2 = v1. The op amp served as a buffer between the two circuits, eliminated the voltage divider problem.

EE 201 op amps – 9

Digital-to-analog converter (DAC)Data is stored as digital information on your phone or computer. Some of that data needs to be converted to analog form in order to use it, i.e. music.

4-bit DAC

The digital bits control the operation of the switches – one bit per switch. Si = 0, switch is open, Si = 1 switch is closed.

So only closed: vo = –VREF/8. S1 only closed: vo = –VREF/4.

S2 only closed: vo = –VREF/2. S3 only closed: vo = –VREF.

S vo (V)0000 00001 0.1250010 0.250011 0.3750100 0.50101 0.6250110 0.750111 0.8751000 11001 1.1251010 1.251011 1.3751100 1.51101 1.6251110 1.751111 2

R3 = RF R2 = RF / 2R1 = RF / 4 Ro = RF / 8

VREF = -1 V

–+

VREF

–vo+

Ro

R1

R2

R3S3

S2

S1

S0

RF

EE 201 op amps – 10

Cascading amps

The various types of circuits can serve as building blocks for more complicated circuits.

–+

R3

R4

vo1

–+

vo2

R1

R2

–+R6

R7

vo3

R5va

vb

inverting

non-inverting

summing

YR� = �5�5�

YD

YR = �5�5�

YR� � 5�5�

YR�

YR =5�5�

5�5�

YD � 5�5�

��+

5�5�

�YE

YR� =

��+

5�5�

�YE

EE 201 op amps – 11

10 k!

1 k!

20 k!

8 k!

2 k!

non-inverting

YR� =

��+

5�5�

�YR�

= �YR�

summing

YR� = �5�5�

Y6 � 5�5�

YR�

want vo2 / vS.

–+

vo2

R3

R4

–+

R1R2

vS

R5

vo1

= ���Y6 � �.�YR�

YR�� = ���Y6 � �.�YR�

�.�YR� = ���Y6YR�Y6

= ���.��

Feedback loop around feedback loops !!