Operational amplifiers (Op amps)
v+
Ri
v–
+
vi
–
+ Av
i
–
Ro
+
vo
–
View it as an ideal amp.
Take the properties to the extreme: Ri → ∞, Ro = 0, A → ∞. ?!?!?!?!
v+
v–
+
vi
–
+
+ Av
i
–
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 – 1
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.
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 – 2
Ideal op amp
v+
v–
+
–
+
vo
–
v+ → non-inverting input
v_ → inverting input
When using an op amp in a circuit:
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.
EE 201
op amps – 3
Non-inverting amplifier
+
–
vS +
–
R1
+
vo
–
R2
Write a node equation at
the inverting terminal.
=
+
v– = v+ = vs (virtual short due to feedback)
i– = 0 (infinite input resistance)
=
=
EE 201
=
+
=
+
(Worth memorizing.)
op amps – 4
Inverting amplifier
R2
R1
–
+
vS +
–
Write a node equation at
the inverting terminal.
+
vo
–
=
+
v– = v+ = 0 (virtual ground!)
i– = 0 (infinite input resistance)
=
=
=
Note the negative sign!
(Also worth memorizing.)
EE 201
op amps – 5
Summing amp (weighted summer)
RF
R3
vS3
R2
+
–
+
–
vS2
=
–
+
R1
+ v
– S1
At the inverting terminal:
v– = v+ = 0 (virtual
+ ground). Write a node
vo equation there. (Or use
– superposition.)
+
+
+
+
=
Use another inverter if you
don’t like the negative sign.
The “virtually grounded” inverting terminal becomes a summing node.
EE 201
op amps – 6
Difference amp
We would like to amplify only the difference between va and vb.
Anything this applied in common to both, will not be amplified.
R2
At the inverting input:
R1
vb
va
–
+
R3
=
R4
=
+
At the non-inverting input:
+
+
if
EE 201
=
=
+
v+ = v–
=
+
=
then
=
(
+
)
+
Difference only!
(Check it.)
op amps – 7
Unity gain buffer
vs
+
–
vo
Non-inverting amp with R2 = 0 and R1 → ∞ . So G = 1, meaning vo = vs.
What good is that?
v1 +
–
Ro1
Ri2
Connecting two circuits. If
Ro1> Ri2, then much of the
voltage is lost in the
voltage divider, vi2 << v1.
EE 201
v1 +
–
Ro1
+
–
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.
op amps – 8
Digital-to-analog converter (DAC)
VREF = -1 V
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.
S
vo (V)
0000
0
0001
0.125
0010
0.25
0011
0.375
0100
0.5
0101
0.625
0110
0.75
0111
0.875
1000
1
1001
1.125
1010
1.25
1011
1.375
1100
1.5
1101
1.625
So only closed: vo = –VREF/8. S1 only closed: vo = –VREF/4.
1110
1.75
S2 only closed: vo = –VREF/2. S3 only closed: vo = –VREF.
1111
2
4-bit DAC
VREF
S3
R3 = RF R2 = RF / 2
R1 = RF / 4 Ro = RF / 8
R3
S2
RF
R2
S1
R1
S0 Ro
–
+
+
vo
–
The digital bits control the operation of the switches
– one bit per switch. Si = 0, switch is open, Si = 1
switch is closed.
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op amps – 9
Cascading amps
va
The various types of circuits can serve as building blocks for more
complicated circuits.
R4
=
R3
R5
v
R7
–
o1
+
inverting
summing
–
vo3
+
non-inverting R
vb
6
+
–
vo2
=
R1
R2
+
=
=
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+
op amps – 10
20 k!
R5
10 k!
R2
R1
vS
–
+
1 k!
vo1
+
–
vo2
8 k!
summing
2 k!
=
=
=
+
=
.
want vo2 / vS.
=
.
.
Feedback loop around
feedback loops !!
=
=
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R3
R4
non-inverting
.
op amps – 11