Ideal Opamps

1
Ideal Opamps
What happens if
In that case, (Vni Vinv) 0 V, so Vout 0.
The common-mode Voltage gain, Acm, is zero for an
ideal opamp, and the common-mode input Voltage is
(Vni Vinv) /2, or 10 V in this case. The
common-mode input Voltage may not be greater than
Esupply or less than Esupply, so if Esupply
15 V and Esupply is less than -15 V, Vni Vinv
10V is OK. If the common mode input Voltage
were 16 V, the opamp would malfunction. The
opamp would malfunction if either Vni or Vinv
were greater than Esupply or less than Esupply
Esupply and Esupply are sometimes called the
power supply rails, or just rails. Theyre
boundaries which Vni, Vinv and Vout must always
stay between.
Esupply
Vni

Vout
-
Vinv
-Esupply
2
Ideal Opamps
Below is the schematic symbol for an operational
amplifier. It has two power supply terminals,
Esupply and Esupply two inputs, Vni (the
noninverting input) and Vinv (the inververting
input), and one output Vout. The opamp is a
differential voltage amplifier. That means its
output voltage is the difference in Voltage
between Vni and Vinv, multiplied by a constant
called the gain or amplification factor
Adm is the amplifier gain. The subscript dm
stand for differential mode. The quantity in
parentheses, which Herrick calls error Voltage,
is more commonly called the differential input
Voltage, and its amplified by the differential
mode gain, Adm. As has been pointed out before,
an ideal opamp has
Esupply
Vni

Vout
-
Vinv
-Esupply
3
Ideal Opamps
The power supply Voltages, Esupply and Esupply,
are referred to as the power supply rails, rail
voltages, or just rails, because Vni, Vinv and
Vout must stay within the limits imposed by
Esupply and Esupply as if they were guardrails
on a highway. Neither Vni, Vinv nor Vout can go
above Esupply or below Esupply. In fact, Vout
can only approach the rails, perhaps within a
Volt or two. If Esupply 15 V and -Esupply
-15 V, and if Vout cant go below -14 V or above
13 V, then -14 V and 13 V are the lower and upper
rails. If Vout reaches either rail, then it
cant increase (if
it hit the upper rail, or it cant decrease if it
hit the lower rail). The opamp is said to be
saturated under that condition. If, with 15 V
and -15 V supplies, the upper and lower rails are
13 V and -14 V respectively, the opamp requires 2
V of headroom at the upper rail, and 1 V of
headroom at the lower rail.
Esupply
Vni

Vout
-
Vinv
-Esupply
4
Ideal Opamps
Opamp manufacturers publish highly detailed
specifications for each type of opamp. One of
the quantities specified is always the maximum
supply voltage. The difference between the two
supply voltages must not exceed the maximum
supply Voltage, which for the LM324 is 32 Volts.
For that opamp, -16 V and 16 V would be OK, but
-17 V and 17 V would not.
The output of the opamp is, in effect, a Voltage
source. Of course, it is a nonideal Voltage
source, so it looks like an ideal Voltage source
with a resistance in series. This resistance
limits the current can be supplied by the opamp
output, even if its short-circuited. An opamp
output is therefore current-limited. The LM324
has a short circuit current of 20 mA.
Esupply
Vni

Vout
-
Vinv
-Esupply
5
Opamp Comparator
Because Adm is very high (infinite for an ideal
opamp), if the differential input Voltage (Vni
Vinv) is even slightly positive, Vout will
saturate at the positive rail. If the
differential input Voltage is even slightly
negative, Vout will saturate at the negative
rail. This means an opamp can be used to compare
two voltages if Vni is greater than Vinv, then
Vout will be at the positive rail, and if Vni is
less than Vinv then Vout will be at the negative
rail.
Some opamps require little or no headroom, and
can operate at common digital power supply
Voltages. For example, the LM324 will operate
with -Esupply 0 V (i.e., grounded) and
Esupply 5 V. Under these conditions, Vout
5V for Vni gt Vinv, and Vout 0 V for Vni lt
Vinv. Such a comparator can interface an analog
signal, which can have any Voltage between 0 V
and 5 V, and a digital circuit where all Voltages
must be 0V or 5 V.
Esupply
Vni

Vout
-
Vinv
-Esupply
6
Opamp Comparator
Heres a circuit using an LM324 opamp as a
comparator, comparing Vin to an adjustable
reference Voltage, Vref. If Vin gt Vref, Vout 5
V. If Vin lt Vref, Vout 0 V.
What are the minimum and maximum Voltages Vref
can be adusted to? How much current does the
Voltage divider draw? How much power does it
dissipate? What should the power ratings of R1,
R2 and R3 be? If Vref 2.5 V and Vin 2.6 V,
what is Vout?
5 V
Vin
Esupply
R1
2.2 KW
Vout

Vref.
-
2.2 KW
R2
-Esupply
2.2 KW
R2
7
Opamp Transfer Characteristic
An op-amp has a transfer characteristic, which
maps input Voltage (Vd Vni Vinv, the
differential input Voltage) to output Voltage,
similar to the way the I-V characteristic of a
diode maps Voltage to current. For an ideal
opamp, Vd gt 0 results in Vout saturated at the
upper rail, and Vd lt 0 results in Vout saturated
at the lower rail.
Vout
Esupply
For an ideal opamp, Adm infinity so the slope
of the transition is undefined (vertical).
Esupply
Vni
Vd Vni - Vinv

Vout
-
Vinv
-Esupply
-Esupply
8
Opamp Transfer Characteristic
A non-ideal op-amp has differential mode gain
which is less than infinity, so the slope of its
transfer characteristic in the transition region
from the lower rail to the upper rail is not
vertical. The slope of the transition is equal
to Adm, and the transition begins at
Vout
and ends at
Vupper rail
slope Adm
Esupply
Vni
Vd Vni - Vinv

Vout
-
Vinv
-Esupply
-Vlower rail
9
Opamp Transfer Characteristic
The crossover point is the point at which the
transfer characteristic crosses over the
horizontal axis (the point at which Vout 0).
Vout
and ends at
Vupper rail
slope Adm
Esupply
Vni
Vd Vni - Vinv

Vout
-
Vinv
crossover
-Esupply
-Vlower rail
10
Opamp Transfer Characteristic
The comparator below has its crossover point at
Vin Vref, but Vout never crosses 0 V. In such
cases, the crossover is the midpoint of the
transition from the lower to upper rail.
5 V
Vout
Vin
3.5 V
Esupply
R1
2.2 KW
Vout

Vref.
LM324
-
2.15 V
2.2 KW
R2
-Esupply
crossover
2.2 KW
R2
0.8V
Vd
11
Opamp Transfer Characteristic
This circuit is a noninverting comparator,
because Vin is connected to the noninverting
input. In this case, Vout is at the upper rail
(logic high, 1 or true) when Vin gt Vref, and
Vout is at the lower rail (logic low, 0 or
false) when Vin lt Vref.
5 V
Vout
Vin
3.5 V
Esupply
R1
2.2 KW
Vout

Vref.
LM324
-
2.15 V
2.2 KW
R2
-Esupply
2.2 KW
R2
0.8V
Vd
12
Opamp Transfer Characteristic
This is an inverting comparator. Vin is
connected to the inverting input, so Vout is at
the upper rail (logic high, 1 or true) when Vin
lt Vref, and Vout is at the lower rail (logic
low, 0 or false) when Vin gt Vref. This is
equivalent to a noninverting comparator followed
by an inverter.
5 V
Vout
Vin
3.5 V
Esupply
R1
2.2 KW
Vout
-
Vref.
LM324

2.15 V
2.2 KW
R2
-Esupply
2.2 KW
R2
Vd
0.8V
13
Opamp Transfer Characteristic
Consider a noniverting opamp circuit, like the
one shown below In the real world, Vin is never
a true, absolutely constant, DC Voltage. Even
when connected to a DC source, such as a battery,
Vin varies in a random fashion above and below
the nominal source voltage.
5 V
Vout
Vin
3.5 V
Esupply
R1
2.2 KW
Vout

Vref.
LM324
-
2.15 V
2.2 KW
R2
-Esupply
crossover
2.2 KW
R2
0.8V
Vd
14
Opamp Transfer Characteristic
These random variations, which usually average
out over time to the nominal source voltage, are
called noise. Noise is always present in some
amount at every node in any real-world circuit.
An example of a noisy DC Voltag is shown here
5 V
Vin
Esupply
R1
2.2 KW

Vref.
LM324
-
2.2 KW
R2
-Esupply
2.2 KW
R2
15
Opamp Transfer Characteristic
Suppose, in the circuit below, the potentiometer
is adjusted so Vref 2.5 V and the Vin is the
noisy DC voltage shown. What does Vout do?
5 V
Vin
Esupply
R1
2.2 KW

Vref.
LM324
-
2.2 KW
R2
-Esupply
2.2 KW
R2
16
Positive Feedback Comparator
Use positive feedback to shift the threshold
Voltage at which the opamp output switches from
one rail to the other. This is called
hysteresis. Consider the circuit below
First, lets assume that Vref 2.5 V and Vout
5 V. An ideal opamp has infinite input
impedance, so no current can flow into or out of
Vin or Vinv. This means the feedback current If
flows
5 V
If
Vin
Ri
Rf
If
0
Esupply
R1
2.2 KW

through both Ri and Rf, so those two resistors
are essentially in series. The Voltage divider
rule says
Vout
Vref.
LM324
-
2.2 KW
R2
-Esupply
2.2 KW
R2
17
Positive Feedback Comparator
In effect, Vin is shifted upward by the amount
5 V
so to cross the threshold to switch Vout from 5 V
to 0 V, Vin must satisfy
If
Vin
Ri
Rf
If
0
Esupply
R1
2.2 KW

Vout
Vref.
LM324
-
2.2 KW
R2
-Esupply
2.2 KW
R2
18
Positive Feedback Comparator
Let Ri 1KW, and let Rf 1KW. If Vout is at
the upper rail (5 V), then to switch to the lower
rail Vin must drop to
5 V
If
Vin
Ri
Rf
After Vout switches to the lower rail, to switch
back to the upper rail Vin must rise to
If
0
Esupply
R1
2.2 KW

Vout
Vref.
LM324
-
2.2 KW
R2
-Esupply
2.2 KW
R2
19
Positive Feedback Comparator
If we plot the transfer characteristic of this
comparator circuit, as Vin goes from the lower
rail (0 V) to the upper rail (5 V), we see that
Vout switches from 0 V to 5 V when Vin 2.75 V
Vout
5 V
Vin
0 V
2.75 V
2.5 V
20
Positive Feedback Comparator
As Vin returns from the upper rail to the lower
rail, Vout switches when Vin crosses 2.25 V. The
switching threshold depends on whether Vin is
crossing the threshold toward the lower or upper
rail.
Vout
5 V
Vin
0 V
2.25 V
2.5 V
21
Positive Feedback Comparator
The complete transfer characteristic shows how
the switching threshold changes depending on
Vout. This effect is called hysteresis, and it
provides a degree of noise immunity. Notice that
the amount of noise present here would not be
enough to cause jitter.
Vout
5 V
0 V
2.25 V
Vin
2.5 V
22
Positive Feedback Comparator
Lets take another look at the comparator, after
simplifying the circuit. The Voltage divider
simply establishes Vref, but doesnt supply any
current to the opamp, so we dont neet to show it
5 V
If
Vin
Ri
Rf
Ri
Rf
Vin
If
0
Esupply
R1
If
2.2 KW
0
Esupply
Vni.


Vref.
LM324
-
Vout
Vref.
LM324
2.2 KW
-
Vout
R2
-Esupply
-Esupply
2.2 KW
R2
23
Positive Feedback Comparator
The opamp is assumed to be ideal, so its input
resistance is infinite. The current flowing into
Vni is zero, so Ri and Rf are essentially
connected in series. The opamp output looks like
a Voltage source, equal to Vout, so we can remove
the opamp
Ri
Rf
Ri
Rf
Vin
Vin
Vout
If
0
If
0
Esupply
Vni.

Vni
Vref.
LM324
-
Vout
-Esupply
24
Positive Feedback Comparator
The opamp switches, in either direction, when Vni
Vref. If Vref 2.5 V, and Vout is initially
at the upper rail (5 V), then at the switching
point
Ri
Rf
Ri
Rf
Vin
Vin
Vout
If
If
0
0
Esupply
Vni.

Vni
Vref.
LM324
Vout
-
-Esupply
25
Positive Feedback Comparator
Because the resistors are in parallel, the same
current flows in each, so
Similarly, if Vout 0V, Vout switches to 5 V when
Ri
Rf
Ri
Rf
Vin
Vin
Vout
If
If
0
0
Esupply
Vni.

Vni
Vref.
LM324
Vout
-
-Esupply
26
Positive Feedback Comparator
Similarly, if Vout 0V, Vout switches to 5 V when
Ri
Rf
Ri
Rf
Vin
Vin
Vout
If
If
0
0
Esupply
Vni.

Vni
Vref.
LM324
Vout
-
-Esupply
27
Inverting Comparator
An inverting comparator with hysteresis is shown
below. Note that its the same as the
noninverting comparator with hysteresis, except
Vin and Vref are interchanged.
Vref.
Ri
Rf
If
Ri
Rf
0
Vref
Esupply
Vni.

Vout
If
0
LM324
Vout
-
Vin
Vni
-Esupply
28
Inverting Comparator
As before, Vout switches in either direction when
Vni Vinv, only this time that occurs when Vin
Vni. In this circuit, when Vout 5 V, Vni is
given by
Vref.
Ri
Rf
Ri
Rf
Vref
Vout
If
If
0
0
Esupply
Vni.

Vni
LM324
Vout
-
Vin
-Esupply
29
Inverting Comparator
Vref.
Ri
Rf
Ri
Rf
Vref
Vout
If
If
0
0
Esupply
Vni.

Vni
LM324
Vout
-
Vin
-Esupply
30
Inverting Comparator
When Vout 0 V, Vni is given by
Vref.
Ri
Rf
Ri
Rf
Vref
Vout
If
If
0
0
Esupply
Vni.

Vni
LM324
Vout
-
Vin
-Esupply
31
Inverting Comparator
Vref.
Ri
Rf
Ri
Rf
Vref
Vout
If
If
0
0
Esupply
Vni.

Vni
LM324
Vout
-
Vin
-Esupply
32
Negative FeedbackInverting Amplifier
The output of a Voltage amplifier should not be
at one rail or the other, most (or all) of the
time it should be between the rails. The output
Voltage should be a copy of the input Voltage,
but magnified by the amplifier gain A.
Vout
Vin
Amplifier Gain A
Vin
Vout
33
Negative FeedbackInverting Amplifier
An amplifier can be built using an ideal opamp
with negative feedback. This is a little like a
cars cruise control. If you set the cruise
control for a speed of 65 mph, it adjusts the
throttle as required to keep the speed at 65 mph.
If the car comes to a hill, which causes the
speed to decrease, the cruise control changes the
throttle setting in the direction which causes an
increase. This increase compensates for the
decrease in speed caused by the hill.
The throttle setting is an input which controls
the cars speed. When the speed increases from
the setpoint (65 mph) due to external conditions,
the cruise control uses negative feedback to
reduce the throttle setting and compensate for
the unwanted change in speed.
Desired Speed
Throttle Setting
Actual Speed

Car
S
-
34
Negative FeedbackInverting Amplifier
If too much feedback is used, the cruise control
will overcompensate for the hill. The
overcompensation makes the cars speed overshoot
the setpoint. As the car accelerates above the
setpoint, the cruise control reduces the throttle
setting, but overcompensates again, causing the
speed to drop to low, and the process repeats.
Depending on several factors, the speed of the
car may continue to oscillate, or the
oscillations may get smaller or larger over time.
This effect can be reduced or eliminated by
reducing the amount of feedback. The signal fed
back is probably a Voltage which is proportional
to speed, so a potentiometer (an adjustable
Voltage divider) could be used to adjust the
amount of negative feedback and prevent
overcompensation or undercompensation.
Desired Speed
Throttle Setting
Actual Speed

Car
S
-
35
Negative FeedbackInverting Amplifier
In the case of the cruise control, the signal
labeled Actual Speed is really a Voltage
proportional to the cars speed, and the signal
labeled Desired Speed is a voltage input. The
Actual Speed, and therefore the Voltage
proportional to the Actual Speed, is controlled
by the Desired Speed. Increasing Desired Speed
increases Actual Speed reducing Desired Speed
reduces Actual Speed.
Suppose instead of the speed of a car, were
actually trying to control the output Voltage of
an amplifier?
Desired Speed
Throttle Setting
Actual Speed

Car
S
-
36
Negative FeedbackInverting Amplifier
In this circuit,
But Vfb is actually a portion of Vout, derived
using a Voltage divider, so
where k is the division ratio of the Voltage
divider. The feedback system adjusts the Vout as
necessary to satisfy the above relationships.
These relationships may be combined into a single
equation by substituting kVout for Vfb
Vin-Vfb

Vin
Then we can solve for Vout
S
Vout
A
-
Vfb
37
Negative FeedbackInverting Amplifier
Remember, the feedback system adjusts Vout as
necessary so
Next, suppose the gain A is very, very large
approaching infinity. If thats the case, then
So
Vin-Vfb
If A is infinite but Vout is finite, then

Vin
S
Vout
A
-
Vfb
The feedback system adjusts Vout as necessary to
make Vfb Vin.
38
Negative FeedbackInverting Amplifier
Heres the same feedback system, implemented
using an ideal opamp. Because the ideal opamp
has infinite input resistance, the current
flowing into the inverting input is zero. Rf
and Ri are effectively in series, forming a
Voltage divider, so
Recall that
Vout
Vin.

-
so
Rf
Vfb.
Ri
39
Negative FeedbackInverting Amplifier
For this circuit,
Recall that
Vout
Vin.

-
so
Rf
Vfb.
Ri
40
Negative FeedbackInverting Amplifier
An amplifier can be built using an ideal opamp
with negative feedback. Unlike the comparator
circuits, in which a portion of the output
Voltage was derived with a Voltage divider and
fed back to the noninverting input of the opamp,
in a negative feedback amplifier a portion of the
output Voltage is fed back to the inverting
input. This is negative feedback, which tends to
stabilize the circuit. An ideal opamp has
infinite gain. Since the output Voltage is given
by
Vin.
Ri
Rf
a nonzero differential input Voltage would result
in infinite ouput Voltage. In real life, the
output Voltage would be at one rail or the other.
The only way for the output to not be saturated
is for the differential input Voltage to be zero.
Vni and Vinv must be at the same potential.
If
0
Esupply
Vni.
-
LM324
Vout

-Esupply
41
Negative FeedbackInverting Amplifier
In this circuit, Vni is grounded. Since Vinv
must be at the same potential, it is at ground
potential 0 V. However, unlike a real ground,
no current can flow into it because of the ideal
opamps infinite input resistance. In this
circuit, the noninverting input is said to be a
virtual ground. As with the comparator,
Vref.
Ri
Rf
a nonzero differential input Voltage would result
in infinite ouput Voltage. In real life, the
output Voltage would be at one rail or the other.
The only way for the output to not be saturated
is for the differential input Voltage to be zero.
Vni and Vinv must be at the same potential.
If
0
Esupply
Vni.
-
LM324
Vout

-Esupply