Introduction to Operational Amplifiers (M. LATINA) p. 1
Differential amplifiers
A differential (or difference) amplifier is a circuit
used for amplifying a voltage difference between
two input signals while rejecting signals that are
common to both inputs.
DC Analysis:
Loop 1:
VBE VE = 0
VE = VBE = 0.7V
and IE1 = IE2
since both currents combine in RE,
IE1 = IE2 = IRE/2
Loop 2:
IRERE + VEE + VE = 0
IRE =
based on approximation IC IE
then IC1 = IC2 = IRE/2
therefore, VC1 = VC2 = VCC IC1R1
Modes of Signal Operation:
Single-ended input input signal is applied to
either input with the other input connected to
ground
Differential or double-ended input two
opposite polarity input signals are applied.
Common-mode input same signal is applied to
both inputs.
Single-ended input:
Introduction to Operational Amplifiers (M. LATINA) p. 2
Differential input:
Common-mode input:
Common-mode signal:
signal that drives both inputs of a
differential amplifier equally.
these are interference, static and other
kinds of undesirable signals picked-up by the
circuit.
Common Mode Rejection Ratio(CMRR)
measure of an amplifiers ability to reject
common-mode signals.
Example: A certain differential amplifier has a
differential voltage gain of 2000 and a common-
mode gain of 0.2. Determine the CMRR and
express in dB.
Differential Gain:
vin(d) = vin1 vin2
From Loop 1:
vin1 ie1re (ie1 + ie2)RE = 0
vin1 = ie1(re + RE) + ie2RE (1)
vin2 ie2re (ie1 + ie2)RE = 0
vin2 = ie2(re + RE) + ie1RE (2)
express in terms of the current:
from (2) ie2 =
substitute ie2 in equation 1:
vin1 = ie1(re + RE) + RE
Introduction to Operational Amplifiers (M. LATINA) p. 3
which makes
ie1 =
do the same to compute for ie2
ie2 =
at the output side:
vout(d) = vc1 vc2
= RC (ic1 ic2)
= RC (ie1 ie2)
= RC
Simplify to obtain
Av(d) = =
*true for balanced output
(vout(d) = vc1 vc2)
Av(d) = =
*true for unbalanced output
(vc1 or vc2 only)
Common-mode gain:
for common-mode, emitter currents ie1 = ie2
since the two transistors are matched, only one-
half of the circuit may be considered:
Example: For the circuit shown, calculate:
(a) ICQ and VCEQ
(b) Av(d) and Acm
(c) CMRR
Solution:
VE = 0.7V
IRE = = = 1.378mA
IE = IRE/2 = 0.689mA = ICQ
VCQ = VCC ICQRC = 9.726V
VCEQ = VCQ VEQ = 9.726 (0.7) = 10.426V
re = 25mV/IE = 36.28W
Av(d) = = 90.95
Acm = = 0.2
CMRR = 90.95/0.2 453
Introduction to Operational Amplifiers (M. LATINA) p. 4
The operational Amplifier (Op-amp) The operational amplifier is a direct coupled high
gain amplifier and is used to perform a wide variety
of linear as well as non-linear functions. This circuit
was originally used for carrying out mathematical
operations such as summation, differentiation, and
integration on input signals. Now, operational
amplifiers are used for functions other than
mathematical operations such as dc as well as ac
amplification, rectification, waveform generation,
filtration, non-linear waveshaping, etc.
Block diagram of an op-amp
Input stage - this stage provides most of the voltage
gain and also establishes the input resistance of the
OPAMP.
Intermediate stage - another differential amplifier
which is driven by the output of the first stage.
Level shifting circuit - used to shift the dc level at the
output downward to zero with respect to ground.
Output stage - increases the output voltage swing and
raise the current supplying capability of the OPAMP,
also provides low output resistance.
Symbols and Terminals
The standard operational amplifier symbol is shown
below. It has two input terminals, the inverting input (-)
and the non-inverting input (+), and one output
terminal. The typical op-amp operates with two dc
supply voltages, one positive and the other negative.
The ideal op-amp characteristics:
- An ideal op-amp draws no currents at the
input I1=I2=0, thus its impedance is infinite.
Any source can drive it and there is no
loading on the driver stage.
- The gain of an ideal op-amp is infinite,
hence the differential input vd=v1-v2 is
essentially zero for the finite output voltage
Vo.
- The output voltage Vo is independent of the
current drawn from the output terminals.
Thus, its output impedance is zero and
hence output can drive an infinite number
of other circuits.
- Infinite bandwidth - amplifies signals from
0 to a hertz without attenuation.
Introduction to Operational Amplifiers (M. LATINA) p. 5
The practical op-amp:
1. Very high voltage gain (~105)
2. Very high input impedance (~2MW)
3. Very low output impedance (~75W)
4. Wide bandwidth (0 1MHz)
5. Very high differential gain (~80dB)
6. Large CMRR (~80dB)
Open-loop Configuration:
Since the inherent open-loop gain of a typical op-
amp is very high, usually > 100,000, or more, an
extremely small difference in the two input
voltages drives the op-amp into its saturated
output states.
VinAol = (1mV)(100,000)
= 100V
Negative Feedback:
Negative feedback is the process whereby a
portion of the output voltage of an amplifier is
returned to the input with a phase angle that
opposes (or subtracts from) the input signal. This
method helps stabilize the gain and reduce
distortion. It can also increase the input resistance.
Vout = AOL Vin (1)
Vin = Vin Vout (2)
Substituting (2) in (1)
Vout = AOL(Vin Vout)
Vout (1 + ) = AOLVin
ACL = =
Closed-loop voltage gain is the voltage gain of an
op-amp with negative feedback
Advantages of Negative Feedback:
1. Decreased voltage gain
2. Decreased output impedance
Introduction to Operational Amplifiers (M. LATINA) p. 6
3. Increased/decreased input impedance
depending on circuit
4. Decreased distortion
5. Increased bandwidth
Concept of Virtual ground:
When finding the gain, assume there is
infinite impedance at the input (i.e.
between the inverting and non-inverting
inputs). Infinite input impedance implies
zero current at the input.
If there is no current at the input
impedance, there is no voltage drop
between the inverting and non-inverting
inputs. Thus, the voltage at the inverting
input is zero. The zero at the inverting input
is referred to as virtual ground.
The Inverting Amplifier
The inverting amplifier has the output fed back to
the inverting input for gain control. The gain for the
inverting op-amp can be determined by the
formula below.
V1 = V2 = 0
I1 = If
The Non-inverting Amplifier:
The closed loop gain for a non-inverting amplifier
can be determined by the formula below.
V1 = V2 = Vin
I1 + If = 0
(I)
f
cl
i
RA
R
Closed-loop voltage gain is
determined by circuit
components.
011
f
out
i R
VV
R
V
0 outinifin VVRRV
in
i
f
in
fi
out
outifiin
VR
RV
R
RRV
VRRRV
)1(
)(
(NI) 1f
cl
i
RA
R
Introduction to Operational Amplifiers (M. LATINA) p. 7
The Voltage Follower:
The voltage-follower amplifier configuration has
all of the output signal fed back to the inverting
input. The voltage gain is 1. This makes it useful as
a buffer amp since it has high input impedance and
low output impedance.
Effects of Negative Feedback on Open-loop Gain:
ACL = =
since 1>ZoutIout
then Vout Aol(VinVf)
substituting Vout for Vf
Vout Aol (Vin Vout)
Vout AolVin AolVout
AolVin Vout + AolVout
(1 + Aol) Vout
since output impedance Zout(NI) = Vout/Iout
AolVin (1+Aol) IoutZout(NI)
dividing both sides by Iout:
AolVin/Iout (1+Aol) Zout(NI)
AolVin/Iout (1+Aolb) Zout(NI)
since AolVin = Vout and Vout/Iout =Zout
then
Zout = (1+Aolb) Zout(NI)
thus,
Introduction to Operational Amplifiers (M. LATINA) p. 8
Example: What are the input and output
resistances and the gain of the non-inverting
amplifier? Assume the op amp has Aol = 100,000,
Zin = 2 M, and Zout = 75 .
Input impedance for the inverting amplifier:
Recall that negative feedback forces the inverting
input to be near ac ground for the inverting
amplifier. For this reason, the input impedance of
the inverting amplifier is equal to just the input
resistor, Ri. That is, Zin(I) = Ri.
Output impedance for the inverting amplifier:
The equation for the output impedance of the
inverting amplifier is essentially the same as the
non-inverting amplifier:
Example: What is the input impedance and the
gain of the inverting amplifier? Assume the op-amp
has Aol = 100,000, Zin = 2 M, and Zout = 75 .
Voltage Follower Input and Output impedance:
The voltage-follower is a special case of the non-
inverting amplifier in which Acl = 1. The input
impedance is increased by negative feedback and
the output impedance is decreased by negative
feedback. This makes it an ideal circuit for
interfacing a high-resistance source with a low
resistance load.
Zin(NI) = (1 + AolB)Zin
Op-amp Parameters:
Input Bias Current
The input bias current is the dc current required by
the inputs of the amplifier to properly operate the
first stage. By definition, the input bias current is
the average of input currents and is calculated as
follows:
Introduction to Operational Amplifiers (M. LATINA) p. 9
Input Offset Current
Ideally, the two input bias currents are equal, and
thus their difference is zero. In a practical op-amp,
however, the bias currents are not exactly equal.
The input offset current, Ios, is the difference of the
input bias currents expressed as an absolute value.
Input Offset Voltage (VOS)
It is desired that the dc voltage at the output is
zero with no input voltage. But because of the
unequal amount of current drawn by the input
transistors of the first differential amplifier due to
unbalance in the circuit, the output voltage will not
become zero. Input offset voltage is the voltage
required between the inputs to force the
differential output to zero volts. Typical values are
in the range of 2mV or less.
Input Offset Voltage Drift with Temperature
The input offset voltage drift is a parameter related
to Vos that specifies how much change occurs in
the input offset voltage for each degree change in
temperature. Typical values range anywhere from
about 5mV per degree Celsius to about 50mV per
degree Celsius. Usually, an op-amp with a higher
nominal value of input offset voltage exhibits a
higher drift.
Example: What is the input offset voltage of the
LM741A at 750C?
Input Impedance
Differential input impedance is the total resistance
between the inverting and the non-inverting
inputs. It is measured by determining the change in
bias current for a given change in differential input
voltage.
Common-mode input impedance is the resistance
between each input and ground and is measured
by determining the change in bias current for a
given change in common-mode input voltage.
Output Impedance
The output impedance is the resistance viewed
from the output terminal of the op-amp.
OS 1 2I I I
Introduction to Operational Amplifiers (M. LATINA) p. 10
Input Voltage Range
All op-amps have limitations on the range of
voltages over which they will operate. The input
voltage range is the range of input voltages which,
when applied to both inputs will not cause clipping
or other output distortion.
Maximum Output Voltage Swing (Vo(pp))
The output voltage of an op-amp cannot be higher
than the positive dc power supply voltage (+VDC),
and cannot be lower than the negative dc power
supply voltage (-VDC). Vo(pp) also varies with the
load connected and increases directly with load
resistance.
Open-Loop Voltage Gain, Aol
The open-loop voltage gain of an op-amp is the
internal voltage gain of the device and represents
the ratio of output voltage to input voltage when
there are no external components. The open-loop
voltage gain is set entirely by the internal design.
Open-loop voltage gain can range up to 200,000
and is not a well-controlled parameter. Data sheets
often refer to the open-loop voltage gain as the
large-signal voltage gain.
Example: Refer to the op-amp specifications. If
741C is to be used in a non-inverting amplifier,
what is the input impedance if Rf = 500 k and Ri =
2.5 k? Use typical values.
Common-mode Rejection Ratio (CMRR)
The CMRR is a measure of an op-amps ability to
reject common-mode signals. A good op-amp
should have a very high value of CMRR, this
enables the op-amp to virtually eliminate
interference signals from the output.
Slew Rate
The slew rate of an op-amp is the maximum rate of
change of the output voltage in response to a step
input voltage. It is dependent upon the high-
frequency response of the amplifier stages within
the op-amp.
The slew rate is measured using a circuit given
below:
-the output voltage cannot change instantaneously
when a high frequency, large amplitude signal is
applied at the input side.
Example: What is the slew rate for the output
signal shown in response to a step input?
( )CMRR
v d ol
cm cm
A A
A A
Introduction to Operational Amplifiers (M. LATINA) p. 11
Frequency Response
Ideally, an op-amp should have infinite bandwidth.
This means the gain of an op-amp must remain the
same for all frequencies from 0 to infinite. Practical
op-amps however decreases its gain at higher
frequencies. The dependence of gain on frequency
is due primarily to the presence of capacitive
component in the equivalent circuit of the op-amp.
Maximum Operating Temperature. The maximum
temperature is the highest ambient temperature at
which the device will operate according to
specifications with a specified level of reliability.
Minimum Operating Temperature. The lowest
temperature at which the device operates within
specification.
Output Short-Circuit Duration. This is the length of
time the op-amp will safely sustain a short circuit
of the output terminal. Many modern op-amps can
carry short circuit current indefinitely.
Bias Current Compensation:
Effect of an Input Bias Current
Ideally, if the input voltage is zero, there should be
zero current coming into the inverting input of the
op-amp. However, there is a small bias current, I1,
that goes through Rf.
This current creates a voltage at the output equal
to I1Rf known as the error voltage.
If we look at the voltage follower circuit shown, it is
easy to see that the output error voltage is I1Rs.
Bias current compensation in a voltage-follower
Introduction to Operational Amplifiers (M. LATINA) p. 12
Bias current compensation in the non-inverting
and inverting configurations
Effect of Input Offset Voltage
The output voltage of an op-amp should be zero
when the differential input is zero. However, there
is always a small output error voltage present
whose value typically ranges from microvolt to
millivolts. This is due to unavoidable imbalances
within the internal op-amp transistors aside from
the bias currents previously discussed.
VOUT(error) = AclVIO
since Acl for the voltage follower is 1,
VOUT(error) = VIO
Input Offset Voltage Compensation
Op-amp Frequency Response
Frequency Dependence of Op-amp Gain
Gain vs. Frequency Limitations
The internal RC circuit of an op-amp limits the gain
at frequencies higher than the cutoff frequency.
The gain of an open-loop op-amp can be
determined at any frequency by the formula
below:
Example. Determine Aol for the following values of
f: (a) f = 0 Hz (b) f = 10 Hz (c) f = 100 Hz. Assume
fc(ol) = 100 Hz and Aol(mid) = 100,000.
3db Open-loop Bandwidth
The bandwidth of an AC Amplifier is the frequency
range between the points where the gain is 3dB
less than the midrange gain.
Introduction to Operational Amplifiers (M. LATINA) p. 13
In general, the bandwidth equals the upper
critical frequency (fCU) minus the lower critical
frequency (fCL).
Since fCL for an op-amp is zero, the bandwidth is
simply equal to the upper critical frequency.
BW = fC(OL)
Unity Gain Bandwidth
In the bode plot of the Open-loop amplifier, the
gain steadily decreases to a point where it is equal
to 1 (0 dB).
The value of the frequency at which this unity
gain occurs is the unity gain bandwidth.
Phase Shift
An RC Network causes a propagation delay from
input to output, thus creating a phase shift
between the input signal and the output signal.
An RC lag (low pass) network such as found in an
op-amp stage causes the output signal voltage to
lag the input.
Phase Shift () is expressed as:
= -tan-1(f/fC)
The negative sign indicates that the output
lags the input.
The math expression shows that the phase
shift increases with frequency and
approaches -90 as f becomes much greater
than fC.
Example. Calculate the phase shift for an RC lag
circuit for each of the following frequencies, and
then the curve of phase shift versus frequency: (a)
f = 1 Hz (b) f= 10 Hz (c) f = 100 Hz (d) f = 1000 Hz
(e) f = 10,000 Hz. Assume fc = 100 Hz
Introduction to Operational Amplifiers (M. LATINA) p. 14
Example: A certain op-amp has three internal
amplifier stages with the following gains and
critical frequencies:
Stage 1: Av1 = 40dB , fc1 = 2000Hz
Stage 2: Av2 = 32dB , fc2 = 40kHz
Stage 3: Av3 = 20dB , fc3 = 150kHz
Determine the open-loop midrange gain in decibels
and the total phase lag when f = fcl.
Closed-Loop Response
Op-amps are normally used in a closed-loop
configuration with negative feedback in order to
achieve precise control of the gain and the
bandwidth.
The closed-loop critical frequency of an op-amp is:
fC(CL) = fC(OL) ( 1 + A (mid) )
The bandwidth of a closed loop amplifier is:
BW(CL) = BW(OL) ( 1 + A (mid) )
ACL = =
Closed-Loop vs Open-loop Response
Gain-Bandwidth Product
An increase in closed loop gain causes a decrease
in the bandwidth and vice versa, such that product
of gain and bandwidth is constant.
Condition is true as long as the roll-off rate is fixed
at -20dB/decade.
The gain bandwidth product is always equal to
the frequency at which the op-amps open loop
gain is unity (unity gain bandwidth).
AC(OL) fOL = AC(CL) fCL = unity gain bandwidth
Example. Determine the BW of each of the
amplifiers below. Both op-amps have an open-loop
gain of 100dB and a unity-gain bandwidth of 3MHz.
Positive Feedback
With negative feedback , the signal fed back to the
input of an amplifier is out of phase with the input
signal, thus subtracting from it and effectively
reducing the voltage gain. As long as the feedback
is negative, the amplifier is stable.
When the signal fed back from output to input is in
phase with the input signal, a positive feedback
condition exists and the amplifier can oscillate.
Oscillation is an unwanted voltage swing on the
output when there is no signal present on the
input.