### lecture6

```Analog Electronics
Lecture 6
Op amp Stability Analysis and Opamp Circuits
Electronic Devices, 9th edition
Thomas L. Floyd
Lecture:
 Stability analysis and compensation of op-amps
 Op-amp Circuits
Electronic Devices, 9th edition
Thomas L. Floyd
Stability analysis and compensation of op-amps
 Op-amp’s gain is so high that even a slightest input signal
would saturate the output.
Negative feedback is used to control the gain.
A classic form of feedback equation
Electronic Devices, 9th edition
Thomas L. Floyd
Loop Gain
Feedback equation:
The term AolB( or AB for simplicity) is called ‘Loop
Gain’
When loop gain is higher i.e. AB >> 1
o The system gain is determined by only feedback factor B.
o Feedback factor is implemented by stable passive components
o Thus in ideal conditions the closed loop gain is predictable and stable
because B is predictable and stable.
Electronic Devices, 9th edition
Thomas L. Floyd
Stability analysis and compensation of op-amps
System output heads to infinity as fast as it can
when 1+ AB approaches to zero.
Or |AB| =1 and ∠AB = 180o
If the output were not energy limited the system
would explode the world.
System is called unstable under these conditions
Electronic Devices, 9th edition
Thomas L. Floyd
Feedback equation for Op-amp feedback systems
Non-inverting amplifier
+
Vout
–
Vin
Rf
Vf
Feedback
circuit
Ri
Non-inverting amplifier
AB  A
Electronic Devices, 9th edition
Thomas L. Floyd
R
i
R R
i
f
Feedback equation for Op-amp feedback systems
Inverting amplifier
Rf
Ri
–
Vout
Vin
+
AB  A
R
i
R R
i
f
Replace ZF with Rf and ZG with Ri
Loop Gain for both inverting and non inverting amplifier
circuits are identical, hence the stability analysis is identical.
Electronic Devices, 9th edition
Thomas L. Floyd
Bode plot analysis of Feedback circuits
Bode plots of loop gain are key to understanding Stability:
Stability is determined by the loop gain,
when AB = -1 = |1| ∠180o
instability or oscillation occurs
Electronic Devices, 9th edition
Thomas L. Floyd
Loop gain plots are key to understanding Stability:
Notice that a one pole can only accumulate 90° phase shift, so when a
transfer function passes through 0 dB with a one pole, it cannot
oscillate.
A two-slope system can accumulate 180° phase shift, therefore a
transfer function with a two or greater poles is capable of oscillation.
Electronic Devices, 9th edition
Thomas L. Floyd
Op-amp transfer function
The transfer function of even the
simplest operational amplifiers will
have at least two poles.
At some critical frequency, the phase
of the amplifier's output = -180°
compared to the phase of its input
signal.
The amplifier will oscillate if it has a gain of one or more at this critical
frequency.
This is because:
(a) the feedback is implemented through the use of an inverting input that adds
an additional -180° to the output phase making the total phase shift -360°
(b) the gain is sufficient to induce oscillation.
Electronic Devices, 9th edition
Thomas L. Floyd
Phase Margin, Gain Margin
Phase Margin = ΦM
Phase margin is a measure of the difference in the
actual phase shift and the theoretical 180° at gain
1 or 0dB crossover point.
Gain Margin = AM
The gain margin is a measure of the
difference of actual gain (dB) and 0dB at
the 180° phase crossover point.
For Stable operation of system:
ΦM > 45o or AM > 2 (6dB)
Electronic Devices, 9th edition
Thomas L. Floyd
Phase Margin, Gain Margin
The phase margin is very small, 20o
So the system is nearly stable
A designer probably doesn’t want a 20°
phase margin because the system
Increasing the loop gain to (K+C) shifts
the magnitude plot up. If the pole
locations are kept constant, the phase
margin reduces to zero and the circuit will
oscillate.
Electronic Devices, 9th edition
Thomas L. Floyd
Compensation Techniques:
 Dominant Pole Compensation (Frequency Compensation)
Gain Compensation
Electronic Devices, 9th edition
Thomas L. Floyd
Dominant Pole Compensation (Frequency Compensation)
Dominant Pole Compensation is
implemented by modifying the gain and
phase characteristics of the amplifier's open
loop output or of its feedback network, or
both, in such a way as to avoid the
This is usually done by the internal or
external use of resistance-capacitance
networks.
 A pole placed at an appropriate low frequency in the open-loop
response reduces the gain of the amplifier to one (0 dB) for a frequency
at or just below the location of the next highest frequency pole.
Electronic Devices, 9th edition
Thomas L. Floyd
Dominant Pole Compensation (Frequency Compensation)
 The lowest frequency pole is called the
dominant pole because it dominates the effect
of all of the higher frequency poles.
Dominant-pole compensation can be
implemented for general purpose operational
amplifiers by adding an integrating capacitance.
The result is a phase margin of ≈ 45°, depending on the proximity of still
higher poles.
Electronic Devices, 9th edition
Thomas L. Floyd
Gain Compensation
 The closed-loop gain of an op-amp circuit is related to the loop gain. So
the gain can be used to stabilize the circuit.
AB  A
R
i
R R
i
f
Gain compensation works for both
inverting and non-inverting op-amp
circuits because the loop gain
equation contains the closed-loop
gain parameters in both cases.
As long as the application can stand the higher gain, gain
compensation is the best type of compensation to use.
Electronic Devices, 9th edition
Thomas L. Floyd
Gain Compensation
Electronic Devices, 9th edition
Thomas L. Floyd
It consists of putting a zero in the loop transfer function to cancel out
one of the poles.
The best place to locate the zero is on top
of the second pole, since this cancels the
negative phase shift caused by the second
pole.
Electronic Devices, 9th edition
Thomas L. Floyd
Electronic Devices, 9th edition
Thomas L. Floyd
Lecture:
 Stability analysis and compensation of op-amps
 Op-amp Circuits
Electronic Devices, 9th edition
Thomas L. Floyd
Comparators
A comparator is a specialized nonlinear op-amp circuit that
compares two input voltages and produces an output state that
indicates which one is greater.
Comparators are designed to be fast and frequently have other
capabilities to optimize the comparison function.
Electronic Devices, 9th edition
Thomas L. Floyd
Comparator with Hysteresis
Noise contaminated signal may cause an unstable output.
To avoid this, hysteresis can be used.
Hysteresis is incorporated by adding regenerative (positive) feedback, which
creates two switching points:
The upper trigger point (UTP) and the lower trigger point (LTP).
After one trigger point is crossed, it becomes inactive and the other one
becomes active.
VUTP
Vin 0
t
VLTP
+Vout (max)
–Vout(max)
Electronic Devices, 9th edition
Thomas L. Floyd
Schmitt Trigger
A comparator with hysteresis is also called a Schmitt trigger. The
trigger points are found by applying the voltage-divider rule:
VUTP 
R2
R1  R 2
 V
out ( m ax )

V LT P 
and
R2
R1  R 2
What are the trigger points for the circuit
if the maximum output is ±13 V?
Vin
 V
out ( m ax )

–
Vout
VUTP 
R2
R1  R 2
 V out ( m ax )  
10 k 
47 k  + 10 k 
 + 13 V 
+
R1
47 k
R2
10 k
= 2.28 V
By symmetry, the lower trigger point = 2.28 V.
Electronic Devices, 9th edition
Thomas L. Floyd
Output Bounding
Some applications require a limit to the output of the
comparator (such as a digital circuit). The output can be
limited by using one or two Zener diodes in the feedback
circuit.
The circuit shown here is bounded as a positive value equal to
the zener breakdown voltage.
Vin
Ri
0V
+VZ
–
0
+
Electronic Devices, 9th edition
Thomas L. Floyd
–0.7 V
Comparator Applications
Simultaneous or flash analog-to-digital
converters use 2n-1 comparators to
value for processing. Flash ADCs are a
series of comparators, each with a
slightly different reference voltage.
The priority encoder produces an
output equal to the highest value input.
VREF
R
Vin
(analog)
Op-amp
comparators
+
–
R
+
–
R
R
R
–
(7)
(6)
+
(5)
(4)
–
(3)
(2)
+
(1)
(0)
–
In IC flash converters, the priority
encoder usually includes a latch that
holds the converter data constant for a
period of time after the conversion.
Electronic Devices, 9th edition
Thomas L. Floyd
R
+
–
R
Priority
encoder
+
D2
D1
D0
Binary
output
Enable
input
+
–
R
Comparator Applications
Over temperature sensing circuit:
o R1 is temperature sensing resistor with a negative temperature
coefficient.
o R2 value is set equal to the resistance of R1 at critical temperature
oAt normal conditions R1 > R2 driving op-amp to low
At R1=R2, balance bridge
creates high op-amp output,
energizes relay, activates
alarm.
An over
temperature
sensing circuit
Electronic Devices, 9th edition
Thomas L. Floyd
Summing Amplifier
A summing amplifier has two or more inputs; normally all inputs have
unity gain. The output is proportional to the negative of the algebraic sum
of the inputs.
Electronic Devices, 9th edition
Thomas L. Floyd
Example
Summing Amplifier
What is VOUT if the input voltages are +5.0 V, 3.5 V and +4.2 V and all
resistors = 10 k?
Rf
R1
VOUT = (VIN1 + VIN2 + VIN3)
= (+5.0 V  3.5 V + 4.2 V)
= 5.7 V
Electronic Devices, 9th edition
Thomas L. Floyd
VIN1
R2
VIN2
10 k
–
R3
VIN3
VOUT
+
Averaging Amplifier
An averaging amplifier is basically a summing amplifier with the gain
set to Rf /R = 1/n (n is the number of inputs). The output is the negative
average of the inputs.
What is VOUT if the input voltages are +5.0 V, 3.5 V and +4.2 V? Assume
R1 = R2 = R3 = 10 k and Rf = 3.3 k?
Rf
R1
VIN1
R2
VOUT = ⅓(VIN1 + VIN2 + VIN3)
= ⅓(+5.0 V  3.5 V + 4.2 V)
= 1.9 V
Electronic Devices, 9th edition
Thomas L. Floyd
VIN2
3.3 k
–
R3
VIN3
VOUT
+
A scaling adder has two or more inputs with each input having a different
gain.
It is useful when one input has higher weight than the other.
The output represents the negative scaled sum of the inputs.
Assume you need to sum the inputs from three microphones. The first two
microphones require a gain of 2, but the third microphone requires a gain
of 3. What are the values of the
Rf
input R’s if Rf = 10 k?
R1
VIN1
R2
R1  R 2  
R3  
Rf
Av 3
Electronic Devices, 9th edition
Thomas L. Floyd
Rf

10 k 
Av 1

10 k 
3
2


VIN2
5.0 k
10 k
–
R3
VIN3
VOUT
+
3.3 k
An application of a scaling adder is the D/A converter circuit shown
here. The resistors are inversely proportional to the binary column
weights. Because of the precision required of resistors, the method is
useful only for small DACs.
+V
8R
20
Rf
4R
21
2R
–
VOUT
2
2
+
R
23
Electronic Devices, 9th edition
Thomas L. Floyd
A more widely used method for D/A conversion is the R/2R ladder. The
gain for D3 is 1. Each successive input has a gain that is half of previous
one. The output represents a weighted sum of all of the inputs (similar to
Inputs
only two values of
resistors are required
to implement the
circuit.
Electronic Devices, 9th edition
Thomas L. Floyd
D0
D1
D2
D3
R3
2R
R6
R5
2R
R7
2R
R2
R1
2R
R4
2R
R
R
R8
Rf = 2 R
–
R
Vout
+
The Integrator
An op-amp integrator simulates mathematical
integration, a summing process that determines
total area under curve.
The ideal integrator is an inverting amplifier
that has a capacitor in the feedback path. The
output voltage is proportional to the negative
integral (running sum) of the input voltage.
Electronic Devices, 9th edition
Thomas L. Floyd
IC
C
R
–
Vin
Ii
Vout
+
Ideal
Integrator
The Integrator
Capacitor in the ideal integrator’s feedback is
open to dc.
Rf
C
This implies open loop gain with dc offset.
The practical integrator overcomes these
issues– the simplest method is to add a
relatively large feedback resistor.
R
Vin
–
Vout
+
Practical
Integrator
Rf should be large enough
Electronic Devices, 9th edition
Thomas L. Floyd
Example
The Integrator
If a constant level is the input, the current is constant. The capacitor
charges from a constant current and produces a ramp. The slope of the
V in
output is given by the equation:  V out

t
Ri C
Sketch the output wave:
Rf
+2.0 V
Vin
 V out
t
220 k
t (ms)
0V
0.0
2.0 V
0.5

V in
RiC
1.0

1.5
2.0
2 V
10
k    0.1 μF 
C
 2 V /m s
Vin
Ri
0.1mF
–
10 k
Vout
+1.0 V
+
Vout
Electronic Devices, 9th edition
Thomas L. Floyd
0V
0.0
1.0 V
t (ms)
0.5
1.0
1.5
2.0
The Differentiator
R
C
An op-amp differentiator simulates
mathematical differentiation, a process to
determine instantaneous rate of change of a
function.
Vin
–
Vout
+
Ideal
Differentiator
The ideal differentiator is an inverting amplifier that has a capacitor in
the input path. The output voltage is proportional to the negative rate of
change of the input voltage.
Electronic Devices, 9th edition
Thomas L. Floyd
Example
The Differentiator
The output voltage is given by
V 
V out    C  R f C
 t 
+1.0 V
Sketch the output wave: Vin
0V
0.0
1.0 V
t (ms)
0.5
1.0
1.5
2.0
V 
V out    C  R f C
 t 
Rf
 1 V 
 
  10 k    0.1 μF   2 V
 0.5 m s 
10 k
Vin
Rin
C
–
220  0.1mF
Vout
+
+2.0 V
Vout
Electronic Devices, 9th edition
Thomas L. Floyd
Rc
0V
0.0
2.0 V
t (ms)
0.5
1.0
1.5
10 k
2.0
References
Slides by ‘Pearson Education’ for Electronic Devices
by Floyd
‘Op.amp for every one’ by Ron Mancini
’Stability Analysis for volatge feedback op-amps’,
Application Notes byTexas Instruments (TI)
’Feedback amplifiers analysis tool’ by TI
‘Feedback, Op Amps and Compensation’ Application
Note 9415 by Intersil