Solving Op Amp Stability Issues
Part 2
(For Voltage Feedback Op Amps)
Tim Green & Collin Wells
Precision Analog Linear Applications
1
Stability Analysis - Method 2
(Aol and1/b Technique)
(CF Compensation)
RI 180kOhm
VIN
V2 18V
10.00m
T
RF 180kOhm
+
Large Input Resistance &
Input Capacitance
+
VOUT
+
U1 OPA140
V1 18V
VIN
-10.00m
27.04m
Do you want this hidden in
your product - in production?
VOUT
-26.95m
990.00u
1.01m
1.03m
Time (s)
1.05m
3
VFB
T
RI 180kOhm
CT 1TF
RF 180kOhm
+
Aol and 1/b
LT 1TH
140
Aol = Vout/VFB
1/ = 1/VFB
Loop Gain (Aol ) = Vout
Aol and 1/
Aol
VG1
V2 18V
120
+
100
Vout
+
U1 OPA140
V1 18V
Voltage (V)
80
60
40
20
Rate-of-Closure
40dB/decade
1/
fcl
0
fz1
104kHz
-20
STABLE
-40
1
10
100
1k
10k
Frequency (Hz)
100k
1M
10M
4
Op Amp Input Capacitance
VEE 18V
INCcm- 7pF
Cdiff 10pF
IN+
+
U1 OPA140
VOUT
+
Ccm+ 7pF
OPA140 - Input Capacitance
VCC 18V
5
Equivalent Input Capacitance and b
RF 180kOhm
VFB
VOUT
1 VOUT

b
VFB
+
RI 180kOhm
b
VIN
V1 18V
Ccm- 7pF
Cdiff 10pF
+
VOUT
+
U2 OPA140
VOUT
RF 180kOhm
VFB
V2 18V
Ccm+ 7pF
Cin_eq 17pF
RI 180kOhm
VFB
RI 180kOhm
RF 180kOhm
Cin_eq 17pF
V3 18V
+
VOUT
+
U3 OPA140
V4 18V
6
Equivalent Input Capacitance and 1/b
VOUT
(Set to 1V)
RF 180kOhm
b
1/β Computation :
1 RF (RI // X

β
RI // X
Cin_eq
VFB
)
Cin_eq 17pF
RI 180kOhm
Cin_eq




1
s 
  Cin _ eq  RF  RI
 RF  RI  

Cin _ eq  


1
RF  RI  


 (after simplifica tion)
β
RI
1
RF  RI
RF
180k
DC 
 1
 1
 2  6dB
β
RI
RI
180k
1
1
1
zero: fz 1 

 104kHz
β
2π  Cin_eq  (RF // RI) 2π  17pF  (180k // 180k)
7
CF Compensation Design Steps
1) Determine fz1 in 1/b due to Cin_eq
A) Measure in SPICE
OR
B) Compute by Datasheet CDIFFand CCM and Circuit RF and RI
2) Plot 1/b with fz1 on original Aol
3) Add Desired fp1 on 1/b for CF Compensation
A) Keep fp1 < 10*fz1
B) Keep fp1 < 1/10 * fcl
4) Compute value for CF based on plotted fp1
5) SPICE simulation with CF for Loop Gain (Aolb) Magnitude and Phase
6) Adjust CF Compensation if greater Loop Gain (Aolb) phase margin desired
7) Check closed loop AC response for VOUT/VIN
A) Look for peaking which indicates marginal stability
B) Check if closed AC response is acceptable for end application
8) Check Transient response for VOUT/VIN
A) Overshoot and ringing in the time domain indicates marginal stability
8
1),2),3) Plot Aol, 1/b,
Add fp1 in 1/b for Stability
T
For fp1:
fp1 < 10 * fz1
fp1 < 1/10 * fcl
140
Aol and 1/
Input Capacitance Compensation
Aol
120
100
Gain (dB)
80
60
40
Add fp1
316kHz
1/
20
Hi-f = 15dB
Lo-f = 6dB
0
New fcl
fz1
104kHz
-20
-40
1
10
100
1k
10k
Frequency (Hz)
100k
1M
10M
9
CF 2.7pF
4) Compute value of CF
RI 180kOhm
RF 180kOhm
+
VOUT
CF 2.7pF
VIN
RF 180kOhm
-
1/β Computation :
1 (RF// X )  (RI // X

RI // X
β
CF
V3 18V
Cin_eq 17pF
VFB
Cin_eq
)
Cin_eq 17pF
RI 180kOhm
Cin_eq
+
VOUT
+
U3 OPA140
V4 18V




1
Cin_eq  CF  s 

RI

RF



Cin_eq  CF



1
 RF  RI 
 (after simplifica tion)

1 
β

CF  s 
 RF  CF 
180k
RF
RF  RI
1
 2  6dB
 1
 1
DC 
180k
RI
RI
β
1
1
1
 89.77kHz

zero : fz1 
2  (RF // RI)  Cin_eq // CF 2  (180k // 180k )  17pF // 2.7pF
β
1
1
1
 327.48kHz

pole : fp1 
2π  CF  RF 2π  2.7pF  180k
β
10
CF 2.7pF
VFB
CT 1TF
RF 180kOhm
Aol = Vout/VFB
1/ = 1/VFB
Loop Gain (Aol ) = Vout
+
RI 180kOhm
LT 1TH
5), 6) Loop Gain Check
Vtest
V2 18V
+
Vout
+
U1 OPA140
V1 18V
Phase Margin at fcl = 68 degrees
11
CF 2.7pF
7) VOUT/VIN AC Response
RF 180kOhm
+
RI 180kOhm
VIN
V2 18V
+
VOUT
+
U1 OPA140
V1 18V
T
0
VOUT/VIN
CF Compensation
VOUT
-3dB=394.5kHz
Gain (dB)
-20
-40
-60
180
Phase [deg]
135
90
45
0
1
10
100
1k
10k
Frequency (Hz)
100k
1M
10M
12
CF 2.7pF
8) Transient Analysis
RF 180kOhm
+
RI 180kOhm
VIN
V2 18V
+
T
VOUT
+
U1 OPA140
10.00m
V1 18V
VOUT / VIN
Transient Analysis
VIN
-10.00m
10.11m
VOUT
-9.98m
0
500u
1m
Time (s)
2m
2m
13
Stability
Tricks and Rules-of-Thumb
Loop Gain Bandwidth Rule: 45 degrees for f < fcl
Aolβ (Loop Gain) Phase Plot
180
fcl
90o
Phase Shift
at fcl
fp1
 (degrees)
135
Frequency
(Hz)
90
10
45o
“Phase Buffer”
away from
180o Phase Shift
45
0
-45
100
1k
fp2
10k
fz1
100k
1M
10M
90o
Phase Margin
at fcl
Loop Stability Criteria: < -180 degree phase shift at fcl
Design for: < -135 degree phase shift at all frequencies < fcl
Why?:
Because Aol is not always “Typical”
Power-up, Power-down, Power-transient  Undefined “Typical” Aol
Allows for phase shift due to real world Layout & Component Parasitics
Prevent excessive ringing due to phase margin dip near fcl
15
Frequency Decade Rules for Loop Gain
For 45O Phase Buffer away from 180O Phase Shift
fp1
fp1
fp2
fz1
fp3
Aol
pole
pole
---------
1/b
--------zero
pole
Loop Gain
pole
pole
pole
zero
100
Aol
A (dB)
80
Loop Gain View: Poles: fp1, fp2, fz1; Zero: fp3
Rules of Thumb for Good Loop Stability:
 Place fp3 within a decade of fz1
fp1 and fz1 = -135° phase shift at fz1
fp3 < decade will keep phase from dipping further
 Place fp3 at least a decade below fcl
Allows Aol curve to shift to the left by one decade
60
fcl
40
Rn
fp3
1/Beta
20
Cn
RF
fp2
fz1
+
0
VIN
1
10
100
1k
10k
100k
1M
10M
VOUT
RI
+
CL
-
Frequency (Hz)
VOUT/VIN
Note locations of poles in zeroes in Aol and 1/β plots
16
Frequency Decade Rules for Loop Gain
Phase Plot Prediction
Phase Shift (deg)
Phase Shift (deg)
Individual Pole & Zero Plot
+90
fp3
+45
0
fp1
-45
-90
fz1
1
10
fp1
fp2
fz1
fp3
+180
Aol
pole
pole
---------
100
1/b
--------zero
pole
Loop Gain
pole
pole
pole
zero
fp2
1K
10K
Frequency (Hz)
100K
1M
fcl
At fcl:
Phase Shift = 135O
Phase Margin = 45O
100K
1M
Final Plot
10M
+135
+90
+45
0
1
10
100
1K
10K
45O
Frequency (Hz)
“Phase Buffer”
Note locations of poles in zeroes in Aol and 1/β plots
10M
17
Op Amp Circuits & Second Order Systems
100
fp1
80
RI
RF
4.7k
4.7k
Aol
A (dB)
VOUT
60
+
+
VIN
40
-
1/Beta
fcl
20
fp2
0
1
10
100
1k
10k
100k
1M
10M
Frequency (Hz)
Most Op Amp Circuits are
adequately analyzed,
simulated, and real world
tested using well-known
second order system
response behavior.
Most Op Amps are dominated by Two Poles:
Aol curve shows a low frequency pole, fp1
Aol curve also has a high frequency pole, fp2
Often fp2 is at fcl for unity gain
This yields 45 degrees phase margin at unity gain
18
Control Loop - Second Order System
R(s)
G(s)
C(s)
2
C(s)
ωn
 2
R(s) s  2ωn s  ωn 2
w here:
ωn  naturalfre que ncy(rad/s )
ζ  dam pingratio
Clos e dLoopRe s pons e(Ste p and AC) :
Unde rdam ped : 0  ζ  1
M arginallyStable : ζ  0
CriticallyDam pe d: ζ  1
Ove rdam pe d: ζ  1
19
Indirect Phase Margin Measurements
Phase Margin can be measured indirectly on closed-loop circuits!
T 15.00m
PM = 30°
PM = 60°
Gain (dB)
0.00
-10.00
-20.00
7.50m
-30.00
0.00
3.75m
Phase [deg]
Voltage (V)
11.25m
PM = 30°
PM = 60°
T 10.00
-90.00
-180.00
0.00
2.00u
3.50u
Time (s)
Time Domain  Percent Overshoot
5.00u
-270.00
100.00k
1.00M
10.00M
Frequency (Hz)
AC Gain/Phase  AC Peaking
20
Closed Loop Peaking in
AC Frequency Sweep vs Phase Margin
T 10.00
6dB
Gain (dB)
0.00
-10.00
-20.00
-30.00
100.00k
6dB AC peaking  29° phase margin
1.00M
Frequency (Hz)
10.00M
21
Transient Real World Stability Test
RI
RF
Volts
+VS
VOUT
VOUT
VOffset
-VS
VIN
1kHz
50mVPP
+
RL
IOUT
+
time
-
Test Tips:
 Choose test frequency << fcl
“Small Signal” AC Output Square Wave (1kHz usually works well)
 Adjust VIN amplitude to yield output <50mVpp
 Worst case is usually when VOffset = 0  Largest Op Amp RO (IOUT = 0)
 Use VOffset as desired to check all output operating points for stability
 Set scope = AC Couple & expand vertical scope scale to look for amount of
overshoot, undershoot, ringing on VOUT small signal square wave
 Use X1 Scope Probe on VOUT for best resolution
22
2nd Order Transient Curves
T 15.00m
12.50m
14.3mV
Voltage (V)
10.00m
7.50m
5.00m
2.50m
0.00
2.00u
43% overshoot  29° phase margin
3.50u
Time (s)
23
When RO is really ZO!!
OPA627 has RO
OPA2376 has ZO
Capacitive
Inductive
Resistive
OPA627
OPA2376
Resistive
Note: Some op amps have ZO characteristics other than pure
resistance – consult data sheet / manufacturer / SPICE Model
24
Open Loop Output Impedance – SPICE Measurement
SPICE Zo Test :
Run SPICE AC Analysis
L1 1TH
IG1 is AC Current Generator
Vee 15V
IG1 DC Value  0A for unloaded Zo
C1 1TF
Zo  Vout
-
Convert Vout (dB) to Vout (Logarithm ic)
+
Zo (ohms)  Vout (Logarithm ic)
T
Vout
+
Vcc 15V
IG1
U1 OPA627E
55.5
Ro (ohms)
55.0
54.5
54.0
53.5
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
100M
25
Summary for Stability
For Stability Loop Gain Analysis all we need is:
1) Aol – from op amp data sheet or macromodel
2) Zo – Op Amp open loop, small signal AC output impedance
from op amp data sheet or macromodel
3) 1/b – basic by application, modified for stability
4) Z_Load – given by application
Stability General Comments:
1)
2)
3)
4)
5)
6)
7)
8)
Stability by modifying 1/b will decrease closed loop bandwidth
Stability compensation can slow large signal response (charging of caps) – check it
Simulate AC Transfer function (Closed Loop AC Response) as final check
Simulate Small Signal Transient Response as final check
DC operation in the lab does not guarantee stability
Marginal stability can cause undesired overshoot and ringing
DC circuits can get real world transient inputs from supplies, inputs, or output
That ringing in your circuit is not your Grandmother’s dial telephone
T

64.94m
VOUT
33.64m
100u
150u
200u
Time (s)
250u
300u
26
Additional Design Resources:
http://www.ti.com/ww/en/analog/precision-designs/
TI Precision Designs
Three design levels from the desks of our analog experts.
TI Precision Designs
Hub blog
Get tips, tricks and techniques from TI precision analog experts
27
Technical Support: http://e2e.ti.com/
28
Acknowledgements
A special thanks to Jerald Graeme, whom we honorably dub
“The Godfather of 1/b” for his work at Burr-Brown Corporation, et ali, in research
and writing about Op Amp Stability using 1/b.
Jerald Graeme Brief Biography:
From: http://electronicdesign.com/analog/jerald-graeme
When ICs and op amps were separate devices, Jerald Graeme was among the first to develop a combined IC
op amp while at Burr-Brown, in a 1968 team effort with Motorola. He designed many more op amps and video
amplifiers whose precision, high speed, or low drift amplification made them a very useful component in a
variety of analog applications. Nine U.S. patents and numerous foreign counterparts resulted from these
designs. The internationally acknowledged authority on electronic amplifiers wrote five very popular books about
op amps, the latest being Photodiode Amplifiers: Op Amp Solutions and Optimizing Op Amp Performance. The
latter, subtitled "A new approach for maximizing op amp behavior in circuit designs without extensive
mathematical analysis," offers design equations and models that reflect real-world op amp behavior and makes
analysis of difficult-looking configurations easy. Graeme's earlier books are: Op Amps: Design and Application,
Designing with Operational Amplifiers, and Amplifier Applications of Op Amps. He expects signal processing
with op amps to be the domain of digital devices, but they will still require an analog interface to integrate with
real-world items like process control or avionics.
Jerald Graeme Books:
http://www.amazon.com/Jerald-G.-Graeme/e/B001HO9X60
29
Appendix
30
Appendix Index
1) Op Amp Output Impedance
2) Pole and Zero: Magnitude and Phase on Bode Plots
3) Dual Feedback Paths and 1/b
4) Non-Loop Stability Problems
5) Stability: Riso (Output Cload)
6) Stability: High Gain and CF (Output Cload)
7) Stability: CF Non-Inverting (Input Cload)
8) Stability: CF Inverting (Input Cload)
9) Stability: Noise Gain Inverting & Non-Inverting (Output Cload)
10) Stability: Noise Gain and CF (Output Cload)
11) Stability: Output Pin Compensation (Output Cload)
12) Stability: Riso w/Dual Feedback (Output Cload)
– Zo, 1/b, Aol Technique
13) Stability: Riso w/Dual Feedback (Output Cload)
–1/b, Loaded Aol Technique
14) Stability: Riso w/Dual Feedback plus RFx (Output Cload)
–1/b, Loaded Aol Technique
15) Stability: Discrete Difference Amplifier (Output Cload)
31
Appendix Index
Appendix No.
Title
1
Op Amp Output Impedance
Pole and Zero:
2
Magnitude and Phase on Bode Plots
3
Dual Feedback Paths and 1/β
4
Non-Loop Stability Problems
Description/Stability Technique
Zo vs Zout difference and datasheet curves
Closed loop magnitude and phase shifts of a signal
frequency due to poles and zeroes on a Bode Plot
How to avoid problems when using dual feedback
paths for stability compensation
Oscillations and causes not seen in loop gain stability
simulations
5
Riso (Output Cload)
Stability: Isolation resistor with feedback at op amp
6
High Gain and CF (Output Cload)
Stability : High gain circuits and a feedback capacitor
7
CF Non-Inverting (Input Cload)
Stability : Non-inverting gain and feedback capacitor
Output Cload, closed loop gain >20dB
Input Cload, non-inverting gain, large value
input resistor
Stability: Inverting gain and feedback capacitor
Input Cload, non-inverting gain, large value
input resistor, photodiode type circuits
8
When to use the Stability Technique
Zo is a key parameter for stability analysis
Magnitude and phase shift at a frequency of
interest for closed loop poles and zeroes
Key tool in analyzing op amp circuits that use
dual feedback for stability
Check all designs to avoid oscillations that do
not show up in SPICE simulation
Output Cload, Note: accuracy of output is
dependent upon load current
9
CF Inverting (Input Cload)
Noise Gain Inverting and
Non-Inverting (Output Cload)
10
Noise Gain and CF (Output Cload)
Stability: Noise Gain added by input R-C network
Stability: Noise Gain (input R-C) and feedback
capacitor
11
13
Output Pin Compensation (Output Cload)
Riso w/Dual Feedback (Output Cload)
- Zo, 1/β, Aol Technique
Riso w/Dual Feedback (Output Cload)
- 1/β, Loaded Aol Technique
Stability: Series R-C on op amp output to ground
Stability: Isolation resistor with two feedback paths analysis by Zo, 1/β, and Aol technique
Stability: Isolation resistor with two feedback paths analysis by 1/β, and Loaded Aol technique
14
Riso w/Dual Feedback plus RFx (Output Cload) Stability: Isolation resistor with two feedback paths - 1/β, Loaded Aol Technique
analysis by 1/β, and Loaded Aol technique
Output Cload, closed loop gain <20dB
Output Cload, Loaded Aol has a second pole
located >20dB
Output Cload, no access to -input, monolithic,
integrated difference amplifiers, complex
feedback where not practical to use -input
Output Cload, some additional Vdrop across
isolation resistor is okay, accurate Vout at load
Output Cload, some additional Vdrop across
isolation resistor is okay, accurate Vout at load
Output Cload, some additional Vdrop across
isolation resistor is okay, accurate Vout at
load. RFx can provide wider BW control at
output load.
15
Discrete Difference Amplifier (Output Cload)
Output Cload, difference amp configuration,
32
any closed loop gain
12
Stability: Balanced use of noise gain (series R-C)
1) Op Amp Output Impedance
Open Loop (ZO)
&
Closed Loop (ZOUT)
Op Amps and “Output Resistance”
Definition of Terms:
RO = Op Amp Open Loop Output Resistance
ROUT = Op Amp Closed Loop Output Resistance
RF
RI
RO
-IN
RDIFF
VFB
xAol
+
VE
VOUT
VO
IOUT
-
1A
+
+IN
Op Amp Model
ROUT = RO / (1+Aolβ)
ROUT = VOUT/IOUT
34
From: Frederiksen, Thomas M. Intuitive Operational Amplifiers.
McGraw-Hill Book Company. New York. Revised Edition. 1988.
Derivation of ROUT
(Closed Loop Output Resistance)
1) b = VFB / VOUT = [VOUT (RI / {RF + RI})]/VOUT = RI / (RF + RI)
2) ROUT = VOUT / IOUT
3) VO = -VE Aol
4) VE = VOUT [RI / (RF + RI)]
5) VOUT = VO + IOUTRO
6) VOUT = -VEAol + IOUTRO Substitute 3) into 5) for VO
7) VOUT = -VOUT [RI/(RF + RI)] Aol+ IOUTRO Substitute 4) into 6) for VE
8) VOUT + VOUT [RI/(RF + RI)] Aol = IOUTRO Rearrange 7) to get VOUT terms on left
9) VOUT = IOUTRO / {1+[RIAol/(RF+RI)]} Divide in 8) to get VOUT on left
10) ROUT = VOUT/IOUT =[ IOUTRO / {1+[RIAol / (RF+RI)]} ] / IOUT
Divide both sides of 9) by IOUT to get ROUT [from 2)] on left
11) ROUT = RO / (1+Aolβ) Substitute 1) into 10)
ROUT = RO / (1+Aolβ)
35
ROUT vs RO
RO does NOT change when Closed Loop feedback
is used
ROUT is the effect of RO, Aol, and β controlling VO
 Closed Loop feedback (β) forces VO to increase or
decrease as needed to accommodate VO loading
 Closed Loop (β) increase or decrease in VO appears at
VOUT as a reduction in RO
 ROUT increases as Loop Gain (Aolβ) decreases
36
When RO is really ZO!!
OPA627 has RO
OPA2376 has ZO
Capacitive
Inductive
Resistive
OPA627
OPA2376
Resistive
Note: Some op amps have ZO characteristics other than pure
resistance – consult data sheet / manufacturer
37
With Complex ZO, Accurate Models are Key!
L1 1TH
SPICE Te s tof Op Am pM acro Zo :
V2 2.5V
C1 1TF
4
3
2
-
Vout
+
+
1
5
U1 OPA2376
V1 2.5V
T
IG1 0
Run SPICE AC Analys is
IG1 is AC Curre ntGe ne rator
IG1 DC Value  0A for unloade d Zo
Zo  Vout
Conve rt Vout (dB) to Vout (Logarithmic)
Zout (ohm s ) Vout (Logarithmic)
1.00k
Impedance (ohms)
OPA2376
100.00
10.00
400uA Load
1.00
100.00m
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
38
Some Data Sheets Specify ZOUT NOT ZO
Recognize ROUT instead of RO:
ROUT inversely proportional to Aol
ROUT typically <100 at high frequency
ROUT is Inve rs eof Aol :
RO
1  Aolb
1

Aol
ROUT 
ROUT
This is really
ZOUT or ROUT!
TLC082
1
TLC082
3
2
2
3
1
39
Some Data Sheets Specify ZOUT NOT ZO
Point frequency Aol
---------(Hz)
(dB)
1
10M
0
2
100k
40
3
10k
60
ROUT (AV=1)
Datasheet
(ohms)
100
2
0.2
ROUT (AV=1)
Computed
(ohms)
RO = 200
2
0.2
TLC082
Com puteR O fromR OUT w hereAolb  1 (0dB) :
1
2
R OUT  100Ω for A V  1, f  10MHz, Aolb  1 (0dB)
RO
1  Aolβ
R
100  O  RO  200
1 1
ROUT 
3
40
2) Pole and Zero:
Magnitude & Phase on Bode Plots
41
Formulae for Pole and Zero Calculations
Ze roM agnitudefor f
w he re:
Pole M agnitudefor f
w he re:
f  fre que ncyof s ignal
f  fre que ncyof s ignal
fz  fre que ncyof pole
fp  fre que ncyof pole
Magnitude( dB)  20  LOG10
Pole Phase Shift for f
w here:
f  frequencyof signal
fp  frequencyof pole
Phase( deg )   tan1(
f
)
fp
1
f2
1 2
fp
Magnitude( dB)  20  LOG10 1 
f2
fz2
Ze roPhas e Shift for f
w he re:
f  fre que ncyof s ignal
fz  fre que ncyof pole
f
Phase( deg )   tan1( )
fz
42
Closed Loop Gain: Magnitude and Phase
Clos e dLoopM agnitudeand Phas e for fre que ncyf :
w he re:
f  fre que ncyof s ignal
fp1  fre que ncyof pole 1
fp2  fre que ncyof pole 2
fz1 fre que ncyof ze ro1
fz2  fre que ncyof ze ro2
K  DC Gain in V/V
Magnitude( dB)  20  LOG10 K  20  LOG10
Phase(deg)   tan1(
1
1
f2
f2
 20  LOG10
 20  LOG10 1  2  20  LOG10 1 
f2
f2
f z1
f z22
1
1

f p12
f p22
f
f
f
f
)  tan1(
)  tan1( )  tan1( )
f p1
f p2
f z1
f z1
43
Closed Loop Gain: Magnitude and Phase
1
1

 10.00974485Hz
2  R1 C1 2  1k  15.9F
1
1
fp2 

 1.004766055kHz
2  RF  CF 299k  1.6nF
fp1 
CF 1.6nF
RF 99kOhm
V2 2.5V
RI 1kOhm
U1 OPA364
Clos e dLoop M agnitudeand Phas e for fre que ncyf :
+
+
VOUT
+
w he re:
f  fre que ncyof s ignal
R1 1kOhm
VIN
V1 2.5V
fp1  fre que ncyof pole 1
fp2  fre que ncyof pole 2
C1 15.9uF
K  DC Gain in V/V
Magnitude( dB)  20  LOG10 K  20  LOG10
Phase(deg)   tan1(
1
1
 20  LOG10
2
f
f2
1
1

fp12
fp22
f
f
)  tan1(
)
fp1
fp2
Magnitude( dB)  20  LOG10 (100)  20  LOG10
Phase(deg)   tan1(
1
1
 20  LOG10
2
f
f2
1
1

(10.00974485)2
(1.004766055e3)2
f
f
)  tan1(
)
10.00974485
1.004766055e3
44
Closed Loop Gain: Magnitude and Phase
T
40
20
[B]
Gain :
0
VOUT B:(3.924588k; -24.084018)
[A]
Gain (dB)
-20
Phase :
VOUT B:(3.924588k; -163.475026)
Gain :
-40
VOUT A:(92.20173; 20.624635)
Phase :
-60
VOUT A:(92.20173; -89.067371)
-80
-100
-120
-140
0.00
Phase [deg]
-45.00
-90.00
-135.00
-180.00
1
10
100
1k
10k
Frequency (Hz)
100k
1M
10M
45
Spice Compared with Calculated Analysis
SPICE AC Analysis:
For best accuracy use highest resolution
i.e. maximum “Number of Points”
f
(Hz)
---------9.22017300E+01
3.92458800E+03
Magnitude
(dB)
Calculated
20.6263744
-23.97775119
Magnitude
(dB)
SPICE
20.624635
-24.084018
Phase
(deg)
Calculated
-89.04706182
-165.49354967
Phase
(deg)
SPICE
-89.067371
-163.475026
Note:
1) SPICE analysis accounts for loop gain effects and closed loop phase shifts
due to op amp Aol.
2) Calculated results do not account for loop gain effects and closed loop phase shifts
due to op amp Aol.
46
Closed Loop Gain: Magnitude and Phase
SPICE Ideal Op Amp & Poles: Equivalent Circuit
DC Gain 100
VG1
f
(Hz)
---------9.22017300E+01
3.92458800E+03
Buffer 1
RF 99kOhm
+
R1 1kOhm
C1 15.9uF
+
+
-
-
Magnitude
(dB)
Calculated
20.6263744
-23.97775119
Magnitude
(dB)
SPICE - Ideal
20.626374
-23.977751
CF 1.6nF
+
+
-
-
Phase
(deg)
Calculated
-89.04706182
-165.49354967
VOUT
Phase
(deg)
SPICE - Ideal
-89.047062
-165.49355
Note:
1) SPICE - Ideal Circuit analysis matches Calculated results.
2) No loop gain effect or closed loop phase shifts due to op amp Aol.
47
Closed Loop Gain: Magnitude and Phase
SPICE Ideal Op Amp & Poles: Equivalent Circuit
T
40
20
[B]
0
Gain (dB)
SPICE Poles-Magnitude and Phase
Ideal Op Amp Circuit
[A]
-20
Gain :
-40
VOUT A:(92.20173; 20.626374)
-60
Phase :
VOUT A:(92.20173; -89.047062)
-80
Gain :
-100
VOUT B:(3.924588k; -23.977751)
-120
Phase :
-140
VOUT B:(3.924588k; -165.49355)
-160
0
Phase [deg]
-45
-90
-135
-180
1
10
100
1k
10k
Frequency (Hz)
100k
1M
10M
48
3) Dual Feedback Paths and 1/b
49
Dual Feedback and 1/β Concept
FB#1
RF
Analogy: Two people are talking in your ear. Which
one do you hear? The one talking the loudest!
CF
FB#2
RI
-
VO RO
VOA
+
VIN
VOUT
Riso
CL
Dual Feedback: Op amp has two feedback paths
talking to it. It listens to the one that feeds back the
largest voltage (β = VFB / VOUT). This implies the
smallest 1/β!
+
-
100
Dual Feedback Networks:
Analyze & Plot each FB#? 1/β
80
A (dB)
Aol
 Smallest FB#? dominates 1/β
60
 1/β = 1 / (β1 + β2)
FB#1
1/b1
FB#2
1/b
40
 1/β relative to VO
Note: VO = Op Amp Aol Output before
Ro for this Dual Feedback Example
1/Beta
[1/(b1 + b2)]
20
0
1
10
100
1k
10k
Frequency (Hz)
100k
fcl
1M
10M
50
Dual Feedback and 1/β
How will the two feedbacks combine?
Large b
Answer:
Small 1/β
The Largest β
(Smallest 1/β) will
dominate!
Small b
Large 1/β
-
51
Dual Feedback and the BIG NOT
WARNING: This can be
hazardous to your circuit!
100
Aol
80
A (dB)
FB#1
1/Beta
60
FB#2
40
1/Beta
20
0
1
10
100
1k
10k
100k
fcl
1M
10M
Frequency (Hz)
Dual Feedback and the BIG NOT:
1/β Slope changes from +20db/decade to -20dB/decade





Implies a “complex conjugate pole ” in the 1/β Plot with small damping ratio, ζ.
Implies a “complex conjugate zero” in the Aolβ (Loop Gain Plot) with small damping ratio, ζ.
+/-90° phase shift at frequency of complex zero/complex pole.
Phase slope from +/-90°/decade slope to +/-180° in narrow band near frequency
of complex zero/complex pole depending upon damping ratio, ζ.
Complex zero/complex pole can cause severe gain peaking in closed loop response.
52
Complex Conjugate Pole Phase Example
From: Dorf, Richard C. Modern Control Systems. Addison-Wesley Publishing Company. Reading, Massachusetts.
Third Edition, 1981.
53
Dual Feedback and 1/β Example
RF 100k
CF 82n
FB#1
FB#2
VEE 12
VO 5V
-
U2 REF02
U1 OPA177E
Vin
Gnd
REF02
+
Trim
Riso 26.7
Vout
5V
+
Out
CL 10u
VREF 5V
VCC 12
Dual Feedback:
FB#1 through RF forces accurate Vout across CL
FB#2 through CF dominates at high frequency for stability
Riso provides isolation between FB#1 and FB#2
54
Zo External Model for Dual Feedback Analysis
RF 100k
CF 82n
VFB 4.999V
VEE 12
-
U2 REF02
U1 OPA177E
Vin
Gnd
REF02
+
Trim
+
Z o External Model
VOA 4.9991V
VCV1
Ro 60
x1
+
+
Out
-
-
VO 4.9991V
Vout
Riso 26.7
4.9991V
CL 10u
VREF 4.999V
VCC 12
Zo External Model:
VCV1 ideally isolates U1 so U1 only provides data sheet Aol
Set Ro to match measured Ro
Analyze with unloaded Ro (largest Ro) which creates worst instability
Use 1/β on Aol stability analysis
1/β, taken from VOA will include the effects of Zo, Riso and CL
55
Dual Feedback, FB#1 And FB#2
RF 100k
CF 82n
VFB 4.999V
FB#2
VEE 12
-
U2 REF02
U1 OPA177E
Vin
Gnd
REF02
+
Trim
+
Z o External Model
VOA 4.9991V
VCV1
Ro 60
x1
+
+
Out
FB#1
-
-
VO 4.9991V
Vout
Riso 26.7
4.9991V
CL 10u
VREF 4.999V
VCC 12
FB#1 and FB#2 1/ β Analysis:
There is only one net voltage fed back as β to the -input of the op amp
β_net = β_FB#1 + β_FB#2
This implies that the largest β will dominate → smallest 1/ β will dominate
Analyze FB#1 with CF = open since it will only dominate at high frequencies
Analyze FB#2 with CL = short since it is at least 10x CF
56
Dual Feedback and 1/β Example
T
160
140
Aol
120
100
FB#1 1/b
Gain (dB)
80
60
1/b FB#2
FB#1
fcl
40
20
fza
fpc
1/b
0
fzx
-20
Vout/Vin
-40
10m
100m
1
10
100
1k
Frequency (Hz)
10k
100k
1M
10M
57
Dual Feedback and 1/β – Create the BIG NOT
VFB 4.999V
LT 1T
Aol = VOA
FB#1: 1/Beta1= VOA / VF B
Loop Gain = VFB
RF 100k
+
CT 1T
VG1
CF 220p
VEE 12
Vtrim 1.2298V
-
U2 REF02
U1 OPA177E
Vin
Gnd
REF02
+
Trim
+
Zo External Model
VOA 4.9991V
Ro 60
VCV1
x1
+
+
Out
-
-
VO 4.9991V
Riso 26.7
Vout
4.9991V
CL 10u
VREF 4.999V
VCC 12
58
Dual Feedback and 1/β – Create the BIG NOT
T
160
140
Aol
120
100
FB#1 1/b
Gain (dB)
80
60
1/b FB#2
FB#1
40
fcl
20
fpc
fza
0
1/b
-20
-40
10m
100m
1
10
100
1k
Frequency (Hz)
10k
100k
1M
10M
59
Dual Feedback and 1/β – Create the BIG NOT
T
160
140
Aol
120
100
???
STABLE
???
Gain (dB)
80
60
fcl
BIG
NOT
40
20
BIG NOT 1/
0
-20
-40
10m
100m
1
10
100
1k
Frequency (Hz)
10k
100k
1M
BIG NOT 1/b: At fcl rate-of-closure rule-of-thumb says circuit is stable but is it?
10M
60
Dual Feedback and 1/β – Create the BIG NOT
???
STABLE
???
BIG NOT Loop Gain:
Loop Gain phase shift >135 degrees (<45 degrees from 180 degree phase shift) for
frequencies <fcl which violates the loop gain phase buffer rule-of-thumb. But is it stable?
61
Dual Feedback and 1/β – Create the BIG NOT
VFB 4.999V
DC=0V
Trans ient=10mVpp
f =100Hz, 10ns rise/f all
CF 220p
VEE 12
Vtrim 1.2298V
-
U2 REF02
Vin
Vin
Gnd
REF02
Trim
Out
RF 100k
+
+
+
Zo External Model
VOA 4.9991V
VCV1
Ro 60
x1
+
+
U1 OPA177E
-
-
VO 4.9991V
Riso 26.7
Vout
4.9991V
CL 10u
VREF 4.999V
VCC 12
62
Dual Feedback and 1/β – Create the BIG NOT
T
5.38
VO
4.34
6.10
VOA
3.04
100.00m
Vin
-100.00m
5.17
STABLE
Vout
4.76
0.00
2.50m
5.00m
Time (s)
7.50m
10.00m
BIG NOT Transient Stability Test:
Excessive ringing and marginal stability are apparent. Real world implementation and
use may cause even more severe oscillations. We do not want this in production!
63