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Abstract— Motivated by course requirements and personal
interest in CMOS design, the following papers describes the
process taken to design a classical two-stage op amp configuration
with PMOS input transistors and Miller compensation with a
nulling resistor. This design takes into consideration
specifications on the minimum channel length, supply voltage,
load capacitance, open loop gain, phase margin, and unity gain
frequency. An analytical approach to solving the design is the
initially prepared. Then minor modifications are made, via the
assistance of simulations tools, to ensure all specifications of the
two stage op-amp are met.
I. INTRODUCTION
HE design of a CMOS op-amp contains multiple stages.
Identifying the basic structure of the op-amp as well as the
compensation type is the first stage. Additionally,
identifying the dc currents and transistor sizes and verifying
the results via simulations (such as SPICE) follows. The
physical implementation of the design, fabrication,
specification verification via measurements, and final
medications are the additional stages which will not be
covered in the following two stage operational amplifier
design. Due to the specifications given for the design, the
classical two-stage op amp configuration (voltage to current
and current to voltage stages) with PMOS input transistors and
Miller compensation with a nulling resistor was chosen (shown
in Appendix A).
The analytical design approach taken for the two stage
operational amplifier begins by indentifying the parameters of
the current mirror. The differential input parameters, second
gain parameters, and active load parameters follow
respectively. The final analytical parameter incorporates the
resultant compensation. With the utilization of SPICE
simulation and the BSIM3 model, the analytical design
undergoes minor modifications to achieve the design
requirements.
II. PARAMETER EXTRACTION
While the supply voltage is given, the supply current,
operating temperature, and temperature range are ignored for
this design. The process specifications can be identified via
simulation of the BSIM3 model in SPICE. The process
specifications which are necessary for this design are Vtn, Vtp,
Kn’, Kp’, λn, and λp. In order to identify Vtn, Vtp, Kn’, and
Kp’, linearly sweep values of Vgs for an N-MOS and a P-
MOS transistor biased in the saturation region. Vtn and Vtp
are identified graphically by determining the value of Vgs
when the drain current through the N-MOS and P-MOS
transistor become zero. Kn’ and Kp’ are identified by the slope
of the √(Id) vs. Vgs curve (slope = √[K’ / 2 * (W/L)]). λn and
Submitted for review no later than December 6th
, 2007 by 13:00 PST
λp are obtained by varying the Vds and identifying the current
through an N-MOS and P-MOS transistor biased in the
saturation region. Once the currents are obtained, λn and λp
can be identified from equation one
(eq.1: 1 1
2 2
1
1
d ds
d ds
I V
I V
λ
λ
   + ×
=   
+ ×   
).
III. DESIGN OF THE OPERATIONAL AMPLIFIER
While there are multiple approaches in analytically
designing an operational amplifier, the subsequent method
utilized consists of selecting the preferred drain currents for
each gain stage, identifying the current mirror parameters to
achieve such values, and finally designing the gain stages to
achieve the op-amp’s specifications. The following describes
such process; In Appendix B is the detailed calculations listed
in respective order with the proceeding process.
A. Current Mirrors
The segments of the two stage op-amp which pertains to the
current mirror includes; Iref, M5, M7, and M8. The purpose of
the current mirrors is to supply a drain current to each of the
two gain stages. The reference current (Iref) as well as the
current for each of the two stages must be determined. In this
design, the reference current was arbitrarily chosen to be
20µA. While not in the requirements of the design, a larger
reference current has not been selected to minimize the effect
of reference current error. Based upon the requirements of the
op-amp, the gain-bandwidth (GB) as well as the open-loop
gain (Avo) can be utilized in order to identify the current
margin of the first stage. (If there were maximum power
dissipation, input common mode range, and/or slew rate
requirements, they would be taken into account in selecting the
current of the first stage.) The chosen current must be large
enough to adhere to the GB requirement (eq.2: 1m
C
G
GB
C
= )
while small enough to maintain the requirement for open loop
gain (eq.3:
( )
1 2
2
5 6,7
2 m m
vo
d d N P
G G
A
I I λ λ
× ×
=
× × +
). The current of
the first gain stage was chosen to be 300µA. From equation 3
and 4 (eq.4:
'
2 6,7
6
2m d N
W
G I K
L
 
= × × × 
 
), the current
for the second stage can be chosen as a balance between
increasing the trans-conductance without sacrificing the open-
loop gain. For simplicity of design, the current of the first and
second stage was chosen to be the same (300µA).
Transistor 8 is in a diode-connected configuration (drain is
directly connected to the gate); therefore the device will
Design of a Two-Stage Single Ended CMOS Op-Amp
(November 2007)
STEVEN G. ERNST
T
operate in either the saturation or cut-off region. Transistor 8
plays a direct role in determining the VdSAT for the other
biasing transistors. With this in mind, the (W/L) ratio for
transistor 8 was chosen to be 8. With equation 5, the (W/L)
ratio for M5 is calculated (eq. 5: 5
5 88
IW W
L I L
   
= ×   
   
).
Additionally with equation 6, the (W/L) ratio for M7 is
calculated (eq. 6: 7
7 88
IW W
L I L
   
= ×   
   
).
B. Differential Input Stage
As shown in equation 2, the trans-conductance of the first
gain stage can be calculated if the gain-bandwidth and the
compensation capacitor (CC) are known. To ensure the two
stage op-amp achieves the minimum gain-bandwidth
(determined by the unity gain frequency), a gain-bandwidth
buffer of 5 is included in equation 2. From Allen [1], when a
phase margin of 60o
is required, the compensation capacitor
must be chosen to be greater than 0.22 times the capacitance
seen as the output. With such condition, an arbitrarily chosen
value of 5pF for the compensation capacitor and a gain-
bandwidth buffer of 5, the gain of the first stage is identified
and utilized in equation 7 (eq. 7:
2
1
'
1,2 5
m
d P
GW
L I K
 
= 
× 
) to
calculate the (W/L) ratio of the differential input stage.
C. Second Gain Stage
The second pole of the two stage op-amp is determined
directly by the gain of the second stage and inversely by the
load capacitance (eq. 8: 2
2
m
L
G
P
C
= ). In order to obtain a
sufficient phase margin (PM), the second pole of the two stage
op-amp must be located past the unity gain frequency. From
Allen [1], the second pole should be extended by at least 1.73
times the gain-bandwidth. For a margin of safety, the value of
2.2 is recommended. With equation 2, equation 8, and the
buffer of 2.2, the trans-conductance of the second stage can be
calculated. Furthermore, the (W/L) ratio of the transistor in the
second gain stage (M6) can be calculated from equation 9 (eq.
9:
2
2
'
6 62
m
d N
GW
L I K
 
= 
× × 
).
D. Active Load Stage
To ensure a reduced input offset voltage, the systematic
offset condition must apply; the drain voltages of M3 and M4
must remain the same. This allows the current for the first gain
stage to equally split between M1 and M2. With the systematic
offset condition applied, the (W/L) ratio for M3 and M4 can
be calculated from equation 10 (eq.
10: 5
3,4 66
1
2
d
d
IW W
L I L
    
= × ×    
    
).
E. Compensation
Included in series with the Miller compensation (previously
estimated in the differential input stage) on the second gain
stage is a nulling resister. The purpose of the nulling resister is
to eliminate or move the right hand plane (RHP) zero to the
left plane. From Hershenson, Boyd, and Lee [2], the value of
the nulling resistor can be estimated by equation 11 (eq.
11:
2
1
M
R
G
= ).
F. Defining Widths and Lengths
Due to channel length modulation, the minimum channel
length should not be chosen for an op-amp design. From
Kartikeya Mayaram (November 2007) [3], the channel length
is best chosen if it is at least 3 to 4 times the minimum
allowable length. For the intended operation of the differential
input and active load stage, transistors M1, M2, M3, and M4
must have matching widths and matching lengths.
Additionally, the biasing transistors (M5, M7, and M8) must
have matching lengths. For the above reasons and due to
simplicity, all lengths are set equal (1um). All transistor widths
are calculated from equation 12 (eq. 12: i i
i
W
W L
L
 
= × 
 
)
where Li is the same for every transistor.
IV. SIMULATION
Once the design parameters were identified, the two stage
op-amp is ready for the last stage of modification. For this
design, PSPICE was chosen as the preferred simulation device.
The analytical result of the op-amp with the BSIM3 model will
not always meet all the specifications. For this reason, minor
modifications of the device parameters. With these
modifications, the design parameters are fully identified, and
the performance of the op-amp can be simulated. The net-list
for such simulation can be found in Appendix C.
A. Offset Voltage
The input common mode range is determined by grounding
the non-inverting input, connecting the inverting input to the
output, and measuring the dc operating point on the inverting
input/output node. This dc operating point is the offset voltage.
B. Open-Loop Gain
The open loop gain is achieved by plotting the magnitude of
the output vs. the input (in dB) of the op-amp while sweeping
the input with an AC source and visually identifying the
magnitude at low frequencies.
C. Unity-Gain Frequency
The unity-gain frequency is achieved similar to that of the
open-loop gain. Simply plot the magnitude of the output vs.
the input of the op-amp (in dB) while sweeping the input with
an AC source and visually identify the frequency at which the
gain is 0dB (1V/V).
D. Phase Margin
The phase margin is achieved by plotting the phase of the
output vs. input (in degrees) of the op-amp and identifying the
magnitude of the value of the phase that corresponds with the
unity-gain frequency. The phase margin is then calculated by
subtracting the corresponding value from 180 degrees.
E. Input Common Mode Range
The input common mode range is determined by applying
the offset voltage from the non-inverting input to the inverting
input, applying a voltage source from the non-inverting input
to ground, and sweeping dc values of the voltage source by the
range of the supply voltages. From a plot of the output
voltages, both edges of the linear region identify VICMAX and
VICMIN.
F. Input Referred Noise Voltage
Within the simulation software, PSPICE, there is a prebuilt
function specifically designed to evaluate the input referred
noise voltage (.NOISE output input). As the frequency
of the input to the op-amp is swept, this function will identify
the input referred noise voltage. Therefore, one can probe the
desired frequency to distinguish the noise voltage referred to
the input.
G. Power Supply Rejection Ratio
As there are positive and negative power supplied to the op-
amp, the power supply rejection ratio (PSRR) can be viewed
from two viewpoints. The positive PSRR can be determined by
equation 13 (eq. 13:
( 0)
( 0)
V dd dd
dd in out
A V V
PSRR
A V V
+ =
= =
=
). The
negative PSRR can be determined by equation 14 (eq.
14:
( 0)
( 0)
V dd ss
ss in out
A V V
PSRR
A V V
− =
= =
=
). In order to simulate the
PSRR, ground the non-inverting input and tie the inverting
input to the output.
V. RESULTS
Table I includes a comprehensive list of the design
specifications and the performance of the two stage op-amp. It
can be seen that the performance of the two stage op-amp
exceeds each of the specifications.
TABLE I
SPECIFICATIONS AND PERFORMANCE
Constraints / Variables Specifications Performance
Open-loop gain > 70 dB 74.46dB
Unity-gain frequency > 10 MHz 79.28MHz
Phase margin 60o
64.3o
ICMR 0.75V
VICMAX 0V
VICMIN -0.75V
IR Noise Voltage @ 1 kHz 1.8mV
IR Noise Voltage @ 4 MHz 11µV
PSRR (Vdd) @ 1kHz 78.5dB
PSRR (Vdd) @ 10MHz 44.9dB
PSRR (Vss) @ 1kHz 77.9dB
PSRR (Vss) @ 10MHz 15.4dB
Table II includes the design parameters of the two stage op-
amp.
TABLE II
DESIGN PARAMETERS
Constraints / Variables Specifications Design Parameters
W1 = W2 260µm
W3 = W4 45µm
W5 120µm
W6 90µm
W7 120µm
W8 8µm
L1-8 ≥ 0.18µm 1µm
Iref 20µA
CL 4pF 4pF
CC 5pF
R 800
VI. CONCLUSION
When the open-loop gain, phase margin, unity gain
frequency, power supply, and load capacitance are the only
constraints in designing a two stage operational amplifier, the
design parameters listed in table II can be determined via the
approach taken in this paper. Identifying the current mirror
through educated analytical methods, then categorizing the
remaining parameters to adhere to the specifications is one
approach which, in this case, yields desired results.
ACKNOWLEDGMENTS
In addition to the listed references, information used
throughout the paper was taken from general knowledge
obtained by the author thorough out his educational career,
industrial career, and hobbyist projects.
REFERENCES
[1] P.E. Allen. “Chapter 6 – CMOS Operational Amplifiers.” CMOS
Analog Circuit Design. 2005.
http://aicdesign.org/scnotes/2004notes/Chap06_2UP_5_2_04_.pdf
[2] Maria del Mar Hershenson, Stephen P. Boyd, and Thomas H. Lee.
“Optimal Design of a CMOS Op-Amp via Geometric Programming.”
IEEE. January 2001.
http://www.stanford.edu/~boyd/papers/pdf/opamp_tcad.pdf
[3] Kartikeya Mayaram. Professor. School of Electrical Engineering and
Computer Science. Oregon State University.
[4] Kartikeya Mayaram. “Lecture notes for ECE 522.” Fall 2007. Oregon
State University
Steven G. Ernst Born in Salem, Oregon in
1984. He became a member of IEEE in 2007.
Currently pursuing a master’s of science degree
in Electrical Engineering at Oregon State
University in Corvallis, Oregon. The
anticipated date of graduation is June 2009. He
received his bachelor’s of science degree in
Electrical Engineering at Oregon State
University in Corvallis, Oregon in March of
2007. His interests include Power Electronics
and Power Systems. His research focuses on
Energy Systems.
He has worked for Intel Corporation as a
JUNIOR DESIGN ENGINEER in 2006 –
2007. He has worked for Siltronic Corporation as a FACILITIES ENGINEER
in 2005. He currently is working for the Army Corps of Engineers as a
STUDENT ENGINEER in Portland, Oregon since 2007.
APPENDIX
A. Two Stage CMOS Operational Amplifier Configuration
B. Calculations
( ) ( )
2 2
'
2
2 2 8.4575 3
143.059
1
1
N
slope E A
K
W V
L
× × − µ
= = =
   
   
   
( ) ( )
2 2
'
2
2 2 4 3
32
1
1
P
slope E A
K
W V
L
× × − µ
= = =
   
   
   
1 1
2 2
1
1
d P ds
d P ds
I V
I V
λ
λ
   + ×
=   
+ ×   
1 52.93 4
0.010791
2.9 4 1 4
P
P
P
E
E
λ
λ
λ
 + ×− 
→ = ∴ =  
− + ×   
1 1
2 2
1
1
d N ds
d N ds
I V
I V
λ
λ
   + ×
=   
+ ×   
1 44.3 4
0.010922
4.39 4 1 6
N
N
N
E
E
λ
λ
λ
 + ×− 
→ = ∴ =  
− + ×   
Set Iref = 20µA
Set I5 = 300µA
Set I6 = 300µA
Set (W/L)8 = 8
5
5 88
300 8
120
20 1
IW W
L I L
µ
µ
     
= × = × =     
     
7
7 88
300 8
120
20 1
IW W
L I L
µ
µ
     
= × = × =     
     
Set buffer = 5
Set CC = 5pF
1 2
5 2 10 5 15.7m C
mA
G C GB buffer p M
V
= × × = × ×Π× × =
( )
22
1
'
1,2 5
15.7
256.76
300 32
m
d P
mGW
L I K µ µ
 
= = = 
× × 
1
2
2.2
2.2 m
m L L
C
G
G C GB C
C
×
= × × = ×
2
2.2 15.7
4 2.765
5
m mA
p
p V
×
= × =
( )
22
2
'
6 6
2.765
89.0659
2 2 300 143.059
m
d N
mGW
L I K µ µ
 
= = = 
× × × × 
5
3,4 66
1
2
d
d
IW W
L I L
    
= × ×    
    
( )
1 300
89.0659 44.5329
2 300
µ
µ
 
= × × = 
 
2
1 1
361.669
2.765M
R
G m
= = = Ω
( )1,2 256.76 1 256.76W m mµ µ= × =
( )3,4 44.5329 1 44.5329W m mµ µ= × =
( )5,7 60 1 60W m mµ µ= × =
( )6 89.0659 1 89.0659W m mµ µ= × =
( )8 8 1 8W m mµ µ= × =
C. PSPICE Code
*Two Stage Op-Amp Design
M1 5 0 4 4 CMOSP W = 260u L = 1u AD = 154.2p AS = 154.2p
PD = 515.2u PS = 515.2u
M2 6 8 4 4 CMOSP W = 260u L = 1u AD = 154.2p AS = 154.2p
PD = 515.2u PS = 515.2u
M3 5 5 3 3 CMOSN W = 45u L = 1u AD = 27p AS = 27p
PD = 91.2u PS = 91.2u
M4 6 5 3 3 CMOSN W = 45u L = 1u AD = 27p AS = 27p
PD = 91.2u PS = 91.2u
M5 4 2 1 1 CMOSP W = 120u L = 1u AD = 72p AS = 72p
PD = 241.2u PS = 241.2u
M6 7 6 3 3 CMOSN W = 90u L = 1u AD = 53.4p AS = 53.4p
PD = 179.2u PS = 179.2u
M7 7 2 1 1 CMOSP W = 120u L = 1u AD = 72p AS = 72p
PD = 241.2u PS = 241.2u
M8 2 2 1 1 CMOSP W = 8u L = 1u AD = 4.8p AS = 4.8p
PD = 17.2u PS = 17.2u
R1 6 10 800
Cc 10 7 5pF
CL 7 0 4pF
Iref 2 3 20uA
Vdd 1 0 0.9v
Vss 3 0 -0.9v
Vi 8 0 DC 0 AC 1V
* BSIM3 model inserted here i.e. .MODEL CMOSN NMOS LEVEL=7
* BSIM3 model inserted here i.e. .MODEL CMOSP PMOS LEVEL=7
.PROBE
.AC DEC 10 1 200MEG
.OP
.END
****************************DC*OPERATING*POINTS********************************
NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE
( 1) .9000 ( 2) .1713 ( 3) -.9000 ( 4) .5619
( 5) -.3006 ( 6) -.3006 ( 7) .0620 ( 8) 0.0000
( 10) -.3006
D. KN’ and VTN Parameters Extraction
E. KP’ and VTP Parameters Extraction
F. Magnitude and Phase of the Two Stage Op-Amp
Vgs 2 0 1
Vds 1 0 0.1V
M1 1 2 0 0 CMOSN W=1u L=1u
*BSIM3 MODEL
.DC Vgs 0 1.5 0.01
.PROBE
.OP
.END
*RESULTS:
*VTN = 0.4V
*KPN = 143.059uA/V^2
Vg 2 0 1
VdS 1 0 -0.1V
M1 0 2 1 1 CMOSP W=1u L=1u
*BSIM3 MODEL
.DC Vg -1 1 0.1
.PROBE
.OP
.END
*RESULTS:
*VTP = -0.29V
*KPN = 32uA/V^2
G. Open-Loop Gain
H. Unity-Gain Frequency & Phase Margin
I. ICMR
M1 5 9 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M2 6 8 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u
M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u
R1 6 10 800
Cc 10 7 5pF
CL 7 0 4pF
Iref 2 3 20uA
Vdd 1 0 0.9v
Vss 3 0 -0.9v
Vic 8 0 DC 0
Vos 8 9 11.69E-06
*BSIM3 MODELS
.PROBE
.DC Vic -1.1 1.1 .01
.OP
.END
*RESULTS:
*Vicmax = 0V
*Vicmin = -0.75V
*ICMR = 0.75V
J. Input Referred Noise Voltage
K. Power Supply Rejection Ratio (Vdd)
L. Power Supply Rejection Ratio (Vss)
M1 5 8 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M2 6 8 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u
M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u
R1 6 10 800
Cc 10 7 5pF
CL 7 0 4pF
Iref 2 3 20uA
Vdd 1 0 0.9v
Vss 3 0 -0.9v
Vi 8 0 DC 11.69E-06 AC 1
*BSIM3 MODELS
.PROBE
.AC DEC 10 0.1 200MEG
.OP
.NOISE V(7) Vi
.END
*RESULTS: 1.8mV, 11uV
M1 5 7 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M2 6 0 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u
M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u
R1 6 10 800
Cc 10 7 5pF
CL 7 0 4pF
Iref 2 3 20uA
Vdd 1 0 0.9v AC 1V
Vss 3 0 -0.9v
*Vi 8 0 DC 0
*BSIM3 MODELS
.PROBE
.AC DEC 10 1 200MEG
.OP
.END
*RESULTS: 78.5dB, 44.9dB
M1 5 7 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M2 6 0 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u
M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u
M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u
M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u
M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u
R1 6 10 800
Cc 10 7 5pF
CL 7 0 4pF
Iref 2 3 20uA
Vdd 1 0 0.9v
Vss 3 0 -0.9v AC 1V
*BSIM3 MODELS
.PROBE
.AC DEC 10 0.5 200MEG
.OP
.END
*RESULTS: 77.9dB, 15.4dB

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Design of a Two-Stage Single Ended CMOS Op-Amp

  • 1. Abstract— Motivated by course requirements and personal interest in CMOS design, the following papers describes the process taken to design a classical two-stage op amp configuration with PMOS input transistors and Miller compensation with a nulling resistor. This design takes into consideration specifications on the minimum channel length, supply voltage, load capacitance, open loop gain, phase margin, and unity gain frequency. An analytical approach to solving the design is the initially prepared. Then minor modifications are made, via the assistance of simulations tools, to ensure all specifications of the two stage op-amp are met. I. INTRODUCTION HE design of a CMOS op-amp contains multiple stages. Identifying the basic structure of the op-amp as well as the compensation type is the first stage. Additionally, identifying the dc currents and transistor sizes and verifying the results via simulations (such as SPICE) follows. The physical implementation of the design, fabrication, specification verification via measurements, and final medications are the additional stages which will not be covered in the following two stage operational amplifier design. Due to the specifications given for the design, the classical two-stage op amp configuration (voltage to current and current to voltage stages) with PMOS input transistors and Miller compensation with a nulling resistor was chosen (shown in Appendix A). The analytical design approach taken for the two stage operational amplifier begins by indentifying the parameters of the current mirror. The differential input parameters, second gain parameters, and active load parameters follow respectively. The final analytical parameter incorporates the resultant compensation. With the utilization of SPICE simulation and the BSIM3 model, the analytical design undergoes minor modifications to achieve the design requirements. II. PARAMETER EXTRACTION While the supply voltage is given, the supply current, operating temperature, and temperature range are ignored for this design. The process specifications can be identified via simulation of the BSIM3 model in SPICE. The process specifications which are necessary for this design are Vtn, Vtp, Kn’, Kp’, λn, and λp. In order to identify Vtn, Vtp, Kn’, and Kp’, linearly sweep values of Vgs for an N-MOS and a P- MOS transistor biased in the saturation region. Vtn and Vtp are identified graphically by determining the value of Vgs when the drain current through the N-MOS and P-MOS transistor become zero. Kn’ and Kp’ are identified by the slope of the √(Id) vs. Vgs curve (slope = √[K’ / 2 * (W/L)]). λn and Submitted for review no later than December 6th , 2007 by 13:00 PST λp are obtained by varying the Vds and identifying the current through an N-MOS and P-MOS transistor biased in the saturation region. Once the currents are obtained, λn and λp can be identified from equation one (eq.1: 1 1 2 2 1 1 d ds d ds I V I V λ λ    + × =    + ×    ). III. DESIGN OF THE OPERATIONAL AMPLIFIER While there are multiple approaches in analytically designing an operational amplifier, the subsequent method utilized consists of selecting the preferred drain currents for each gain stage, identifying the current mirror parameters to achieve such values, and finally designing the gain stages to achieve the op-amp’s specifications. The following describes such process; In Appendix B is the detailed calculations listed in respective order with the proceeding process. A. Current Mirrors The segments of the two stage op-amp which pertains to the current mirror includes; Iref, M5, M7, and M8. The purpose of the current mirrors is to supply a drain current to each of the two gain stages. The reference current (Iref) as well as the current for each of the two stages must be determined. In this design, the reference current was arbitrarily chosen to be 20µA. While not in the requirements of the design, a larger reference current has not been selected to minimize the effect of reference current error. Based upon the requirements of the op-amp, the gain-bandwidth (GB) as well as the open-loop gain (Avo) can be utilized in order to identify the current margin of the first stage. (If there were maximum power dissipation, input common mode range, and/or slew rate requirements, they would be taken into account in selecting the current of the first stage.) The chosen current must be large enough to adhere to the GB requirement (eq.2: 1m C G GB C = ) while small enough to maintain the requirement for open loop gain (eq.3: ( ) 1 2 2 5 6,7 2 m m vo d d N P G G A I I λ λ × × = × × + ). The current of the first gain stage was chosen to be 300µA. From equation 3 and 4 (eq.4: ' 2 6,7 6 2m d N W G I K L   = × × ×    ), the current for the second stage can be chosen as a balance between increasing the trans-conductance without sacrificing the open- loop gain. For simplicity of design, the current of the first and second stage was chosen to be the same (300µA). Transistor 8 is in a diode-connected configuration (drain is directly connected to the gate); therefore the device will Design of a Two-Stage Single Ended CMOS Op-Amp (November 2007) STEVEN G. ERNST T
  • 2. operate in either the saturation or cut-off region. Transistor 8 plays a direct role in determining the VdSAT for the other biasing transistors. With this in mind, the (W/L) ratio for transistor 8 was chosen to be 8. With equation 5, the (W/L) ratio for M5 is calculated (eq. 5: 5 5 88 IW W L I L     = ×        ). Additionally with equation 6, the (W/L) ratio for M7 is calculated (eq. 6: 7 7 88 IW W L I L     = ×        ). B. Differential Input Stage As shown in equation 2, the trans-conductance of the first gain stage can be calculated if the gain-bandwidth and the compensation capacitor (CC) are known. To ensure the two stage op-amp achieves the minimum gain-bandwidth (determined by the unity gain frequency), a gain-bandwidth buffer of 5 is included in equation 2. From Allen [1], when a phase margin of 60o is required, the compensation capacitor must be chosen to be greater than 0.22 times the capacitance seen as the output. With such condition, an arbitrarily chosen value of 5pF for the compensation capacitor and a gain- bandwidth buffer of 5, the gain of the first stage is identified and utilized in equation 7 (eq. 7: 2 1 ' 1,2 5 m d P GW L I K   =  ×  ) to calculate the (W/L) ratio of the differential input stage. C. Second Gain Stage The second pole of the two stage op-amp is determined directly by the gain of the second stage and inversely by the load capacitance (eq. 8: 2 2 m L G P C = ). In order to obtain a sufficient phase margin (PM), the second pole of the two stage op-amp must be located past the unity gain frequency. From Allen [1], the second pole should be extended by at least 1.73 times the gain-bandwidth. For a margin of safety, the value of 2.2 is recommended. With equation 2, equation 8, and the buffer of 2.2, the trans-conductance of the second stage can be calculated. Furthermore, the (W/L) ratio of the transistor in the second gain stage (M6) can be calculated from equation 9 (eq. 9: 2 2 ' 6 62 m d N GW L I K   =  × ×  ). D. Active Load Stage To ensure a reduced input offset voltage, the systematic offset condition must apply; the drain voltages of M3 and M4 must remain the same. This allows the current for the first gain stage to equally split between M1 and M2. With the systematic offset condition applied, the (W/L) ratio for M3 and M4 can be calculated from equation 10 (eq. 10: 5 3,4 66 1 2 d d IW W L I L      = × ×          ). E. Compensation Included in series with the Miller compensation (previously estimated in the differential input stage) on the second gain stage is a nulling resister. The purpose of the nulling resister is to eliminate or move the right hand plane (RHP) zero to the left plane. From Hershenson, Boyd, and Lee [2], the value of the nulling resistor can be estimated by equation 11 (eq. 11: 2 1 M R G = ). F. Defining Widths and Lengths Due to channel length modulation, the minimum channel length should not be chosen for an op-amp design. From Kartikeya Mayaram (November 2007) [3], the channel length is best chosen if it is at least 3 to 4 times the minimum allowable length. For the intended operation of the differential input and active load stage, transistors M1, M2, M3, and M4 must have matching widths and matching lengths. Additionally, the biasing transistors (M5, M7, and M8) must have matching lengths. For the above reasons and due to simplicity, all lengths are set equal (1um). All transistor widths are calculated from equation 12 (eq. 12: i i i W W L L   = ×    ) where Li is the same for every transistor. IV. SIMULATION Once the design parameters were identified, the two stage op-amp is ready for the last stage of modification. For this design, PSPICE was chosen as the preferred simulation device. The analytical result of the op-amp with the BSIM3 model will not always meet all the specifications. For this reason, minor modifications of the device parameters. With these modifications, the design parameters are fully identified, and the performance of the op-amp can be simulated. The net-list for such simulation can be found in Appendix C. A. Offset Voltage The input common mode range is determined by grounding the non-inverting input, connecting the inverting input to the output, and measuring the dc operating point on the inverting input/output node. This dc operating point is the offset voltage. B. Open-Loop Gain The open loop gain is achieved by plotting the magnitude of the output vs. the input (in dB) of the op-amp while sweeping the input with an AC source and visually identifying the magnitude at low frequencies. C. Unity-Gain Frequency The unity-gain frequency is achieved similar to that of the open-loop gain. Simply plot the magnitude of the output vs. the input of the op-amp (in dB) while sweeping the input with an AC source and visually identify the frequency at which the gain is 0dB (1V/V).
  • 3. D. Phase Margin The phase margin is achieved by plotting the phase of the output vs. input (in degrees) of the op-amp and identifying the magnitude of the value of the phase that corresponds with the unity-gain frequency. The phase margin is then calculated by subtracting the corresponding value from 180 degrees. E. Input Common Mode Range The input common mode range is determined by applying the offset voltage from the non-inverting input to the inverting input, applying a voltage source from the non-inverting input to ground, and sweeping dc values of the voltage source by the range of the supply voltages. From a plot of the output voltages, both edges of the linear region identify VICMAX and VICMIN. F. Input Referred Noise Voltage Within the simulation software, PSPICE, there is a prebuilt function specifically designed to evaluate the input referred noise voltage (.NOISE output input). As the frequency of the input to the op-amp is swept, this function will identify the input referred noise voltage. Therefore, one can probe the desired frequency to distinguish the noise voltage referred to the input. G. Power Supply Rejection Ratio As there are positive and negative power supplied to the op- amp, the power supply rejection ratio (PSRR) can be viewed from two viewpoints. The positive PSRR can be determined by equation 13 (eq. 13: ( 0) ( 0) V dd dd dd in out A V V PSRR A V V + = = = = ). The negative PSRR can be determined by equation 14 (eq. 14: ( 0) ( 0) V dd ss ss in out A V V PSRR A V V − = = = = ). In order to simulate the PSRR, ground the non-inverting input and tie the inverting input to the output. V. RESULTS Table I includes a comprehensive list of the design specifications and the performance of the two stage op-amp. It can be seen that the performance of the two stage op-amp exceeds each of the specifications. TABLE I SPECIFICATIONS AND PERFORMANCE Constraints / Variables Specifications Performance Open-loop gain > 70 dB 74.46dB Unity-gain frequency > 10 MHz 79.28MHz Phase margin 60o 64.3o ICMR 0.75V VICMAX 0V VICMIN -0.75V IR Noise Voltage @ 1 kHz 1.8mV IR Noise Voltage @ 4 MHz 11µV PSRR (Vdd) @ 1kHz 78.5dB PSRR (Vdd) @ 10MHz 44.9dB PSRR (Vss) @ 1kHz 77.9dB PSRR (Vss) @ 10MHz 15.4dB Table II includes the design parameters of the two stage op- amp. TABLE II DESIGN PARAMETERS Constraints / Variables Specifications Design Parameters W1 = W2 260µm W3 = W4 45µm W5 120µm W6 90µm W7 120µm W8 8µm L1-8 ≥ 0.18µm 1µm Iref 20µA CL 4pF 4pF CC 5pF R 800 VI. CONCLUSION When the open-loop gain, phase margin, unity gain frequency, power supply, and load capacitance are the only constraints in designing a two stage operational amplifier, the design parameters listed in table II can be determined via the approach taken in this paper. Identifying the current mirror through educated analytical methods, then categorizing the remaining parameters to adhere to the specifications is one approach which, in this case, yields desired results. ACKNOWLEDGMENTS In addition to the listed references, information used throughout the paper was taken from general knowledge obtained by the author thorough out his educational career, industrial career, and hobbyist projects. REFERENCES [1] P.E. Allen. “Chapter 6 – CMOS Operational Amplifiers.” CMOS Analog Circuit Design. 2005. http://aicdesign.org/scnotes/2004notes/Chap06_2UP_5_2_04_.pdf [2] Maria del Mar Hershenson, Stephen P. Boyd, and Thomas H. Lee. “Optimal Design of a CMOS Op-Amp via Geometric Programming.” IEEE. January 2001. http://www.stanford.edu/~boyd/papers/pdf/opamp_tcad.pdf [3] Kartikeya Mayaram. Professor. School of Electrical Engineering and Computer Science. Oregon State University. [4] Kartikeya Mayaram. “Lecture notes for ECE 522.” Fall 2007. Oregon State University Steven G. Ernst Born in Salem, Oregon in 1984. He became a member of IEEE in 2007. Currently pursuing a master’s of science degree in Electrical Engineering at Oregon State University in Corvallis, Oregon. The anticipated date of graduation is June 2009. He received his bachelor’s of science degree in Electrical Engineering at Oregon State University in Corvallis, Oregon in March of 2007. His interests include Power Electronics and Power Systems. His research focuses on Energy Systems. He has worked for Intel Corporation as a JUNIOR DESIGN ENGINEER in 2006 – 2007. He has worked for Siltronic Corporation as a FACILITIES ENGINEER in 2005. He currently is working for the Army Corps of Engineers as a STUDENT ENGINEER in Portland, Oregon since 2007.
  • 4. APPENDIX A. Two Stage CMOS Operational Amplifier Configuration B. Calculations ( ) ( ) 2 2 ' 2 2 2 8.4575 3 143.059 1 1 N slope E A K W V L × × − µ = = =             ( ) ( ) 2 2 ' 2 2 2 4 3 32 1 1 P slope E A K W V L × × − µ = = =             1 1 2 2 1 1 d P ds d P ds I V I V λ λ    + × =    + ×    1 52.93 4 0.010791 2.9 4 1 4 P P P E E λ λ λ  + ×−  → = ∴ =   − + ×    1 1 2 2 1 1 d N ds d N ds I V I V λ λ    + × =    + ×    1 44.3 4 0.010922 4.39 4 1 6 N N N E E λ λ λ  + ×−  → = ∴ =   − + ×    Set Iref = 20µA Set I5 = 300µA Set I6 = 300µA Set (W/L)8 = 8 5 5 88 300 8 120 20 1 IW W L I L µ µ       = × = × =            7 7 88 300 8 120 20 1 IW W L I L µ µ       = × = × =            Set buffer = 5 Set CC = 5pF 1 2 5 2 10 5 15.7m C mA G C GB buffer p M V = × × = × ×Π× × = ( ) 22 1 ' 1,2 5 15.7 256.76 300 32 m d P mGW L I K µ µ   = = =  × ×  1 2 2.2 2.2 m m L L C G G C GB C C × = × × = × 2 2.2 15.7 4 2.765 5 m mA p p V × = × = ( ) 22 2 ' 6 6 2.765 89.0659 2 2 300 143.059 m d N mGW L I K µ µ   = = =  × × × ×  5 3,4 66 1 2 d d IW W L I L      = × ×          ( ) 1 300 89.0659 44.5329 2 300 µ µ   = × × =    2 1 1 361.669 2.765M R G m = = = Ω ( )1,2 256.76 1 256.76W m mµ µ= × = ( )3,4 44.5329 1 44.5329W m mµ µ= × = ( )5,7 60 1 60W m mµ µ= × = ( )6 89.0659 1 89.0659W m mµ µ= × = ( )8 8 1 8W m mµ µ= × = C. PSPICE Code *Two Stage Op-Amp Design M1 5 0 4 4 CMOSP W = 260u L = 1u AD = 154.2p AS = 154.2p PD = 515.2u PS = 515.2u M2 6 8 4 4 CMOSP W = 260u L = 1u AD = 154.2p AS = 154.2p PD = 515.2u PS = 515.2u M3 5 5 3 3 CMOSN W = 45u L = 1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M4 6 5 3 3 CMOSN W = 45u L = 1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M5 4 2 1 1 CMOSP W = 120u L = 1u AD = 72p AS = 72p PD = 241.2u PS = 241.2u M6 7 6 3 3 CMOSN W = 90u L = 1u AD = 53.4p AS = 53.4p PD = 179.2u PS = 179.2u M7 7 2 1 1 CMOSP W = 120u L = 1u AD = 72p AS = 72p PD = 241.2u PS = 241.2u M8 2 2 1 1 CMOSP W = 8u L = 1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u R1 6 10 800 Cc 10 7 5pF CL 7 0 4pF Iref 2 3 20uA Vdd 1 0 0.9v Vss 3 0 -0.9v Vi 8 0 DC 0 AC 1V * BSIM3 model inserted here i.e. .MODEL CMOSN NMOS LEVEL=7 * BSIM3 model inserted here i.e. .MODEL CMOSP PMOS LEVEL=7 .PROBE .AC DEC 10 1 200MEG .OP .END ****************************DC*OPERATING*POINTS******************************** NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE NODE VOLTAGE ( 1) .9000 ( 2) .1713 ( 3) -.9000 ( 4) .5619 ( 5) -.3006 ( 6) -.3006 ( 7) .0620 ( 8) 0.0000 ( 10) -.3006
  • 5. D. KN’ and VTN Parameters Extraction E. KP’ and VTP Parameters Extraction F. Magnitude and Phase of the Two Stage Op-Amp Vgs 2 0 1 Vds 1 0 0.1V M1 1 2 0 0 CMOSN W=1u L=1u *BSIM3 MODEL .DC Vgs 0 1.5 0.01 .PROBE .OP .END *RESULTS: *VTN = 0.4V *KPN = 143.059uA/V^2 Vg 2 0 1 VdS 1 0 -0.1V M1 0 2 1 1 CMOSP W=1u L=1u *BSIM3 MODEL .DC Vg -1 1 0.1 .PROBE .OP .END *RESULTS: *VTP = -0.29V *KPN = 32uA/V^2
  • 6. G. Open-Loop Gain H. Unity-Gain Frequency & Phase Margin I. ICMR M1 5 9 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M2 6 8 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u R1 6 10 800 Cc 10 7 5pF CL 7 0 4pF Iref 2 3 20uA Vdd 1 0 0.9v Vss 3 0 -0.9v Vic 8 0 DC 0 Vos 8 9 11.69E-06 *BSIM3 MODELS .PROBE .DC Vic -1.1 1.1 .01 .OP .END *RESULTS: *Vicmax = 0V *Vicmin = -0.75V *ICMR = 0.75V
  • 7. J. Input Referred Noise Voltage K. Power Supply Rejection Ratio (Vdd) L. Power Supply Rejection Ratio (Vss) M1 5 8 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M2 6 8 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u R1 6 10 800 Cc 10 7 5pF CL 7 0 4pF Iref 2 3 20uA Vdd 1 0 0.9v Vss 3 0 -0.9v Vi 8 0 DC 11.69E-06 AC 1 *BSIM3 MODELS .PROBE .AC DEC 10 0.1 200MEG .OP .NOISE V(7) Vi .END *RESULTS: 1.8mV, 11uV M1 5 7 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M2 6 0 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u R1 6 10 800 Cc 10 7 5pF CL 7 0 4pF Iref 2 3 20uA Vdd 1 0 0.9v AC 1V Vss 3 0 -0.9v *Vi 8 0 DC 0 *BSIM3 MODELS .PROBE .AC DEC 10 1 200MEG .OP .END *RESULTS: 78.5dB, 44.9dB M1 5 7 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M2 6 0 4 4 CMOSP W = 260u L =1u AD = 154.2pAS = 154.2pPD = 515.2uPS = 515.2u M3 5 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M4 6 5 3 3 CMOSN W = 45u L =1u AD = 27p AS = 27p PD = 91.2u PS = 91.2u M5 4 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M6 7 6 3 3 CMOSN W = 90u L =1u AD = 53.4p AS = 53.4p PD = 179.2uPS = 179.2u M7 7 2 1 1 CMOSP W = 120u L =1u AD = 72p AS = 72p PD = 241.2uPS = 241.2u M8 2 2 1 1 CMOSP W = 8u L =1u AD = 4.8p AS = 4.8p PD = 17.2u PS = 17.2u R1 6 10 800 Cc 10 7 5pF CL 7 0 4pF Iref 2 3 20uA Vdd 1 0 0.9v Vss 3 0 -0.9v AC 1V *BSIM3 MODELS .PROBE .AC DEC 10 0.5 200MEG .OP .END *RESULTS: 77.9dB, 15.4dB