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Titlemathcad_-_opamp_1974
Tags Field Effect Transistor Mosfet Cmos Operational Amplifier
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Page 1

An Operational Amplifier for a CMOS VLSI Design
by Marc Bernier, Consultant
Middleboro, MA

Recent advances in Very Large Scale Integration (VLSI) technologies have made possible the
realization of complete systems on a single chip. Since complete systems often include analog
devices as well as digital devices, there has been a reemergence of interest in Metal-Oxide
Semiconductor (MOS) analog circuits. Examples of these types of integrated circuits include:
Digital-to-Analog Converters, Analog-to-Digital Converters, Voltage-Controlled Oscillators,
Analog Input/Output Pads, and amplifiers. All these devices can be based on a simple element
in MOS design, the MOSFET transistor.

CMOS VLSI Design

Complementary Metal-Oxide-Semiconductor (CMOS) technology is circuit implementation using
both pMOS and nMOS transistors on the same silicon chip. CMOS designs typically offer high
gain and speed at low power consumption. In addition, CMOS scales well to smaller devices
without drastic changes in performance. The effect of device dimensions on FET operation
becomes clearer in the design calculations below.

Several design parameters affect MOS device performance, including the following: doping
(level of semiconducting impurities added) of the substrate, doping of the source, doping of
the drain, oxide thickness, channel width, and channel length. In practice, all but two of these
parameters are often determined by the manufacturing process and are out of the designer's
control. The two remaining parameters are channel width, w, and channel length, L, as shown
in the drawing. CMOS design becomes a simple process of connecting together "rectangles"
(transistors) of varying dimensions. Typical topography for such a circuit is included at the
end of this article.

Additionally, certain device characteristics will not change when the transistor is scaled.
Thus, for a preliminary design, a width-to-length ratio, or "aspect ratio" can be used in
place of the two dimensions. Performance factors such as current handling, noise, and
actual circuit size are affected by a transistor's dimension, so specific transistor size is an
important consideration in later design stages. Transistors designed in this way can be
arranged into larger devices that can be addressed as a single unit on a chip.

Operational Amplifier Design Using Mathcad

One of the most common analog circuit elements is the operational amplifier. It will
compare two inputs (V+ an V- in the schematic on the next page) and amplify their
difference. This device is commonly used to amplify small signals, to add/subtract
voltages, and in active filtering. It must have high gain, low current draw (high

Page 2

input/output impedance), and should function over a variety of frequencies. This
article considers the design of such a device using Mathcad.

The op-amp will meet or exceed the
following specifications:
1) open loop gain of at least 1000
2) total current drain < 20
3) gain-bandwidth product > 106

The "garden-variety" operational amplifier
circuit shown at left was used as a basis for
this design. The maroon transistors (with
circles on the gates) are p-type, and the
purple transistors (no circles) are n-type.

Transistor 2 functions as a constant current source, and transistors 1, 3, and 8 function as two
current mirror 'pairs'. The network consisting of transistors 4, 5, 6, and 7 is the differential
amplifier, and transistor 9 is an output amplifier stage (a very simple one!). Some constants
and parameters required for the calculation are listed below. These will be used along with
the specifications above to design the size of the transistors required for this circuit.

Specified by overall VLSI circuit requirements:

Vdd 5 volt Supply voltage

Specified by doping levels:

Channel length modulation for n-type and p-type devices

n 2.752 10
3

volt
1

p 1.427 10
2

volt
1

Intrinsic transconductance for n-type and p-type devices

Kn 5.518 10
5 amp

volt
2

Kp 2.226 10
5 amp

volt
2

Specified by doping levels and oxide thickness:

Threshold voltage for n-type and p-type devices

Vtn .77 volt Vtp .77 volt

The parameters , K, and Vt are determined by the manufacturing process. First, transistor 1's
aspect ratio is calculated. It will be a function of the required voltage drop across the transistor to
make sure it is operating above the threshold voltage, Vt. A minimal gate-to-source voltage of
Vtp +Vsafety is used to insure the device would turn on despite any variations in processing. It's
important to make the voltage as small as possible, since the output voltage in the circuit should
not be clipped by the supply voltage. Since the current consumption is limited to 20 A, a
somewhat arbitrary current of 5 A is drained through this leg.

A 10
6

amp Vsafety 0.5 volt

Page 3

Vgs1 Vtp Vsafety Ids1 5 10
6

amp

The formula for AR1 is derived from the following formula [1]:

Ids
1

2
K AR Vgs Vt= AR1

2 Ids1

Kp Vgs1 Vtp
2

AR1 1.797

Since Vgs1 has been fixed, Vgs2 is also fixed, since they both carry the same current. Transistor
2's aspect ratio can be calculated:

Vgs2 Vdd Vgs1 AR2
2 Ids1

Kp Vgs2 Vtp
2

AR2 0.051

Vgs2 3.73volt

The difference amplifier can now be designed, starting at the first current mirror (transistors 1
and 3). Like Vgs1, a minimal Vgs3 is desired for the same reason. The current through
transistor 3 is to be half of Ids1.

Vgs3 Vgs1 Idsd
1

2
Ids1 AR3

2 Idsd

Kp Vgs3 Vtp
2

AR3 0.898

Note that AR3 is exactly half that of AR1. This is because the following equation applies:

AR1

AR3

Ids1

Idsd
= and Idsd was chosen to be one half of the current in the first leg.

The derivation of this formula isn't shown here, although it is relatively easy. The name 'current
mirror' is quite fitting. Now the differential amplifier can be designed.

To obtain a balanced output, transistors 4 and 6 should be matched, as well as transistors 5
and 7. Because of this balancing, the current flowing through transistors 4 and 5 is the same
as the current flowing through transistors 5 and 7 (almost no current is drained through the
gates of transistors 5 and 7). Like transistor 1, transistor 5's (and 7's) Vgs is to be as small as
possible.

Vgs5 Vtn Vsafety AR5

2
Idsd

2

Kn Vgs5 Vtn
2

AR5 0.181 AR7 AR5

Some additional parameters must be determined to calculate the
high-frequency characteristics. The drain conductance and
transconductance parameters are needed for ac small signal analysis.

Page 4

gd3 p Idsd gd3 3.567 10
8

mho (drain conductance)

gm3 2 Idsd Kp AR3 gm3 1 10
5

mho (transconductance)

gd5 n
Idsd

2
gd5 3.44 10

9
mho

mho
1

ohm
gm5 2

Idsd

2
Kn AR5 gm5 5 10

6
mho

gd4 p
Idsd

2
gd4 1.784 10

8
mho

Like Vgs1, Vgs4 should be as small as possible while still turning on the transistor:

Vgs4 Vtp Vsafety AR4

2
Idsd

2

Kp Vgs4 Vtp
2

AR4 0.449

which allows calculation of gm4:

gm4 2
Idsd

2
Kp AR4 gm4 5 10

6
mho

The drain conductances and transconductances that have been calculated enter into the
following equation for the gain of the differential amplifier. The derivation of this equation
(omitted) is somewhat lengthy. In a nutshell, an ac circuit analysis is performed on the
differential amplifier using the appropriate small signal models for the transistors [2].
Some reasonable approximations (such as gm4 + gd4 gm4) were made to greatly
simplify the equation.

The differential stage gain is given by

Ad
1

2

gm4

gd4 gd5
Ad 117.495

This is the gain of the differential amplifier stage. Thus, a gain of 10 or so is needed at the
output amplifier stage. Now for the output stage: as with the differential amplifier, the current is
set to half of the mirror's Ids:

Vgs8 Vtp Vsafety Ids8
1

2
Ids1

AR8
2 Ids8

Kp Vgs8 Vtp
2

AR8 0.898

Page 5

gd8 p Ids8 gd8 3.567 10
8

mho

At several points in this design, it was mentioned that certain Vgs's should be as small as
possible. The reason for this is apparent at the output stage: Vout cannot be above
Vdd-Vgs8 or below Vgs9. Vout will be clipped if such a situation does occur. Because
Vgs1=Vgs3=Vgs8, these voltages also reflect this consideration. Similarly, Vgs5 and
Vgs7 are limited to prevent clipping at the output of the differential amplifier.

Vgs9 Vtn Vsafety

gd9 n Ids8 gd9 6.88 10
9

mho

AR9
2 Ids8

Kn Vgs9 Vtn
2

AR9 0.362

gm9 2 Ids8 Kn AR9 gm9 1 10
5

mho

The equation for the gain of the output stage is shown below:

Ao
gm9

gd9 gd8
Ao 235

Thus, the total open-loop gain for the amplifier is:

Required gain
is > 1000.Av Ad Ao Av 27610

It would appear that the amplifier was over-engineered by a factor of 10. The circuit was
simulated (with CAzM) and an open-loop gain of 2680 was determined. Two reasons could
account for the design/simulation difference. First, to determine the open-loop gain, an ac
voltage is applied to the inputs in the simulation. This ac voltage is biased with a dc voltage
determined by the simulation's transfer characteristic curve of the circuit (i.e., Vin versus Vout).
The determination of the bias voltage is, at best, an inaccurate procedure. Unfortunately, a
small change in the bias voltage greatly affects the open-loop gain, so the optimal open-loop
gain may not have been found. Second, the simulation takes into account all sorts of parasitic
capacitances not included here, which all affect the gain adversely.

The final part of the design calls for selection of a compensation
capacitor, C. The capacitor acts simply as a low pass filter to
ensure that the op-amp will be stable. Stability is indicated by the
condition where the gain of the circuit vs. frequency crosses unity
before the phase crosses 0 . A little trial-and-error with the
simulator found a value of .1 pF that fit the requirement rather well.

C .1 10
12

farad

Hz
1

sec

Page 6

Rout s( )
1

s C gd9 gd3 s
10

3

2
Hz

10
4

2
Hz

10
7

2
Hz

100 1 10
3

1 10
4

1 10
5

1 10
6

1 10
7

0

1 10
7

2 10
7

3 10
7

Rout s( )

sg

s

Use the marker on
the graph to help find
a guess for the 3 dB
point:

sg 1 10
5

sec
1

The dc output impedance is:

Rout 0 sec
1

2.35 10
7

ohm

The -3db point is found by the following root finding equation, using the guess value on the
previous page:

s3dB root Rout sg Rout 0 sec
1 1

2
sg

So the -3db point occurs at the frequency

s3dB 1.763 10
5

sec
1

Finally, the Gain-Bandwidth product can be determined:

GBW Av s3dB GBW 4.867 10
9

sec
1

The above information along with the simulation
results give the following specifications:

Current drain:
Power consumption:
Bias Voltage:
Open-Loop Gain:
Bandwidth:
Gain-bandwidth product:


Simulated
10 amp
50 watt
2.5001 volt
2680
2.2 kHz
3 MHz

Calculated
10 mamp
50 mwatt
2.5 volt
27610
1.8 MHz
4.9 GHz

Specified
< 20
< 100 watt
2.5 volt
>1000
>1 kHz
> 1 GHz

On the following page is a schematic of the op-amp layout. Note that transistor 9 has been
split in half. Two transistors in parallel with identical lengths, L, and different widths, W1 and
W2, are equivalent to one transistor with the same length, L , but with a width of W1+W2
(similar property to parallel capacitors). This property makes design a bit easier, because
very wide transistors can be made into two or three not-so-wide transistors which fit more

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