Simulations of a typical CMOS amplifier circuit using the Monte Carlo method
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This paper presents simulations of a typical CMOS amplifier circuit using the Monte Carlo method to analyze the effects of manufacturing imperfections on circuit performance. It emphasizes the importance of computational tools in microelectronics and details the influence of component tolerances and threshold voltage variations on the operating point and overall circuit behavior. The findings highlight that incorporating real-world uncertainties leads to more efficient design and simulation processes in microelectronics.
![Borges, Jacques Cousteau da Silva. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 11, ( Part -3) November 2016, pp.73-75
www.ijera.com 73 | P a g e
Simulations of a typical CMOS amplifier circuit using the Monte
Carlo method
Borges, Jacques Cousteau da Silva*
*(Department of Physics, IFRN, Natal-RN-Brazil)
ABSTRACT
In the present paper of Microelectronics, some simulations of a typical circuit of amplification, using a CMOS
transistor, through the computational tools were performed. At that time, PSPICE® was used, where it was
possible to observe the results, which are detailed in this work. The imperfections of the component due to
manufacturing processes were obtained from simulations using the Monte Carlo method. The circuit operating
point, mean and standard deviation were obtained and the influence of the threshold voltage Vth was analyzed.
Keywords: Monte Carlo simulation, analog electronics, amplifiers, instrumentation, manufacturing processes
I. INTRODUCTION
The modeling of circuits, in
microelectronics, is an essential step in the
elaboration of a project, due, mainly, to the high cost
employed in the manufacturing processes. In this
way, we find in this work with a simple circuit, but
with large applications, being the best known
polarization configuration by means of a voltage
divider. Used mainly in the amplification of small
signals.
Therefore, we used the PSPICE®
simulator, where the circuit was described and the
simulations performed, including the Monte Carlo
simulation, which takes into account the
uncertainties of the values of the related physical
quantities. These uncertainties, which are inherent in
the manufacturing processes, but still need to be
taken into consideration before a real integrated
circuit design.
The importance of the computational
simulation can be realized, and in the work will be
shown in detail the main tools used and the results
achieved.
II. DESCRIPTION OF THE CIRCUIT
The simulated circuit can be presented in Fig 01, and
details, in addition to the schematic configuration,
the values of the elements associated to the
transistor. The knots of the circuit are also visible:
M1
Vdd
5V
Vdd
5V
R1
30kΩ
R2
100kΩ
C1
100uF
0
Vdd
G
0
Rd
500Ω
Vdd
Vout
Vin in
out
Fig.1. Schematic diagram of the analyzed Circuit
Another information, not present in the
diagram, is the characteristics of the transistor being:
W = 15μ and L = 0.5μ. Recalling that the transistor
model adopted was the model ami05.mod.In
SPICE® language the circuit can be described
according to the descriptions below:
R2 dd g 100K
R1 g 0 30K
Rd dd out 500
C1 in g 100u
MN1 out g 0 0 n
w=15u L=0.5u
Vdd dd 0 5v
Vin in 0 AC 1
This configuration of voltage divider
polarization provides greater independence of the
currents that circulate through the transistor, an
important fact, mainly because the amplification
values of the transistors are more inaccurate and
quite sensitive to the increase in temperature. In
some references, this factor is placed as a β factor
[1].
The capacitor placed between the input
voltage Vin and the node g is used to separate the
possible DC levels from Vin (the capacitor behaves
as an open circuit for the DC voltages). This
configuration has an operating point, which will be
described next..
III. OPERATING POINT
The first paragraph under each heading or
subheading should be flush left, and subsequent
paragraphs should have a five-space indentation. A
colon is inserted before an equation is presented, but
there is no punctuation following the equation. All
equations are numbered and referred to in the text
solely by a number enclosed in a round bracket (i.e.,
(3) reads as "equation 3"). Ensure that any
RESEARCH ARTICLE OPEN ACCESS](https://image.slidesharecdn.com/k0611037375-161128114010/85/Simulations-of-a-typical-CMOS-amplifier-circuit-using-the-Monte-Carlo-method-1-320.jpg)
![Borges, Jacques Cousteau da Silva. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 11, ( Part -3) November 2016, pp.73-75
www.ijera.com 74 | P a g e
miscellaneous numbering system you use in your
paper cannot be confused with a reference [4] or an
equation (3) designation.
According to Boylestad, in amplification
with transistors, the current and the DC voltage
resulting from the polarization, establish an
operating point in the characteristic curves, which
defines the region that will be used in the
amplification of the signal. As this point is usually
fixed in the curve, it is given the name of point Q,
which comes from the term quiescent point, which
means rest, or stable [1].
To find the point of operation it is
necessary to observe how the voltages Vds and Ids
behave, and these values depend essentially on the
values of the resistors R1 and R2.
Although calculations to find the operating
point are not complicated, they can become long and
laborious in more complex circuits, which increases
the risk of errors throughout the process.
In this first step, the values of the resistors
are considered constant (R1 = 30 kΩ and R2 = 100
kΩ), and then the simulation with ELDO® will be
used, through the [.op]. As a result of the simulation,
we have the value of Ids = 75.55 mA.
IV. INFLUENCE OF THE DIVIDER IN
POLARIZATION
As previously commented, the polarization
depends essentially on the values of R1 and R2. By
changing the values of these resistors, the operating
point is also changed. Thus, even small variations in
R1 and R2 may promote small variations in the
current Ids.
It is well known that the values of the
resistors are not exact, since the manufacturing
process is not perfect. Given this, we can note that
variations due to imperfections can be taken into
account when dealing with microelectronics
projects.
In this case, we can deal with this problem
from a Monte Carlo simulation. The Monte Carlo
method (MMC) is a statistical method used in
stochastic simulations with several applications in
areas such as physics, mathematics and biology [2].
The Monte Carlo method has been used for a long
time as a way of obtaining numerical
approximations of complex functions. This method
typically involves generating observations of some
probability distribution and using the sample
obtained to approximate the function of interest.
According to Hammerseley [3] the name
"Monte Carlo" arose during the Manhattan project in
World War II. In the design and construction of the
atomic bomb, Ulam, von Neumann and Fermi
considered the possibility of using the method,
which involved the direct simulation of probabilistic
problems related to the neutron diffusion coefficient
in certain materials. Although it had attracted the
attention of these scientists in 1948, the logic of the
method had been known for some time.
In summary, the higher the number of
samples analyzed, the closer to the actual (or
average) values the results will be. And
consequently, the lower the standard deviation
found. For our circuit, we will make the values of
the resistors R1 and R2 have a variation of 10%,
simulating an imprecision of 10%, due to the
manufacturing process.
For the Monte Carlo simulation, a sample
of 1000 units was defined. At the end of the
simulation, we analyzed the values of Ids:
.MC 1000
.Extract Label= “Ids” i (rd)
The results of the simulation can be seen
below. The sample was analyzed from a total of
1000. Recalling that the larger the number of
samples, the lower the standard deviation and the
closer the average will be to the nominal / actual
value.
Rage [5.1626 E-4 1.0419 E-3]
Valor Nominal = 7.5515 E-4
Average Value = 7.5791 E-4
δ = 1.0905 E-4
δ = 1.0908 E-4
V. INFLUENCES OF VTH
Although the uncertainties in the resistivity
values in the resistors are common due to
fabrication, these are not the only elements of the
circuits that can suffer this kind of variation.
Elements such as capacitors and even transistors can
also suffer from the "imperfect" manufacturing
process and are therefore susceptible to this type of
uncertainty.
Recalling that the transistor is simulated
following a pre-established model, with tens (and
sometimes hundreds) of diverse variables. In order
to simulate these variations, it is necessary to modify
the variables in the transistor model itself, which in
this case will be the Vth.
Proceeding to file ami.05.mod, it was
determined that the Vth should have a Gaussian
variation, of standard deviation equal to 5% in the
value of the threshold voltage Vth0.
For analysis, the EZwave® was used, with
the frequency response, by means of a log scale,
similar to a Bode diagram. The result of this analysis
can be observed in Fig 02, which details the
behavior of the circuit in front of this variation in the
transistor.](https://image.slidesharecdn.com/k0611037375-161128114010/85/Simulations-of-a-typical-CMOS-amplifier-circuit-using-the-Monte-Carlo-method-2-320.jpg)
![Borges, Jacques Cousteau da Silva. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 11, ( Part -3) November 2016, pp.73-75
www.ijera.com 75 | P a g e
Fig.2. Frequency response
VI. CONCLUSION
Once again we can see the importance of
computational tools in microelectronics projects.
One can also see, through these simulations, how the
uncertainties of values, inserted through
manufacturing processes, can not be overlooked,
especially in micrometric (and even nanometric)
mechanisms such as microelectronic circuits. The
presentation to new mathematical tools such as
Monte Carlo simulation and the Gaussian
distribution inserted in Vth provided an increase in
the range of possibilities in statistical analysis of the
circuits.
Finally, it can be concluded that the
simulations can not only be a representation of the
idealized, but an approximation of the real, the
imperfect. The more real features (such as
imperfections) are inserted in the computational
process, the more efficient the design and simulation
process, and consequently the pre-fabrication
process [4]
REFERENCES
[1] Boylestad, Robert L. Introduction to circuit
analysis. Pearson Education, 2012.
[2] J. Hromkovic, Algorithms for hard
problems: introduction to combinatorial
optimization, randomization,
approximation, and heuristics. [S.l.]:
Springer-Verlag, London - Berlin -
Heidelberg - New York, 2001.
[3] J.M. Hammersley, D. C. Handscomb.
“Monte Carlo Methods”Editora: Wiley.
1ed, 1964.
[4] [A. Maxim and M. Gheorghe,"A novel
physical based model of deep-submicron
CMOS transistors mismatch for Monte
Carlo SPICE simulation," Circuits and
Systems, 2001. ISCAS 2001. The 2001
IEEE International Symposium on,Syd ney,
NSW, 2001, pp. 511-514 vol. 5.](https://image.slidesharecdn.com/k0611037375-161128114010/85/Simulations-of-a-typical-CMOS-amplifier-circuit-using-the-Monte-Carlo-method-3-320.jpg)
Simulations of a typical CMOS amplifier circuit using the Monte Carlo method
- 1. Borges, Jacques Cousteau da Silva. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 6, Issue 11, ( Part -3) November 2016, pp.73-75 www.ijera.com 73 | P a g e Simulations of a typical CMOS amplifier circuit using the Monte Carlo method Borges, Jacques Cousteau da Silva* *(Department of Physics, IFRN, Natal-RN-Brazil) ABSTRACT In the present paper of Microelectronics, some simulations of a typical circuit of amplification, using a CMOS transistor, through the computational tools were performed. At that time, PSPICE® was used, where it was possible to observe the results, which are detailed in this work. The imperfections of the component due to manufacturing processes were obtained from simulations using the Monte Carlo method. The circuit operating point, mean and standard deviation were obtained and the influence of the threshold voltage Vth was analyzed. Keywords: Monte Carlo simulation, analog electronics, amplifiers, instrumentation, manufacturing processes I. INTRODUCTION The modeling of circuits, in microelectronics, is an essential step in the elaboration of a project, due, mainly, to the high cost employed in the manufacturing processes. In this way, we find in this work with a simple circuit, but with large applications, being the best known polarization configuration by means of a voltage divider. Used mainly in the amplification of small signals. Therefore, we used the PSPICE® simulator, where the circuit was described and the simulations performed, including the Monte Carlo simulation, which takes into account the uncertainties of the values of the related physical quantities. These uncertainties, which are inherent in the manufacturing processes, but still need to be taken into consideration before a real integrated circuit design. The importance of the computational simulation can be realized, and in the work will be shown in detail the main tools used and the results achieved. II. DESCRIPTION OF THE CIRCUIT The simulated circuit can be presented in Fig 01, and details, in addition to the schematic configuration, the values of the elements associated to the transistor. The knots of the circuit are also visible: M1 Vdd 5V Vdd 5V R1 30kΩ R2 100kΩ C1 100uF 0 Vdd G 0 Rd 500Ω Vdd Vout Vin in out Fig.1. Schematic diagram of the analyzed Circuit Another information, not present in the diagram, is the characteristics of the transistor being: W = 15μ and L = 0.5μ. Recalling that the transistor model adopted was the model ami05.mod.In SPICE® language the circuit can be described according to the descriptions below: R2 dd g 100K R1 g 0 30K Rd dd out 500 C1 in g 100u MN1 out g 0 0 n w=15u L=0.5u Vdd dd 0 5v Vin in 0 AC 1 This configuration of voltage divider polarization provides greater independence of the currents that circulate through the transistor, an important fact, mainly because the amplification values of the transistors are more inaccurate and quite sensitive to the increase in temperature. In some references, this factor is placed as a β factor [1]. The capacitor placed between the input voltage Vin and the node g is used to separate the possible DC levels from Vin (the capacitor behaves as an open circuit for the DC voltages). This configuration has an operating point, which will be described next.. III. OPERATING POINT The first paragraph under each heading or subheading should be flush left, and subsequent paragraphs should have a five-space indentation. A colon is inserted before an equation is presented, but there is no punctuation following the equation. All equations are numbered and referred to in the text solely by a number enclosed in a round bracket (i.e., (3) reads as "equation 3"). Ensure that any RESEARCH ARTICLE OPEN ACCESS
- 2. Borges, Jacques Cousteau da Silva. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 6, Issue 11, ( Part -3) November 2016, pp.73-75 www.ijera.com 74 | P a g e miscellaneous numbering system you use in your paper cannot be confused with a reference [4] or an equation (3) designation. According to Boylestad, in amplification with transistors, the current and the DC voltage resulting from the polarization, establish an operating point in the characteristic curves, which defines the region that will be used in the amplification of the signal. As this point is usually fixed in the curve, it is given the name of point Q, which comes from the term quiescent point, which means rest, or stable [1]. To find the point of operation it is necessary to observe how the voltages Vds and Ids behave, and these values depend essentially on the values of the resistors R1 and R2. Although calculations to find the operating point are not complicated, they can become long and laborious in more complex circuits, which increases the risk of errors throughout the process. In this first step, the values of the resistors are considered constant (R1 = 30 kΩ and R2 = 100 kΩ), and then the simulation with ELDO® will be used, through the [.op]. As a result of the simulation, we have the value of Ids = 75.55 mA. IV. INFLUENCE OF THE DIVIDER IN POLARIZATION As previously commented, the polarization depends essentially on the values of R1 and R2. By changing the values of these resistors, the operating point is also changed. Thus, even small variations in R1 and R2 may promote small variations in the current Ids. It is well known that the values of the resistors are not exact, since the manufacturing process is not perfect. Given this, we can note that variations due to imperfections can be taken into account when dealing with microelectronics projects. In this case, we can deal with this problem from a Monte Carlo simulation. The Monte Carlo method (MMC) is a statistical method used in stochastic simulations with several applications in areas such as physics, mathematics and biology [2]. The Monte Carlo method has been used for a long time as a way of obtaining numerical approximations of complex functions. This method typically involves generating observations of some probability distribution and using the sample obtained to approximate the function of interest. According to Hammerseley [3] the name "Monte Carlo" arose during the Manhattan project in World War II. In the design and construction of the atomic bomb, Ulam, von Neumann and Fermi considered the possibility of using the method, which involved the direct simulation of probabilistic problems related to the neutron diffusion coefficient in certain materials. Although it had attracted the attention of these scientists in 1948, the logic of the method had been known for some time. In summary, the higher the number of samples analyzed, the closer to the actual (or average) values the results will be. And consequently, the lower the standard deviation found. For our circuit, we will make the values of the resistors R1 and R2 have a variation of 10%, simulating an imprecision of 10%, due to the manufacturing process. For the Monte Carlo simulation, a sample of 1000 units was defined. At the end of the simulation, we analyzed the values of Ids: .MC 1000 .Extract Label= “Ids” i (rd) The results of the simulation can be seen below. The sample was analyzed from a total of 1000. Recalling that the larger the number of samples, the lower the standard deviation and the closer the average will be to the nominal / actual value. Rage [5.1626 E-4 1.0419 E-3] Valor Nominal = 7.5515 E-4 Average Value = 7.5791 E-4 δ = 1.0905 E-4 δ = 1.0908 E-4 V. INFLUENCES OF VTH Although the uncertainties in the resistivity values in the resistors are common due to fabrication, these are not the only elements of the circuits that can suffer this kind of variation. Elements such as capacitors and even transistors can also suffer from the "imperfect" manufacturing process and are therefore susceptible to this type of uncertainty. Recalling that the transistor is simulated following a pre-established model, with tens (and sometimes hundreds) of diverse variables. In order to simulate these variations, it is necessary to modify the variables in the transistor model itself, which in this case will be the Vth. Proceeding to file ami.05.mod, it was determined that the Vth should have a Gaussian variation, of standard deviation equal to 5% in the value of the threshold voltage Vth0. For analysis, the EZwave® was used, with the frequency response, by means of a log scale, similar to a Bode diagram. The result of this analysis can be observed in Fig 02, which details the behavior of the circuit in front of this variation in the transistor.
- 3. Borges, Jacques Cousteau da Silva. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 6, Issue 11, ( Part -3) November 2016, pp.73-75 www.ijera.com 75 | P a g e Fig.2. Frequency response VI. CONCLUSION Once again we can see the importance of computational tools in microelectronics projects. One can also see, through these simulations, how the uncertainties of values, inserted through manufacturing processes, can not be overlooked, especially in micrometric (and even nanometric) mechanisms such as microelectronic circuits. The presentation to new mathematical tools such as Monte Carlo simulation and the Gaussian distribution inserted in Vth provided an increase in the range of possibilities in statistical analysis of the circuits. Finally, it can be concluded that the simulations can not only be a representation of the idealized, but an approximation of the real, the imperfect. The more real features (such as imperfections) are inserted in the computational process, the more efficient the design and simulation process, and consequently the pre-fabrication process [4] REFERENCES [1] Boylestad, Robert L. Introduction to circuit analysis. Pearson Education, 2012. [2] J. Hromkovic, Algorithms for hard problems: introduction to combinatorial optimization, randomization, approximation, and heuristics. [S.l.]: Springer-Verlag, London - Berlin - Heidelberg - New York, 2001. [3] J.M. Hammersley, D. C. Handscomb. “Monte Carlo Methods”Editora: Wiley. 1ed, 1964. [4] [A. Maxim and M. Gheorghe,"A novel physical based model of deep-submicron CMOS transistors mismatch for Monte Carlo SPICE simulation," Circuits and Systems, 2001. ISCAS 2001. The 2001 IEEE International Symposium on,Syd ney, NSW, 2001, pp. 511-514 vol. 5.