Optimal design of symmetric switching CMOS inverter using symbiotic organisms search algorithm

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 1, February 2020, pp. 171~179
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i1.pp171-179  171
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Optimal design of symmetric switching CMOS inverter using
symbiotic organisms search algorithm
Nihad Dib1
, Umar Al-Sammarraie2
1
Department of Electrical Engineering, Jordan University of Science and Technology, Jordan
2
Department of Social Sciences, Technische Universität Kaiserslautern, Germany
2
Department of Electrical and Communications Engineering, German Jordanian University, Jordan
Article Info ABSTRACT
Article history:
Received Feb 25, 2019
Revised Jul 31, 2019
Accepted Aug 29, 2019
This paper investigates the optimal design of symmetric switching CMOS
inverter using the Symbiotic Organisms Search (SOS) algorithm. SOS has
been recently proposed as an effective evolutionary global optimization
method that is inspired by the symbiotic interaction strategies between
different organisms in an ecosystem. In SOS, the three common types of
symbiotic relationships (mutualism, commensalism, and parasitism) are
modeled using simple expressions, which are used to find the global
minimum of the fitness function. Unlike other optimization methods, SOS
has no parameters to be tuned, which makes it an attractive and easy-to-
implement optimization method. Here, SOS is used to design a high speed
symmetric switching CMOS inverter, which is considered the most
fundamental logic gate. SOS results are compared to those obtained using
several optimization methods, like particle swarm optimization (PSO),
genetic algorithm (GA), differential evolution (DE), and other ones,
available in the literature. It is shown that the SOS is a robust straight-
forward evolutionary algorithm that can compete with other well-known
advanced methods.
Keywords:
CMOS inverter
Evolutionary optimization
Switching characteristics
Symbiotic organisms search
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Nihad Dib,
Department of Electrical Engineering,
Jordan University of Science and Technology,
P. O. Box 3030, Irbid 22110, Jordan.
Email: nihad@just.edu.jo
1. INTRODUCTION
Needless to say, the inverter is the most basic logic gate that performs a Boolean operation on
a single input. It is considered as the main component in the design of more complex digital integrated
circuits (ICs). A CMOS inverter consists of NMOS and PMOS transistors, where the transistors’ geometries
(channel width and length) have to be carefully chosen. The switching characteristics of the inverter are
the fundamental parameters to characterize the integrated circuits technology. Thus, to ensure the reliability
and performance of a CMOS circuit, the switching speed of the circuit must be optimized very early in
the design phase. The objective of this paper is to optimally design a symmetric switching CMOS inverter,
which is of importance to the CMOS IC industry. The right and effective choice of the transistors’ channel
widths (W) and lengths (L) and the output load capacitance (CL) is very important and essential in designing
a CMOS inverter with symmetric switching characteristics, i.e, symmetric output voltage waveform.
Performing such a task by hand turns out to be tedious and time consuming. Recently, to make the design
more reliable and flexible, different evolutionary optimization algorithms have been used in the design of
CMOS inverter with symmetric switching characteristics [1-8]. In [1, 2], particle swarm optimization (PSO)
was used in the design of a CMOS inverter having symmetrical waveform of output voltage with equal rise
time (tr) and fall time (tf) and equal propagation delay times (tpHL and tpLH). In [3-8], B. P. De et al applied

 ISSN: 2088-8708
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172
several nature-inspired optimization methods on the design of symmetric switching CMOS inverter.
Specifically, PSO with constriction factor and inertia weight approach (PSOCFIWA), differential evolution
(DE), craziness based particle swarm optimization (CRPSO), hybrid harmony search with differential
evolution (HS-DE), PSO with aging leader and challenger (ALC-PSO), and firefly algorithm (FA) were
applied in [3-8], respectively. In each paper, it was shown that the used optimization method gives better
results than those presented in [1, 2] and those obtained using the real-coded genetic algorithm (RGA).
In this paper, the optimal transient behavior of CMOS inverter is investigated using a new
optimization method, the symbiotic organisms search (SOS) [9]. Specifically, SOS is used to find
the optimum values of the widths and lengths of the channels of the NMOS and PMOS transistors along with
the load capacitor that give symmetric switching characteristics of the inverter. Since its invention, SOS has
been used in several optimization problems in engineering. In [10, 11], SOS has been applied to solve
the optimal power flow and the economic emission load dispatch problems. Moreover, SOS has been
successfully applied in the optimal design of linear and circular antenna arrays [12, 13], in solving the load
frequency control problem [14], in structure optimization problems [15], in solving unconstrained function
optimization [16], in the design of an improved 3D Turbo Code [17], in the optimal design of analog active
filters [18], in the synthesis of elliptical antenna arrays [19], and other engineering areas [20-24]. The results
indicated that SOS gives very good results and is very competitive with the state of the art for the solution of
these problems, which motivated this work. The SOS results will be compared with those obtained using
different optimization methods available in the literature [3-7]. Moreover, SOS results will be compared with
those obtained by the PSPICE simulator.
The paper is divided as follows: SOS algorithm is briefly described in Section 2. The interested
reader may consult the above references for more details on the SOS algorithm. In Section 3, the formulation
of the problem is presented, while numerical results and discussion are presented in Section 4. Section 5
concludes the paper.
2. SOS ALGORITHM
Symbiotic Organisms Search (SOS) has been recently proposed in [9] as an effective evolutionary
global optimization method. It is based on the symbiotic interaction strategies between different organisms in
an ecosystem. Symbiosis describes the interaction between different species living together. In SOS, the three
main types of symbiotic relationships (mutualism, commensalism, and parasitism) are modeled using simple
mathematical expressions. An example of mutualism, in which both organisms benefit from such
a relationship, is the interaction between bees and flowers. An example of commensalism, where one species
benefits from the interaction, while the other does not gain or lose anything, is the relationship between
remora fish and sharks. An example of parasitism, where the relationship benefits one species while the other
species is harmed, is the interaction between the plasmodium parasites and humans. One of the main
advantages in using SOS is the fact that it is free from any tuning parameters. In its implementation, SOS
requires only common controlling parameters like population size (number of organisms) and maximum
number of iterations.
Similar to most evolutionary optimization methods, the first step in SOS is to randomly generate
an initial population of organisms (which is called the ecosystem). The number of generated organisms N is
called the population size. Thus, the ecosystem is an NxD matrix, where D is the dimension of the problem
(i.e., the number of design variables). An organism (individual) Xi (i=1, …, N) within the ecosystem
(population) is a real-valued vector with D elements which represents a single possible solution to a specific
optimization problem. The second step is to associate each organism Xi of the ecosystem (i.e., each row in
the ecosystem matrix) with a certain value of the fitness function which imitates the degree of adaptation to
the desired objective. Afterwards, new solutions are created by simulating the above mentioned three
symbiotic interactions between the organisms in the ecosystem. Each organism in the ecosystem randomly
interacts with other organisms through all these three phases and new organisms are created which have
better fitness function value. This process is repeated until a specific termination criterion (e.g., maximum
number of iterations) is fulfilled. In each iteration, every organism Xi (i=1, …, N) goes through the following
three phases:
2.1. Mutualism phase
Firstly, another organism Xj is randomly selected to “mutually” interact with Xi. Then, Xi and Xj are
updated according to the following expressions [9]:
(1)

Int J Elec & Comp Eng ISSN: 2088-8708 
Optimal design of symmetric switching CMOS inverter using symbiotic organisms search … (Nihad Dib)
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(2)
where the (Mutual Vector) is the average between the two interacting organisms, i.e.,
(Mutual Vector) = ( Xi +Xj ) / 2.
In the above equations, Xbest is the best organism (that is the one that resulted in the minimum value
of the fitness function), and rand(0,1) is a random number in the range [0, 1]. The parameters BF1 and BF2
are called benefit factors which represent the level of benefit between the organisms as the organisms may
fully or partially benefit from such a relationship. They are randomly determined as either 1 or 2. At the end
of the mutualism phase, the new organisms are accepted, and taken to the next phase, if they give better
(that is less) fitness function values than their pre-interaction fitness values.
2.2. Commensalism phase
Similar to the first phase, an organism Xj is randomly selected to interact with Xi. However, in this
phase, Xi tries to benefit from the interaction, while Xj itself is not affected at all. Then, Xi is updated using
the following expression [9]:
(3)
The term (Xbest – Xj ) reflects the benefit provided by Xj to Xi to increase its degree of adaptation
(to the highest degree represented by Xbest) such that it can better survive in the ecosystem. The new
organism Xi (new) is adopted, and taken to the third phase, if it results in a better fitness function value. If not,
then Xi remains the same, which means that it didn’t gain anything from such an interaction with Xj.
2.3. Parasitism phase
In this last third phase, a new vector called “Parasite Vector” is created by mutating Xi in a random
dimension using a random number. Then, an organism Xj is randomly selected to serve as a host to
the parasite vector, and the fitness function for both organisms is evaluated. If the parasite vector has a better
fitness, then Xj is removed and its position is taken over by the parasite vector. On the other hand, if Xj has
a better fitness value, then it will be immune from the parasite vector, and will not be affected by its
existence. In this case, the parasite vector is simply removed from the ecosystem. A pseudo-code and
a detailed description of the SOS algorithm can be found in [9].
3. PROBLEM FORMULATION
One of the most important issues in the CMOS inverters is the symmetric switching speed.
In CMOS inverters circuits, the switching speed is affected by the output resistance of the high state of
the P-MOSFET, the capacitive load, the length and width of the PMOS and the NMOS channels and on both
the input voltage Vin and the power supply. The switching speed can be improved by decreasing the output
resistance and this can be done by increasing the power supply. However, the larger power supply will result
in increased power dissipation. In this paper, the CMOS inverter is optimized to operate at 2.5 V, and
the switching operation is analyzed to determine the rise and fall times (tr and tf ) and propagation delays (tpHL
and tpLH ). The PSPICE schematic diagram of the CMOS inverter is shown in Figure 1, while the input and
the output waveforms are shown in Figure 2.
As well known, the fall time (tf ), is the time required for the output voltage to drop from V90% of
the output voltage level down to V10%, and the rise time (tr ) is defined as the time required for the output
voltage to rise from V10% to V90%. They can be computed as follows [3-8]:
*
( )
+ (4)
[
( ( ))
] (5)
On the other hand, the high-to-low propagation delay time (tpHL ) is defined as the time delay
between the V50% transition of the rising input voltage and the V50% transition of the falling output voltage,

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and can be computed as [3-8]:
*
( )
+ (6)
Lastly, the low-to-high propagation delay time (tpLH ) is defined as the time delay between the V50%
transition of the falling input voltage and the V50% transition of the rising output voltage, and can be
computed as [3-8]:
[
( ( | |))
] (7)
Figure 1. CMOS inverter circuit (as drawn in PSIPICE)
Figure 2. Input and output voltage waveforms of the CMOS inverter
The cost (fitness) function to be minimized can be written as [3-8]:
(8)

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where
|( ( ( ) )) | |( ) |
(9)
Subject to the following constraints:
( )
( )
Thus, the objective of the optimization is to get, as much as possible, a symmetric response of
the inverter, such that the fall and rise times are equal, and the propagation delays are equal. There are three
variables in the optimization problem, namely, the load capacitance CL, and the ratio between the widths and
the lengths of the channels, and . Thus, the dimension of the optimization problem is 3.
The minimum and maximum values of the above constraints are shown in Table 1, where similar to [3-8],
8 different design sets are considered in this paper. Moreover, similar to [1-8], TSMC 0.25 micron fabrication
technology is considered, where: VDD= 2.5 V, Vtn = 0.3655 V, Vtp = 0.5466 V, = 243.6 A/V2
, and
= 51.6 A/V2
.
Table 1. Constraints values for the 8 design sets [3-8]
Design set no. CL (pF) (W/L)n (W/L)p tf (ns) tr (ns) tpHL (ns) tpLH (ns)
1 0.2–4 1.1–6.1 2.8–19.3 1.1–13 1.1–13 0.5–10 0.5–10
2 0.1–5.1 1.6–7.1 1.8–18 1.1–15 1.1–15 0.5–8 0.5–8
3 0.47–2 1.4–6.7 3.2–38 0.5–12 0.5–12 0.2–9 0.2–9
4 0.1-1.1 1.2–7 1.5-17.5 0.5–5 0.5–5 0.2–4 0.2–4
5 0.2–14 1.9–5.0 2.7–17 0.7–6 0.7–6 0.4–5 0.4–5
6 0.3–3.6 1.3–3.5 3.5–16.2 0.25–7 0.25–7 0.3–4.5 0.3–4.5
7 0.2–4.9 1.1–5.8 2.2–25.3 0.5–6.6 0.5–6.6 0.2–7.7 0.2–7.7
8 0.2–3.5 0.3–7.6 1.3–39 0.3–6.6 0.3–6.6 0.1–4.4 0.1–4.4
4. NUMERICAL RESULTS
For each design set in Table 1, SOS code with population size=10 and maximum number of
iterations=500 is run for 50 independent times. Each run took around 8 seconds on an HP ProBook 6550b
Laptop with “Intel Core i7 CPU M 620 @ 2.67 GHz” and “4 GB RAM”. Table 2 shows the best results
obtained using SOS for each design set, where it can be seen that a symmetric output waveform is obtained
with equal fall and rise times and almost equal propagation delays. In Table 3, SOS results are compared with
the best-known results found in the literature. Namely, SOS results are compared with those obtained using
particle swarm optimization with constriction factor and inertia weight approach (PSOCFIWA) [3],
differential evolution (DE) [4], craziness based particle swarm optimization (CRPSO) [5], hybrid harmony
search with differential evolution (HS-DE) [6], and particle swarm optimization with aging leader and
challenger (ALC-PSO) [7]. It should be mentioned that we took the values of the input parameters (CL, ,
) from the tables in [3-7] and computed the corresponding CF values ourselves, as it is believed that
the reported CF values in [3-7] are not right. Even though SOS does not get the least error in all design sets,
it does compete very well with other well-developed methods. It should be also mentioned that, unlike other
methods, SOS is free of tuning parameters; one has to set the values of the population size and the maximum
number of iterations only. Moreover, the basic SOS algorithm is used here without any modifications,
in contrary to [3, 5, 7] where PSO was modified to overcome some of its inherent drawbacks. Table 4 shows
some statistical measures of the SOS results. It can be clearly seen that the SOS is a robust optimization
method as it obtains a very small value for the standard deviation (compared, for example, with the one
reported in [5] using CRPSO). Figure 3 shows the convergence curve for the design set number 8 which

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obtains the least value for CF. Very good convergence is reached within only 100 iterations. Figure 4 shows
box-and-whisker plot for the same design set. In this figure, median (middle of dataset) is represented by
the red line, i.e., 50% of the data are larger than this value. Upper and lower ends of the box represent
the 75th and 25th percentiles, i.e., 25% of the data are larger than the upper end, while 25% of the data are
less than the lower end. The difference between the upper and lower ends of the box (i.e., the length of
the box) is known as the “interquartile range (IQR)”. The whiskers are lines extending from the ends of
the box to show the extent of the rest of the data. The end of the upper whisker indicates the largest value in
the dataset (excluding upper outliers), while the end of the lower whisker indicates the smallest value in
the dataset (excluding lower outliers). Outliers are data with values beyond the ends of the whiskers,
and are indicated with red “+” sign. An outlier is any value that lies more than one and a half times the length
of the box from either end of the box.
Table 2. SOS results
Design set no. CL (pF) (W/L)n (W/L)p tf (ns) tr (ns) tpHL (ns) tpLH (ns) CF (ps)
1 0.7428 3.49783 18.67556 1.1785 1.1785 0.5 0.51835 18.35
2 0.71593 3.3713 18 1.1785 1.1785 0.5 0.51835 18.35
3 0.60366 6.7 35.77255 0.5 0.5 0.21213 0.21992 7.79
4 0.10812 1.2 6.40702 0.5 0.5 0.21213 0.21993 7.79
5 0.32279 1.9 10.14445 0.9428 0.9428 0.4 0.41469 14.68
6 0.3866 3.03417 16.2 0.7071 0.7071 0.3 0.31101 11.01
7 0.42687 4.73782 25.29611 0.5 0.5 0.21213 0.21992 7.79
8 0.39487 7.30448 39 0.3 0.3 0.12728 0.13195 4.67
Table 3. Comparison of SOS-obtained CF values (in ps) with other methods in the literature.
The least CF value is in boldface
Design set no. SOS CRPSO
[5]
RGA
[5]
HS-DE
[6]
PSO-CFIWA
[3]
DE
[4]
ALC-PSO
[7]
1 18.35 31.84 65.77 17.74 18.26 37.26 17.83
2 18.35 35.06 75.09 17.92 18.17 39.5 17.85
3 7.79 15.37 77.54 7.81 7.79 16.01 7.8
4 7.79 15.12 36.83 7.79 7.79 12.81 13.68
5 14.68 30.04 69.28 14.19 14.46 42.13 14.19
6 11.01 20.59 70.92 22.95 11.53 23.58 10.96
7 7.79 8.84 69.25 7.81 7.9 15.97 7.79
8 4.67 7.91 68.78 4.67 4.67 9.35 4.67
Table 4. Statistics of SOS CF values out of 50 runs for each set
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8
Set 8
(CRPSO) [5]
Minimum (ps) 18.3538 18.3538 7.78695 7.78695 14.683 11.01227 7.78695 4.67217 5.2594
Maximum (ps) 18.537 18.8628 7.847 7.88833 14.8548 11.04983 7.97586 4.76056 15.3432
Mean (ps) 18.3853 18.3815 7.7943 7.7983 14.6958 11.01903 7.81208 4.68196 10.0572
Standard deviation (x 10-14
) (s) 4.1548 7.9977 1.3174 1.9595 2.66413 0.89952 3.684 1.53946 108.15
The 0.25 micron CMOS inverter circuit is simulated using the new version of PSPICE (V 17.7).
The results of eight different design set are obtained and shown in Table 5, considering the same SOS
optimized values as shown in Table 2 of the important parameters, output load capacitor and the geometry of
the transistor, as inputs for PSPICE simulation. Figures 5 and 6 shows the timing diagrams for the rise and
fall times and the propagation delay times, respectively, for design set number 8 using PSPICE simulation.
A reasonable similarity between the SOS results in Table 2 and PSPICE results in Table 5 for the rise and fall
times can be observed. However, larger group delays are predicted using PSPICE. This is due to the fact that
more complex expressions are used in PSPICE to evaluate the rise and fall times and the propagation delays,
while simple approximate expressions are used here assuming long channel transistors and an ideal step input
voltage. Similar observation has been noted in [3-7]. Clearly, these approximate expressions underestimate
the propagation delays, and thus, we are now exploring the use of more complex expressions, like
the improved ones very recently proposed in [25].
It is worth mentioning that PSPICE results are obtained using V17.7 with the following settings:
a. Simulation profile:
Analysis type = Transient
TSTOP: 8 seconds

177
Start saving data after: 0 seconds
Maximum step size: 1ms
SKIPBP: true
All other simulation parameters are left to their default values (RELTOL, VNTOL, ABSTOL, CHGTOL,
GMIN, ITL1, ITL2, ITL4, TNOM, THREADS, Speed level, SKIPBP).
b. CMOS technology used: MbreakP and MbreakN to simulate the 0.25 μm TSMC fabrication technology
with the parameters mentioned in the paper (μn Cox, Vtn, Vtp, (W/L)n, (W/L)p).
c. Input source: (VPULSE) source with the following characteristics:
V1 = 0; V2 = 2.5 = VDD; TD = 0; TR = 1 n; TF = 1 n; PW = 1; PER = 2
Figure 3. Convergence curve for the SOS results for
the design set number 8
Figure 4. Box-and-whisker plot for the SOS results
for design set number 8
Table 5. PSPICE results using Vin “High” = VDD
Design set no. CL (pF) (W/L)n (W/L)p
tf
(ns)
tr
(ns)
tpHL
(ns)
tpLH
(ns)
CF (ns)
1 0.7428 3.49783 18.67556 1.35524 0.96916786 1.1966 1.1166 0.46607
2 0.71593 3.3713 18 1.35524 0.96916741 1.0971 1.1627 0.45167
3 0.60366 6.7 35.77255 0.76703569 0.64575389 0.887348 0.602467 0.40616
4 0.10812 1.2 6.40702 0.76704227 0.64575767 0.914830 0.627832 0.40828
5 0.32279 1.9 10.14445 1.15790 0.83511820 1.0645 0.715316 0.67197
6 0.3866 3.03417 16.2 1.00256 0.70468631 0.961739 0.673224 0.58639
7 0.42687 4.73782 25.29611 0.7670353 0.64575389 0.878689 0.566099 0.43387
8 0.39487 7.30448 39 0.679782 0.668853 0.795644 0.512026 0.29455
Figure 5. Fall time and rise time timing diagrams obtained using PSPICE for design set 8
0 50 100 150 200 250 300 350 400 450 500
-114
-113
-112
-111
-110
-109
-108
-107
-106
-105
-104
X: 97
Y: -113.3
iteration
fitnessfunctionvalue,J
4.67
4.68
4.69
4.7
4.71
4.72
4.73
4.74
4.75
4.76
Case 3, set 8
CFvalue(ps)

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5. CONCLUSION
In this paper, SOS algorithm was used in the optimal design of a CMOS inverter with symmetric
switching characteristics. SOS results were compared with those obtained using other evolutionary
optimization methods available in the literature. The comparison showed that SOS is a robust, simple, and
fast converging method. For all 8 design sets, SOS method performed very well, which clearly shows
the effectiveness of the SOS method. As mentioned before, compared to other methods, SOS has
the advantage of being free of tuning parameters; one just needs to set the values of the population size and
the maximum number of iterations.
ACKNOWLEDGEMENTS
This research was part of the sabbatical leave of the first author which was supported by Jordan
University of Science and Technology, Irbid, Jordan. The authors acknowledge the help of late Dr. B. El-Asir
with the PSPICE simulations.
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