International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 6, June 2012)
Fuzzy Modelling of Voltage Standing Wave Ratio using
Fuzzy Regression Method
T. D. Dongale1, T .G. Kulkarni2, R. R. Mudholkar3
1,2,3
Department of Electronics, Shivaji University, Kolhapur
The fuzzy reasoning is broadly classified into two
methods: first the Direct method and second the Indirect
method. The direct method includes Mamdani’s method,
Takagi and Sugeno’s fuzzy modeling and Simplified
method. The direct methods are popular due to their
simplicity. The indirect methods are based on truth value
space. The indirect method has complex reasoning
mechanism. In the fuzzy modeling of VSWR we have
employed the fuzzy reasoning by Takagi-Sugeno
reasoning, where consequence part of the rule is
represented by linear function.
Abstract— The transmission line or waveguide exhibit the
nonlinear characteristics of VSWR (Voltage Standing Wave
Ratio) pattern. An attempt is made to model the VSWR
pattern characteristics using method of Fuzzy Reasoning and
Fuzzy Regression Method. The very idea of fuzzy reasoning
by linear function has been explored in modelling the VSWR
pattern. The results show a great resemblance between
practical results and that obtained using fuzzy based VSWR
model. The VSWR pattern of any transmission line or
waveguide consists of many other parameters which
profoundly affect the transmitting and receiving signal. These
include the parameters such as Return Loss, Reflection
Coefficient and mismatch loss. The modelling of VSWR
pattern is demonstrated by using Sugeno Reasoning Method.
This reasoning method consists of output membership
function as linear function. The fuzzy regression is used for
the reducing the output linear equation. Fuzzy Regression
Method reduces the output linear equations from 49 to 7. This
also demonstrates the feasibility of using Fuzzy reasoning in
the microwave device modelling exhibiting non-linear
behaviour.
Keywords— Fuzzy Logic, VSWR pattern,
Reasoning, Fuzzy Regression, linear function
Sugeno
I. INTRODUCTION
The fuzzy logic in broad sense is the theory that includes
fuzzy sets, fuzzy logic, fuzzy reasoning, fuzzy measure,
fuzzy relation etc. Its main objective lies in the modeling of
complex, nonlinear dependency that exists between input
and output variables of a system. Further it aims to
represent system operation knowledge in the form of
linguistic rules [1]. The basic elements of fuzzy system are
shown in figure.1. Amongst them reasoning forms a key
element.
Reasoning is a process by which you can reach a
conclusion after consulting all the facts i.e. reasons behind
the decision. This paper demonstrates how fuzzy model of
VSWR can be realized based on fuzzy reasoning. Fuzzy
reasoning is a process in which given a value of input
antecedent variable, yields the values of consequent output
variable based on imprecise and non-linear dependency
formulated in the rule base [2, 3].
Figure.1. Fuzzy Inference System
II. FUZZY REASONING USING LINEAR FUNCTION
The Direct method of fuzzy reasoning has following
difficulties such as,
1. As premise variable increases, the number of rules
increases exponentially.
2. As number of rule increases the time required to
construct the rule also increases along with
complexity.
3. If number of premises is too large then it becomes
difficult to keep track of mapping between input
and output premises.
To overcome these problems the fuzzy reasoning by
linear function is adopted for consequence part.
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 6, June 2012)
This greatly reduces the number of fuzzy rules and
accelerates the reasoning process. However it needs a
skillful off-line modeling work to identify the functional
relationship between input and output variables forming the
consequence part of fuzzy rule [1, 3].
A. VSWR Model - A Fuzzy Approach
The VSWR is dependent of the reflection coefficient,
mismatch loss, return loss, the dependency however is
nonlinear. Hence for modelling the VSWR a Sugeno
method of Fuzzy reasoning has been used. This approach
involves the construction of fuzzy sets for antecedent part
and linear function for consequence part to model nonlinear mapping of VSWR with reflection coefficient,
mismatch loss and the return loss. The fuzzy model of
VSWR relies on the linear equation of the form:
y = mx + c
(1)
The whole range of VSWR has been partitioned into
seven fuzzy sets labelled as VERY LOW, LOW,
MEDIUM, MEDIUM HIGH, HIGH, VERY HIGH and
BIG. The ANFIS EDIT tool box of MATLAB creates the
required number of input-out membership functions, rules
and linear equations. For modelling of VSWR fuzzy rule
by triangular membership function for antecedent part and
linear function for consequence part has been employed.
The rules are defined here on universe of discourses with
respect to reflection coefficient, mismatch loss and the
return loss. The observations of VSWR used for modelling
are shown in figure 2, 3 and 4.
Figure.3.Relationship between Reflection Coefficients (Γ) and
VSWR pattern
Figure.4.Relationship between Mismatch loss and VSWR pattern
VSWR is the ratio of the maximum voltage to the
minimum voltage in the standing wave on a transmission
line [7]. Standing waves are the result of reflected RF
energy. When the reflections on the line approaches zero
the maximum power may be transmitted. Reflections occur
at any place where the impedance of the transmission line
changes. Usually this phenomenon happens on a radio or
radar transmission line. If all the energy gets reflected (for
example, by an open or short circuit) at the end of the line,
then none gets absorbed producing a perfect ‘standing
wave’ on the line. This is undesirable situation. If the
reflected wave is not as strong as the forward wave, then
some ‘standing wave’ pattern will be observed, but the
nulls will not be as deep or the peaks as high as for a
perfect reflection (or complete mismatch). Therefore, any
‘standing wave’ is an indication of an imperfect condition,
with part of the power meant for radiation being returned
because of a mismatch.
Figure.2.Relationship between Return loss and VSWR pattern
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 6, June 2012)
If there is no mismatch with only a forward travelling
wave, then there is no standing wave; i.e., the voltage at
any point on the line is the same as it is everywhere else.
The intensity of the standing wave is referred to as the
‘Voltage Standing Wave Ratio’ abbreviated as VSWR. [5]
There are four quantities that describe the effectiveness
of transferring power from a line to a load or antenna: the
VSWR, the reflection coefficient, the mismatch loss and
the return loss [6]. These are all inter-related with each
other. The voltage component of a standing wave in a
uniform transmission line consists of the forward wave
(with amplitude, Vf) superimposed on the reflected wave
(with amplitude, Vr). Reflections occur as a result of
discontinuities, such as an imperfection in an otherwise
uniform transmission line, or when a transmission line is
terminated with other than its characteristic impedance. If
the feed line has no loss, and matches both the transmitter
output impedance and the generator input impedance, then
the maximum power will be delivered to the load. In this
case the VSWR will be 1:1 and the voltage and the current
will be constant over the whole length of the feed line.
Return loss is a measure in dB of the ratio of power in the
incident wave to that in the reflected wave, and we define it
to have a negative value. The higher the return loss,
minimum is the less power loss.
Return loss = 10 log10 [Pr/Pi] =20 log10 [Er/Ei]
The output variable is VSWR. Figure.5 depicts snap shot
of FIS-Editor window for design structure of VSWR
model.
Figure.5. FIS editor for VSWR model
(2)
Figure.6.Input membership function (Return loss) editor for VSWR
model
Also of considerable interest is the mismatch loss. This
is a measure of how much the transmitted power is
attenuated due to reflection. It is given by the following
relation,
Mismatch Loss = 10 log (1 -ρ 2)
(3)
The reflection coefficient Γ is equal to,
Γ= (ZL-Z0)/ (ZL l+Z0)
(4)
The load impedances ZL and characteristic impedance Zo
of the transmission line are in ohm. The reflection
coefficient magnitude, |Γ| or ρ, is the ratio of the amplitude
of the reflected wave to the amplitude of the incident wave
at the junction of a transmission line and the terminating
impedance. |Γ| has a value lying between 0 and 1. The |Γ| =
0 means the line is perfectly matched, and a value of 1
means that the line is either shorted or open-circuit. [6-7]
B. Fuzzy Inference System (FIS) for VSWR modelling
The fuzzy inference systems associated with input and
output variables create a fuzzy mapping between the design
variables. For simple consequence part Sugeno method is
good choice. The input variables are return loss, reflection
coefficient and mismatch loss.
For a better result of modelling, we have chosen five
triangular and two trapezoidal membership functions. The
knowledge base pertaining to the VSWR modelling is
formulated in terms of fuzzy inference rules, which are
supported by the data-base. The knowledge-base thus
comprises a Data-base and a Rule-base. The data-base
provides the required information to fuzzification, rule-base
and defuzzification modules. The information includes, the
membership functions representing meanings of linguistic
values of input and output variables in addition to Labels,
shapes, slopes and domain-ranges. The Figure.6.depicts
snap shot of Membership function-Editor window for input
variable ‘Return Loss’. The ranges of membership function
considered for Return loss are as followsμVS (x) = L (x,0,1.8,2.2)
μSM (x) = ^ (x,3.6,7,10.4)
μM (x) = ^ (x,7,10.4,14)
μL (x) = ^ (x,10.4,14,17.8)
μVL (x) = ^ (x,14,17.8,21)
μB (x) = Γ (x,18,20,20)
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 6, June 2012)
The symbol ‘L’ stands for left bounded membership
function, the ‘Γ’ for right bounded membership function
and the ‘^’ symbol indicate that membership function
centred triangular membership function.
μSM (x) = ^ (x,1.3,2.5,3.8)
μM (x) = ^ (x,2.5,3.8,5)
μL (x) = ^ (x,3.8,5,6.2)
μVL (x) = ^ (x,5,6.2,7.5)
μB (x) = Γ (x,6.4,7,7.5)
This modelling is based on TSK method in which
consequence part in the form of linear equation of the form:
Y= mx+C. For modelling Adaptive Neuro-Fuzzy Inference
System (ANFIS) tool from MATLAB is being used. The
Neuro-adaptive learning method works similarly to that of
neural networks. Neuro-adaptive learning techniques
provide a method for the fuzzy modelling procedure to
learn based on the information about a data set. Fuzzy
Logic Toolbox software computes the membership
function parameters that best allow the associated fuzzy
inference system to track the given input/output data. The
Fuzzy Logic Toolbox function that accomplishes this
membership function parameter adjustment is called
ANFIS. The ANFIS function can be accessed either from
the command line or through the ANFIS Editor GUI as the
functionality of the command line function ANFIS and the
ANFIS Editor GUI is similar in nature. In this tool the
decision making such as rule formulation, number of rule
etc. is done automatically according to data provided in the
form of .MAT format. Even though it can create output
linear equation, but we can edit this equation according to
need based tuning [8].
For modelling of data on VSWR, first of all the data set
is converted into linear equation of the form (1). If ‘y’ is a
function of more than one independent variable, the matrix
equations that express the relationships among the variables
are expanded to accommodate the additional data. This is
called multiple regressions. A model of this data is formed,
where multiple regressions solve the modelling equations
for unknown coefficients β1, β2 and β3 by minimizing the
sum of the squares of the deviations of the data from the
model (least-squares-fit technique). By Constructing and
solving the set of simultaneous equations by forming the
Vander monde matrix [X], and solving for the parameters
Figure.7.Input membership function (Reflection Coefficient) editor
for VSWR model
The Figure.7.depicts snap shot of Membership functionEditor window for input variable ‘Reflection Coefficient’.
The ranges of membership function considered for
Reflection Coefficient are as followsμVS (x) = L (x,0,0.1,0.14)
μS (x) = ^ (x,0,0.15,0.30)
μSM (x) = ^ (x,0.15,0.30,0.45)
μM (x) = ^ (x,0.30,0.45,0.60)
μL (x) = ^ (x,0.45,0.60,0.75)
μVL (x) = ^ (x,0.60,0.75,0.90)
μB (x) = Γ (x,0.78,0.85,0.90)
computed as flows-[8]
X = [ones (size(x1)) x1 x2 x3]
The coefficient vector is:
Figure.8.Input membership function (mismatch loss) editor for
VSWR model
The Figure.8.depicts snap shot of Membership functionEditor window for input variable ‘Mismatch Loss’. The
ranges of membership function considered for Mismatch
loss are as followsμVS (x) = L (x,0,0.8,1.2)
μS (x) = ^ (x,0,1.3,2.5)
a = X/y
β1 = 0.0548
β2 = -7.9246
β3 =3.2493
Here x1, x2, x3 are the data set of Return loss, Reflection
coefficient and Mismatch loss respectively.
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 6, June 2012)
From the data set we get a one linear equation which is
given by least-squares method. The least-squares fit model
of the data is as given by equation (5),
Y= β0 + β1 x1 + β2 x2+ β3 x3
Y= 1.5396 + 0.0548 x1 – 7.9246 x2 + 3.2493 x3
(5)
Where, β0 is called as intercept and β1, β2, and β3are
called as coefficient of linear equation. To validate the
model, we find the maximum of the absolute value of the
deviation of the data from the actual model.
Y = X*a;
MaxErr = max (abs (Y - y))
MaxErr = 0.13
Where, y is called as observed value and Y called as
estimated value. The error is 0.13 between observed and
estimated value which is very small. It means that the
equation (5) models the system’s behaviour. The intercept
part and coefficient of equation gives the actual behaviour
of small set of data. Hence instead of using different output
linear equations for individual data in TSK method [10],
we can use only one regression equation for set of data. By
using the Fuzzy Regression Method, we can eliminate
unnecessary linear equations. But only one equation may
give some sort of error. For minimizing the error, we can
create suitable linear equation by fuzzy regression method
through number of iterations based on trial and fit process.
The seven different linear equations (6 to 12) give zero
error for data set. Hence it is suitable for modelling of
VSWR pattern. By fuzzy regression method only seven
linear equations instead of forty nine equations are
sufficient to model the VSWR. The linear equations and
rule base are as follows which satisfies the actual behaviour
of observed data.
If VSWR is VERY LOW Then Y= 0.9996 - 0.020 x1 +
2.5791 x2 + 0.0865 x3
(6)
If VSWR is LOW Then Y= -1.3292 + 0.1473 x1+4.1237 x2
+ 1.1046 x3
(7)
If VSWR is MEDIUM Then Y= 9.9692 - 0.7692 x1 –
9.2308 x2 + 1.5385 x3
(8)
If VSWR is MEDIUM HIGH Then Y=-3.5205 + 0.6849
x1– 1.3699 x2+2.7397 x3
(9)
If VSWR is HIGH Then Y= -31.00 + 0.00 x1 +50.00 x2 +
0.00 x3
(10)
If VSWR is VERY HIGH Then Y= -23.7059 + 5.8824 x1 –
5.8824 x2 + 5.8824 x3
(11)
If VSWR is BIG Then Y= -29.97 + 5.8455x1 – 1.7745x2 +
6.1587x3
(12)
Figure.9.Output membership function editor for
VSWR model
Figure.10.Rule viewer for VSWR model
III. CONCLUSIONS
The process of data reconstructed pertaining to VSWR
characteristics using Sugeno Fuzzy Inference is shown in
figure 10. Comparison with actual data and observed rule
viewer graph shows resemblance between practical VSWR
characteristics and characteristics generated by fuzzy
model of VSWR. Idea of fuzzy modeling of VSWR is
though simple but demonstrates powerful application of
fuzzy logic in device modeling based on imprecise data.
The same idea can be extended to highly non-linear and
complex electronic devices.
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 6, June 2012)
The Fuzzy Regression is very useful for creating linear
equation when set of data is very large. Instead of using
different output linear equations for individual data in TSK
method, we can use only one regression equation for set of
data. By using this Fuzzy Regression Method we can
eliminate unnecessary linear equations. These sets of linear
equation for VSWR pattern are discussed from equation
number 6 to equation number 12. Also fuzzy regression
gives unique linear equation for whole system in some
tolerable error of 0.13. This validates the successful
implementation of Fuzzy Reasoning by TSK method and
Fuzzy Regression Method in modeling the VSWR.
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[5] Zouhair Benmoussa and Don Barrick, (April 2006), the Effects of
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[6] R.E. Collien, (1987), Fundamental of Microwave Engineering,
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[7] S.Y. Liao, (2006), Microwave Device and Circuit, page no. 79-100
[8] http://www.mathworks.com
[9] http://www.microwave101.com/microwave.calculator/VSWR
Calculator
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