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A survey on the application of the Elliptical Trigonometry in industrial
electronic systems using controlled waveforms with modeling and
simulating of two functions Elliptic Mar and Elliptic Jes-x
CLAUDE BAYEH
Faculty of Engineering II
Lebanese University
LEBANON
Email: claude_bayeh_cbegrdi@hotmail.com
Abstract: - In industrial electronic systems, power converters with power components are used. Each controlled
component has its own control circuit. In this paper, the author proposes an original control circuit for each
function in order to replace the different existing circuits. The proposed circuit is the representation of an
elliptical trigonometric function as “Elliptic Mar” and “Elliptic Jes-x” that are particular cases of the elliptical
trigonometry. Thus, with one function, by varying the values of its parameters, the output waveform will
change and can describe more than 12 different waveforms. Finally for each function, a block diagram, a model
of the circuit and a programming part are treated using Matlab/Simulink. The results of the studied circuit are
presented and discussed.
Key-words: - Power electronics, power converters, mathematics, trigonometry, multi form signal.
1 Introduction
In motor drives, robotics, or other industrial
electronic applications, the use of power converters
is essential to improve the control and, therefore, the
efficiency of the studied system [17],[18]. Power
converters and power electronics circuits are
generally composed of power components with
different characteristics [17]. These components are
divided in two categories: the controlled
components and the uncontrolled components [17],
[19],[20]. The controlled components that are based
on semi-conductors like thyristors, Triac and
transistors need controlled signals with specified
waveforms in order to be applied on their controlled
terminals (base or gate) [17],[19]. Thus, each
component has its own control source. This paper
underlines the importance of the elliptical
trigonometric functions in generating different
waveforms by varying some parameters of a single
function. In a particular case, the Elliptic Mar and
the Elliptic Jes-x functions are chosen to be treated.
In fact, the elliptical trigonometry is an original
study introduced with new concepts [1],[2]. The
existed trigonometry (Circular trigonometry) is a
particular case of the elliptical trigonometry [6],[7].
The traditional trigonometry has an enormous
variety of applications in all scientific domains
[6],[7],[8],[9]. It can be considered as the basis and
foundation of many domains as electronics, signal
theory, astronomy, navigation, propagation of
signals and many others… [10],[11],[12],[13].
Particularly, the mathematical topics of Fourier
series and Fourier transforms rely heavily on
knowledge of trigonometric functions [10],[11] and
find application in a number of areas, including
statistics [12],[13].
In this paper, the new concept of the elliptical
trigonometry is introduced and few examples are
shown and discussed briefly. Figures and results are
drawn and simulated using Matlab/Simulink and
AutoCAD. A survey on the applications of this
trigonometry in the power electronics domains
presented in section 2. In the third section, the
angular functions are defined, these functions have
enormous applications in all domains, and it can be
considered as the basis of this trigonometry [1],[2].
The definition of the Elliptical trigonometry is
presented and discussed briefly in section 4. In the
fifth section, a survey on the Elliptical
Trigonometric functions is discussed and two
principal functions are presented. In the sections 6
and 7, two functions are studied and discussed
briefly with simulation on Simulink/Matlab, their
block diagrams are presented, their programming
parts and so their modeling circuits. Finally, a
conclusion about the elliptical trigonometry is
presented in the section 8.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
859
Issue 11, Volume 5, November 2010
2 A survey on the application of the
elliptical trigonometry in engineering
domain
The used controlled circuits for power transistors
(MOSFET, IGBT, etc) differ from those used for
Thyristors (GTO, Triac, etc). Designing and
modeling circuits for all these controlled
components taking into account their different
characteristics, take time, and realizing them
practically, take time and money. In this paper, for
each elliptical trigonometric function, one electronic
circuit is proposed to be used in simulation
(Labview, Matlab, Simulink etc). For a particular
case, in order to control the different existed power
components for the function Elliptic Mar (figure 1),
two parameters ′′ and ′′ are used as variable
inputs. The output will be the studied elliptical
trigonometry function.
Thus, the main goal of the Elliptical Trigonometry is
to produce a huge number of multi form signals
using a single function and by varying some
parameters of this function. For a particular case,
more than 12 different output signals can be
obtained by varying two parameters of the Elliptic
Mar function.
Fig. 1: Electronic circuit of the function ,()
with its inputs and output.
3 The angular functions
In order to make a review on the elliptical
trigonometry, it is necessary to introduce the
definition of the angular functions. In fact, angular
functions are new mathematical functions that
produce a rectangular signal, in which period is
function of angles. Similar to trigonometric
functions, the angular functions have the same
properties as the precedent, but the difference is that
a rectangular signal is obtained instead of a
sinusoidal signal [14],[15],[16] and moreover, one
can change the width of each positive and negative
alternate in the same period. This is not the case of
any other trigonometric function. In other hand, one
can change the frequency, the amplitude and the
width of any period of the signal by using the
general form of the angular function.
In this section three types of angular functions are
presented, they are used in this trigonometry; of
course there are more than three types, but in this
paper the study is limited to three functions.
3.1 Angular function ()
The expression of the angular function related to the
(ox) axis is defined, for ∈ ℤ, as:
((+))=
+1 (4−1)
−≤≤(4+1)
−
−1 (4+1)
−<<(4+3)
− (1)
Fig. 2: The ((+)) waveform.
For = 1 and = 0, the expression of the
angular function becomes:
()=+1 cos ()≥0
−1 cos()<0
3.2 Angular function ()
The expression of the angular function related to the
(oy) axis is defined, for ∈ ℤ, as:
((+))=
+1 2/−≤≤(2+1)/−
−1 (2+1)/−<<(2+2)/−
(2)
Fig. 3: The ((+)) waveform.
For = 1 and = 0, the expression of the
angular function becomes:
()=+1 sin()≥0
−1 sin()<0
3.3 Angular function ()
α (called firing angle) represents the angle width of
the positive part of the function in a period. In this
case, we can vary the width of the positive and the
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
860
Issue 11, Volume 5, November 2010
negative part by varying only α. The firing angle
must be positive.
(+)=
⎩
⎪
⎨
⎪
⎧
+1
(2−)/−≤≤(2+)/−
−1
(2+)/−<<(2(+1)−)/−
(3)
Fig. 4: The ((+)) waveform.
4 Definition of the Elliptical
Trigonometry
In order to study the proposed circuits, their block
diagrams and their simulations, it is necessary to
introduce the mathematical definition of the
elliptical trigonometry and its functions.
4.1 The Elliptical Trigonometry unit
The Elliptical Trigonometry unit is an ellipse with a
center O (x = 0, y = 0) and has the equation form:
(/) + (/)=1 (4)
With:
‘a’ is the radius of the ellipse on the (x’ox) axis,
‘b’ is the radius of the ellipse on the (y’oy) axis.
Fig. 5: The elliptical trigonometry unit.
It is essential to note that ‘’ and ‘’ must be
positive. In this paper, ‘’ is fixed to 1. One is
interested to vary only a single parameter which is
‘’.
4.2 Intersections and projections of different
elements of the Elliptical Trigonometry on
the relative axes
From the intersections of the ellipse with the
positive parts of the axes () and (), define
respectively two circles of radii [] and [].
These radii can be variable or constant according to
the form of the ellipse.
The points of the intersection of the half-line [)
(figure 5) with the internal and external circles and
with the ellipse and their projections on the axes
() and () can be described by many functions
that have an extremely importance in creating plenty
of signals and forms that are very difficult to be
created in the traditional trigonometry.
Definition of the letters in the Figure 5:
: Is the intersection of the ellipse with the positive
part of the axe () that gives the relative circle of
radius "". It can be variable.
: Is the intersection of the ellipse with the positive
part of the axe () that gives the relative circle of
radius "". It can be variable.
: Is the intersection of the half-line [) with the
circle of radius .
: Is the intersection of the half-line [) with the
ellipse.
: Is the intersection of the half-line [) with the
circle of radius .
: Is the projection of the point on the axis.
: Is the projection of the point on the axis.
: Is the projection of the point on the axis.
: Is the projection of the point on the axis.
: Is the projection of the point on the axis.
: Is the projection of the point on the axis.
: Is the angle between the () axis and the half-
line [).
: Is the center (0, 0).
4.3 Definition of the Elliptical Trigonometric
functions ()
The traditional trigonometry contains only 6
principal functions: Cosine, Sine, Tangent, Cosec,
Sec, Cotan. [15],[16]. But in the Elliptical
Trigonometry, there are 32 principal functions and
each function has its own characteristics. These
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
861
Issue 11, Volume 5, November 2010
functions give a new vision of the world and will be
used in all scientific domains and make a new
challenge in the reconstruction of the science
especially when working on the economical side of
the power of electrical circuits, the electrical
transmission, the signal theory and many other
domains [15],[17].
The functions (),(),() and
(), which are respectively equivalent to
cosine, sine, tangent and cotangent. These functions
are particular cases of the “Circular Trigonometry”.
The names of the cosine, sine, tangent and cotangent
are replaced respectively by Circular Jes, Circular
Mar, Circular Ter and Circular Jes-y.
()⇔(); ()⇔()
()⇔(); ()⇔().
The Elliptical Trigonometric functions are denoted
using the following abbreviation “()”:
-the first letter “E” is related to the Elliptical
trigonometry.
-the word “()” represents the specific function
name that is defined hereafter: (refer to Figure 5).
• Elliptical Jes functions:
El. Jes: ()=
=
(5)
El. Jes-x: ()=
=()
() (6)
El. Jes-y: ()=
=()
() (7)
• Elliptical Mar functions:
El. Mar: ()=
=
(8)
El. Mar-x: ()=
=()
() (9)
El. Mar-y: ()=
=()
() (10)
• Elliptical Ter functions:
El. Ter: ()=()
() (11)
El. Ter-x:
()=()
()=()∙() (12)
El. Ter-y: ()=()
()=()
() (13)
• Elliptical Rit functions:
El. Rit: ()=
=
=()
() (14)
El. Rit-y: ()=
=()
() (15)
• Elliptical Raf functions:
El. Raf: ()=
=().() (16)
El. Raf-x: ()=
=()
() (17)
• Elliptical Ber functions:
El. Ber: ()=()
() (18)
El. Ber-x:
()=()
()=()∙() (19)
El. Ber-y: ()=()
()=()
() (20)
4.4 The reciprocal of the Elliptical
Trigonometric function
() is defined as the inverse function of
(). (−1()=1/()). In this
way the reduced number of functions is equal to 32
principal functions.
E.g.: ()=1
()
4.5 Definition of the Absolute Elliptical
Trigonometric functions
()
The Absolute Elliptical Trigonometry is introduced
to create the absolute value of a function by varying
only one parameter without using the absolute value
“| |”. The advantage is that one can change and
control the sign of an Elliptical Trigonometric
function without using the absolute value in an
expression. Some functions are treated to get an idea
about the importance of this new definition. To
obtain the Absolute Elliptical Trigonometry for a
specified function (e.g.: () ) we must multiply
it by the corresponding Angular Function (e.g.:
() with ∈ℕ) in a way to obtain the
original function if is even, and to obtain the
absolute value of the function if is odd (e.g.:
|()|).
If the function doesn’t have a negative part (not
alternative) it will be multiplied by ((−
)) to obtain an alternating signal which form
depends on the value of the frequency “” and the
translation value “ ”. By varying the last
parameters, one can get a multi form signals.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
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Issue 11, Volume 5, November 2010
• ()=()∙() (21)
=()∙()=|()| =1
()∙()=() =2
• ,()=(−)∙() (22)
=(−)∙() =1
() =2
• ,()=(2)∙() (23)
=(2)∙()=|()| =1
() =2
• ()=()∙() (24)
• ,()=(2)∙() (25)
• ,()=(−)∙() (26)
• ()=()∙() (27)
And so on…
5 A survey on the Elliptical
Trigonometric functions
As previous sections, a brief study on the Elliptical
Trigonometry is given. Two functions of 32 are
treated; the others functions can be easily interpreted
using formulae from (5) to (20).
Elliptic cosine and Elliptic sine that appear in the
previous articles [1] and [2], are particular cases of
the Elliptic Jes and Elliptic Mar respectively.
For this study the following conditions are taken:
- =1
- >0 the radius of the ellipse on the ′ axis.
- ∈ℕ
5.1 Determination of the Elliptic Jes function
The Elliptical form in the figure 5 is written as the
equation (4). Thus, given (5), the Elliptic Jes
function can be determined using the following
method. In fact:
()=
=
, it is significant to replace the
equation =(). in that defined in (4).
+()
=
1+()
=1⇒
()=±
()
Therefore:
•()=
() for −
≤≤
;
≥0
•()=
() for
<<
;
<0
Thus, the expression of the Elliptic Jes can be
unified by using the angular function expression (1),
therefore the expression becomes:
()=()
()⇒
()=()
() (28)
• Expression of the Absolute Elliptic Jes:
,()=()
()∙() (29)
The Absolute Elliptic Jes is a powerful function that
can produce more than 12 different signals by
varying only two parameters and . Similar to the
cosine function in the traditional trigonometry, the
Absolute Elliptic Jes is more general than the
precedent.
5.2 Determination of the Elliptic Mar
function
The elliptical form in the figure 5 is written as the
equation (4). Thus, given (8), the Elliptic Mar
function can be determined using the following
method. In fact:
()=
=
⇒=
(), it is significant
to replace the equation =
() in that defined in
(4). Thus, (
∙()) + (/)=1⇒
()=
=±
()
()
⇒()=
=±
Cter ()
1+
Cter ()2
Therefore:
• ()=
()
() for 0≤<
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
863
Issue 11, Volume 5, November 2010
• ()=
()
() for
<≤
• ()=
()
() for ≤<3
• ()=
()
() for 3
≤≤2
Thus, the expression of the elliptic Mar can be
unified by using the angular function expression (1),
therefore the expression becomes:
()=
()()
() (30)
• Expression of the Absolute Elliptic Mar:
,()=()∙() (31)
The Absolute Elliptic Mar is a powerful function
that can produce more than 12 different signals by
varying only two parameters and . Similar to the
sine function in the traditional trigonometry, the
Absolute Elliptic Mar is more general than the
precedent.
5.3 Original formulae of the Elliptical
Trigonometry
In this sub-section, a brief review on some
remarkable formulae formed using the elliptical
trigonometric functions.
• ()+()=1 (32)
In fact: ()+()=
()
()+
()()
()=
()+
()
() =
()
() =
()
()=1
•
()()+
()
()=1 (33)
In fact:
()+()=()
()+()
()
=
()⇒
()()=()
And
()+()=()
()+()
()
=
()⇒1
2()+
2()=()
Therefore:
()()+
()
()=
()+()=cos()+sin()=1
6 Studying the function
,()
In this section, the Absolute Elliptic Mar, which is
defined in equation (31), is chosen to be treated. The
main goal of this study is to model and simulate the
function using Matlab and Simulink.
6.1 The block diagram of
,()
The block diagram of the function ,()is
illustrated in figure 6. There are three inputs
connected to this diagram, two variable parameters
"" and "", and one sinusoidal waveform (Circular
Mar or “sine”). "" is chosen to be a constant with
=1, so it can’t be considered as an input.
Fig. 6: Block diagram of the function ,()
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
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Issue 11, Volume 5, November 2010
6.2 Modeling and simulating the function
,() using Matlab-Simulink
Consider =1, "" and "" are the controlled
inputs, The Absolute Elliptic Mar function is
obtained as an output. With three inputs, one can
obtain a single output, which can produce more than
12 different signals.
Fig. 7: the circuit model of the function ,()
6.2.1 First case (≪)
From equation (30), two configurations are studied:
1. Consider ≠ and ≪∙().
In this case, ()≠0. As "" is too small
number but non zero, therefore:
()≫1. Thus the equation (30) becomes:
()=
()()
()≈
()()
()
≈±
()()
()≈±()=±1
Therefore: ()=
+1 2<<(2+1)
−1 (2+1)<<(2+2)
2. Consider = and ≫∙().
In this case, ()=0. As "" is too small
number but non zero, therefore:
()=0. Thus the equation (30) becomes:
()=
()()
()=0
Consequently: ()=
0 =
+1 2<<(2+1);≠
−1 (2+1)<<(2+2);≠
Thus, a rectangular signal is obtained.
Figures 8.a and 8.b represent the waveforms of the
function ,() for =0.001≪1,
(rectangular waveform for =2 and continuous
signal for =1).
a) =2;=0.001 b) =1;=0.001
Fig. 8: the waveforms of the function ,()
for =0.001≪1
6.2.2 Second case (<1)
In this case, an elliptic swollen form is obtained for
=2 (figure 9.a), and a rectified elliptic swollen
form is obtained for =1 (figure 9.b). The
importance of this signal is by obtaining an average
value greater than that of an absolute value of the
sinusoidal signal. Hence, the average of the signal
can be increased by varying only the value of one
parameter "".
a) =2;=0.4 b) =1;=0.4
Fig. 9: the waveforms of the function ,()
for =0.4<1
6.2.3 Third case (=)
When ==1, the ellipse equation defined in (4)
becomes:
()+()=1 (34)
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
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Issue 11, Volume 5, November 2010
This is the equation of a circle of radius =1.
• The Elliptic Mar function defined in (8) becomes:
()=
=
=()=sin()
• The Elliptic Jes function defined in (5) becomes:
()=
=
=()=cos()
Therefore, ,()=sin()∙() (35)
Figures 10.a and 10.b represent the waveforms of
the function ,() for =1, (sinusoidal
signal for =2 and rectified sinusoidal signal for
=1).
a) =2;=1 b) =1;=1
Fig. 10: the waveforms of the function ,()
for =1
6.2.4 Fourth case (>1)
In this case, an elliptic deflated form is obtained for
=2 (figure 11.a), and a rectified elliptic deflated
form is obtained for =1 (figure 11.b). The
importance of this signal is by obtaining an average
value smaller than that of an absolute value of the
sinusoidal signal. Hence, the average of the signal
can be decreased by varying only the value of one
parameter "".
a) =2;=6 b) =1;=6
Fig. 11: the waveforms of the function ,()
for =6>1
6.2.5 Fifth case (≫)
From equation (30), two configurations are studied:
1. Consider ≠()
and ≫∙().
In this case, ()≠±∞. As "" is too large
number but non infinite, therefore:
()=≪1. It is much smaller than the
unit. Thus the equation (30) becomes:
()=
()()
()≈√∙()
√ , by
using Taylor development for the first degree
()=√∙()1−
≈√∙()
Therefore:
()=0 2<<(2+1)
0 (2+1)<<(2+2)
2. Consider =()
and ≪∙().
In this case, ()=±∞. As "" is too large
number but non infinite, therefore:
()≫1. Thus the equation (30) becomes:
()=
()()
()≈
()()
()
≈±
()()
()≈±()=±1
Consequently: ()=
⎩
⎪
⎪
⎨
⎪
⎪
⎧
0 2<<(2+1);≠()
0 (2+1)<<(2+2);≠()
+1 2<<(2+1);=()
−1 (2+1)<<(2+2);=()
Thus, an impulse signal is obtained.
Figures 12.a and 12.b represent the waveforms of
the function ,() for =100≫1, (impulse
train with positive and negative part for =2 and
impulse train with positive part only for =1).
a) =2;=100 b) =1;=100
Fig. 12: the waveforms of the function ,()
for =100≫1
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
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ISSN: 1991-8763
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Issue 11, Volume 5, November 2010
6.2.6 Determining the value of "" for a quasi-
triangular signal
The main objective of this part is to obtain a signal
closed to a triangular one. Therefore, the following
method is proposed in order to calculate the value of
"" for which the error between the desired signal
and the obtained one is minimal. This study is
limited to the interval 0;
.
1. consider the equation of a straight line:
=+ (36)
This line is supposed to contain the following two
points (= 0; =0) and (=
;=1).
Therefore, the expression (36) becomes: =
, it
is represented in the red color in the figure 13.
2. For the same interval 0;
, the angular function
() is equal to one, therefore the function (30)
becomes: ()=
()
(), it is
represented in the blue color in the figure 13.
Fig. 13: represents the function () (in blue
color) and the straight line (in red color) in the
considered interval.
Fig. 14: represents the expression |()−|
and two maximal errors to determine.
To obtain the minimal error, the difference between
the two functions must be smaller than a certain
value "", thus ()− =≤.
It is considered that for =
(the center of the
studied interval) the error ε is equal to zero. Thus,
for =
, the error =0,
⇒
−
∙
=0
⇒
(
)
(
)−
=0⇒=3⇒=√3
To calculate errors and presented in the figure
14, the derivative of must be equal to zero,
=()− =0
⇒= 1.784% =0.45085
= 7.375% =1.31307
And the average error = 2.5062%
• The average error can be reduced by using another
method, for example for =√3+0.24782, the
average error is reduced as the following:
= 5.55% =0.6241
= 5.55% =1.381;⇒=2.157%
Figures 15.a and 15.b represent the waveforms of
the function ,() for =√3, (quasi-
triangular signal for =2 and saw signal for =1).
a) =2;=√3 b) =1;=√3
Fig. 15: the waveforms of the function ,()
for =√3
Practically, the value of the average error =
2.5% is insignificant in some applications, therefore
the obtained signal can be considered as a pure
triangular signal for =2 and a pure saw signal for
=1.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
867
Issue 11, Volume 5, November 2010
Tables 1 and 2 represent a summary of different
waveforms obtained using the Elliptic functions
,() and ,().
Absolute Elliptic Mar
"
,
(
)
"
=
=
b<<1 Rectangle signal Continuous signal
b<1 Elliptical swollen
signal Rectified elliptical
swollen signal
b=1 Sinusoidal signal (sine
wave form) Rectified sinusoidal
signal
b=
√
3
Quasi-triangluar signal Saw signal
b>1 Elliptical deflated
signal Rectified elliptical
deflated signal
b>>1 Impulse train with
positive and negative
pulses
Impulse train with
positive part only
Table 1: summary of multi form signals obtained
using the Absolute Elliptic Mar function.
Absolute Elliptic Jes
"
,
(
)
"
=
=
b<<1 Impulse train with
positive part only Impulse train with
positive and
negative pulses
b<1 Elliptical deflated
signal Rectified elliptical
deflated signal
b=
√
3
/
3
Quasi-triangluar
signal Saw signal
b=1 Sinusoidal signal
(cosine wave form) Rectified sinusoidal
signal
b>1 Elliptical swollen
signal Rectified elliptical
swollen signal
b>>1 Rectangle signal Continuous signal
Table 2: summary of multi form signals obtained
using the Absolute Elliptic Jes function.
These types of signals are widely used in power
electronics, electrical generators and in transmission
of analog signals [17].
6.3 First conclusion
As presented previously, the Elliptic Mar function
takes different waveforms by varying the parameter
"". The same analysis can be treated using the
parameter "". Therefore, the same waveforms can
be obtained. Practically, instead of varying the value
of "" from 0 to +∞ in a goal to obtain all
waveforms, by introducing "", one can change the
values of "" or "" form 0 to 1 in a way to obtain
the desired waveform. E.g.:
=
=.
6.4 programming the Elliptic Mar function in
Matlab
As presented and analyzed in the previous section,
the Elliptic Mar function can be also programmed
and written in the Matlab software. Thus, the
elliptical trigonometry functions can be used in any
industrial applications.
The following program represents the detailed steps
in writing the Elliptic Mar function in Matlab.
%-------------------------------------------------------------------
%Programming the Elliptic Mar function in Matlab
%Introduced by Claude Ziad Bayeh
a=1; x=-15:0.0001:15; clc
fprintf('Absolute Elliptic Mar “AEmar”\n');
repeat='y';
while repeat=='y'
b=input('determine the form of the Elliptic
trigonometry: b=');
fprintf('b is a variable can be changed to obtain
different signals \n'); %b is the intersection of the
Ellipse and the axe y'oy in the positive part.
if b<0,
b
error('ATTENTION: ERROR b must be greater than
Zero');
end;
fprintf('AEmar=Emar*(angy(x))^i\n');
i=input('for Absolute Elliptic Mar put 1, for Elliptic
Mary put 2: i=');
if i<0,
i
error('ATTENTION: ERROR i must be greater or
equal to Zero');
end;
Ejes=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x);
Emar=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x).*tan(x
).*a/b; % the Elliptic Mar "Emar"
AEmar=Emar.*(angy(x)).^i; % Absolute Elliptic Mar
plot(x,AEmar); axis([0 4*pi -1.5 1.5]); grid on;
fprintf('Do you want to repeat ?\nPress y for ''Yes'' or
any key for ''No''\n');
repeat=input('Y/N=','s'); %string input
clc; close all
end; %End while
%-------------------------------------------------------------------
7 Studying the function
()
In this section, a brief study on the Absolute Elliptic
Jes-x. The main goal of this study is to model and
simulate the function using Matlab and Simulink.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
868
Issue 11, Volume 5, November 2010
7.1 The Elliptic Jes-x function
The elliptical form in the figure 5 is written as the
equation (4). Thus, given (6), the Elliptical Jes-x
function can be determined. In fact:
()=()
()=()
()∙
() (37)
• Expression of the Absolute Elliptic Jes-x
()=()∙(−) (38)
7.2 The block diagram of
()
The block diagram of the function
()is
illustrated in figure 16. There are four inputs
connected to this diagram, three variable
parameters "", "" and "", and one sinusoidal
waveform (Circular Mar or “sine”). "" is chosen to
be a constant with =1, so it can’t be considered
as an input.
Fig. 16: the block diagram of the function
()
7.3 Modeling and simulating the function
() using Matlab-Simulink
Consider =1, "", "" and ""are the controlled
inputs, The Absolute Elliptic Jes-x function is
obtained as an output. With four inputs, one can
obtain a single output, which can produce more than
12 different signals.
Fig. 17: the circuit model of the function
()
• Multi form signals made by
():
Taking =0 for this example, figures 18 and 19
represent multi form signals obtained by varying
two parameters ( and ). For the figures 18.a to
18.e the value of =2, for the figures 19.a to 19.f
the value of =1.
a) =2;=0.01 b) =2;=0.5
c) =2;=1 d) =2;=1.5
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
869
Issue 11, Volume 5, November 2010
e) =2;=2.9
Fig. 18: multi form signals of the function
() for =2 and for different values
of >0.
a) =1;=0.01 b) =1;=0.5
c) =1;=1 d) =1;=1.5
e) =2;=2.9 f) =2;=2.9; =/2
Fig.19: multi form signals of the function
()
for =1 and for different values of >0.
Important signals obtained using this function:
Impulse train with positive part only, sea waves,
continuous signal, amplified sea waves, impulse
train with positive and negative part, square
waveform, saw signal …
7.4 programming the Elliptic Jes-x function
in Matlab
As presented and analyzed in the previous section,
the Elliptic Jes-x function can be also programmed
and written in the Matlab software. Thus, the
elliptical trigonometry functions can be used in any
industrial applications.
The following program represents the detailed steps
in writing the Elliptic Jes-x function in Matlab.
%-------------------------------------------------------------------
%Programming the Absolute Elliptic Jes-x
%Introduced by Claude Ziad Bayeh
clc; close all; a=1; x=-15:0.004:15;
fprintf('---The Absolute Elliptic Jes-x---\n');
repeat='y';
while repeat=='y'
b=input('determine the form of the Elliptic
trigonometry: b=');
fprintf('b is a variable can be changed to obtain
different signals \n');
%b is the intersection of the Ellipse and the axe y'oy in
the positive part.
if b<0,
b
error('ATTENTION: ERROR b must be greater than
Zero');
end;
fprintf('AEjesx=Ejesx*(angx(x-T))^i\n');
i=input('for Absolute Elliptic Jes-x put 1, for Elliptic
Jes-x put 2: i=');
T=input('put the translation of the period for angx(x-
T), T=');
if i<0,
i
error('ATTENTION: ERROR i must be greater or
equal to Zero');
end;
Ejes=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x);
Emar=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x).*tan(x
).*a/b;
% the Absolute Elliptic Jes-x "AEjesx"
AEjesx=Ejes./cos(x).*(angx(x-T)).^i;
plot(x,AEjesx); axis([-2 8 -3 3]); grid on;
fprintf('Do you want to repeat ?\nPress y for ''Yes'' or
any key for ''No''\n');
repeat=input('Y/N=','s'); %string input
close all
end; %End while
%-------------------------------------------------------------------
8 Conclusion
In this paper, an original study in trigonometry is
introduced. The elliptical unit and its trigonometric
functions are presented and analyzed. In fact the
proposed Elliptical Trigonometry is a new form of
trigonometry that permits to produce multiple forms
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
870
Issue 11, Volume 5, November 2010
of signals by varying some parameters; it can be
used in numerous scientific domains and
particularly in mathematics and in engineering. For
the case treated in this paper, 32 elliptical
trigonometric functions are defined; only two
functions are analyzed and simulated using software
as Matlab-Simulink. In general, a connection cable
with specific transmission data protocol connects
any industrial system to the computer. One can use
the studied functions in order to generate control
signals in need for power components of the
industrial system.
The elliptical trigonometry functions will be widely
used in electronic domain especially in power
electronics. Thus, several studied will be improved
and developed after introducing the new functions
of the elliptic trigonometry. Some mathematical
expressions and electronic circuits will be replaced
by simplified expressions and reduced circuits.
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WSEAS Transactions on Mathematics, Issue 9,
Volume 8, September 2009, pp. 551-560.
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[17] Cyril W. Lander, Power electronics, third
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Glory to Jesus, the God of all
King of kings and lord of lords
All functions in this article are dedicated to:
Jes= God Jesus Mar= Saint Mary
Ter= Sainte Thérèse de l’enfant Jésus Rit= Saint Rita
Raf= Saint Rafka Ber= Saint Bernadette
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
Claude Bayeh
ISSN: 1991-8763
871
Issue 11, Volume 5, November 2010
