CM 1
Numerical methods for approximation of nonlinear characteristics given by
magnetic coils
Dan D. Micu, Andrei Ceclan, and Emil Simion
Technical University of Cluj-Napoca, Electrical Engineering Department
G.Baritu 26, Cluj-Napoca, Romania,
e-mail: Dan.Micu@et.utcluj.ro
Abstract – The paper tends to issue the importance of
approximation characteristics for nonlinear electrical circuit
elements. It is created a robust structure for approximating curves
with analytical polynomial functions based on numerical methods.
Magnetic coil cases are studied as an example.
Keywords— numerical methods, nonlinear characteristics,
magnetic coils, nonlinear electric circuits
I.
APPROXIMATION PROCESS
Nonlinear circuit elements, as magnetic coils, and nonlinear
devices obtained by, take an important place in electrical
engineering.
The characteristics of nonlinear circuit elements are
generally deduced by graphical representation of experimental
data. Only exceptionally, and not on the entire domain, the
characteristics can be analyticaly established by solving
equations, if the approximation process is done on
characteristic sections, when the domain is been divided. [1]
There are distinguished two situations: the first, on which
the nonlinear circuit element characteristics has an analytical
expresion resulted on a theoretical basis, the experimental
work being done just to determine the coefficients; and the
second, on which the characteristics may be obtained only
experimentaly. For this last case it is necessary to find a
function that best fitts the characteristics of the nonlinear coil,
and to determine the coefficients that appear in the analytical
expression of the approximating function.
Not being stated a general approximation method for
experimental nonlinear characteristics, it is necessary to select
by repeated tests the best approximating function.
Usually, the approximation process is realised with the
following types of functions: approximation by polynomial
functions, by power functions, by segments, by exponential
functions, by trigonometric functions and by transcendent
functions. [2]
A convenient fitting of the hysteresis cycle for the
magnetic coils it is done if the characteristic is splited onto the
inferior curve, corresponding to the magnetization process,
and the superior curve, corresponding to the demagnetization
process.
II.
For certain per-unit numerical values, for the above
mentioned situations, there has been selected to do the
approximations a fifth and a ninth order polynomial functions,
with initially unknown coefficients:
n
f ( x) 

k
u k x
n5
n9
.
k 0
The problem is to identify the coefficients of these functions
for each case, when there are no losses, for the first
magnetization, and when there are considered the losses and
appears the hysteresis, both for the uncontrolled and controlled
coil. It is used the numerical calculus program Mathcad and
the determination of the coefficients is performed with two
numerical methods. The first, by appealing the Mathcad inside
function genfit and the second, by using the least square
method. Both of these methodes is to be explained.
When the experimental points of a characterstic are defined
by two vectors Ψ for the flux values, and i for the current
values, the genfit function returns a vector of the unkonwn
coefficients, calculated from an initial approximation given
within a vector u. Thus, the arguments of the called function
are as follows:
C  genfit   i u  F
The fifth order polynomial function deduced for the first
magnetization curve in the case of an uncontroled magnetic
coil has the expresion:
4
i5    9.76 10  0.677   0.012   1.124  
4
5
 0.016   1.507 
2
3
The fitting process of the characteristic is showed below:
SRUCTURE FOR PRACTICAL CASES
The present paper concerns on building up a definite structure
to a numerical method for approximation of nonlinear
characteristics given by magnetic coils. There are studied the
simple and controlled coil without and with losses for the
analytical function i = f(Ψ). [3]
Fig.1. First magnetization without losses
If the magnetic flux is: Ψ(t)=Amaxcos(ωt), then, the electrical
current for a given 50 Hz frequency, with fundamental and
third harmonic component, are presented in figure 2.
CM 2
u m  ( 0.303 0.209 0.049 0.721 0.092 2.305 0.287 2.99 0.153 0.82 )
u dm  ( 0.303 0.209 0.049 0.721 0.092 2.305 0.287 2.99 0.153 0.82 )
Fig.2. Time-dependent current
The next case, for the controlled coil and certain data
experiments, there are been chosen, beside the fifth order
polynomial, also the ninth order polynomial function. In the
figure 3 are shown the analytical curves obtained, with the
same notations Ψ, i and u.
Fig.5. Hysteresis cycle with inferior and superior curve fitting
The controlled coil imposes the use of four functions for the
approximation, two for magnetization in the first and the third
quadrant, and two for demagnetization [5]:
Fig.3. Two curve fitting for controlled first magnetization coil
The analytical expressions of the functions:
9
8
 13
 12
i5    4.67 10  1.798   3.58 10
5
8 4
 3.37 10    2.915 
2
3
   3.671  
i9    1.02 10
 2.401   3.29 10
   12.139  
5
 11 4
 11 6
 1.66 10
   33.859   2.68 10
  
7
 11
 ( 40.913)    1.35 10
2
8
   17.778 
Fig.6. Controlled hysteresis cycle
3
III.
9
For the same expresion of the flux, the time dependent
electrical current has the variation in the figure 4.
Fig.4. Time-dependent current
The least square method consists in defining a functional with
the coefficients of approximating function like unknowns:
last ( )

It has been created a structure for a numerical model of
approximation characteristics of nonlinear circuit elements,
like magnetic coils. Irrespective of the numerical data resulted
from the experiments, this structure is capable to best fit the
selected function with the experiments. The rapid and efficient
approximation of characteristics with polynomial functions
compete to an optimal design of the applications containing
nonlinear elements. The analytical approximation of
characteristics and hysteresis, allows the calculation of the
induced voltage by coupling coils, within acceptable errors, so
as to optimally design coils and transformers.
2 .
 y   u  u  x  u  x 2  u  x 3  u  x 4 

3  j
4  j
 j  0 1 j 2  j

j0 
 u  x 5  u  x 6  u  x 7  u  x 8  u  x 9 
6  j
7  j
8  j
9  j  
  5  j
This functional is partially derived in respect with the
coefficients and each derivative equalled with zero. The
resulted linear system of equations, solved with an inside
Mathcad (Given-Find) numerical method, provides the values
of the coefficients. In the case of the hysteresis for the two
types of coils, the characteristic is divided onto the superior
and the inferior curves. For a ninth order polynomial function
the identified coefficients for magnetization, respectively
demagnetization and the hysteresis cycle are obtained [4]:
G( x y  u) 
CONCLUSIONS
IV.
[1]
[2]
[3]
[4]
[5]
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Electrotehnica, Editura Casa Cartii de Stiinta, Cluj-Napoca, 2002.
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orthogonal polarized feromagnetic core using EMTP software
environement, BP Iasi, EPE, pp. 188- 192, 2004.
D. D. Micu, A. Ceclan, E. Simion, Analytical Technical Curve Fitting.
Method and Implementation, 6th International Power Systems
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