Table of Contents
Nonlinear meaning the characteristic of a system that the output value is not directly proportional to the input.
In algebra, a linear operator or function f(x) is described as follows:
Additivity – f(x + y) = f(x) + f(y)
Homogeneity – f(αx) = αf(x)
If the above assumptions are not met, we have nonlinearity. In nature, most interactions are described by nonlinear functions. However, modelling reality is based on using the simplest possible mathematical tools and it often happens that non-linear phenomena are described by linear functions, e.g. Hooke’s law, where some area for relatively small stresses behaves almost linearly.
The basic measure of dependence is the linear correlation coefficient, which provides a measure of linear relationships between variables. In analytical problems, e.g. in economics, nonlinear relationships are often neglected and treated as neglected.
In nature, manifestations of interdependence are very common. There are also many other ways in which elements depend on each other to form a structure-network. An example is the food chain, which places an organism in a certain environment of other organisms with which it interacts in such a way that it can eat them or be eaten. The investigator of such phenomena must exercise great knowledge and ingenuity to describe nonlinear relationships accurately. He can use measures such as the Manhattan measure, which is the sum of distances between variables in a n-dimensional space. To reproduce hierarchy one can draw minimum range tree or graph with marked paths between objects. An example can be sociogram, which shows relations between people in a group.
In time series analysis, tools based on fractality are also useful. The economic crisis of 2007 caused non-linear models, previously neglected by economists, to enjoy popularity.
What is nonlinear circuit
An electric circuit element is called nonlinear if its characteristic y=f(x) or x=φ(y)is nonlinear, i.e., it cannot be described analytically by a simple equation (y=ax+b).
A nonlinear circuit elements, whether passive or active, is described by providing a continuous (graph) or discrete (table) set of independent variables and function values.
The nonlinear circuit elements in the circuit models are described using graphical symbols and a description of the nonlinear parameter.
Classification of nonlinear elements can be done based on different criteria. Depending on the course of the characteristic y=f(x) we distinguish between nonlinear elements:
(a) symmetrical f(x)=-f(-x),
(b) unsymmetrical f(x)≠-f(-x),
(c) unambiguous Each value x ∈ X corresponds to one and only one value y,
(d) equivocal there are such intervals of the independent variable x∈(x1,x2) that inside these intervals y=f(x) can take more than one value.
Nonlinear circuit analysis
With the characteristics of the nonlinear elements in the circuit, it is possible to analyze the circuit by transfiguration and possibly graphical retransfiguration. Graphical methods of transfiguration of nonlinear circuit are carried out on the basis of Kirchhoff’s laws.
Combined Characteristics nonlinear circuit Analysis Method
For elements connected in series.
Consider a series connection of n resistors with characteristics given by equations.
The series connection results in
This equation defines the characteristics of the new element
If the voltage in the equivalent element of the series circuit is UX, then after obtaining the combined characteristic of RNL, the operating point P can be found on it and then the current in the circuit IX and the voltages on the circuit elements.
For elements connected in parallel
We consider a parallel connection of n resistors with characteristics defined by Eq.
The series connection results in.
If the current in the equivalent element of the parallel circuit is IX, then after obtaining the combined characteristic RNL, the operating point P is determined on it and then the supply voltage UX and the currents in the branches of the circuit.
Characteristic intersection method
For elements connected in series
If the supply voltage is constant and its fixed value does not change, then in order to determine the current Ix (the operating point on the combined characteristic) it is not necessary to determine the combined characteristic. The method of intersecting the characteristic curves, so called “mirror reflection”, can be used.
Nonlinear circuit analysis procedure:
- we plot the characteristic of an element, e.g. Rn1,
- on the U axis measure a given value Ux of the voltage at the system terminals
- for element Rn2 we assume a coordinate system with the origin in point 0′ (distant from point 0 by Ux) and U axis having the opposite direction than for element Rn1,
- we plot the characteristics of RN2 in the new coordinate system,
- the working point of the circuit P is the intersection of the characteristics and its abscissa divides Ux into Un1 and Un2.
For elements connected in parallel.
If the current supplying circuit IX is known and it is known that it will not change or otherwise, only for this value of current we want to determine the voltages and currents in the branches, we can use the “mirroring” method.
Nonlinear circuit analysis procedure:
- plot the characteristics of an element, e.g. Rn1,
- measure a given value of Ix on the I axis,
- for element Rn2 assume a coordinate system with origin in point 0′ (distant from point 0 by Ix) and axis of current I having opposite direction than for element Rn1
- In the new coordinate system we plot the characteristics of Rn2,
- the circuit operating point P is the intersection of the characteristics and its ordinate divides Ix into In1 and In2.
Nonlinear circuit elements
Nonlinear circuit resistive elements
Resistive element:
Linear resistor:
FR(u,i) = u – Ri = 0
U = fR(i) = Ri
FR [u(t), i(t)] = 0
FR is an entangled algebraic function(no derivatives, integrals or any other operations on t.
FR (u,i) = 0
i = fG(u) = Gu
G = 1/R
The excitation can be either current or voltage.
FR(u, i) = 0
There exists a function fR such that u = fR(i), but there is no inverse function to fR. Such an element is called current dependent. The function fR is called the current-voltage characteristic.
The excitation of a current-dependent element can only be a current – with a voltage excitation the system is not solvable (there is no unambiguous solution).
There exists a function fG such that i = fG(u) but there is no inverse function to fG. Such an element is called voltage dependent. The function fG is called the voltage-current characteristic.
The excitation of a voltage dependent element can only be a voltage with current excitation the system is not solvable.
Both u = fR(i) and i = fG(u) can be determined. Then both current-voltage and voltage-current characteristics exist. They are monotonic functions. Such an element is called uncorrelated.
The excitation can be either current or voltage.
If neither current nor voltage can be unambiguously determined from the equation FR(u,i) = 0, then such an element is called a degraded element. It can be described by a system of parametric equations.
Nonlinear circuit capacitive elements
Capacitive element
Fc is an entangled algebraic function with no derivatives, integrals, or any other operations on t, i.e., Fc(u,q) = 0. We will assume that q = fc(u) can be determined.
Linear Condeser:
q = fc(u) = Cu, C = const
i(t) = d/dt (Cu) = C du/dt
Static capacity
Dynamic capacity
Nonlinear inductive element
FL is an entangled algebraic function with no derivatives, integrals, or any other operations on t, i.e.
We will assume that it is possible to determine
Linear Inductor:
Static inductance
Dynamic inductance
An inductor on a ferromagnetic core:
The position of the operating point depends not only on I0 but also on the history of changes in this current.
References:
http://th.if.uj.edu.pl/~gulakov/life_corr/
http://www.zto.ita.pwr.wroc.pl/~deren/Wyklad%20EKEK00018/8.%20Obwody%20nieliniowe.pdf
