Modeling of Nonlinear Circuit using Volterra Series

J. Anilioniene Kaunas University of Technology, Department of Applied Mathematics, Studentų g. 50, LT-51368, Lithuania, phone:+370 37 300318, e-mail: julija.anilioniene@ktu.lt R. Anilionis, D. Andriukaitis Department of Electronics Engineering, Kaunas University of Technology, Studentų str. 50, LT-51368 Kaunas, Lithuania, phone: +370 37 300519, e-mails: romualdas.anilionis@ktu.lt, darius.andriukaitis@ktu.lt


Introduction
Distortion is a key issue in the design of many types of circuits.In modern circuits linearity is a very important parameter, especially with the strict modern telecommunication system standards The method of Volterra series is the most widely method to analyze the nonlinear circuits [1,2].In order to get a low signal distortion, a great focus is made on investigation of nonlinearities The Volterra series method of distortion analysis is presented in the analysis of a common emitter circuit.
For analysis of nonlinear systems differential equations representing nonlinear, frequency and parametrical performances of the system are applied.Unlike numerical simulations which give no information about the source of the distortion, closed form expressions for distortion components in terms of circuit parameters can be found using Volterra series.
In this paper we modeling nonlinearities of a common emitter circuit, analyze their influence on signal distortions, obtain general solutions of Volterra kernels appropriate for engineering calculations.To determine nonlinear behaviour is used the Volterra series, the most widely used method to analyze the nonlinear behavior of analog circuits.

Model and analysis methodology
The Volterra series for a circuit is generally represented as a summation of operators [3] where The Laplace transform of the Volterra kernel is the input,   t y the output.Using the method of Volterra series of distortion analysis a nonlinear system is composed of linear,square,cube,etc.subsystems connected in parallel (Fig. 1).

Fig. 1. Model nonlinear system (H i -appropriate kernel Volterra series)
The first subsystem is linear.The output signal of this system is the following: or a view is as follows where p -complex variable.An output signal of the square subsystem is: An output signal of the cube subsystem is: Only three harmonics will be evaluated as already third harmonic in amplifier is very low.
If a amplifier receives the following signal change constant component of a output signal is defined by analyzing a square system [4]  where w 2  -phase of the second harmonic.
Change in the first harmonic is figured out by analyzing a cube system where w 3  -phase of the third harmonic.

Analytic solutions
The first step befor calculating a Volterra series is to expand the circuit nonlinearities in Taylor series.
The main nonlinearities that determine the signal distortion of amplifier are nonlinear junction currents of an emitter and collector and their nonlinear capacitances [4].
Fig. 2 presents an equivalent circuit for signal distortion analysis of amplifier.
Most analytical amplifier models [5 ]express the emitter current following where e u -junction voltage, φ T -temperature potencial, α -collector current transmission coefficient, i k -current of collector, I eT -heat current where 0 e I -current constant component;  Nonlinear generator current flowing through the collector capacitance where K U -junction voltage of a collector; where K V -constant.Variable component of an output current is -coefficients dependent on circuit parameters and frequency.
After evaluation of amplifier nonlinearities for equivalent circuit formed equations system:  Adressing the equations system by method of Volterra series first found Volterra kernels of linear subsystem,then of square subsystem,after evalution of linear kernells,afterwords of cube subsystem,after evalution kernels of linear, squar subsystems.Kernels of a linear system will be referred to as where   p W -conductance matrix.
Two-dimensional kernels of IT diagram units will be marked respectively   Then change in a constant component of a output current after figuring out a two-dimensional kernel Similarly, change first harmonic output currrent depending on nonlinearities in a amplifier is defined.The first harmonic is observed at the output of linear and cube systems.In order to establish the first harmonic at the output of a cube system, three-dimensional kernels shall be calculated.A matrix equation that is used to figure out the aforementioned kernels is the following

Conclusions
For modeling nonlinearities of nonlinear circuit the method of Volterra series has been offered.Unlike numerical simulations,closed form expressions for distortion components in terms of common emitter circuit parameters has be found..The mathematical model establisched here let to calculate Volterra kernels of the respective linear, square and cube systems and presents dependences of change output carrent component on magnitude of an input signal within a frequency range.

Fig. 2 .
Fig. 2. Equivalent circuit for distortion analysis of amplifier After expanding (12) in a Taylor series, the following is obtained 3 3 2 2 1 Nonlinear generator current flowing through the junction capacitance of an emitter is as follows: where potentials, Z INinput impedance, Z L -load impedance, e Z -emitter impedance, C KB -feedback capacitance, C K -output capacitance, r b -base impedance, Z K -output impedance.

Fig. 3 Fig. 3 . 3 -
Fig. 3 presents dependences change a constant component of a IT output current γ K0 = ΔI K0 /I K0 on magnitude and frequency of an input signal m=U/φ T .

Fig. 4 Fig. 4 .
Fig. 4 presents dependences of change the first harmonic output current γ K1 = ΔI K1 /I K1 of on magnitude and frequency on an input signal m=U/φ T .