The Process Analysis in Domain of Two Variables

A steady-state processes in RLC circuit with power sources having incommensurable frequencies is considered. In such a circuit a periodic steady-state process does not exist. In order to obtain the periodic steady-state behavior an expansion of an ordinary differential equation is considered. This expansion is based on introducing of an additional time variable and therefore on transition from ordinary differential equations to partial differential equations with two time variables. Obtained differential equations are solved by use of the two-dimensional Laplace transform. Using double integrals expressions for a transfer function, active power and frequency responses for domain of two time variables are defined. Voltage and current amplitude-frequency characteristics of RLC circuit in the domain of two variables are presented. Ref. 6, fig. 5.


INTRODUCTION
Processes in inverter circuits are dependent on control signals acting on power transistors.These signals result of frequencies that ratios are mostly irrational [1], [2].In that case a steady-state behavior exists, but is not periodic.In order to obtain the periodic steady-state behavior it is expedient to expand an ordinary differential equation with one time variable to a partial differential equation with two independent time variables [3].Such an approach allows to expand the well-known methods used for analysis of ordinary circuits with ordinary signals on circuits with irrational frequencies in acting signals.
The aim of this article is an expansion of such important circuit characteristics as a transfer function, frequency response and power in the domain of two time variables.

II. MATHEMATICAL MODEL
Let us consider an RLC circuit with two sources connected in series ( ) ( ) shown in Fig. 1.We assume that and are two incommensurable angular frequencies.
The differential equation describing processes in the circuit can be written as (1) where , , is the vector of state variables, RL is a resistance of a choke, .
Let us expand the domain of the differential equation (1) from one independent variable of time t to two independent variables of time t and τ [4] , (2) Електронні системи та сигнали Applying the two-dimensional Laplace transform [5] to (2) and assuming, that initial conditions are equal to zero, i.e. and, one obtains the following equation [3], [6] , ( where , , and are images of the vector , current , voltage and vector . Solving (3) for we get , where is the inverse matrix and I is the unit matrix.
Let us introduce a transfer function.In contrast with an ordinary definition [4], we define the transfer function as the result of the quotient of images and where is an image voltage on the resistor R. From (6) one can see that this expression is periodic in the domain of two time variables, i.e. , where , .
The solution in the domain of one time variable can be obtained equating .
Using the definition of the RMS voltage [7] one obtains the following expression .It is not too difficult to show that .In case of the domain of two time variables an active power can be defined as . ( ) s q X s q AX s q B s q I s q X s q U s q Using expressions for and one obtains the following formula .This result also agreed with the calculations of active power for different components of voltage using the superposition principle.

III. FREQUENCY RESPONSE
In order to obtain a frequency response in the domain of two time variables we assume that the input voltage is sinusoidal .
(8) Applying the two-dimensional Laplace transform one finds the image function where .
Using expressions ( 7), ( 9) and (11) one obtains the active power .One can see that the form of the expressions for active power and is the same.One can see that the magnitude response (Fig. 3) has maximum and then its value decreases with the increasing of frequencies and .

CONCLUSIONS
In this paper the analyses of processes in the RLC circuit with incommensurable frequencies is considered.The periodic steady-state behavior is defined in the domain of two time variables.The solution of partial differential equations with two time variables is obtained by using the two-dimensional Laplace transform.
The transfer function, active power and frequency responses are defined and the magnitude responses for analysed circuit are presented.These characteristics are the response of the circuit to the sinusoidal signals with sum and difference of angular frequencies.Double integrals are used for the calculation of the active power in the domain of two time variables.The example of calculation of the frequency characteristics of current and voltage for the RLC load is presented.

Fig. 1
Fig. 1.Analyzed circuit Since the image function of sources connected in series in the domain of two variables has the form , state voltage can be obtained by calculating the residues at singularity points and of the image function .After the calculation of residues one obtains the voltage , this frequency response the magnitude response can be obtained as follows .In case of the input voltage (10) the output voltage is expressed by the formula where the magnitude and phase .For the input voltage (8) the current through the resistor R is ,

Fig. 3 .
Fig. 3. Magnitude response Ip in domain of two variables IV.RESULTS OF CALCULATION The amplitude-frequency characteristics have been calculated for the circuit with following parameter values: , , , , , .The magnitude responses for the input voltage