Solving the Transients Event in Electric Circuits Using a Mathematical Model of Differential Equations

Electrical circuits are systems that can be described in different ways using differential equations of first, second and upper order. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. If the circuit contains storage elements such as capacitors and inductance, these circuits can be described as integral-differential equations. Analytical solution of such circuits in complex networks is very difficult and lengthy. To calculate the circuit state variables (voltages and currents) can be used those programs in which the electrical circuit is created with a simple drag & drop system "and connecting graphical blocks representing directly individual electrical devices. This works some a desktop application such as the EMTP-ATP or Matlab/Simulink. Advantage of solution lies in the fact, that the algorithm changes are implemented through this application, changing, deleting and adding blocks of the circuit. The disadvantage of this approach is that it avoids the need to create a mathematical model of the circuit. Physical basis for a deeper understanding of the behavior of the circuit in steady state, but also for the transient events is necessary. This paper deals with the calculation of state variables of the electrical circuit using a mathematical model consisting of differential equations. 1. Theoretical part: Circuit differential equations are essentially based on two Kirchhoff laws and their creating may use general methods, but most often used method of loop currents or node-voltage method, or modified nodal voltage method. Creating equations of concrete branches of the circuit is then based on the fundamental relationship between the current state variables at the individual circuit elements: Resistance: = ∗ (1) Capacity: = ∗ = + ∗ , = ∗ ( ) (2) Inductance: = ∗ = + ∗ , = ∗ ( ) (3) Using these relations based on a description above of the complex circuit the state variables are becoming describe as a set of integral-differential equations. Integral equations can be easily converted by derivate to the differential equations. The complex circuit is then described as a system of n linear or nonlinear differential equations with constant coefficients, respectively single differential equation of n-th order [1]: a + a +⋯ . . a + a x = y(t) (4) The solution of the diferential equation consist from the homogenous equation X0(t) and particular solutions Xp(t). X t = X t + X (t) (5) Homogenous equation: a + a +⋯ . . a + a x = 0 (6) Its general solution depends only on the characteristics of the circuit without any independent sources. However, it is fundamentally affected by the initial energy state of the circuit, ie sizes of the accumulated energy in capacitors and coils at the beginning of a solution, at t=0s. The character of the solution equation is the given by the roots λ1, λ2,....λn of so characteristic equation, which is a polynomial equation form: a λ + a λ + a λ +⋯ .+a λ + a = 0 (7) If the polynomial roots are simple and different of each other, the solution of homogeneous differential equations is given by linear combination of exponential functions of type exp (λkt), ie X t = ∑ K e (8) Where K1, K2.....Kn are integration constants whose values determine the specific initial conditions in the system. When operating in the circuit periodic or direct voltage and current sources, the circuit reaches after the transient the stationary or periodic steady state expressed by homogeneous equation X0(t). Steady state is thus expressed precisely by particulate solution. 2. Solving the circuit state variables using differential equation – mathematical model of simply electrical circuit given by linear differential equation 2-th order: The figure (Fig. No. 1) shows the scheme of simple RLC circuit supplying with DC voltage source voltage Us and the equivalent circuit model created in software Matlab / Simulink. At the time t = 0, the switch closes and the circuit is connected to the voltage source. In the circuit we are interested in state variables, the voltage on capacitor uc(t) and the circuit current i(t). Other variables such as voltage for inductance and resistance can be easily determined from known relationships. Sizes of circuit elements are: Us = 1V, R = 15Ω, L = 20mH, C = 333μF. Fig. 1 – Electrical circuit scheme and its equivalent model in Matlab/Simulink After writing II. Kirchhoff’s law and an expression of current via capacity voltage we will obtain a homogeneous linear differential equations 2-th. order with constant coefficients, which reflects the voltage loop circuit: ( ) = − ( ) − u (t) (9) Solving of this differential equation we get the capacitor voltage uc(t). The current circuit is calculated via derivation and multiplying the capacitor voltage and the value of capacitor C. This differential equation is thus a mathematical model of that circuit. The solution of this mathematical model is performed in the software Matlab / Simulink in two ways (Fig. 2). Via gradual integration and using the transfer function G(s) entered in block Transfer Fcn. Fig. 2 – scheme of mathematical model for solving the differential equation (gradual integration and transfer function) Transfer G (s) of general dynamic system (in this case the circuit) is defined as the proportion of the Laplacian image of output signal and Laplacian image of input signal at zero initial conditions: