CIRCUITS and SYSTEMS – part II Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 9 Transient states in electrical circuits – differential equation approach

3 Basic notions Steady state – the sdtate in the circuit when the response of circuit is of the same shape as the excitation. Transient state – the response of the circuit following the commutation in the circuit. In this state the response is of different character than the excitation. Transient response is the superposition of steady state and natural response. Natural response - response of circuit deprived of external excitation, following the nonzero initial conditions) Commutation – arbitrary change in the circuit. - the time point prior to commutation (left side limit) - the time point directly after commutation (right side limit)

4 Commutation laws Commutation law for capacitors Determination of initial conditions:  Calculate the steady state response of circuit before commutation  Write the response in time form  Calculate the currents of inductors and voltages of capacitors at the time t 0 of commutation Commutation law for inductors

5 Example Determine the initial conditions in the circuit. Assume: L=1H, C=0,5F, R=1 , Solution: Complex represenation of elements:

6 Initial conditions Circuit equations in steady state Initial conditions

7 State space decription of the circuit The general differential form description of the linear circuit The variables x of the minimal quantity form the state variables.

8 Matrix form of state description The normal state description A, B – state matrices Response matrix equation y(t) C, D – output matrices.

9 Example Determine the state description of the circuit in normal form From Kirchhof laws and definition of elements we get

10 Example (cont.) Matrix form of state equations State vector x and excitation vector u Assuming: R=2 , L=1H, C=1F we get

11 Solution of transient state using classical method In the first step we transform the system of n first order state space equations into one nth order differential equation of one variable x. The solution of it is composed of two components: the steady state x u and natural response x p. The steady state corresponds to the external excitation and natural response to nonzero initial conditions only.

12 Natural response The natural response corresponds to the solution of the homogenous differential equation (zero excitation) Characteristic equation The roots of this equation s i (i=1, 2,..., n) are the poles of the system.

13 Final solution The general solution of the homogenous differential equation of nth order is in the form A i – constants of integration calculated on the basis of initial conditions (solution of system of linear equations). The final solution of the nonhomogenous differential equation is the sum of steady state and natural response solutions This method is called the classical method of solution of the differential equations. It is very easy in application to the first order differential equations only.

14 Consider the transient response in RL circuit at DC excitation. The steady state current in the circuit Transient in RL circuit at DC excitation

15 Solution of transient state The homogenous differential equation Characteristic equation General form of solution of natural response The final (general) form of transient

16 Solution of transient state (cont.) Commutation law Hence Current of inductor The current in RL circuit at different time constants Time constant of RL circuit

17 Voltage of inductor Transient voltage of the inductor The voltage of the inductor in RL circuit at different time constants

18 Transient in RC circuit at DC excitation Consider the transient response in RC circuit at DC excitation The voltage of capacitor in steady state

19 Solution of transient state After eliminating the source we get the homogenous equation Characteristic equation General solution of natural response Final general form of solution of transient

20 Solution of transient state (cont.) Commutation law Hence Final solution Graphical presentation of capacitor voltage at different time constants Time constant of RC circuit

21 Current of the capacitor Current of capacitor in transient form Graphical presentation of capacitor current at different time constants