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 11 Transient states in electrical circuits – Laplace transformation approach

Laplace representation of basic elements Resistor Inductor Capacitor Any real circuit element has its Laplace model valid in complex frequency space (s-space).

Kirchhoff’s laws for transforms Current law Voltage law These laws are formed identically as for real time currents and voltages.

Transient in the circuit using Laplace transforms Determine the initial values iL(0-) and uC(0-) 2) Determine the steady state in circuit after commutation iLu(0+) and uCu(0+) 3) Calculate the natural responses ucp and iLp of the circuit deprived of external excitations (voltage source - short circuit, current source – open circuit) 4) Final solution is the superposition of both states This is so called method of superposition of states (necessary at sinusoidal excitations).

Calculation of natural response Eliminate the external sources the RLC circuit Determine the initial conditions for natural response Form the Laplace model of the RLC circuit deprived of external sources Using Kirchhoff’s laws find the solution of this circuit in s-space (operator form) Calculate the inverse Laplace transforms (original fuctions) of the currents of inductors and voltages of capacitors.

Example Determine the transient of inductor current after commutation. Assume: R=2 , L=1H, C=1/4F, Solution: Initial conditions

Steady state after commutation

Natural response Laplace model of the circuit for natural response Initial conditions for natural response Solution as Laplace transform

Final solution Because of complex poles we apply the table of trasforms Natural response in time form Total current of the inductor

Transient state in RLC circuit at DC excitation Zero initial conditions Laplace model of the circuit

Laplace form of solution Current in Laplace form Characteristic equation Poles

Three cases of general solution Overdamped (aperiodic) case: Critically damped case Oscillatory (periodic) case Critical resistance

Overdamped case Both poles are real and single. The time form of current Damping coefficient Voltages of capacitor and inductor

Graphical form of solution Examplary transients in RLC circuit for R = 2,3, C = 1F i L = 1H at E = 1V.

Transients of capacitor voltage and current in RC and RLC circuits

Critically damped case Double pole Laplace form of current solution

Time form of solution Current of inductor Voltage of inductor Voltage of capacitor

Comparison of uC(t) at overdamped and critically damped cases

Oscillatory case Both poles are complex. Laplace form of solution Self-oscillation frequency

Time solution Current of inductor Voltage of inductor Voltage of capacitor

Graphical form of solution

Transient uC(t) at different resistances in oscillatory case