1. 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

2. Lecture 11
Transient states in electrical circuits – Laplace transformation approach

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

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

5. 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).

6. 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.

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

8. Steady state after commutation

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

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

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

12. Laplace form of solution
Current in Laplace form
Characteristic equation
Poles

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

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

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

16. Transients of capacitor voltage and current in RC and RLC circuits

17. Critically damped case
Double pole
Laplace form of current solution

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

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

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

21. Time solution Current of inductor Voltage of inductor
Voltage of capacitor

22. Graphical form of solution

23. Transient uC(t) at different resistances in oscillatory case