Transient Analysis DC Steady-State

1. Transient AnalysisDC Steady-State ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall

2. Transient Analysis • Scope of study: • Circuits that contain sources, switches, resistances, inductances, and capacitances • Transients: • Time-varying currents and voltages resulting from sudden application of sources, usually due to switching • Transient Analysis: • Involves using circuit concepts from Chapters 1 & 2 • Current-voltage relationships for inductances and capacitances involve derivatives and integrals • Create circuit equations that are differential equations ELEC 308

3. First-Order RC or RL Circuits • First-Order RC Circuits • Contains DC sources, resistances, and a SINGLE capacitance • First-Order RL Circuits • Contains DC sources, resistances, and a SINGLE inductance • Frequently used in timing applications • Due to time constant ELEC 308

4. First-Order Circuit Algorithm • Apply KCL, KVL, and/or Ohm’s Law to write the circuit equation. • If equation contains integrals, differentiate each term to produce a PURE differential equation. • Assume a solution of the form K1+K2est. • Substitute the solution into the diff. eqn. to determine the values of K1 and s. • Use the initial conditions to determine the value of K2. • Write the final solution. ELEC 308

5. Discharge of a Capacitance ELEC 308

6. Discharge of a Capacitance ELEC 308

7. Discharge of a Capacitance ELEC 308

8. Discharge of a Capacitance ELEC 308

9. Time Constant • Time interval τ = RC is called the time constant of the circuit • After t=5τ, vC(t)≈0 ELEC 308

10. Charging a Capacitance ELEC 308

11. Charging a Capacitance ELEC 308

12. Charging a Capacitance ELEC 308

13. Charging a Capacitance • First term is STEADY-STATE RESPONSE • Or FORCED RESPONSE • Second term is TRANSIENT RESPONSE ELEC 308

14. Charging a Capacitance ELEC 308

15. DC Steady State • Transient terms in the expressions for current and voltages in RLC circuits decay to zero with time • For DC sources, steady-state currents and voltages are CONSTANT • For steady-state conditions with DC sources: • CAPACITANCES behave like OPEN circuits • INDUCTANCES behave like SHORT circuits ELEC 308

16. Capacitance in DC Steady-State • Remember current through a capacitance: • If voltage is constant, current is _________. • CAPACITANCE behaves just like an ____ circuit ELEC 308

17. Inductance in DC Steady-State • Remember voltage across an inductance: • If current is constant, voltage is _________. • INDUCTANCE behaves just like a ______ circuit ELEC 308

18. Steady-State DC Analysis • Find vx and ix for t >> 0 ELEC 308

19. Exercise 4.3 • Find va and ia for t >> 0 ELEC 308

20. RL Transient Analysis • Find i(t) and v(t) ELEC 308

21. RL Transient Analysis • Time interval τ = L/R is called the time constant of the circuit • After t=5τ, i(t)≈2 ELEC 308

22. Example 4.3 • Find i(t) and v(t) ELEC 308

23. Example 4.3 ELEC 308

24. Exercise 4.6 • Find i(t) and v(t) ELEC 308

25. Summary • Transient Analysis • First Order RC Circuits • First Order RL Circuits • DC Steady State ELEC 308