1. First-Order Circuits Part II Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

2. Contents Introduction The Source-Free RC Circuit The Source-Free RL Circuit Singularity Functions Step Response of an RC Circuit Step Response of an RL Circuit First-Order Op Amp Circuits Applications

3. Singularity Functions To aid the understanding of transient analysis To serve as good approximations to the switching signal in circuits Singularity functions are functions that either are discontinuous or have discontinuous derivatives. Three most widely used types are introduced –Unit step function –Unit impulse function –Unit ramp function

4. Unit Step Function

5. Unit Step Function (Cont’d)

7. Unit Impulse Function

8. Unit Impulse Function (Cont’d)

9. (Sampled at t 0 )

10. Unit Ramp Function

11. Unit Ramp Function (Cont’d)

12. Summary

13. Example 1 =

14. Example 1 (Cont’d)

15. Step Response of an RC Circuit = iCiC iRiR

16. Cont’d

17. Forced Response (V 0 =0)

18. Step Response (I) Complete Response = Natural Response (v n ) + Forced Response (v f ) (stored energy) (independent source)

19. Step Response (II) Complete Response = Transient Response + Steady-State Response (temporary part) (permanent part)

20. Short-Cut Method Three items required to describe the response –The initial capacitor voltage v(0) (or v(t 0 ) ) –The final capacitor voltage v(  ) –The time constant  = RC v(0) v()v()

21. Example 1

22. Example 2 t < 0 t > 0

23. Step Response of an RL Circuit iLiL iRiR =

24. Forced Response (I 0 =0)

25. Short-Cut Method Three items required to describe the response –The initial inductor current i(0) (or i(t 0 ) ) –The final inductor current i(  ) –The time constant  = L/R i(0) i()i()

26. Example 1 Find i(t).

27. Example 2 Find i(t).

28. Example 2 (Cont’d)

29. Example: OP AMP Circuits (b) (a)

30. Applications: Delay Circuit t v 70 V III I:II:  = (R 1 +R 2 )C  = RC R

31. Applications: Relay Circuit v +_+_

32. Applications: Ignition Circuit Two steps to work –S is closed to build the inductor current –Open S to force the inductor current to pass through the air gap S