1. ECE201 Lect-51 Phasor Relationships for Circuit Elements (8.4); Impedance and Admittance (8.5) Dr. Holbert February 1, 2006

2. ECE201 Lect-52 Phasor Relationships for Circuit Elements Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. A complex exponential is the mathematical tool needed to obtain this relationship.

3. ECE201 Lect-53 I-V Relationship for a Resistor Suppose that i(t) is a sinusoid: i(t) = I M e j(  t+  Find v(t) R v(t) + – i(t)

4. ECE201 Lect-54 Computing the Voltage

5. ECE201 Lect-55 Class Example Learning Extension E8.5

6. ECE201 Lect-56 I-V Relationship for a Capacitor Suppose that v(t) is a sinusoid: v(t) = V M e j(  t+  Find i(t) C v(t) + – i(t)

7. ECE201 Lect-57 Computing the Current

8. ECE201 Lect-58 Phasor Relationship Represent v(t) and i(t) as phasors: V = V M   I = j  C V The derivative in the relationship between v(t) and i(t) becomes a multiplication by j  in the relationship between V and I.

9. ECE201 Lect-59 Example v(t) = 120V cos(377t + 30  ) C = 2  F What is V? What is I? What is i(t)?

10. ECE201 Lect-510 Class Example Learning Extension E8.7

11. ECE201 Lect-511 I-V Relationship for an Inductor V = j  L I L v(t) + – i(t)

12. ECE201 Lect-512 Example i(t) = 1  A cos(2  9.1510 7 t + 30  ) L = 1  H What is I? What is V? What is v(t)?

13. ECE201 Lect-513 Class Example Learning Extension E8.6

14. ECE201 Lect-514 Circuit Element Phasor Relations (ELI and ICE man)

15. ECE201 Lect-515 Impedance AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law: V = I Z Z is called impedance (units of ohms,  )

16. ECE201 Lect-516 Impedance Resistor:V = I R –The impedance is Z R = R Inductor:V = I j  L –The impedance is Z L = j  L

17. ECE201 Lect-517 Impedance Capacitor: –The impedance is Z C = 1/j  C

18. ECE201 Lect-518 Some Thoughts on Impedance Impedance depends on the frequency,  f Impedance is (often) a complex number. Impedance is not a phasor (why?). Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state.

19. ECE201 Lect-519 Impedance Example: Single Loop Circuit 20k  +–+– 1F1F10V  0  VCVC + –  = 377 Find V C

20. ECE201 Lect-520 Impedance Example How do we find V C ? First compute impedances for resistor and capacitor: Z R = 20k  = 20k  0  Z C = 1/j (377·1  F) = 2.65k  -90 

21. ECE201 Lect-521 Impedance Example 20k  0  +–+– 2.65k  -90  10V  0  VCVC + –

22. ECE201 Lect-522 Impedance Example Now use the voltage divider to find V C :

23. ECE201 Lect-523 Low Pass Filter: A Single Node-pair Circuit Find v(t) for  =2  3000 1k  0.1  F 5mA  0  + – V

24. ECE201 Lect-524 Find Impedances 1k  -j530  5mA  0  + – V

25. ECE201 Lect-525 Find the Equivalent Impedance 5mA  0  + – VZ eq

26. ECE201 Lect-526 Parallel Impedances

27. ECE201 Lect-527 Computing V

28. ECE201 Lect-528 Impedance Summary

29. ECE201 Lect-529 Class Examples Learning Extension E8.8 Learning Extension E8.9