Experiment 5

1. Experiment 5 * AC Steady State Theory * Part A: RC and RL Circuits * Part B: RLC Circuits

2. AC Steady State Theory • Transfer Functions • Phasors • Complex Impedance • Complex Transfer Functions

3. Transfer Functions • The transfer function describes the behavior of a circuit at Vout for all possible Vin.

4. Simple Example

5. More Complicated Example What is H now? • H now depends upon the input frequency (w = 2pf) because the capacitor and inductor make the voltages change with the change in current.

6. Influence of Resistor on Circuit • Resistor modifies the amplitude of the signal by R • Resistor has no effect on the phase

7. Influence of Capacitor on Circuit • Capacitor modifies the amplitude of the signal by 1/wC • Capacitor shifts the phase by -p/2

8. Influence of Inductor on Circuit • Inductor modifies the amplitude of the signal by wL • Inductor shifts the phase by +p/2

9. How do we model H? • We want a way to combine the effect of the components in terms of their influence on the amplitude and the phase. • We can only do this because the signals are sinusoids • cycle in time • derivatives and integrals are just phase shifts and amplitude changes

10. Phasors • Phasors allow us to manipulate sinusoids in terms of amplitude and phase changes • Phasors are based on complex polar coordinates • The influence of each component is given by Z, its complex impedance

11. Phasor References • http://ccrma-www.stanford.edu/~jos/filters/Phasor_Notation.html • http://www.ligo.caltech.edu/~vsanni/ph3/ExpACCircuits/ACCircuits.pdf • http://robotics.eecs.berkeley.edu/~cssharp/Phasors/Phasors.html

12. Phasor Applet

13. Complex Polar Coordinates

14. Review of Polar Coordinates point P is at ( rpcosqp , rpsinqp )

15. Introduction to Complex Numbers • zp is a single number represented by two numbers • zp has a “real” part (xp) and an “imaginary” part (yp)

16. Now we can define Phasors • The real part is our signal. • The two parts allow us to determine the influence of the phase and amplitude changes mathematically. • After we manipulate the numbers, we discard the imaginary part.

17. Magnitude and Phase • Phasors have a magnitude and a phase derived from polar coordinates rules.

18. Euler’s Formula

19. Phasor Applet

20. Manipulating Phasors (1) • Note wt is eliminated by the ratio • This gives the phase change between signal 1 and signal 2

21. Manipulating Phasors (2)

22. Understanding the influence of Phase

23. Complex Impedance • Z defines the influence of a component on the amplitude and phase of a circuit • Resistors: ZR = R • change the amplitude by R • Capacitors: ZC=1/jwC • change the amplitude by 1/wC • shift the phase -90 (1/j=-j) • Inductors: ZL=jwL • change the amplitude by wL • shift the phase +90 (j)

24. Capacitor Impedance Proof Prove:

25. Complex Transfer Functions • If we use phasors, we can define H for all circuits in this way. • If we use complex impedances, we can combine all components the way we combine resistors. • H and V are now functions of j and w

26. Simple Example

27. Simple Example (continued)

28. Using H to find Vout

29. Simple Example (with numbers)

30. Adding Phasors & Other Applets

31. Part A -- RC and RL Circuits • High and Low Pass Filters • H and Filters

32. High and Low Pass Filters High Pass Filter H = 0 at w ® 0 H = 1 at w ®¥ H = 0.707 at wc wc=2pfc fc Low Pass Filter H = 1 at w ® 0 H = 0 at w ®¥ H = 0.707 at wc wc=2pfc fc

33. Corner Frequency • The corner frequency of an RC or RL circuit tells us where it transitions from low to high or visa versa. • We define it as the place where • For RC circuits: • For RL circuits:

34. Corner Frequency of our example

35. H(jw), wc, and filters • We can use the transfer function, H(jw), and the corner frequency, wc, to easily determine the characteristics of a filter. • If we consider the behavior of the transfer function as w approaches 0 and infinity and look for when H nears 0 and 1, we can identify high and low pass filters. • The corner frequency gives us the point where the filter changes:

36. Our example at low frequencies

37. Our example at high frequencies

38. Our example is a low pass filter What about the phase?

39. Our example has a phase shift

40. Taking limits • At low frequencies, (ie. w=10-3), lowest power of w dominates • At high frequencies (ie. w =10+3), highest power of w dominates

41. Capture/PSpice Notes • Showing the real and imaginary part of the signal • in Capture: PSpice->Markers->Advanced • ->Real Part of Voltage • ->Imaginary Part of Voltage • in PSpice: Add Trace • real part: R( ) • imaginary part: IMG( ) • Showing the phase of the signal • in Capture: • PSpice->Markers->Advanced->Phase of Voltage • in PSPice: Add Trace • phase: P( )

42. Part B -- RLC Circuits • Band Filters • Another example • A more complex example

43. Band Filters Band Pass Filter H = 0 at w ® 0 H = 0 at w ®¥ H = 1 at w0=2pf0 f0 Band Reject Filter H = 1 at w ® 0 H = 1 at w ®¥ H = 0 at w0 =2pf0 f0

44. Resonant Frequency • The resonant frequency of an RLC circuit tells us where it reaches a maximum or minimum. • This can define the center of the band (on a band filter) or the location of the transition (on a high or low pass filter). • We are already familiar with the equation for the resonant frequency of an RLC circuit:

45. Another Example

46. At Very Low Frequencies At Very High Frequencies

47. At the Resonant Frequency if L=10uH, C=1nF and R=1K W w0=10M rad/secf0=1.6M Hz |H|=0.1

48. Our example is a low pass filter Phase f = 0 at w ® 0 f = -180 at w ®¥ -90 Magnitude H = 1 at w ® 0 H = 0 at w ®¥ 0.1 f0=1.6M Hz

49. A more complex example • Even though this filter has parallel components, we can still handle it. • We can combine complex impedances like resistors

50. Determine H