1. Lesson 11 Lesson 11 AC Circuits AC Ciruits Power
Maximum and Instantaneous voltage drops and current
Phasor Diagrams
Phase angle
RLC Circuits
Resonance frequency
High and Low pass filters
Step up and Step down Transformers 1

2. AC Generator
AC Generator 2

3. AC emf source 3

4. ò ò rms current ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Effective ( Integrated
values of I and V
rms current ( ) i t = ( ) I
sin w t
max 2 p w = 2 p f = ; T
is the period of oscillation T
Instantaneous power = ( ) ( ) i t v t = ( ) 2 = 2 2 ( )
Heat dissipated /
Power used in load i t R I R
sin w t m
Average power over one cycle ò T ò T 1 1 I 2 R P = I 2 ( R
sin 2 w ) t dt = I 2 R
sin 2 ( w ) t dt = m
ave T m m T 2
Define I P = I 2 R Þ I = m
ave
eff
eff 2 4

5. ac-R circuit
Veff and v(t)
Ieff and i(t)
t) 5

6. Phasor diagram for R Phasor Diagram iR(t) IRmsin(t)= iR(t) t
Load Resistance 6

7. Phasor diagram for R cont.
vR(t)
VRmsin(t)=
vR(t) t
Load Resistance 7

8. Instantaneous current and voltage8

9. ac-C circuit
vC(t)
iC(t)
t)
iC(t)
vC(t) t 9

10. Instantaneous current and voltage10

11. ac-L circuit
vL(t)
iL(t)
t)
vL(t) t
iL(t) 11

12. Instantaneous current and voltage12

13. Phase angle between current
Summary
Phase angle between current
and voltage
resistor is 0 rad
capacitor is - rad
( Current Ahead)
inductor is + rad
(Current Behind) 13

14. Inductive and Capacitive Reactance14

15. Instantaneous current and voltage in R,L, C circuits15

16. series ac-RLC circuit
Series RLC circuit 16

17. Instantaneous current
Current through all elements is the same
Thus the instantaneous PD’s must be out of phase 17

18. Total Potential Drop across R, L & C.
Picture
Total Potential Drop across R, L & C. 18

19. Instantaneous current and voltage in RLC circuit19

20. Phasor Diagram for RLC circuit I
vL(t)
vR(t)
vC(t) 20

21. Instantaneous PD 21

22. Phasor Diagram for RLC circuit II
vL(t)
vR(t)
vC(t)
vRLC(t)  22

23. Instantaneous PD as projection onto y-axis
v(t2)
vRLC(t)
v(t1) 23

24. Phase Angle  vRLC(t)  Phase Angle ( ) V - V tan f = V - - I X ILm Cm V Rm - - I X I X X X = m L m C = L C I R R
Phase Angle m æ ö X - X f =
tan - 1 ç L C ÷ è ø R 24

25. series ac-RLC graph 25

26. Impedance 26

27. Table of definitions 46

28. Impedance and reactance28

29. Generalized Ohm's Law. 29

30. Power Factor

31. Angular frequency dependence
Power and current depend on angular frequency of circuit
Angular frequency dependence

32. Max I ; Min Z ( ) e ( ) ( ) ( ) ( ) ( ) V w = = I w w Z Z w = + - Z R2 + - 2 Z R X X L C æ ö æ ö 2 w 2 - 2 1 LC 1 = R 2 + ç w L - ÷ = R 2 + ç ÷ è w ø è w C C ø ( ) Z w
is a minimum when w 2 LC - 1 =
which occurs when 1 w = w = LC Û = X X L C

33. ( ) ( ) Power as a function of  ( ) ( ) ( ) 1 1 V R w = w = P I R w 2m
ave m ( ) w 2 2 2 Z 2 2 1 V R 1 V R = m = m æ ö 2 2 2 1 2 L ( ) 2 2 + ç w - ÷ 2 + w 2 - w 2 R L R è w ø w 2 C 2 w 2 1 V R = m ( ) 2 2 2 w 2 + 2 w 2 - w 2 R L

34. Resonance
Circuit uses most power / current when it is in RESONANCE with applied frequency

35. Imax and Pave versus  Im
Pave   33

36. Quality of circuit
Width of Power curve is a measure of the QUALITY of the circuit
Small width - High Quality
Sharpness of response to external frequency

37. RC Filters
RC Filters I
Vout
Vin

38. Low Pass Filter

39. RC Filters
RC Filters II
Vout
Vin

40. High Pass Filter

41. Step up and Step down Transformers
Transformers I 44

42. Transformers II F d = - V N dt F d = - V N dt Fluxes are the same N =B 1 1 dt F d = - V N B 2 2 dt
Fluxes are the same N = V 2 V 2 1 N 1