1. Physics 102: Lecture 12
AC Circuits L R C 1

2. Review: Generators and EMF
Voltage across generator: 1
= w A B sin(q)
= w A B sin(wt)
= Vmax sin(wt) w • q v v r 2 x e
Vmax
-Vmax
Frequency = How fast its spinning
Amplitude = Maximum voltage t 20

3. AC Source Example V(t) = Vmax sin(t)=Vmax sin(2pf t)
Vmax = maximum voltage
f = frequency (cycles/second)
V(t) = 24 sin(8p t)
+24
-24
2pf t = 8pt
f = 4 Hz
T=(1/4)seconds/cycle
0.25
0.5
Ave V = 0
Ave V^2: Vmax/2
RMS: Root Mean Square Vrms=Vmax/√2

4. RMS? V(t) = Vmax sin(2pf t) RMS: Root Mean Square Vrms=Vmax/√2 +Vmax
Ave V = 0
Ave V^2: Vmax/2
square Root:
Vmax / √2
RMS: Root Mean Square Vrms=Vmax/√2

5. Preflight 12.1, 12.2 I(t) = 10 sin(377 t) Find Imax Find Irms L R C
Well… We know that the maximum value
sine is 1. So the maximum current is 10!
Imax = 10 A
85% and 78% correct respectively
Just like Vrms=Vmax/√2 …
Irms=Imax/√2
=10/√2 A = 7.07 A

6. Resistors in AC circuit
VR = I R always true – Ohm’s Law
VR,max = ImaxR R
Voltage across resistor is “IN PHASE” with current.
VR goes up and down at the same times as I does. I t VR
Frequency
Resistance (R)
Frequency does not affect Resistance!

7. Capacitors in AC circuit
VC = Q/C always true
VC,max = ImaxXC
Capacitive Reactance: XC = 1/(2pfC) C
Voltage across capacitor “LAGS” current.
VC goes up and down just after I does. I t
Frequency
Reactance (XC)
Frequency does affect Reactance!
Changing voltage tries to charge/discharge capacitor.
Low frequency limit: open circuit
High frequency limit: closed circuit (capacitor never has time to build up charge)
Lag: charge (voltage drop) lags behind current. t VC

8. Inductors in AC circuit
VL = -L(DI)/(Dt) always true
VL,max = ImaxXL
Inductive Reactance: XL = 2pfL L
Voltage across inductor “LEADS” current.
VL goes up and down just before I does. I t
Frequency
Reactance (XL)
Frequency does affect Reactance!
Low frequency: no voltage drop…nothing is changing
High frequency: large rate of change of current implies large emf (as per Faraday)
Voltage leads current. (rate is initially large, current then grows as rate drops) t VL

9. ACT/Preflight 12.4, 12.5 The capacitor can be ignored when…
(a) frequency is very large
(b) frequency is very small
The inductor can be ignored when…
(a) frequency is very large
(b) frequency is very small
“can be ignored” means “behaves like a wire”…no voltage drop

10. AC Circuit Voltages Example
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH has a current I(t) = 0.5 sin(8p t) amps. Calculate the maximum voltage across R, C, and L.
VR,max = Imax R L R C
= 0.5  2 = 1 Volt
VC,max = Imax XC
= 0.5  1/(8p0.015) = 1.33 Volts
VL,max = Imax XL
= 0.5  8p0.03 = 0.38 Volts

11. ACT: AC Circuit Voltages
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH has a current I(t) = 0.5 sin(8p t) amps. Calculate the maximum voltage across R, C, and L. L R C
Now the frequency is increased so I(t) = 0.5 sin(16p t).
Which element’s maximum voltage decreases?
1) VR,max
2) VC,max
3) VL,max

12. Summary so far… I = Imaxsin(2pft) VR = ImaxR sin(2pft)L R C
I = Imaxsin(2pft)
VR = ImaxR sin(2pft)
VR in phase with I VR I
VC = ImaxXC sin(2pft–p/2)
VC lags I t VL VC
VL = ImaxXL sin(2pft+p/2)
VL leads I 1

13. Kirchhoff: generator voltage
Vgen
Write down Kirchhoff’s Loop Equation:
Vgen(t) = VL(t) + VR(t) + VC(t) at every instant of time I t VL VC VR
However …
Vgen,max  VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C
We solve this using phasors

14. a A reminder about sines and cosines q+p/2 q q-p/2y q
q+p/2
q-p/2 a
Recall: y coordinates of endpoints are
asin(q + p/2)
asin(q)
asin(q – p/2) x
whole system rotates, theta=2pi*ft 1

15. Graphical representation of voltages
q+p/2
ImaxXL
I = Imaxsin(2pft) (q = 2pft)
VL = ImaxXL sin(2pft + p/2)
VR = ImaxR sin(2pft)
VC = ImaxXC sin(2pft – p/2) L R C q
ImaxR
q-p/2
ImaxXC 1

16. Phasor Diagrams: A Detailed Example
I = Imaxsin(2pft)
VR = VR,maxsin(2pft)
t = 1 f=1/12
2pft = p/6
VR,max
VR,maxsin(p/6)
p/6
Phasor Animation
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

17. Phasor Diagrams I = Imaxsin(p/3) VR = VR,maxsin(p/3) t = 2 2pft = p/3
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

18. Phasor Diagrams I = Imaxsin(p/2) VR = VR,maxsin(p/2) t = 3 2pft = p/2
VR,maxsin(p/2)=V0
p/2
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

19. Phasor Diagrams I = Imaxsin(4p/6) VR = VR,maxsin(4p/6) t = 4
2pft = 4p/6
VR,max
VR,maxsin(4p/6)
4p/6
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

20. Phasor Diagrams I = Imaxsin(p) VR = VR,maxsin(p) t = 6 2pft = p
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

21. Phasor Diagrams I = Imaxsin(8p/6) VR = VR,maxsin(8p/6) t = 8
2pft = 8p/6
8p/6
VR,max
VR,maxsin(8p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

22. Phasor Diagrams I = Imaxsin(10p/6) VR = VR,maxsin(10p/6) t = 10
2pft = 10p/6
10p/6
VR,maxsin(10p/6)
VR,max
Length of vector = Vmax across that component
Vertical component = instantaneous value of V

23. AC circuit summary Kirchoff’s Loop Equation always holds true:
Vgen = VL + VR + VC
However, Vgen,max  VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C I VR L R C
VR in phase with I
VC lags I
VL leads I t VL VC
Phasors represent instantaneous voltages 1