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

2. Mutual Inductance AC Generator Changing current in P
Primary
Coil
Secondary
AC Generator
Changing current in P
Changing B-field thru P
Changing B-field thru S
Changing  thru S
S proportional to IP:
“Mutual Inductance”
Demo 67
Induced EMF (voltage) in S
Recall Faraday’s law:
Demo 10

3. Self Inductance – Single Coil
AC Generator
Changing current
Changing B-field
Changing 
 proportional to I:
“Inductance”
Induced EMF (voltage)
Recall Faraday’s law:
Direction
Given by Lenz’s Law
Opposes change in current
Units: [L] = [] [t] / [I]
1 H = 1V-sec/amp 13

4. Physical Inductor Inductor resists current change! Energy stored:
Recall: =NBA A l
Recall: B=monI
Derivation is blank for them to fill in. Energy Eqn. is also blank.
(# turns) = (# turns/meter) x (# meters)
N = n l
Energy stored:
U = ½ LI2 16

5. ACT
Compare the inductance of two solenoids, which are identical except solenoid 2 has twice as many turns as solenoid 1.
1) L2=L1 2) L2 = 2 L1 3) L2 = 4 L1 18

6. 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

7. 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
Example
2pf t = 8pt
f = 4 Hz
T=(1/4)seconds/cycle
Ave V = 0
Ave V^2: Vmax/2
0.25
0.5
RMS: Root Mean Square Vrms=Vmax/√2 23

8. 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 23

9. 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
72% correct
85% and 78% correct respectively
Just like Vrms=Vmax/sqrt(2)…
Irms=Imax/sqrt(2)
=10/√2 A = 7.07 A
60% correct 26

10. 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! 29

11. 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 33

12. 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 36

13. ACT/Preflight 12.4, 12.5 The capacitor can be ignored when…
(a) frequency is very large
(b) frequency is very small w XC
very large w gives very small XC
“can be ignored” means “behaves like a wire”…no voltage drop
The inductor can be ignored when…
(a) frequency is very large
(b) frequency is very small w XL
very small w gives very small XL 42

14. Generators in AC circuitL R C
Generators in AC circuit
VG + VL + VR + VC = 0 always true
- VG,max = ImaxZ
- impedance
- Voltage across generators sometimes leads and
sometimes lags current…depends on XL – XC
…see next week’s discussion of phasors.

15. Example: 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.
VR,max = Imax R
Example 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
ACT: 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
Stays same: R doesn’t depend on f
Decreases: XC = 1/(2pfC)
Increases: XL = 2pf L 50

16. 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 I VR
VC = ImaxXC sin(2pft-p/2)
VC lags I t VL
VL = ImaxXL sin(2pft+p/2)
VL leads I VC 1

17. Time Dependence in AC CircuitsL R C
Time Dependence in AC Circuits
Write down Kirchoff’s Loop Equation:
VG + VL + VR + VC = 0 at every instant of time I t VL VC VR
However …
VG,max  VL,max+VR,max+VC,max
Maximum reached at different times for R,L,C 5

18. 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

19. 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

20. See you next lecture.