3/5/07 U.W. Physics 104, Spring 2007 1
Lecture 12: Inductors, AC
Power, Circuits
• AC Power
– Transformers
– Transmission
• Inductors
– RL Circuits
– LCR Circuits• Impedance
• Resonance
3/5/07 U.W. Physics 104, Spring 2007 2
Generators
vv
•
x
!"
r
Flux is decreasing at moment shown
3/5/07 U.W. Physics 104, Spring 2007 3
Generators and EMF
#1 = B L v sin(!)
v = "rvv
•
x
!"
r=w/2
#1 = "r B L sin(!) = #
2
# = #1
+ #2
= 2"r B L sin(!)
# = A B " sin(!)
# = A B " sin("t) #t
1
2
"AB
$"AB
Two of the sides (width w) do not contribute
but two with lengths (L) do
N loops: # = NAB " sin("t)
3/5/07 U.W. Physics 104, Spring 2007 4
Direct versus Alternating Current
• Chemical power
– Battery
– Constant voltagedrop
– Constant current
• Circular motion
– Generator
– Sinusoidal timevariation
3/5/07 U.W. Physics 104, Spring 2007 5
Sinusoidal Pattern
• Circular motion
– Sinusoidalpattern
– Frequency, f
– Period, T = 1 / f
– Goes through 0twice in oneperiod
3/5/07 U.W. Physics 104, Spring 2007 6
AC Power
• Voltage, current and power depend on time
• We are interested in averages (f = 60Hz)
3/5/07 U.W. Physics 104, Spring 2007 7
AC Source
V(t) = V0 sin(2(f t)
– V0 = maximum voltage
– f = frequency (cycles/second)
• V(t) = 24 sin(8( t)
v
t
+24
-240.25 0.52(f t = 8(t
f = 4 Hz
(1/4) seconds/cycle
RMS: Root Mean Square Vrms=Vmax/)23/5/07 U.W. Physics 104, Spring 2007 8
Power Transmission
A generator produces 1.2 Giga watts of power, which it transmits to a
town 7 miles away through power lines with a total resistance 10-5
ohms. How much power is lost in the lines if the energy is
transmitted at 120 Volts?
P = IV Power delivered by generator through lines
I = P/V = 1.2x109 W/120 V = 107 Amps in lines!
P = I2R Power lost in lines
= 1014 10-5 = 1 Giga Watt Lost in Lines!
Large current is the problem. Since P=IV, use high
voltage and low current to deliver power.
If V = 12,000 Volts, only 1 Kilo Watt will be lost!
3/5/07 U.W. Physics 104, Spring 2007 9
Transformers
• Key to Modern electrical system
• Produce from 120 volts AC
– Arbitrarily small voltages.
– Arbitrarily large voltages.
• Nearly 100% efficient
3/5/07 U.W. Physics 104, Spring 2007 10
Transformers
Increasing current in primary createsan increase in flux throughprimary and secondary.
%#
(primary) (secondary)
iron
VsVp
Same &'/&t
Energy conservation: IpVp = IsVs
R
3/5/07 U.W. Physics 104, Spring 2007 11
Transformers Question?
(primary) (secondary)
iron
VsVp R
A 12 Volt battery is connected to a transformertransformer that has a 100 turn primary coil, and200 turn secondary coil. What is the voltageacross the secondary after the battery has beenconnected for a long time?
1) Vs = 0 2) Vs = 6 3) Vs = 12 4) Vs = 24
Transformers depend on a
change in flux so they only
work for alternating
currents!
3/5/07 U.W. Physics 104, Spring 2007 12
RL Circuit• Upon turn-on current rise is inhibited by
L, but eventually rises to V/R
• Upon turn-off current fall to zero is
slowed down by presence of L
– Time constant, * = L/R
3/5/07 U.W. Physics 104, Spring 2007 13
Resistors in AC circuit• Ohm’s Law Vr = I R always true.
– Vmax = ImaxR
I
t
vr
t
R
Frequency
Re
sis
tan
ce
– Voltage across resistor is “IN PHASE” with
current through the resistor.
3/5/07 U.W. Physics 104, Spring 2007 14
Capacitors in AC circuit• Vc = Q/C always true.
– However, voltage & current are not constant
– Vmax = ImaxXc
– Capacitive Impedance: XC = 1/(2(fC)
vC
t
I
t
C
Frequency
Imp
ed
an
ce
– When capacitor is uncharged there is no voltage across it
– When capacitor is charged, no current flows
– Voltage across capacitor
“Lags” current.
3/5/07 U.W. Physics 104, Spring 2007 15
Inductors in AC circuit
I
t
vL
t
L
• VL = -L(&I)/(&t) always true.
– Vmax = ImaxXL
– Inductive Impedance: XL = 2(f L
Frequency
Imp
ed
an
ce
– Voltage across inductor
“Leads” current.
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Impedance
R is independent of f
R
3/5/07 U.W. Physics 104, Spring 2007 17
Impedance
R is independent of f
XL increases with f
R
XL
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Impedance
XC increases
with 1/f
RXL
XC
R is independent of f
XL increases with f
3/5/07 U.W. Physics 104, Spring 2007 19
ImpedanceR is independent of f
XL increases with f
XC increases
with 1/f
Z: XL and XC subtractR
XL XC
Z
f0
3/5/07 U.W. Physics 104, Spring 2007 20
I0(XL-XC)
Impedance Triangle
• Voltage across source must
equal sum of voltage across all of
the elements at all times.
• VL leads by 90o and VC lags by
90o, and are 180o out of phase
and they tend to cancel
I0 XL=VL
I0 XC=VC
I0R
V generator=I 0
Z
• V0 (t) = VR (t) +VC (t) +VL (t)
• Impedance Z = sqrt(R2 +(XL – XC)2)
• Peak V0 = I0 Z
• tan(+) = (XL – XC)/R
+
3/5/07 U.W. Physics 104, Spring 2007 21
Generators in AC circuit
• VG+VL+VR+VC = 0 always true.
– Vmax = Imax Z
– Total Impedance: Z = (R2+ (XL2- XC
2))1/2
– Voltage across generators sometimes leads and
sometimes lags current.
– When impedance is
smallest, i.e., XL = XC,
power is largest
L
R
C
Frequency
Po
we
r
f0
3/5/07 U.W. Physics 104, Spring 2007 22
Power
• The voltage generator supplies power,and the resistor dissipates it.
• Capacitor and Inductor store andrelease energy.
• P = IV varies with values of R, C, L.
– Purely resistive R=Z, only dissipation in R
– For R!Z, energy stored/extracted from LC
• Average power, P = Irms Vrms cos(+)where cos(+)=R/Z
3/5/07 U.W. Physics 104, Spring 2007 23
AC Summary
Resistors: VR=IR In phase
Capacitors: VC=I XC: Xc = 1/(2(f C) Lags
Inductors: VL=I XL: XL = 2(f L Leads
Generator V0 = IZ: Z=sqrt(R2 +(XL-XC)2)
Can lead or lag tan(+) = (XL-XC)/R
Power is only dissipated in resistor:
P=IV0 cos(+)
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Preflight 1• A power outlet puts out 60 Hz AC. Which of the
following statements is true
– Voltage goes to zero only 60 times per second
– Current goes to zero only 60 times per second
– Power output goes to zero only 60 times per s
– Voltage , current and power output stay constant
– Voltage, current and power go to zero 120 times per second
One period is a complete 2(revolution - sine or cosine go
to zero twice, so does sin2.
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Preflight 2• Household electric outlet puts out 120 V 60 Hz AC
that is used to power a light bulb. The peak voltage
difference across the bulb is:
– 84 V
– 120 V
– 170 V
– Depends on the resistance of the bulb.
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Preflight 3• Power is transferred from the power plant to your
house through high voltage power lines because:
– Generators at power plants operate at high voltages.
– It will decrease power loss
– The power company wants to discourage people from
climbing on the lines
3/5/07 U.W. Physics 104, Spring 2007 27
Preflight: LCR 1An alternating current circuit consists of a resistor and
capacitor and inductor connected in series to a generator.
Which of the following statements is true.
– As the frequency of the generator becomes infinitely
large, the capacitor can be ignored
– As the frequency of the generator is decreased to zero,
the capacitor can be ignored.
L
R
C
The current flows through the
capacitor as it is being charged
and discharged. If the charging
and discharging are happening at
rapid rate, the current is freely
flowing through the capacitor -
equivalent to a wire!
3/5/07 U.W. Physics 104, Spring 2007 28
Preflight: LCR 2An alternating current circuit consists of a resistor and
capacitor and inductor connected in series to a generator.
Which of the following statements is true.
– As the frequency of the generator becomes infinitely
large, the inductor can be ignored
– As the frequency of the generator is decreased to zero,
the inductor can be ignored.
L
R
C
The current flow through the
inductor is inhibited due to self
inductance which is proportional
to the rate of change of current. If
the current flow is almost
constant the inductor simply acts
like a wire!
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