Eddy Currents l Suppose I have a pendulum with a copper plate on the end that swings through a magnetic field l Do I expect anything to happen? u there’s a change in magnetic flux through the copper as it swings into the magnetic field, so an emf will be induced u and copper is a good conductor u eddy currents u which direction? u where would it be useful to use eddy current braking?
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Eddy Currentsl Suppose I have a pendulum
with a copper plate on the endthat swings through amagnetic field
l Do I expect anything tohappen?
u there’s a change inmagnetic flux through thecopper as it swings intothe magnetic field, so anemf will be induced
u and copper is a goodconductor
u eddy currentsu which direction?u where would it be useful to
use eddy current braking?
Search on eddy current braking on google
l Ignore links to:u Mary Baker Eddy
u Fish’s Eddy
lIn addition to these generator brakes, the braking function of the vehicle is assured by the modular eddy-current brakes. The individual eddy-currentbraking magnets act on the guidance rails of the guideway and guarantee the braking of the vehicle.
Energy stored in inductor L
As I increases, I is flowingfrom + to – side of L, soenergy is being extractedfrom circuit (battery).Thus inductor is gainingenergy that is being storedin the B field.
e
+
-
R
LI
+
_
2L IL
21
U ⋅=Energy store in inductor
Energy store in capacitor 2C VC
21
U D⋅=
“Discharging” an inductor
e
+
-
R
LI
“Discharging” an inductor
e
+
-
R
LI
RL
I
“Discharging” an inductor
e
+
-
R
LI
R
LI
I will decay asUL dissipated asheat in R
0tI
<DD
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“Discharging” an inductor
e
+
-
R
LI
R
LI
I will decay asUL dissipated asheat in R
0tI
<DD
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+
_
R
LI
“Discharging” an inductor
e
+
-
R
LI
R
LI
I will decay asUL dissipated asheat in R
0tI
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\
+
_
R
LI
Lenz’s law says that inductor willhave an emf across itself to opposethe reduction of I.
“Discharging” an inductor
I
t
t-e= /te
RI
RL
=t
e
+
-
R
LI
Zap!
e
+
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R
LI
Zap!
e
+-
RS
L
e
+
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R
LI
Zap!
e
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RS
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0 ¨ I
e
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Zap!
e
+-
RS
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0 ¨ I
large very tI
DD
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Zap!
tI
LL DD
=e
e
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0 ¨ I +
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large very tI
DD
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Zap!
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spark
Back to our simple circuit
l The battery is the first sourceof emf that we’ve come across
l The circuit on the right has anelectric current I=e/R that isconstant with time
l And we remember that anelectric current really consistsof the electrons in theconductor moving with aspeed vd (a few tenths of amm/sec) opposite thedirection of the electric field
A new type of circuit: an AC circuit
e = emax sin wtemax = N B A w
This is somethingnew. An emf thatchanges with time.not only in magni-tude but in direction.If I turn the generator at
a rate of 60 hz, then f is 60 hz
An electric field that changes direction
l As the emf traces out itssinusoidal path, the electricfield inside the conductor isconstantly changingmagnitude and direction
l This means the electrons arealso changing direction
u 120 times a second for a60 hz current
l We thought vd was small, butnow they’re really not gettinganywhere because they’reconstantly changing direction
l Yet an AC current must stillaccomplish something useful
u remember that the electronsdidn’t have to travel all of theway to the end of theconductor to light a light-bulb
E
Current and voltagel Consider the current and
voltage in the simple circuitshown to the right
l The voltage produced by thegenerator (or electrical outlet)appears across the resistor
u DvR = Dvl The current in the resistor is
the same as the current in thebattery
u iR = Imax sin 2pftl Note that I’m using small
letters for quantities that varywith time, and capital lettersfor quantities that are constant
u I’m going to keep doing that
Current and voltagel Let’s consider something
obviousu whenever the voltage is zero,
the current is zerou whenever the voltage is at a
maximum the current is at amaximum
u whenever the voltagechanges direction, so doesthe current
u I say that the current andvoltage are in phase witheach other
l Why consider the obvious?u because the above points
won’t be true whenever westart to add other circuitelements (such as capacitorsand inductors) into the mix
Current and voltage
l Consider the AC currentgoing through theresistor as shown
l Does the resistor gethot? Yes.
u the electrons are stillcolliding with the atoms inthe resistor no matterwhich way they’re going.
l Does the resistor carethat the electrons can’tmake up their mindswhich way to go? No.
l Can AC currents do justas useful stuff as DCcurrents? Yes.
Fig. 21.3, p.654
Current and powerWhat’s the average current? iav=0
What’s the average power? P=i2R Pav=(i2)avR (i2)av=1/2I2
max
rms current
l (i2)av=1/2I2max
l Note that an AC currentof Imax is not the same asa DC current of Imax
l Let me define a newquantity, the rms current
u The rms current is thedirect current that woulddissipate the same amountof energy in a resistor asdissipated by the actualalternating current
u Irms=sqrt[(i2)av] = Imax/sqrt(2)u P = I2rmsR
l I can also define otherrms quantities
u Vrms = Vmax/sqrt(2)
So when we say that there’s an outlet voltage of 120 V, that refersto the rms valueVpeak = 170 V
Notation
Ohm’s law still applies, to the relevant quantitiesDv = i RDVmax = Imax RDVrms = Irms R
Fig. 21.4, p.656
And now for something completely different
l An AC circuit with acapacitor
Note that the current and thevoltage across the capacitor are no longer in phase.
How can we understand this behavior?
And now for something completely different
l I can’t have any voltageacross the capacitor until Iaccumulate some charge
l And for that, the current has toflow from some time
l When the current goes to 0,the voltage is at a maximum
l When the current reversesdirection, the capacitor startsto discharge
l And eventually will charge upin the reverse direction
…so the voltage across the capacitoralways lags behind the current by 90o
Capacitive reactancel Note that the presence of a
capacitor in the circuitimpedes the flow of current,just as the presence of aresistor would
l We define the impedance fora capacitor as
u XC = 1/(2pfC)l Something new; a frequency
dependent impedancel Capacitors like high
frequencies; don’t like lowfrequencies
l At high frequencies, less timefor charge to accumulate onthe capacitor; less oppositionto current flow
Version of Ohm’s lawDVC,rms = Irms XC
Fig. 21.6, p.657
We’re on a roll
l An AC circuit with aninductor
Now the current and voltage are out of phase but in a different way.
How can we understand thisbehavior?
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