Download AC_2014mar10

Tesla’s Alternating
Current
Dr. Bill Pezzaglia
Updated
2014Mar10
2
Tesla & AC
A. AC Signals
B. Impedance
C. Resonance
D. References
Nikola Tesla (1856-1943)
A. AC Signals
1) AC vs DC
2) Phase, Interference
3) Power Supplies
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1. AC vs DC
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b. Peak Voltage
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c. RMS Voltage
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2. Phase and Power Factor
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b. Power Factor
This is “real power”. Note: PGE gets upset with you if your
electric motors pull the current out of phase with the voltage.
They will charge you for this “reactive” power as well.
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2c. Interference
•
Beats Demo: http://www.animations.physics.unsw.edu.au/jw/beats.htm#sounds
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2c. (iii) FM Modulation
FM Demo http://www.youtube.com/watch?v=ens-sChK1F0
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3. Rectifiers and Power Supplies
Half-Wave Rectifier
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3b Full Wave Rectifier
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3c DC Power Supplies
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B. Impedance
1. Inductive Reactance
2. Capacitive Reactance
3. Impedance (Resistance+Reactance)
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1. Inductors and Reactance
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c. RL Circuit (AC Signal) - DETAILS -
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d. RL as low pass filter
- DETAILS -
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2. Capacitors
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2c. RC Circuit (AC Signal) - DETAILS -
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2d. RC Low Pass Filter
- DETAILS -
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3. Impedance
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C. Resonance
When a system at a stable equilibrium is displaced, it
will tend to oscillate. An Inductor combined with
Capacitor will tend to oscillate if hit with an electrical
impulse (i.e. a square wave).
1. Oscillations
2. Natural Response (RLC Circuit)
3. Force Response (RLC Circuit)
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1. Oscillations
The mechanical analogy would be a mass on a spring.
If you hit the mass, it will oscillate.
The equation of motion for that system would be:
v
m
 kx
t
x
v
t
k
0 
 2 f 0
m
where “k” is the Spring constant, “m” the mass, “v” the
velocity, “x” the displacement and “0” is the resonant
angular frequency (radians per second) as opposed to
the cyclic frequency “f” (units of Hertz or cycles per
second).
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2. Natural Response
(a) LC Circuit
• Lord Kelvin in 1853 first provided a good theory
of the electrical oscillations of RLC circuits. The
role of the spring (potential energy) is played by
the capacitor, as it stores charge. The role of
kinetic energy is played by the inductor, as it has
“inertia” of flowing charge (current).
• Hence replace the mechanical quantities by their
corresponding electrical ones (Voltage plays role
of Force, Inductance as mass, 1/C as spring
constant, charge “Q” as displacement, current “I”
as velocity):
• The square wave signal generator “shocks” the
system regularly when the voltage abruptly
changes polarity. The system responds by
oscillating (imagine hitting a pendulum with a
hammer, it will respond by oscillating).
I 1
L
 Q
t C
Q
I
t
1
0 
 2 f 0
LC
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b. Damping
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All real systems will not oscillate forever as energy
will be dissipated due to friction. Oscillations tend to
die out over time exponentially. At the right, the
oscillation period is 1 second, with a time constant of
=2.5 seconds for the decay.
• The constant “A” is the initial amplitude of the
oscillation.
• The constant  is called the decay constant
or damping constant (the inverse of the time
constant) with units of inverse time.
• Note that the presence of damping makes the
oscillating frequency  to be less than the
resonant frequency 0.
• If the friction in the system is higher 
increases (system decays faster). If   0
then oscillations will not occur (critical
damping).
x(t )  Aet cost 
1


  0 2   2
2c. LRC Series Circuit
• In an electrical circuit, the analogy of friction is
resistance. There is always resistance in a circuit
(e.g. resistance in connecting wires)
• Real inductors often have significant resistance
(RL) because they contain many meters of wire in
their coils.
• The signal generator as well has some resistance
inside it (Rs is approximately 50 ohms). In our
circuit, Rd represents variable decade resistor.
• In the formulas, the resistance “R” represents the
total resistance in the circuit.
• The damping constant  can be expressed in
terms of the “Q” or “Quality factor”, which in turn
is a measure of the average energy stored in the
system divided by the energy lost per cycle. A
system with a big Q will oscillate for a long time
(relative to its period of oscillation).
R  Rd  RL  Rs
R 0


2 L 2Q
Q
1 L L0

R C
R
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3. Forced Response (Driven Oscillations)
• Now lets use a sine wave to force the system to
oscillate at the frequency we set. When you are
close to resonance, the system will oscillate at its
maximum amplitude.
• The output (voltage across the resistor) will be
the same frequency of the input.
• As you move either above or below resonance,
however the amplitude will drop off.
• At resonance, the voltage across the inductor (or
capacitor) will be a factor of “Q” times bigger than
across the resistor!
• However, the voltage across the combination of
inductor plus capacitor will be ZERO! This is
because their voltages are 180 out of phase!
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3b Bandwidth
• The “fatness” (i.e. the “bandwidth”) of the curve is
described by the “Q” factor. For a big Q the
resonance is very narrow. For a low Q value its
quite broad. The exact shape of the curve is
quite messy:
V( f ) 
V0
 f
f 
1  Q 2   0 
f 
 f0
2
• to determine the Q factor from a resonant curve,
you determine the frequencies below (f1) and
above (f2) where the amplitude drops to 70% of
the value at resonance (aka as the “half power
points”).
• The “bandwidth” f is defined to be the
difference of these frequencies, and it is simply
related to the Q factor.
f  f 2  f1
f0
Q
f
3b Bandpass and Bandreject Filter
• What we have here is a “bandpass filter”, that
passes frequencies near the resonance, and
blocks the frequencies out of range.
• Other configurations:
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3c RLC in Parallel
• Band pass or band reject?
• Band pass or band reject?
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References
•War on Currents: http://en.wikipedia.org/wiki/War_of_Currents
•Spark gap http://en.wikipedia.org/wiki/Spark-gap_transmitter
•http://en.wikipedia.org/wiki/Spark_gap
•http://en.wikipedia.org/wiki/Tesla_coil
•http://en.wikipedia.org/wiki/Nikola_Tesla
•http://hyperphysics.phy-astr.gsu.edu/hbase/electric/serres.html#c1
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Things to Do
•FM Movie (1960) http://www.youtube.com/watch?v=gfz1FbIOMbs