Tesla’s Alternating Current

Dr. Bill Pezzaglia

Updated2014Mar10

Tesla & AC

A. AC Signals

B. Impedance

C. Resonance

D. References

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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 4

b. Peak Voltage 5

c. RMS Voltage 6

2. Phase and Power Factor 7

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 9

• Beats Demo: http://www.animations.physics.unsw.edu.au/jw/beats.htm#sounds

2c. (iii) FM Modulation 10

FM Demo http://www.youtube.com/watch?v=ens-sChK1F0

3. Rectifiers and Power Supplies 11

Half-Wave Rectifier

3b Full Wave Rectifier 12

3c DC Power Supplies 13

B. Impedance 14

1. Inductive Reactance

2. Capacitive Reactance

3. Impedance (Resistance+Reactance)

1. Inductors and Reactance 15

c. RL Circuit (AC Signal) 16- DETAILS -

d. RL as low pass filter 17

- DETAILS -

2. Capacitors 18

2c. RC Circuit (AC Signal) 19- DETAILS -

2d. RC Low Pass Filter 20

- DETAILS -

3. Impedance 21

C. Resonance 22

1. Oscillations

2. Natural Response (RLC Circuit)

3. Force Response (RLC Circuit)

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 23

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:

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).

00 2 fm

k

t

xv

kxt

vm

2. Natural Response 24

(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).

00 21

1

fLC

t

QI

QCt

IL

b. Damping 25

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.

220

1

cos)(

tAetx t

• 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).

2c. LRC Series Circuit 26

• 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

L

C

L

RQ

QL

R

RRRR sLd

0

0

1

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3. Forced Response (Driven Oscillations) 27

• 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!

3b Bandwidth 28

• 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:

• 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.

2

0

0

2

0

1

)(

f

f

f

fQ

VfV

f

fQ

fff

0

12

3b Bandpass and Bandreject Filter 29

• What we have here is a “bandpass filter”, that passes frequencies near the resonance, and blocks the frequencies out of range.

• Other configurations:

3c RLC in Parallel 30

• Band pass or band reject?

• Band pass or band reject?

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

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