Download AC Circuits Summary

AC Circuits Summary
Average values (root mean square)
Vrms  avg (V 2 ) 
Vmax
2
avg for sin2 or cos2 = 1/2
Transformers:
V2 N 2

V1 N1
Energy conservation
Power:
I1 V2 N 2


I 2 V1 N1
P1  P2
I1V1  I 2V2
Ohm’s Law: V = I Z Z = impedence, combined resistance and reactance.
Reactance, ω dependent:
Capacitors X C 
1
C
Inductors: X L   L
Phase difference means combination requires PythagoreanTheorem:
2
1 

Z  R  L 
in Ohms
C 

2
Phase difference also means that the amplitude of the power is less than the
(voltage amplitude) · (current amplitude); Prms  I rmsVrms cos( phase difference)
Filters:
Low Pass
High Pass
VC high at low ω
(RC > 1/ω to
charge)
VL high at high ω
(V ~ ΔI/Δt ~ ω)
Oscillator

1
LC
Approach 1: Remember the spring and mass oscillator.
When the mass is at the maximum or minimum spring extension, the stored
potential energy in the spring is a maximum and the mass is momentarily at
rest.
When the mass passes through the equilibrium position, the velocity is at
maximum and the potential energy is zero.
At max or min position, the spring exerts a force to move the mass.
At equilibrium position, the force is zero, but the momentum carries the
mass past that point to continue the oscillations.
For the LC circuit, the capacitor stores potential energy as charge builds up
on the plates. The inductor resists changes in the current, so it keeps
current flowing when the charge (and therefore the voltage) on the
capacitor is zero.
Energy
Force
Continue motion
Spring & mass
Potential, Kinetic
Spring
momentum
LC circuit
E field, B Field
Voltage on Capacitor
Inductor
Approach 2: Follow the change in electrical values.
Vc
+ max
+, decreasing
0
-, increasing
-max
Current
direction
changing
Current
direction
changing
-,decreasing
0
This is plotted below:
I
0
ΔI/Δt
+max
VL
-max
+ increasing
+max
+ decreasing
0
+ decreasing
0
- decreasing
-max
- decreasing
0
+ decreasing
+max
-increasing
-max
- decreasing
0
+decreasing
0
Notice that the voltage across the inductor is a quarter cycle ahead of the
current and the current is a quarter cycle ahead of the voltage on the
capacitor. When we used to use E for voltage, the mnemonic for
remembering this was “ELI the ICE man” {E leads I for L; I leads E for C}
Animation of current flow: http://www.falstad.com/circuit/index.html
Shows fields oscillating http://www.walter-fendt.de/ph11e/osccirc.htm
Circuit info. and animation http://www.allaboutcircuits.com/vol_2/chpt_6/5.html