Alternating Current
Another way to express: Phasors
Time dependence of voltage & current (e.g. generator)
ω always counter
clockwise
Ö it is sinusoidal
Ö periodic reversals
Ö wall socket in the United States 60 cycles per second
Root mean square (rms)
Goal: Describe a sinusoidal quantity with a “average value”.
Calculating a average current IAV gives always zero because of the
symmetric sinusoidal function, so we use the root mean square value.
1. square the function
x = ….
(e.g. I2 = (Imax sin(ωt))2
V=Vmax sin(ωt)
ω = angular frequency (=2πf)
f= frequency
I = V/R = Vmax/R sin(ωt)
I = Imax sin(ωt)
2. xAV2= ½ xmax2
3. xrms= √(xAV)2 = xmax /√2
When both, voltage and current intercept the time axis at the same
time we say “Current and voltage are in phase”
Example: Power
P = I2R = (Imax sin(ωt))2 R
PAV = ½ Imax² R = Irms2 R
Other relationships: Irms= Vrms/R ; Imax= Vmax/R
1
2
RC Circuits
We know so far
Vmax,R= ImaxR
Vmax,C= ImaxXC
The total voltage is not the sum
like for two resistors, because
Vmax,R and Vmax,C are not in
phase.
Capacitors in AC
The relation between rms current and
capacitance can be calculated using
calculus
Irms= ωCVrms or Irms= Vrms / XC
Capacitive Reactance XC= 1/ωC [Ω, Ohm]
Resistors have the same resistance at all frequencies of voltage – a
capacitor does not
Ö depends on frequency
high frequency Irms small
low frequency Irms high
Vmax = Vmax,R + Vmax,C = I max,R R 2 + I max,C X C2 = I max R 2 + X C2
2
Z = R 2 + X C2
2
3
2
Impedance in a RC circuit
 1 
Z = R2 + 

 ωC 
Current and voltage are not in phase.
“The voltage lags the current by 90° ” (ωt = π/2 = 90° )
Power (P = I V) has an average value of zero
2
2
4
Angle between Imax and
Vmax is called the phase
angle
Inductors in AC circuits
If a resistance only limited the
current a tremendous current
would flow when we when a
high voltage was applied
I R R
cos φ = max,R =
I max,C Z Z
Both, the magnitude and
the phase angle between
the voltages are important
for the power consume
Same story:
Vmax = Imax XL
With calculus we get XL = ωL [Ω, Ohm] (Inductive Reactance)
Also here: Resistors have the same resistance at all frequencies of
voltage – A inductor does not
PAV = I2rms R
PAV = Irms(Vrms/Z) R
PAV = IrmsVrmscosφ
Voltage leads the current by
90°
PAV= IrmsVrms cosφ
PAV= IrmsVrms cos90°= 0!
5
6
RLC Cirquits
RL - circuits
Vmax = Vmax,R + (Vmax,L − Vmax,C )2 = I max,R R2 + (I max,L X L − I max,C X C )2
2
Vmax = I max R2 + ( X L − X C )2
Vmax = Vmax,R + Vmax,L = I max,R R 2 + I max,L X L2 = I max R 2 + X L2
2
2
Z = R 2 + X L2
2
2
Z = R2 + ( X L − X C ) 2
Impedance in a RL circuit
1 

Z = R2 + ωL − 
ωL 

Z = R 2 + (ωL )
2
cos φ =
2
I max,R R R
=
I max,L Z Z
Impedance in a RLCcircuit
2
Phaseangleφ
I (X − XC ) X L − XC
tanφ = max L
=
I max,L R
R
Powerangle cosφ
R
cosφ =
Z
7
8
Resonance
LC circuit without a generator but a charged capacitor is an oscillator
(e.g. Pendulum)
If nothing dissipates energy (like friction) the oscillation would be
forever.
Natural frequency
Mass-spring ω = k / m
1
LC circuit ω =
= 2πf
LC
Resonance
Magnitude of oscillation
becomes maximum at the
natural frequency.
Value of R does not
change the natural
frequency
Application:
Radio, Television
9