Download Reactance and Impedance in AC Circuits

Alternating-Current Circuits
Physics 2212
Alternating current
is commonly used
everyday in homes
and businesses
throughout the
word to power
various electrical
appliances
Voltage in AC Circuits
v  Vmax sin t
2
  2f 
T
The voltage supplied by an AC source is sinusoidal with a period T
Resistors in AC Circuits
v  iR R  0
iR 
v Vmax

sin t  I m ax sin t
R
R
Vmax
I m ax 
R
RMS Current & RMS Voltage
• Average value of current is referred as RMS Current &
average value of voltage is referred as RMS Voltage
• The notation rms stands for root-mean-square, which
means the square root of the mean (average) value
I rms  (i 2 ) avg
i2 
I rms
Pavg
1 2
I max
2
I
 m ax  0.707 I m ax
2
 I 2 rms R
Vrms 
Vm ax
2
 0.707Vm ax
Inductors in an AC Circuit
v  vL  0
Instantaneous current and voltage are out of phase by /2
diL
v  L
0
dt
Vmax
diL
I m ax 
v  L
 Vmax sin t
L
dt
Vmax
L is defined as inductive reactance XL
diL 
sin tdt
L
Vmax
V
X L  L
iL 
sin tdt   max cos t

L
t
Vmax

iL 
sin(t  )
t
2
Capacitors in an AC Circuit
v  vC  0
q
0
C
q  C Vmax sin t
v 
Instantaneous current and voltage are out of phase by /2
dq
ic 
 C Vmax cos t
dt

ic  C Vmax sin(t  )
2
Im ax  CVmax 
Vmax
(1/ C )
(1/C) is defined as capacitive reactance XC
X C  1/ C
The RLC Series Circuit
vR  I m ax R sin t  VR sin t

vL  I m ax X L sin(t  )  VL cos t
2

vC  I m ax X C sin(t  )  Vc cos t
2
Vmax  VR 2  (VL  VC ) 2  (I m ax R) 2  (I m ax X L  I m ax X C ) 2
Vmax  I m ax R 2  ( X L  X C ) 2
I m ax 
I m ax 
Vmax
R 2  ( X L  X C )2
Vmax
Z
Z is called impendence of circuit
Resonance in a Series RLC Circuit
I rms 
Vrms
Z
I rms 
Vrms
R 2  ( X L  X C )2
The angular frequency at which XL-XC=0 is called the resonance frequency of the circuit
X L  XC  0
X L  XC
0 L  1/ 0C
0 
1
LC