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Chapter 33
Alternating Current Circuits
1
Capacitor
V
 
Resistor
Q  CV
V  I R
I

V
R


V

Inductance
dI
Vab  L
dt
2
AC power source
 The AC power source provides an alternative voltage, v (t)
 Notation
- Lower case symbols will indicate instantaneous values
- Capital letters will indicate fixed values
• The output of an AC power source is sinusoidal
Δv = ΔVmax sin ωt
• Δv is the instantaneous voltage
• ΔVmax is the maximum output voltage of the source
• ω is the angular frequency of the AC voltage
3
AC voltage
v  Vmax cos (ωt  φ)
• The angular frequency is
2π
ω 2π 
T
– ƒ is the frequency of the source
– T is the period of the source
• The voltage is positive during one half of
the cycle and negative during the other half
• The current in any circuit driven by an AC
source is an alternating current that varies
sinusoidally with time
• Commercial electric power plants in the US
use a frequency of 60 Hz
4
Resistor in AC circuit
• Consider a circuit consisting of an
AC source and a resistor
• The AC source is symbolized by
• Δv = ΔvR = ΔVmaxsin wt
• ΔvR is the instantaneous voltage
across the resistor
• The instantaneous current in the
resistor is
Vmax
v
iR  R 
sin ωt  I max sin ωt
R
R
5
Resistor in AC circuit
v  v R  Vmax sin(ωt)
iR 
v R Vmax

sin ωt  I max sin ωt
R
R
 The current and the voltage are in phase
 Resistors behave essentially the same way
in both DC and AC circuits
6
Resistor in AC circuit: Phasor diagram
v  v R  Vmax sin(ωt)
iR 
v R Vmax

sin ωt  I max sin ωt
R
R
 A phasor is a vector whose length is proportional to the
maximum value of the variable it represents
 The vector rotates at an angular speed equal to the angular
frequency associated with the variable
 The projection of the phasor onto the vertical axis represents
the instantaneous value of the quantity it represents
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rms current and voltage
iR  I max sin ωt
• The average current in one cycle is zero
• rms stands for root mean square
1/ 2
1

I rms    i R2 dt 
T 0

T
 1 2

I max
 2π
2π

0
1/ 2
1 2

2
  I max
sin
(
ω
t
)
dt


T
0


T
1/ 2

sin 2 (τdτ
) 


I max
2
 0.707 I max
• Alternating voltages can also be
discussed in terms of rms values
Vrms
Vmax

 0707
.
Vmax
2
8
rms current and voltage: power
• The rate at which electrical energy is dissipated in the
circuit is given by
P=i2R
• where i is the instantaneous current
• The average power delivered to a resistor that carries
an alternating current is
Pav  I
2
rms
R
9
Inductors in AC circuit
v  v L  0 , or
di
v  L  0
dt
di
v  L  Vmax sin ωt
dt
Vmax
Vmax
iL 
sin ωt dt  
cos ωt

L
ωL
Vmax
Vmax
π

iL 
sin  ωt  
I max 
ωL
2
ωL

This shows that the instantaneous current iL in the inductor and the
instantaneous voltage ΔvL across the inductor are out of phase by
(π/ 2) rad = 90o.
10
Inductors in AC circuit
v  Vmax sin ωt
π

i L  Imax sin  ωt  
2

Vmax
I max 
ωL
11
Inductors in AC circuit
v  Vmax sin ωt
π

i L  Imax sin  ωt  
2

I max 
Vmax
ωL
• The phasors are at 90o with
respect to each other
• This represents the phase
difference between the
current and voltage
• Specifically, the current lags
behind the voltage by 90o
12
Inductors in AC circuit
v  Vmax sin ωt
π

i L  Imax sin  ωt  
2

I max 
Vmax
ωL
• The factor ωL has the same units as resistance and is
related to current and voltage in the same way as resistance
• The factor is the inductive reactance and is given by:
XL = ωL
– As the frequency increases, the inductive reactance increases
I max
Vmax

XL
13
Capacitors in AC circuit
Δv + Δvc = 0
and so
Δv = ΔvC = ΔVmax sin ωt
– Δvc is the instantaneous voltage
across the capacitor
• The charge is
q = CΔvC =CΔVmax sin ωt
• The instantaneous current is given by
dq
iC 
 ωC Vmax cos ωt
dt
π

iC  ωC Vmax sin  ωt  
2

• The current is (π/2) rad = 90o out of phase with the voltage14
Capacitors in AC circuit
vC  Vmax sin ωt
π

iC  ωC Vmax sin  ωt  
2

15
Capacitors in AC circuit
vC  Vmax sinωt
π

iC  ωC Vmax sin  ωt  
2

• The phasor diagram shows that for
a sinusoidally applied voltage, the
current always leads the voltage
across a capacitor by 90o
– This is equivalent to saying the
voltage lags the current
16
Capacitors in AC circuit
vC  Vmax sinωt
π

iC  ωC Vmax sin  ωt  
2

• The maximum current
Imax  ωC Vmax
Vmax

(1 / ωC )
• The impeding effect of a capacitor on the current in an
AC circuit is called the capacitive reactance and is
given by
XC 
1
ωC
and Imax 
Vmax
XC
17
v  Vmax sinωt
v  Vmax sinωt
vC  Vmax sinωt
iL  Imax sinωt
π

i L  Imax sin  ωt  
2

π

iC  Imax sin  ωt  
2

I max 
Vmax
R
I max 
Vmax
ωL

Vmax
XL
Imax 
Vmax
Vmax

(1 / ωC )
XC
18
RLC series circuit
• The instantaneous voltage would
be given by
Δv = ΔVmax sin ωt
• The instantaneous current would be
given by
i = Imax sin (ωt - φ)
– φ is the phase angle between
the current and the applied
voltage
• Since the elements are in series,
the current at all points in the circuit
has the same amplitude and phase
19
RLC series circuit
• The instantaneous voltage
across the resistor is in phase
with the current
• The instantaneous voltage
across the inductor leads the
current by 90°
• The instantaneous voltage
across the capacitor lags the
current by 90°
20
RLC series circuit
• The instantaneous voltage across each of the three
circuit elements can be expressed as
v R  Imax R sin ωt  VR sin ωt
v L  Imax
π

X L sin  ωt    VL cos ωt
2

vC  Imax
π

XC sin  ωt    VC cos ωt
2

21
RLC series circuit
v R  Imax R sin ωt  VR sin ωt
v L  Imax
π

X L sin  ωt    VL cos ωt
2

vC  Imax
π

XC sin  ωt    VC cos ωt
2

• In series, voltages add and the instantaneous voltage
across all three elements would be
Δv = ΔvR + ΔvL + ΔvC
– Easier to use the phasor diagrams
22
RLC series circuit
i  Imax sin ωt
v R  Imax R sin ωt  VR sin ωt
v L  Imax
π

X L sin  ωt    VL cos ωt
2

vC  Imax
π

XC sin  ωt    VC cos ωt
2

v  v R  v L  vC 
 VR sin ωt  VL cos ωt  VC cos ωt 
 Vmax sin (ωt  φ)
Easier to use the phasor diagrams
23
RLC series circuit
The phasors for the individual elements:
• The individual phasor diagrams can
be combined
• Here a single phasor Imax is used to
represent the current in each element
– In series, the current is the same
in each element
24
RLC series circuit
• Vector addition is used to combine the
voltage phasors
• ΔVL and ΔVC are in opposite
directions, so they can be combined
• Their resultant is perpendicular to ΔVR
25
RLC series circuit
• From the vector diagram, ΔVmax can be calculated

Vmax  V  VL  VC
2
R
)
2
 ( Imax R )   Imax X L  Imax XC )
2
2
Vmax  Imax R   XL  XC )
2
2
26
RLC series circuit
Vmax  Imax R   XL  XC )
2
2
• The current in an RLC circuit is
I max 
Vmax
R 2   X L  XC )
2

Vmax
Z
• Z is called the impedance of the circuit and it plays
the role of resistance in the circuit, where
Z  R   X L  XC )
2
2
27
RLC series circuit
Imax
Vmax

Z
Z  R   X L  XC )
2
2
impedance triangle
28
RLC series circuit: impedance triangle
Z  R   X L  XC )
2
2
• The impedance triangle can also be used to
find the phase angle, φ
 X L  XC 
φ tan 

R


• The phase angle can be positive or negative
and determines the nature of the circuit
1
• Also, cos φ =
i  Imax sin ωt
R
Z
v  Vmax sin (ωt  φ)
29
RLC series circuit
Z  R   X L  XC )
2
2
 X L  XC 
φ tan 

R


1
30
Power in AC circuit
Imax
Vmax

Z
Irms 
Imax
2
• The average power delivered by the generator is
converted to internal energy in the resistor
– Pav = ½ Imax ΔVmax cos φ = IrmsΔVrms cos φ
– cos φ is called the power factor of the circuit
• We can also find the average power in terms of R
2
2
Vmax
1 2
1  Vmax 
R
2
Pav  I rms R  I max R  
R

2
2 Z 
2 R 2   X L  X C )2
31
Resonances in AC circuit
2
2
Vmax
R Vmax
R
Pav 

2
2 Z
2 R 2   X L  X C )2
• Resonance in Pav (ω) occurs at the frequency ωo
where the current has its maximum value
• To achieve maximum current, the impedance must
have a minimum value
– This occurs when XL = XC or
X L  ωL
0  XC 
1
ωC
0
– Solving for the frequency gives
ωo  1
LC
• The resonance frequency also corresponds to
the natural frequency of oscillation of an LC circuit
32
Resonances in AC circuit
2
2
2
Vmax
Vmax
R Vmax
R
Pav 


2 Z2
2 R 2   X L  X C )2
2
ωo  1
LC
R
1 

R   ωL 
ωC 

2
2
2
Vmax
Pav (ω0 ) 
2R
• Resonance occurs at the same
frequency regardless of the value of R
• As R decreases, the curve becomes
narrower and taller
• Theoretically, if R = 0 the current would
be infinite at resonance
– Real circuits always have some
resistance
33