Download Lecture Notes Y F Chapter 31

Alternating Current (AC) Circuits
DC Electromotive Force
AC Electromotive Force
v = Constant
v = V0 cos(ωt)
Lower case is instantaneous value
Upper case is Maximum value or Amplitude
v = V0 cos(ωt)
AC Average Values
v = V0 cos(ωt)
2π
1
v=
V0 ∫ cos( x)dx = sin( 2π ) + sin(0) = 0
2π 0
Average
Average rectified
vr =
π
2
2
V0 ∫ cos( x)dx = sin(π ) − sin(0) = V0 = 0.64V0
2
π
π
2
0
“Root Mean Square” - Square Root of the Average value of the Square
v
2
rms
=
1
π
V
∫
π
2
0
cos ( x)dx
2
but
0
π
2
vrms
cos(2 x) = cos 2 ( x) − sin 2 ( x)
1
cos 2 ( x) = [1 − cos(2 x)]
2
π
V02 ⎡ 1 1
V02 π V02
V02 π V02
⎤
cos(2 x)dx =
+
=
=
− cos(2 x)⎥ dx =
∫
∫
⎢
π 0 ⎣2 2
π 2 2π 0
π 2 2
⎦
2
vrms
V02
=
2
vrms =
V0
≈ 0.707V0
2
V
V0 = 170V
Vrms=120V
Vpeak-peak = 340V
Representation of Simple Harmonic Motion through Circular Motion
x = A sin (ωt)
y = A cos (ωt)
Uniform Circular Motion
x = A sin (ωt)
Simple Harmonic Motion
Representation of Sinusoidal Motion Using Rotation of a “Phasor”
Phasor for Pure Resistance
i = I R cos(ωt )
vR = iR = I R R cos(ωt ) = VR cos(ωt )
Voltage and Current are “in Phase”
Phasor for Capacitor
dq
= I cos(ωt )
i=
dt
q = ∫ idt = I ∫ cos(ωt )dt =
I
ω
sin(ωt )
q
I
=
sin(ωt )
C ωC
I
vc =
cos(ωt − 90°)
ωC
vc =
Voltage and Current are “Put of Phase”
Current “leads” Voltage
Phasor for Inductor
dq
= I cos(ωt )
dt
d
di
vL = L = L I cos(ωt ) = − LIω sin(ωt )
dt
dt
vL = LIω cos(ωt + 90°)
i=
Voltage and Current are “Put of Phase”
Current “lags” Voltage
Resistance and Reactance
For a Resistor:
iR = I R cos(ωt )
vR = iR r = I R r cos(ωt ) = VR cos(ωt )
VR = I R R
For a Capacitor: iC = I C cos(ωt )
IC
cos(ωt − 90°) = VC cos(ωt − 90°)
ωC
1
VC =
IC
ωC
1
X
=
VC = I C X C
Define C
as the Capacitative “Reactance”
ωC
vc =
For an Inductor:
iL = I L cos(ωt )
vL = I L Lω cos(ωt + 90°) = VL cos(ωt + 90°)
VL = I L L ω
VL = I L X L
Define X C = Lω as the Inductive “Reactance”
Frequency Dependence of Resistance and Reactance
Putting it all together – The “Driven” L-R-C Circuit
i = I cos(ωt )
v
What is voltage?
The Instantaneous Current is the Same for all Elements,
The Voltages will NOT be in phase for all elements
V 2 = VR2 + (VL − VC )
2
V 2 = IR 2 + (IX L − IX C )
2
I ⎞
⎛
2
2
V = IR + ⎜ IωL −
⎟
ωC ⎠
⎝
2
I ⎞
⎛
V = I R 2 + ⎜ ωL −
⎟
ω
C
⎠
⎝
V = IZ
Z is defined as the “Impedance” of the Circuit
Impedance and Resonance
Impedance of L-R-C circuit is a minimum when XL=XC
ωC = ωL
ω = 1 LC
1
1/LC is the “Resonant” Frequency of the Circuit
AC Power in a Resistor
Instantaneous Power :
P = iv = [I cos(ωt )][V cos(ωt )]
P = IV cos 2 (ωt )
Average Power :
1
2
IV
cos
(ωt )dt
∫
T
1
⎧1
⎫
P = ∫ IV ⎨ [1 − cos(2 x)]⎬dt
T
⎩2
⎭
1 1
1
P = IV ∫ dt − IV ∫ cos(2 x)dt
T 2
T
IV
I V
P=
=
= I rmsVrms
2
2 2
P=
Average Power in a Resistor is the product of The RMS Voltage and Current
Power in AC Circuits
Power in arbitrary AC Circuits
Example: The Light Dimmer
Example: The Light Dimmer
1
P = IV cos ϕ
2
i
vR
φ
vC
120V AC
12 DC
Step 1. Reduce the 120V AC to lower voltage AC using a transformer
Step 2. “Rectify” the AC by cutting out the negative voltage
Step 3. Filter the AC using an RC Circuit
The Transformer
dΦ B2
and
dt
dΦ B1 dΦ B 2 dΦ
=
=
Iron Core serves to “confine” magnetic flux
dt
dt
dt
ε 1 = − N1
dΦ B1
dt
ε 1 N1
=
ε 2 N2
ε 2 = −N2
Vsec ondary
V primary
=
N sec ondary
N primary
“Rectified” AC
“Bridge” Rectifier
“Filtered” AC
RC >>
1
ω