AC circuits
Physics 122
11/2/12
Lecture XIX
1
LC circuit
•  Two forms of energy:
–  Electric – in a capacitor
–  Magnetic - in solenoid
•  Oscillator!
•  When energy in the
capacitance is at its maximum,
energy in the inductance is at
its minimum
•  No energy is lost, it is just
changing its form
CV 2 LI 2
U total =
+
= const
2
2
11/2/12
Lecture XIX
2
LC circuit
•  Total drop of voltage over a
closed circuit must be 0:
Q
dI 1
dI
0 = VL +VC = + L = ∫ I dt + L
C
dt C
dt
•  Take derivative over time:
0=
1
d 2I
I+L 2
C
dt
d 2I
1
=−
I
2
dt
LC
•  Second derivative of function
is proportional to the negative
function I = I 0 sin(ω t)
11/2/12
ω = 1 / LC
Lecture XIX
Q0
-Q0
3
1
LCR circuit
•  Resistors dissipate energy
–  Convert electrical energy
into thermal energy
•  Resistor acts like friction
for a weight on a spring
•  Damped oscillator!
11/2/12
Lecture XVIII
4
Inductance
•  Magnetic field in a solenoid
B∝I
•  It creates a magnetic flux through itself
dΦ dB dI
Φ = BA
∝
∝
dt
dt
dt
•  Changing magnetic flux generates emf (voltage gained) =
-drop of Voltage (voltage lost)
emf = −VL = − N
11/2/12
Lecture XIX
dΦ
dI
= −L
dt
dt
5
AC circuits
•  External source of alternating
voltage V(t)=V0sinωt – Forced
oscillator
•  Frequency f, cyclic frequency ω=2πf
•  Active elements
–  Inductors
–  Capacitors
•  Passive elements
–  Resistors
11/2/12
Lecture XIX
6
2
Phasor diagram
•  Graphical way to present current
and voltage in AC circuit
y
•  Present current and voltage in a
certain element (resistor,
I (t ) = I 0 ⋅ sin(ω t )
inductance, capacitance) as vectors
rotating in (x,y) plane
•  Vector length is equal to maximum
possible value of I or V: I0 and V0
•  Vector projection on y-axis is equal
to the value of current or voltage at
the moment t
11/2/12
I0
ωt
x
Lecture XIX
7
Current and voltage in AC circuit
y
I0
I (t ) = I 0 ⋅ sin(ω t )
VR (t ) = V0 sin(ω t ) Vo
ωt
x
V0 = RI 0
Drop of voltage over
resistor (V) follows I
Current and voltage in phase
11/2/12
Lecture XIX
8
Current and voltage in AC circuit
dI
dt
d sin(ω t )
VL = LI 0
= Lω I 0 cos(ω t )
dt
VL = Lω I 0 sin(ω t + 90 0 )
VL = L
V0 = X L I 0
X L = Lω
11/2/12
y
I0
I (t ) = I 0 ⋅ sin(ω t )
VR (t ) = V0 cos(ω t )
Vo
ωt
x
Lecture XIX
Inductor: current lags voltage
9
3
Current and voltage in AC circuit
Q 1
= ∫ I dt
C C
1
1
VC = ∫ I 0 sin(ω t)dt = −
I 0 cos(ω t)
C
ωC
y
1
I
(
t
)
=
I
⋅
sin(
ω
t)
0
V0 =
I 0 = XC I 0
ωC
V
VC =
o
XC =
1
ωC
I0
ωt
x
VR (t ) = −V0 cos(ω t )
Capacitor: voltage lags current
11/2/12
Lecture XIX
10
LCR circuit
y
I0
I (t ) = I 0 ⋅ sin(ω t )
VL
VR
ωt
VC
x
V0 R = I 0 R,
R = R,
Ohm’s law!!
V0 L = I 0 X L ,
X L = ω L,
V0C = I 0 X C
XC =




"V " = "VR "+"VL "+"VC " = ZI
1
ωC
Z − impedance
11/2/12
Lecture XIX
11
Impedance
I0
y
V0
VL
VR
VC
V0 =| Z | I 0
| Z |= R 2 + (X L − XC )2
tan φ = (X L − XC ) / R
11/2/12
x
This equation relates maximum
voltage to maximum current.
Mind, that they do not happen at the
same time. There is a phase shift
between Imax and Vmax -φ
Lecture XIX
12
4
Phase angle and power factor
X L − XC
R
R
cos ϕ =
Z
tan ϕ =
V0
VL
φ
I0
VR
VC
Vrms = ZI rms
 
P ="V ⋅ I " = Vrms I rms cos φ
x
Cosφ – power factor
NB. V and I are not real vectors, only in phasor diagram representation.
11/2/12
Lecture XIX
13
Energy in AC circuit
Total average power in AC
P = Vrms I rms cos φ
Power dissipated only in resistors:
2
P = I rms R
11/2/12
Lecture XIX
14
Resonance in LCR circuit
I rms
V
= rms
Z
Z = R 2 + ( X L − X C )2
• Largest current when impedance is the smallest
• XL, XC are functions of frequency
• At a certain frequency X = X
L
C
ωL=
ω=
11/2/12
1
ωC
1
LC
Lecture XIX
f =
1
2π LC
15
5