lc circuit - university of rochesterregina/phy122/l19.pdf3 11/2/12 lecture xix 7 phasor diagram •...
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11/2/12 Lecture XIX 1
Physics 122
AC circuits
11/2/12 Lecture XIX 2
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
constLICVUtotal =+=22
22
11/2/12 Lecture XIX 3
LC circuit • Total drop of voltage over a
closed circuit must be 0:
• Take derivative over time:
• Second derivative of function is proportional to the negative function
0 =VL +VC =QC+ L dI
dt=1C
I dt∫ + L dIdt
0 = 1CI + L d
2Idt2
d 2Idt2
= −1LC
I
I = I0 sin(ωt)
ω =1/ LC
Q0 -Q0
2
11/2/12 Lecture XVIII 4
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 XIX 5
• Magnetic field in a solenoid
• It creates a magnetic flux through itself
• Changing magnetic flux generates emf (voltage gained) = -drop of Voltage (voltage lost)
Inductance
IB ∝
dtdI
dtdB
dtdBA ∝∝Φ
=Φ
dtdIL
dtdNVemf L −=Φ
−=−=
11/2/12 Lecture XIX 6
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
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11/2/12 Lecture XIX 7
Phasor diagram • Graphical way to present current
and voltage in AC circuit • Present current and voltage in a
certain element (resistor, 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
tωx
y I0 )sin()( 0 tItI ω⋅=
11/2/12 Lecture XIX 8
Current and voltage in AC circuit
00 RIV =
Drop of voltage over resistor (V) follows I
Current and voltage in phase
tωx
y I0 )sin()( 0 tItI ω⋅=
Vo )sin()( 0 tVtVR ω=
11/2/12 Lecture XIX 9
tωx
y I0 )sin()( 0 tItI ω⋅=
Current and voltage in AC circuit
ω
ωω
ωωω
LXIXV
tILV
tILdt
tdLIV
dtdILV
L
L
L
L
L
=
=
+=
==
=
00
00
00
)90sin(
)cos()sin(
Vo )cos()( 0 tVtVR ω=
Inductor: current lags voltage
4
11/2/12 Lecture XIX 10
Current and voltage in AC circuit
VC =QC=1C
I dt∫
VC =1C
I0 sin(ω t)dt =∫ −1ωC
I0 cos(ω t)
V0 =1ωC
I0 = XCI0
XC =1ωC
Capacitor: voltage lags current
tωx
y I0 )sin()( 0 tItI ω⋅=
Vo
)cos()( 0 tVtVR ω−=
11/2/12 Lecture XIX 11
LCR circuit
V0R = I0R, V0L = I0XL, V0C = I0XC
R = R, XL =ω L, XC =1ωC
"V "= "
VR "+"
VL "+"
VC "= ZI
Z − impedance
tωx
y
I0 )sin()( 0 tItI ω⋅=
VR VL
VC
Ohm’s law!!
11/2/12 Lecture XIX 12
Impedance
V0 =| Z | I0| Z |= R2 + (XL − XC )
2
tanφ = (XL − XC ) / R
y
x
I0
VR VL
VC
V0
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
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11/2/12 Lecture XIX 13
Phase angle and power factor
ZRRXX CL
=
−=
ϕ
ϕ
cos
tan
φcos"" rmsrms
rmsrms
IVIVP
ZIV
=⋅=
=
Cosφ – power factor NB. V and I are not real vectors, only in phasor diagram representation.
x
I0
VR VL
VC
V0
φ
11/2/12 Lecture XIX 14
Energy in AC circuit Total average power in AC Power dissipated only in resistors:
RIP rms2=
φcosrmsrmsIVP =
11/2/12 Lecture XIX 15
Resonance in LCR circuit
LCf
π21
=
22 )( CL
rmsrms
XXRZZVI
−+=
=
• Largest current when impedance is the smallest • XL, XC are functions of frequency • At a certain frequency
LC
CL
XX CL
1
1
=
=
=
ω
ωω