lec14-AC Circuits, RC, RL, LC
Transcript of lec14-AC Circuits, RC, RL, LC
PHYS 221 General Physics II
Spring 2015 Assigned Reading:
Lecture
AC Circuits: RC, RL, LC
22.1 – 22.5 14
0 20 40 60 80 1000
20
40
60
80
100
Cou
nt
percent
mean = 66.67%Mean: 66.7
Exam with your answers can be found on CHIP
Exam 1 results:
RL Circuits
Initially, an inductor acts to oppose changes in current through it. A long time later, it acts like an ordinary connecting wire.
Phys 221 Spring 2014 Lecture 14 3
RL Circuits (EMF on)
I VR
1 eRt/L VR
1 et/ RL
VL VeRt/L Vet /RL
Current
Max I = V/R
63% Max at t=RL=L/R
Voltage on L
Max VL= V/R
37% Max at t=RL=L/R
Phys 221 Spring 2014 Lecture 14 4
RL Circuits
RI
a
b
L
I• Why does RL increase for larger L?
• Why does RL decrease for larger R?
L opposes change in current & slows down the rate of change
Large R decreases final current “easier charge up goal”
Large R dissipates energy quicker, speeds up “discharge of inductor” (speeds up current loss)
Phys 221 Spring 2014 Lecture 14 5
i>Clicker question
Phys 221 Spring 2014 Lecture 14 6
(a) I = 0 (b) I = V/2R (c) I¥ = 2V/R
At t=0 the switch is thrown from position b to position a in the circuit shown:
What is the value of the current I a long time after the switch is thrown?
a
b
R
L
II
R
Sources
Phys 221 Spring 2014 Lecture 14 7
Alternating Current Generators(N = 2 for this coil)
B NBAcos(t)
B
t NBA sin(t)
max NBA
Phys 221 Spring 2014 Lecture 14 8
Alternating Current in a Resistor
VR Vmax sin(t)
I VR
R
Vmax sin(t)R
I Imax sin(t)
=2f measured in rad/s
Phys 221 Spring 2014 Lecture 14 9
Power Dissipated in a Resistor
Average value
Peak value
Pmax Imax2 R
Pave 12
Iave2 R
P I 2R Imax2 Rsin2t
Phys 221 Spring 2014 Lecture 14 10
Root-mean-square (rms) values
This now looks like the DC case !!!!
This now looks like the DC case !!!!
Pave Imax2 R sin2t
ave 1
2Imax
2 R
Irms I 2 ave Imax sint 2
ave
12
Imax2
Irms 12
Imax 0.707 Imax Once we define Irms Pav Irms2 R
Power delivered by the generator:
Pave I av max sint Imax sint
av
max Imax sin2t av
Define: rms 12max 0.707 max Pave
12max Imax rmsIrms
Then : Irms VR, rms
R
Phys 221 Spring 2014 Lecture 14 11
Standard Alternating Voltage in the US
+max
-max
maxmax
2 60
170 1202rms
f f Hz
V V V
Phys 221 Spring 2014 Lecture 14 12
i>Clicker question
Phys 221 Spring 2014 Lecture 14 13
What is the maximum value of an AC voltage whose rms value is 100 V?
(A) zero
(B) 70.7
(C) 141
How “Standard” is 120 VAC?
Phys 221 Spring 2014 Lecture 14 14
AC Power Distribution• AC power can travel at high voltages
and low amps, therefore smaller power loss
• Tesla liked 60 Hz and 240 V• Standard in Europe was defined by a
German company AEG ( monopoly) who chose 50Hz (20% less efficient in generation, 10-15% less efficient in transmission)
• Originally Europe was also 110V, but they changed it to reduce power loss and voltage drop for the same copper diameter
Nikola Tesla
http://www.teslasociety.com/Phys 221 Spring 2014 Lecture 14 15
The Power Grid155,000-765,00 V
<10,000V
Phys 221 Spring 2014 Lecture 14 16
Using rms values: summary
• Using rms values of current and voltage allows you to use the familiar dc formulas, such as V = IR and P = I2 R.
• One ac ampere is said to flow in a circuit if it produces the same joule heating as one ampere of dc current under the same conditions.
• At your house the peak voltage will be 170 V
Phys 221 Spring 2014 Lecture 14 17
Phasors• A phasor is a “vector” whose magnitude is the maximum value
of a quantity (eg V or I) and which rotates counterclockwise in a 2-d plane with angular velocity
• Recall uniform circular motion:The projections of r (on the vertical y axis) execute sinusoidal oscillation.
x
y y
y r sintx r cost
Angular speed:phasors rotate counter clockwise about the origin with an angular speed of .
Length: represents the amplitude of the AC quantityProjection:on the vertical axis represents the value of the AC quantity at time t.Rotation angle: phase of the AC quantity at time t.
Phys 221 Spring 2014 Lecture 14 18
Phasors for R
• V in phase with I
VR RIR Vmax sint
IR Vmax
Rsint
Phys 221 Spring 2014 Lecture 14 19
Capacitors in AC Circuits (Phasors for C)
• V lags I by 90
VC QC Vmax sint
IC Cm sin t 2
Cm cos t
Q CVmax sint
I Qt
Phys 221 Spring 2014 Lecture 14 20
Relationship between Irms & VC,rms
Irms VC , rms
XC
where XC 1C
is the capacitive reactance
1. XC is similar to R in Irms VR, rms
R.
2. SI unit for XC : (ohm)3. Average power delivered to a capacitor in an ac circuit is zero.
, max max , max maxcos sin cos sin
0 for a capacitorC C C
av
P V I V t I t V I t t
P
Phys 221 Spring 2014 Lecture 14 21
Inductors in AC Circuits (Phasors for L)
V leads I by 90
VL LIL
tVmax sint
IL Vmax
Lsin t
2
Vmax
Lcost
Phys 221 Spring 2014 Lecture 14 22
Relationship between Irms & Vrms
Irms VL , rms
XL
where XL L is the inductive reactance.
1. XL is similar to R in Irms VR, rms
R.
2. SI unit for XL : (ohm)3. Average power delivered to an inductor in an ac circuit is zero.
, ,cos sin cos sin
0 for an inductorL L peak peak L peak peak
av
P V I V t I t V I t t
P
Phys 221 Spring 2014 Lecture 14 23
Complex Circuit: Phasor Diagram
• All the elements are in series, so the current is the same through each one
• All the current phasors have the same orientation
• Resistor: current and voltage are in phase
• Capacitor and inductor: current and voltage are 90° out of phase, in opposite directions
Phys 221 Spring 2014 Lecture 14 24
Summary
Symbol Reactance XC
L LR RIR In phase with VR
IC current leads VC
IL current lags VL
1C
Phys 221 Spring 2014 Lecture 14 25
Impedances for L, C, R
R is resistance
XL = L is inductive Reactance
For high , XL grows large and L acts like an open switch.For low , XL grows small and at DC, L acts like a
conducting wire.
is capacitive reactance
For high , XC goes to zero, C acts like a wire. For low , XC grows larger and at DC, C acts like an open
switch
XC 1C
Phys 221 Spring 2014 Lecture 14 26
Summary
I
ωt
ω
t
I
0 VR
VR
Iωt
ω0
I
VCVC
Iωt
ωI
0
VL
VL
R is resistance, in Ohms
is capacitive reactance, in Ohms
XL = ωL is inductiveReactance, in Ohms
VR RIR Vmax sint
IR Vmax
Rsint
VC QC Vmax sint
IC Cm cos t XC
1C
VL LIL
tVmax sint
IL Vmax
Lsin t
2
LC Circuit
• After t = 0, the charge moves from one capacitor plate to the other and current passes through the inductor.
• Eventually, the charge on each capacitor plate falls to zero.• The inductor opposes change in the current, so the induced emf now
acts to maintain the current at a non-zero value.• This current continues to transport charge from one capacitor plate to
the other, causing the capacitor’s charge and voltage to reverse sign.• The charge on the capacitor returns to its original value.
Phys 221 Spring 2014 Lecture 14 29
LC Circuit
• The voltage and current in the circuit oscillate between positive and negative values
• The circuit behaves as a simple harmonic oscillator– Charge: q = qmax cos (2πƒt)– Current: I = Imax sin (2πƒt)
Phys 221 Spring 2014 Lecture 14 30
Energy in a LC Circuit
• For the capacitor,
• For the inductor,
• The energy oscillates back and forth between the capacitor and its electric field and the inductor and its magnetic field.
• The total energy must remain constant
maxcap
qqPE cos ƒtC C
22
21 1 22 2
ind maxPE LI LI sin ƒt 2 2 21 1 22 2
Imax 1LC
qmax
Phys 221 Spring 2014 Lecture 14 31
Energy Conservation
• Capacitors and inductors store energy.– A capacitor stores energy
in its electric field and depends on the charge.
– An inductor stores energy in its magnetic field and depends on the current.
• As the charge and current oscillate, the energies stored also oscillate but the total energy is conserved.
Phys 221 Spring 2014 Lecture 14 32
UB vs UE in LC curcuit:
Phys 221 Spring 2014 Lecture 14 33
LC Circuit Oscillation
Kirchoff’s loop ruleLC
+ +- -
I
QVC VL 0
• In an LC circuit, the instantaneous voltage across the capacitor and inductor are always equal
• Therefore, |VC| = |I XC| = |VL| = |I XL| – Simplifying, XC = XL
• This frequency is the resonant frequency
f ƒ0 1
2 L C
Phys 221 Spring 2014 Lecture 14 34
RC, LR, and LC Circuits
0 1LC
1 / RC 1 / RC
1 / LR R / L
RC Circuit: Charging Rate
LR Circuit: Decay Rate
LC Circuit: Oscillation Freq.
Phys 221 Spring 2014 Lecture 14 35
i>Clicker question
(A) ω2 = 1/2 ω0 (B) ω2 = ω0 (C) ω2 = 2ω0
At t = 0 the capacitor has charge Q0; the resulting oscillations have frequency ω0. The maximum current in the circuit during these oscillations has value I0.
What is the relation between ω0and ω2, the frequency of oscillations when the initial charge = 2Q0?
LC
+ +- -
Q Q= 0
t = 0
Phys 221 Spring 2014 Lecture 14 36