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


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)


i>Clicker question


(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


Alternating Current Generators(N = 2 for this coil)

B NBAcos(t)

B

t NBA sin(t)

max NBA


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


Power Dissipated in a Resistor

Average value

Peak value

Pmax Imax2 R

Pave 12

Iave2 R

P I 2R Imax2 Rsin2t


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


Standard Alternating Voltage in the US

+max

-max

maxmax

2 60

170 1202rms

f f Hz

V V V


i>Clicker question


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?


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


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


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.


Phasors for R

• V in phase with I

VR RIR Vmax sint

IR Vmax

Rsint


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


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


Inductors in AC Circuits (Phasors for L)

V leads I by 90

VL LIL

tVmax sint

IL Vmax

Lsin t

2

Vmax

Lcost


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


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


Summary

Symbol Reactance XC

L LR RIR In phase with VR

IC current leads VC

IL current lags VL

1C


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


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.


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)


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


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.


UB vs UE in LC curcuit:


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


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.


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


Documents

lec14-AC Circuits, RC, RL, LC