chapter 21: alternating current circuits and em waves resistors in an ac circuits homework...

Chapter 21: Alternating Current Circuits and EM Waves

Resistors in an AC CircuitsHomework assignment : 22,25,32,42,63

AC circuits• An AC circuit consists of combinations of circuit elements and an AC generator or an AC source, which provides the alternating current. The alternating current can be expressed by :

tVftVv sin2sin maxmax • The current and the voltage reach their maximum at the same time; they are said to be in phase.

resistor at thecurrent

instaneous theis where2 iRiP

• The power, which is energy dissipated in a resistor per unit time is:

Rms current and voltage

Resistors in an AC Circuits

• rms current

2max

2

2

1Ii av

maxmax 707.02

II

I rms

RIP rmsav2

maxmax 707.02

VV

Vrms

RIV rmsrmsR , RIVR maxmax,

voltage difference at a resitor

• rms voltage

Capacitors in an AC circuit

Capacitors in an AC Circuits

• Phase difference between iC and vC

The voltage across a capacitoralways lags the current by 90o.

When the direction of the currentis reversed, the amount of thecharge at the capacitor starts todecrease.

t

QI

VCQ

At t=0, there is no charge at thecapacitor and the current is freeto flow at i=Imax.Then the current starts to decrease.

Capacitors in an AC circuit

Capacitors in an AC Circuits

• Reactance and VC,rms vs. Irms

Reactance: impeding effect of a capacitor

CfCXC

1

2

1 capacitive

reactance

CrmsrmsC XIV ,

XC in f in HzC in F

Inductors in an AC circuit

Inductors in an AC Circuits

The changing current output of thegenerator produces a back emf thatimpedes the current in the circuit.The magnitude of the back emf is:

t

ILvL

• Phase difference between iL and vL

The effective resistance of coil in anAC circuit is measured by the inductivereactance XL:

LfLX L 2

LrmsrmsL XIV ,

inductive reactance

The voltage across an inductoralways leads the current by 90o.

a: iL/t maximum vL max.

b: iL/t zero vL zeroXL in f in HzL in H

A simple RLC series circuit

RLC Series Circuits

• Phase differences

• Current in the circuit

tIftIi sin2sin maxmax

The instantaneous voltage vR is in phase with the instantaneous current .

The instantaneous voltage vL leads the current by 90o.

The instantaneous voltage vC lags the current by 90o.


RLC Series Circuits

• Phasors

It is convenient to treat a voltageacross each element in a RLC circuitas a rotating vector (phasor) as shownin the phasor diagram on the right.

)sin()2sin( maxmax tVftVv

• Phasor diagram

22max )( CLR VVVV

R

CL

V

VV

tan

max,

max,max, ,

CC

RRLL

VV

VVVV


RLC Series Circuits

• Impedance22

max22

max )()( CLCLR XXRIVVVV

22 )( CL XXRZ impedance

ZIV maxmax in form of Ohm’s law

R

XX CL tan

Note that quantities withsubscript “max” is relatedwith those with “rms”, allthe results in this slide are also applicable to quantities with subscript “rms”

Impedances and phase angles

RLC Series Circuits

Filters : Example

22out

C

RV IR

R X

02 22 1

1

1

out

C

V R

R

0

1

RC

Ex.: C = 1 μf, R = 1Ω

High-pass filter

High-pass filter

0

0.2

0.4

0.6

0.8

1

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06 5.E+06 6.E+06

(Angular) frequency, omega

"tra

nsm

issi

on"

Note: this is ω,2

f

~Vout

RLC Series Circuits

Filters (cont’d)

~Vout

~

ω=0 No currentVout ≈ 0

ω=∞ Capacitor ~ wireVout ≈ ε

~Vout

ω = ∞ No currentVout ≈ 0

ω = 0 Inductor ~ wireVout ≈ ε

ω = 0 No current because of capacitor

ω = ∞ No current because of inductor

outV

0

outV

0

(Conceptual sketch only)

High-pass filter

Low-pass filter

Band-pass filter

outV

0

RLC Series Circuits

Power in an AC circuit

Power in an AC Circuit

• No power losses are associated with pure capacitors- When the current increases in one direction in an AC circuit, charge accumulates on the capacitor and the voltage drop appears across it.- At the maximum value of the voltage, the energy stored in the capacitor is:- When the current reverses direction, the charge leaves the capacitor to the voltage source and the stored energy decreases.- As long as there is no resistance, there is no energy loss.

2max )(

2

1VCPEC

• No power losses are associated with pure inductors- The source must do work against the back emf of an inductor that is carrying a current.- At the maximum value of the current, the energy stored in the inductor is:- When the current starts to decrease, the stored energy returns to the source as the inductor tries to maintain the current in the cuircuit.- As long as there is no resistance, there is no energy loss.

2max )(

2

1ILPEL

Power in an AC circuit (cont’d)

Power in an AC Circuit

The average power delivered by the generator is converted tointernal energy in the resistor. No power loss occurs in an idealcapacitor or inductor.

RIP rmsav2 Average power delivered to the resistor

rmsR IVR /

Rrmsav VIP rmsV

cosrmsR VV

cosrmsrmsav VIP

Resonance

Resonance in a Series RLC Circuit

22 )( CL

rmsrmsrms

XXR

V

Z

VI

• The current in a series RLC circuit

This current reaches the maximum when XL=XC (Z=R).

LC XX

CfLf

00 2

12

LCf

2

10

21(primary) (secondary)

~

NN

iron

V2V1

Transformers Transformers

• AC voltages can be stepped up or stepped down by the use of transformers.

The AC current in the primary circuitcreates a time-varying magnetic field in the iron.

• We assume that the entire flux produced by each turn of the primary is trapped in the iron.

This induces an emf on the secondarywindings due to the mutual inductance ofthe two sets of coils.

Transformers Ideal transformer without a load

1

1

N

V

tturn

No resistance losses All flux contained in iron Nothing connected on secondary

N2N1(primary) (secondary)

iron

V2V1

The primary circuit is just an AC voltagesource in series with an inductor. Thechange in flux produced in each turn is given by:

• The change in flux per turn in the secondary coil is the same as the change in flux per turn in the primary coil (ideal case). The induced voltage appearing across the secondary coil is given by:

11

222 V

N

N

tNV turn

• Therefore, •N2 > N1 -> secondary V2 is larger than primary V1 (step-up) •N1 > N2 -> secondary V2 is smaller than primary V1 (step-down)

• Note: “no load” means no current in secondary. The primary current, termed “the magnetizing current” is small!

Transformers Ideal transformer with a load

R

VI 2

2

21

21 I

N

NI

N2N1(primary) (secondary)

iron

V2V1 R

What happens when we connect a resistive load to the secondary coil?

Changing flux produced by primary coil inducesan emf in secondary which produces current I2

This current produces a flux in the secondary coilµ N2I2, which opposes the change in the originalflux -- Lenz’s law

This induced changing flux appears in the primarycircuit as well; the sense of it is to reduce the emf inthe primary, to “fight” the voltage source. However, V1 is assumed to be a voltage source. Therefore, theremust be an increased current I1 (supplied by the voltagesource) in the primary which produces a flux µ N1I1 which exactly cancels the flux produced by I2.

Transformers Ideal transformer with a load (cont’d)

Power is dissipated only in the load resistor R.

The primary circuit has to drive the resistance R

of the secondary.

Where did this power come from?It could come only from the voltage source in the

primary:

22

222

2dissipated IVR

VRIP

11generated IVP

2

1

1

11

2

1

2

2

1

N

N

V

VNN

V

V

I

I

2

1

21

1

22

1

221

N

N

R

V

N

N

R

V

N

NII

2211 IVIV

Maxell’s Equations Maxwell’s equations

enclenclE

B

encl

AdEdt

dI

dt

dIsdB

AdBdt

d

dt

dsdE

AdB

QAdE

)()(

0

0000

0

Gauss’s law

Gauss’s law for magnetism

Farady’s law

Ampere’s law

Oscillating electric dipole

First consider static electric field produced byan electric dipole as shown in Figs.(a) Positive (negative) charge at the top (bottom)(b) Negative (positive) charge at the top (bottom)Now then imagine these two charge are movingup and down and exchange their position at everyhalf-period. Then between the two cases there isa situation like as shown in Fig. below:

What is the electric filedin the blank area?

EM Waves by an Antenna

Oscillating electric dipole (cont’d)

Since we don’t assume that change propagate instantly once new positionis reached the blank represents what has to happen to the fields in meantime.We learned that E field lines can’t cross and they need to be continuous exceptat charges. Therefore a plausible guess is as shown in the right figure.



What actually happens to the fields based on a precise calculate is shown inFig. Magnetic fields are also formed. When there is electric current, magneticfield is produced. If the current is in a straight wire circular magnetic field isgenerated. Its magnitude is inversely proportional to the distance from thecurrent.


Oscillating electric dipole (cont’d)What actually happens to the fields based on a precise calculate is shown in Fig.


E

B

B is perpendicular to E


This is an animation of radiation of EM wave by an oscillating electric dipoleas a function of time.



A qualitative summary of the observation of this example is:

1) The E and B fields are always at right angles to each other.2) The propagation of the fields, i.e., their direction of travel away from the oscillating dipole, is perpendicular to the direction in which the fields point at any given position in space.3) In a location far from the dipole, the electric field appears to form closed loops which are not connected to either charge. This is, of course, always true for any B field. Thus, far from the dipole, we find that the E and B fields are traveling independent of the charges. They propagate away from the dipole and spread out through space.

In general it can be proved that accelerating electric charges give rise toelectromagnetic waves.


Dipole antenna


At a location far away from the source of the EM wave, the wave becomes plane wave.

++

--

--

++

V(t)=Vocos(t)

• time t=0 • time t=/one half cycle later

XBB

EM Waves by an Antenna Dipole antenna (cont’d)

++

--

x

z

y

Plane EM wave

Properties of EM Waves

y

x

z

Speed of light and EM wave in vacuum

m/s 1099792.21 8

00

c227

0 C/sN 104

)mN/(C 1085419.8 222120

cB

E Speed of light

Light is an EM wave!

EM wave in matter

Maxwell’s equations for inside matter change from those in vacuumby change 0 and 0 to = m0 and 0:

mm

c

00

11

For most of dielectrics the relative permeability m is close to 1 except forinsulating ferromagnetic materials :

mm

c

00

11

mnc

Index of refraction


Intensity of EM wave (average power per unit area)


EM waves carry energy.

0

maxmax

2BE

I intensity of the EM wave

00maxmaxmax / BcBE

2max

00

2max

22B

c

c

EI

Momentum carried by EM wave


Momentum carried by an EM wave: c

Up

Momentum transferred to an area : if the wave is completely absorbed : if the wave is completely reflected :

cUp /cUp /2

p=mv-(-mv)=2mv

Measurement of radiation pressure:

Spectrum of EM waves

Spectrum of EM Waves

fc

c : speed of light in vacuumf : frequency : wavelength

Doppler effect

Doppler Effect for EM Waves

cuc

uff SO

if 1

u : relative speed of the observer with respect to the sourcec : speed of light in vacuum

fo : observed frequency, fS : emitted frequency+ if the source and the observe are approaching each other- if the source and the observer are receding each other

receding approaching

A globular cluster

chapter 21: alternating current circuits and em waves resistors in an ac circuits homework...

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