Electromagnetism - Australian National...
Transcript of Electromagnetism - Australian National...
Electromagnetism
1 ENGN6521 / ENGN4521: Embedded Wireless
Radio Spectrum use for Communications
2 ENGN6521 / ENGN4521: Embedded Wireless
3 ENGN6521 / ENGN4521: Embedded Wireless
Electromagnetism IGauss’s law and Dielectrics
4 ENGN6521 / ENGN4521: Embedded Wireless
Topics
➤ Surface and line integrals
➤ Maxwells equations (DIV and CURL)
➤ Charge and current
➤ Electric and magnetic field
➤ Gauss’ law for E and B
➤ Dielectrics
➤ Ohm’s law
5 ENGN6521 / ENGN4521: Embedded Wireless
The surface integral
‹
AE.dA =
‹
A|E||dA| cos θ (1)
6 ENGN6521 / ENGN4521: Embedded Wireless
The line integral
‰
CE.dl =
‰
C|E||dl| cos θ (2)
7 ENGN6521 / ENGN4521: Embedded Wireless
Maxwell’s equations in integral form
‹
SE · dS =
Q
ǫo[Gauss] (3)
‹
SB · dS = 0 [Gauss] (4)
‰
E · dl = −∂
∂t
ˆ
SB · dS [Faraday] (5)
‰
B · dl = µo
ˆ
Sj · dS+ µoǫo
∂
∂t
ˆ
SE · dS [Ampere] (6)
8 ENGN6521 / ENGN4521: Embedded Wireless
Maxwell’s Equations in differential form
9 ENGN6521 / ENGN4521: Embedded Wireless
Gauss’ Law
For the electric field‹
SE · dS =
Q
ǫo(7)
For the magnetic field‹
SB · dS = 0 (8)
where ǫo is the permittivity of free-space (Farads per m)
10 ENGN6521 / ENGN4521: Embedded Wireless
Charge: Q (positive and negative)
➤ The ultimate source of all EM fields
➤ Measured in Coulombs
➤ Can be postive or negative - note the direction of the electric field
Q =
˚
Vρ · dV (9)
11 ENGN6521 / ENGN4521: Embedded Wireless
Charge as a source of the Electric Field: MAXWELL’s first equation
12 ENGN6521 / ENGN4521: Embedded Wireless
Simple case of a point charge: Coulomb’s Law
‹
SE · dS =
Q
ǫo(10)
Er × 4πr2 =Q
ǫo(11)
Er =Q
4πǫor2(12)
13 ENGN6521 / ENGN4521: Embedded Wireless
Parallel Charged Plates
14 ENGN6521 / ENGN4521: Embedded Wireless
Electrostatics: The Electrostatic Potential (Beware - replace this by Faraday’s law
➤ Consider a situation where there is no changing magnetic field in
Faraday’s law.
➤ Definition: The potential difference between two points x1 and x2 is
defined by,
Φ = −
x2ˆ
x1
E.dl = −
ˆ
γ
E.dl
➤ Since the path γ can be any which connects the points x1 and x2 we may
conclude that E = −∇Φ.
➤ Kirchhoffs voltage law: The sum of the voltage drops in a circuit is
zero - BEWARE: to be ’modified’ by ∂B∂t
15 ENGN6521 / ENGN4521: Embedded Wireless
Parallel Charged Plates
‰
E · dl = 0 [Faraday, no changing B] (13)
16 ENGN6521 / ENGN4521: Embedded Wireless
Parallel Plate Capacitor
➤ Capacitors are reservoirs of charge - a bit like transient batteries
➤ C = QV (Farads)
➤ Since V = Ed, C = ǫ0Ad
17 ENGN6521 / ENGN4521: Embedded Wireless
Actual capacitors
18 ENGN6521 / ENGN4521: Embedded Wireless
The vacuum gap capacitor
19 ENGN6521 / ENGN4521: Embedded Wireless
20 ENGN6521 / ENGN4521: Embedded Wireless
Moving charges → Current
➤ The rate of charge crossing a surface is given by (note that charges are
positive even if electrons are flowing!),
∂Q
∂t=
¨
Sj · dS (14)
➤ Current is measured in Amperes (Coulombs per second)
➤ Charge conservation: the charge flows out of a region as a current Q.
Why the ’-’ sign here?
∂Q
∂t= −
‹
Sj · dS (15)
➤ Charge conservation leads to Kirchhoff’s current rule: currents
entering a circuit node sum to zero
21 ENGN6521 / ENGN4521: Embedded Wireless
Current Density (current per unit area)
22 ENGN6521 / ENGN4521: Embedded Wireless
Kirchhoff’s current law
23 ENGN6521 / ENGN4521: Embedded Wireless
Dielectrics and Conductors
➤ Dielectrics are insulating materials VIZ. do not allow D.C. current to flow
through them. Usually we just call them insulators
➤ In conductors, charge carriers (electrons) are free to move. e.g. metals.
24 ENGN6521 / ENGN4521: Embedded Wireless
Dielectrics 1
➤ Two opposite, cancelling charges separated in space is referred to
collectively as a dipole.
➤ Electrons and nuclei in dielectrics experience opposing forces in the
presence of an imposed electric field.
➤ Electrons move opposite to the field and nuclei move in the direction of the
field. This produces a distribution of dipoles referred to as polarisation.
➤ The charge separation induced by the field acts to reduce the electric field
within the dielectric.
25 ENGN6521 / ENGN4521: Embedded Wireless
Dielectrics 2
26 ENGN6521 / ENGN4521: Embedded Wireless
Dielectrics 3: Polarisation
➤ Polarisation P is the dipole moment per unit volume induced by an imposed, externalelectric field
➤ At the edge of a polarised dielectric there is a charge density left over by the displacementof the dipoles given by σp = P.n̂ where n̂ is the unit vector normal to the surface.
➤ σp belongs to the material. It is not a free charge.
27 ENGN6521 / ENGN4521: Embedded Wireless
Dielectrics 4: Relative dielectric constant
➤ For a slither of dielectric, the surface charge is related to the E-field within the dielectricthat produces it by
σp = P · n̂ = ǫ0(ǫr − 1)E · n̂
➤ ǫr at D.C. is a positive dimensionless number and ǫr > 1 for dielectrics. E.g. ǫr = 2 forteflon, ǫr = 1 for air or vacuum.
➤ When the electric field oscillates, ǫr is a complex function of frequency -lossy
➤ The ratio of the imaginary to real components of ǫr is termed the loss tangent of thedielectric:
tan[δ(ω)] =Im(ǫr(ω))
Re(ǫr(ω))(16)
➤ Even though dielectrics are insulators at DC they can be lossy conductors atradiofrequency.
28 ENGN6521 / ENGN4521: Embedded Wireless
Exercise 1: Compute the capacitance of a parallel plate air-gap capacitor
(E − 0)×A =Q
ǫo=
σ ×A
ǫo=> E =
σ
ǫo(17)
V = E × d => C =Q
V=
ǫoA
d(18)
+ + + + + + + + +
- - - - - - - - -
E = 0
E > 0
Q = σΑ
electricfieldlinesarea = A
enclosedby dots
capacitorplates
d
29 ENGN6521 / ENGN4521: Embedded Wireless
Exercise 2: Compute the capacitance for a gap of dielectric constant ǫr
Use the definition of ǫr and comnpute the E − field inside the dielectric
E =σcap − σp
ǫo=
σp
ǫo(ǫr − 1)(19)
(σcap − σp)(ǫr − 1) = σp => σp =σcap(ǫr − 1)
ǫr(20)
V = E × d =σp × d
ǫo(ǫr − 1)=
σcapd
ǫoǫr(21)
Qcap = σcapA => C =Qcap
V=
ǫrǫoA
d(22)
30 ENGN6521 / ENGN4521: Embedded Wireless
Conductors and Ohm’s law
➤ Ohm’s law: The current density in a conductor is proportional to the electric
field within the conductor.
j = σE (23)
➤ Conductors are completely specified by the conductivity σ.
➤ For copper, σ = 5.80× 107mhos/meter.
➤ Ohm’s law is assumed to be an accurate result for metals at all
radiofrequencies :)
➤ I.E. σ is always a real number and independent of frequency.
31 ENGN6521 / ENGN4521: Embedded Wireless
Resistance
➤ Resistance is obtained by converting j to I and E to V for a short length of
metal.
I
A= σ
V
l(24)
V =l
σAI (25)
➤ From which we define resistance
R =l
σA(26)
➤ We also define resistivity, η from σ
η =1
σ(27)
32 ENGN6521 / ENGN4521: Embedded Wireless
Conductors vs Dielectrics
➤ for dielectrics: σp = ǫ0(ǫr − 1)E · n̂
➤ If σp oscillates as a function of time then, jωσp = jωǫ0(ǫr − 1)E · n̂
➤ since the dipoles are reversing sign at rate ω, we may write, jP · n̂ = jωσp,
where jP is the polarisation current.
➤ Dielectrics obey Ohm’s law with conductivity defined by
σ = jωǫ0(ǫr − 1)
➤ The difference between conductors and insulators is simply that σ is
resistive and frequency independent for a conductor, but is mainly
reactive and frequency dependent for insulators.
➤ As we already know, ǫr is a complex function of frequency and the real and
imaginary frequency functions of ǫr can be computed from each other
(Kramers-Kronig).
33 ENGN6521 / ENGN4521: Embedded Wireless
Charge as a source of the Electric Field: MAXWELL’s first equation
➤ the field can get distorted with complex charge distributions
➤ makes no difference to the integral
The coulomb force...
F = QE (28)
34 ENGN6521 / ENGN4521: Embedded Wireless
Electromagnetism IIMoving Charges. Faraday’s law and Ampere’s law.
Ferrites
35 ENGN6521 / ENGN4521: Embedded Wireless
ELECTROMAGNETISM SUMMARY
➤ RF shielding
➤ Magnetostatics
➤ Ampere’s law and Faraday’s law
➤ Inductance
➤ Maxwell’s equations
36 ENGN6521 / ENGN4521: Embedded Wireless
The Floating Conducting Object
Ohm’s law for metals and Gauss’s law for the electric field give rise to the
concept of electromagnetic shielding
37 ENGN6521 / ENGN4521: Embedded Wireless
The Earthed Conducting Object
38 ENGN6521 / ENGN4521: Embedded Wireless
Images
➤ A charge near a conducting surface attracts charge of the opposite sign to
the nearest point on the surface.
➤ These surface charges arrange themselves so that the tangntial
component of electric field is zero on the surface.
➤ One may compute the electric field outside the conductor by assuming
(mathematically) that there is an image charge within the conductor.
39 ENGN6521 / ENGN4521: Embedded Wireless
Proximity to Floating (coupled) and Earthed Conductors (decoupled)
40 ENGN6521 / ENGN4521: Embedded Wireless
Magnetostatics: The static magnetic field
➤ Gauss’s law for the magnetic field:‹
A
B · dA = 0 (29)
➤ There is no static sink or source of the magnetic field. Also generally true.
➤ However current is the source of a static magnetic field. Using Ampere’s law (with nochaning E-field)
‰
γ
B.dl = µ0I (30)
The line integral of B around a closed circuit γ bounding a surface A is equal to the
current flowing through A. Ampere’s law
41 ENGN6521 / ENGN4521: Embedded Wireless
Important Correction to Ampere’s Law
➤ A time varying Electric field is also a source of the time varying magnetic
field,
‰
γB.dl =
¨
Aµ0
(
j+ ǫ0∂E
∂t
)
.dA (31)
42 ENGN6521 / ENGN4521: Embedded Wireless
Why this Correction to Ampere’s Law? I
➤ Ampere’s law without E contradicts charge conservation
‰
γB.dl =
¨
Aµ0
(
j+ ǫ0∂E
∂t
)
.dA (32)
➤ Consider the new Ampere’s law on a close surface area, A.
43 ENGN6521 / ENGN4521: Embedded Wireless
Why this Correction to Ampere’s Law? II
➤ If we shrink the closed contour γ on the left hand side to zero then we
obtain
0 =
‹
Aµ0
(
j+ ǫ0∂E
∂t
)
.dA (33)
➤ However the two terms on the right hand side are‹
AE.dA =
q
ǫ0(34)
and‹
Aj.dA = −
∂q
∂t(35)
➤ - Q.E.D. In fact it simply confirms how capacitors work!!
44 ENGN6521 / ENGN4521: Embedded Wireless
Faraday’s Law
➤ The law of electromagnetic induction or Lenz’s / Faraday’s law
➤ A time varying magnetic field is also the source of electric field,˛
γ
E.dl = −
ˆ
A
∂B
∂t.dA (36)
45 ENGN6521 / ENGN4521: Embedded Wireless
Inductance I
➤
¸
circuitE · dl = −˜
A∂B∂t · dA = −∂ΦB
∂t
➤ Note that Kirchhoff’s voltage law still works.
V
B field
Iinductor
46 ENGN6521 / ENGN4521: Embedded Wireless
Inductors II Magnetic field of a solenoid
➤ Consider Ampere’s law for a solenoid with a static current ∂I∂t
= 0.
➤ If the solenoid is long and axially uniform then the magnetic field is directed along the axis.
➤ By choosing a curve along the axis of the solenoid, Ampere’s law gives,
ˆ d
0
dlB = µoNI => B =µoNI
d=> B = µoNlI (37)
where µo = 4π × 10−7Henriess/meter is the permeability of free-space, N is the number of turnsenclosed by the contour and Nl is the number of turns per unit length.
47 ENGN6521 / ENGN4521: Embedded Wireless
Inductance III Inductance of a coil
48 ENGN6521 / ENGN4521: Embedded Wireless
Inductance IV: Inductance of a coil
➤ A simple formula defining the inductance of a coil can be found by noting that if there are N turns on a coilthen the total flux linking the coil is Φ = N ×B ×A
➤ From Faraday, the electromotive force (E.M.F.) at frequency ω generated by the changing flux is given byV = jω ×N ×B ×A
➤ Next we use the formula for magnetic field B of an infinite solenoid as an approximation for a coil (a solenoidof finite length)
V = jω ×N × (µoNlI)×A = jω × (µoN2A
d)I (38)
➤ From which we define the inductance, L,
V = jωLI, L =µoN2A
d(39)
➤ More accurate formula (lab2) ...
L(µH) =0.394r2N2
9r +10d(40)
49 ENGN6521 / ENGN4521: Embedded Wireless
Ferromagnetic Materials
➤ Whereas a dielectric reduce E − field with ǫr ≫ 1, magnetic materials known as ferritesthat can amplify an internal magnetic field.
➤ In Ferrites, magnetic domains align with the imposed field (see the figure below).
➤ If the above solenoid were wound on a ferromagnetic core, then the formulae for the B
field and the inductance L would become B = µrµoNlI and L = µrµoN 2Ad
50 ENGN6521 / ENGN4521: Embedded Wireless
Inductors and Ferrites
51 ENGN6521 / ENGN4521: Embedded Wireless
A comparison between ferrites and dielectrics
➤ A dielectric decreases the imposed E-field whereas a ferromagnetic material increasesthe imposed B-field.
➤ ǫr > 1 increases C because it decreases the voltage for a given charge on the capacitorelectrodes.
➤ µr > 1 increases L because it increases the magnetic flux for a given current, thusincreasing the E.M.F.
52 ENGN6521 / ENGN4521: Embedded Wireless
Self Inductance (I)
➤ In practice the magnetic field linking a circuit can either come from current in the circuititself (as we have just discussed) or from some other external current in another circuit.
➤ So far we have just looked at the case of a magnetic field outside the wire which acts backon the circuit through Faraday’s law. This is self-inductance
53 ENGN6521 / ENGN4521: Embedded Wireless
Self Inductance (II)
➤ We have already seen the self-inductance of a coil.
➤ The most obvious case (but definitely not the simplest case) is that of a cylindrical straight conductor wherethe circuit of integration is precisely that above
L = 0.002 l
[
loge
(
2l
a
)
−3
4
]
: l (cm) for L (µH) (41)
54 ENGN6521 / ENGN4521: Embedded Wireless
Mutual Coupling
➤ The magnetic fields of different circuits induce electromotive forces in each other.
➤ Mutual Inductance, M
V1 = MdI2
dt, V2 = M
dI1
dt(42)
55 ENGN6521 / ENGN4521: Embedded Wireless
Transformers
➤ Two neighbouring coils can couple to each other inductively by Faraday’s law
V21 = jωLiii + jωMio (43)
V43 = jωLoii + jωMii (44)
56 ENGN6521 / ENGN4521: Embedded Wireless
A summary of units of measure
The following units can be deduced from the formulae thus far assuming that
S.I. units are used for distance (metres), time seconds), charge (Couombs).
➤ Electric field is Newtons per Coulomb but is measured in Volts per metre
➤ Capacitance is Coulombs per volt but is measured in Farads
➤ Current is Coulombs per second but is measured in Amperes
➤ Magnetic field is measured in Tesla
➤ Magnetic flux is (Teslas×m2) but is measured in Webers
➤ The permittivity of free space, ǫo = 8.85× 10−12 Farads per metre
➤ The permeability of free space, µo = 4π × 10−7 Henries per metre
57 ENGN6521 / ENGN4521: Embedded Wireless
Electromagnetism III
➤ Plane and transverse electromagnetic waves
➤ Power flow
➤ How antennas work
58 ENGN6521 / ENGN4521: Embedded Wireless
EM Waves
➤ Start with Faraday’s law and Ampere’s law in VACUO.
˛
γ
E.dl = −
ˆ
A
∂B
∂t.dA (45)
˛
γ
B.dl =
ˆ
A
µ0ǫ0∂E
∂t.dA (46)
➤ Recall that the line integral along γ is on the perimeter of the surface A
➤ Thus a one dimensional E and B must have E.B = 0.
59 ENGN6521 / ENGN4521: Embedded Wireless
Plane Waves 1
➤ Apply F and A to the diagrams below and note that A points in the direction
of the right hand screw rule with respect to the direction of γ
60 ENGN6521 / ENGN4521: Embedded Wireless
Plane Waves 2
61 ENGN6521 / ENGN4521: Embedded Wireless
Plane Waves 3
➤ Integrating E around γ in the left figure...
[E(z + dz)−E(z)]L = −∂B
∂tL∆z (47)
∂E
∂zL∆z = −
∂B
∂tL∆z (48)
➤ Integrating B around γ in the right figure...
[B(z + dz)−B(z)]L = −ǫ0µ0∂E
∂tL∆z (49)
∂B
∂zL∆z = −ǫ0µ0
∂E
∂tL∆z (50)
62 ENGN6521 / ENGN4521: Embedded Wireless
Plane Waves 4
➤ Two equations for E and B
∂E
∂z= −
∂B
∂t(51)
∂B
∂z= −ǫ0µ0
∂E
∂t(52)
➤ Simultaneous solution gives the wave equation for each field:
∂2E
∂z2= ǫ0µ0
∂2E
∂t2(53)
∂2B
∂z2= ǫ0µ0
∂2B
∂t2(54)
63 ENGN6521 / ENGN4521: Embedded Wireless
Plane Wave Solutions
➤ E = E(X); B = B(X) where X = z ± vt
➤ Propagation speed in VACUO
v = c =
√
1
ǫ0µ0
= 3× 108m/s (55)
➤ For travelling waves , E,B ∝ f(ωt∓ kz) and EB
= ±c
➤ plane waves satisfy E ·B = 0
➤ plane waves propagate in the direction of E×B
➤ plane waves have components Ez = 0 abd Bz = 0 in the direction of propagation of thewave
64 ENGN6521 / ENGN4521: Embedded Wireless
Transverse electromagnetic waves or T.E.M Waves
➤ Consider plane waves propagating along the z − axis perpendicular to the x− y plane
➤ T.E.M. waves are plane waves whose fields also vary in the x− y plane
➤ Like plane waves, T.E.M. waves also have Ez = 0 abd Bz = 0 in the direction ofpropagation of the wave
➤ E = Eo(x, y) exp j(ωt− kz) and B = Bo(x, y) exp j(ωt− kz)
➤ ∇× Eo(x,y) = 0 and ∇×Bo(x,y) = 0
➤ As a result, ∃ a scalar function Φ(x, y) which gives Eo(x,y) = −∇Φ(x, y) and satisfiesLaplace’s equation: ∇2Φ(x, y) = 0
65 ENGN6521 / ENGN4521: Embedded Wireless
Plane and T.E.M Waves
66 ENGN6521 / ENGN4521: Embedded Wireless
Definition of Power
➤ Consider complex vector functions of the form
X,Y = (U(x) + jV(x)) exp jωt (56)
➤ Compute the time average of the product of two complex numbers
< X ·Y > =1
2ℜ[X ·Y∗] (57)
➤ Electromagnetic power dissipation per unit volume (Power density Watts/m2) givenj(A/m2) and E(V/m)
Power density =1
2ℜ[j · E∗] (58)
➤ Only gives dissipated power => Ohmic power
67 ENGN6521 / ENGN4521: Embedded Wireless
What about transmitted or radiated power?
➤ Already seen this with transmission lines, e.g. Pforward = 12ℜ[VfI
∗f ] - But what about free
space E.M. waves??
➤ Approach: relate this definition of power to the T.E.M. B and E fields in the transmissionline to produce a suitable definition of power flux for T.E.M waves
68 ENGN6521 / ENGN4521: Embedded Wireless
Poynting’s Theorem (I)
69 ENGN6521 / ENGN4521: Embedded Wireless