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Massachusetts Institute of Technology RF Cavity and Components for Accelerators 1 Massachusetts Institute of Technology RF Cavity and Components for Accelerators USPAS 2010 • Maxwell’s equations • Wave equations • Plane Waves • Boundary conditions A. Nassiri - ANL Lecture 1- Review

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  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 1Massachusetts Institute of Technology RF Cavity and Components for Accelerators USPAS 2010

    Maxwells equations Wave equations Plane Waves Boundary conditions

    A. Nassiri - ANL

    Lecture 1- Review

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 2

    Maxwells Equations

    The general form of the time-varying Maxwells equations can be written in differential form as:

    0==

    =

    +=

    BD

    BE

    DJH

    t

    t

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 3

    A few other fundamental relationships

    H/m104 F/m,108548with

    ity)(permeabil and ity)(permittiv here

    ips"relationsh veconstituti"

    equation" continuity"

    law" sOhm'"

    70

    120

    00

    ==

    ==

    ==

    =

    =

    .

    HBED

    J

    EJ

    rr

    t

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 4

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 5

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 6

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 7

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 8

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 9

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 10

    A few other fundamental relationships

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 11

    Integral form of the equations

    = D

    =S

    QsdD

    tBE

    =

    =

    SCsd

    tBdE

    0= B

    =S

    sdB 0

    tDJH

    +=

    +=SC

    sdtDJdH

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 12

    Wave Equations

    In any problem with unknown E, D, B, H we have 12 unknowns. To solve for these we need 12 scalar equations. Maxwells equations provide 3 each for the two curl equations. and 3 each for both constitutive relations (difficult task).

    Instead we anticipate that electromagnetic fields propagate as waves. Thus if we can find a wave equation, we could solve it to find out the fields directly.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 13

    Wave equations

    Take the curl of the first Maxwell:

    ( )

    ( )

    2

    2

    t

    tt

    t

    t

    =

    +=

    +=

    +=

    HJ

    HJ

    EJ

    EJH

    ( ) LHS on the use Now 2 HHH 0

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 14

    Wave Equations

    The result is:

    JHH

    =

    2

    22

    t

    Similarly, the same process for the second Maxwell produces

    +

    =

    ttJEE

    2

    22

    Note how in both case we have a wave equation (2nd order PDE) for both E and H with fields to the left of the = sign and sourcesto the right. These two wave equations are completely equivalentto the Maxwell equations.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 15

    Solutions to the wave equations

    Consider a region of free space ( = 0) where there are no sources(J = 0). The wave equations become homogeneous:

    022

    2 =

    tEE

    022

    2 =

    tHH

    Actually there are 6 equations; we will only consider one component:e.g. Ex(z,t)

    2

    00

    22

    2

    22

    2 1 e wher01 cvtE

    vzE xx ===

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 16

    Solutions to the wave equation

    Try a solution of the form f(z-vt) e.g. sin[(z-vt)]. By differentiatingtwice and substituting back into the scalar wave equation, we find that it satisfies!

    f(z) t=0z

    f(z-vt1) t=t1 z

    f(z-vt2) t=t2 z

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 17

    Plane Waves

    First treat plane waves in free space. Then interaction of plane waves with media. We assume time harmonic case, and source free situation.

    0202 =+ EkE

    We require solutions for E and H (which are solutions to thefollowing PDE) in free space

    No potentials here!(no sources)

    Note that this is actually three equations:

    zyxiEkzE

    yE

    xE

    iiii ,,==+

    +

    + 0202

    2

    2

    2

    2

    2

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 18

    How do we find a solution?

    Usual procedure is to use Separation of Variables (SOV). Take one component for example Ex.

    ( ) ( ) ( )

    0

    0

    20

    20

    =+

    +

    +

    =+++

    =

    khh

    gg

    ff

    fghkhfggfhfgh

    zhygxfEx

    Functions of a single variable sum = constant = -k02

    ckkkkk

    khhk

    ggk

    ff

    zyx

    zyx

    ===++

    =

    =

    =

    2 with so and

    020

    222

    222 ;;

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 19

    Mathematical Solution

    We note we have 3 ODEs now.

    zjkz

    yjky

    xjkx

    z

    y

    x

    ehhkdz

    hd

    egkdy

    gd

    effkdx

    fd

    ==+

    ==+

    ==+

    issolution 0

    g issolution 0

    issolution 0

    22

    2

    22

    2

    22

    2

    ( )zkykxkjx

    zyxeAE ++=

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 20

    But, what does it mean physically?

    ( )zkykxkjx

    zyxeAE ++=This represents the x-component of the travelling wave E-field(like on a transmission line) which is travelling in the direction of the propagation vector, with Amplitude A. The direction ofpropagation is given by

    zkykxkk zyx ++=

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 21

    Physical interpretation

    The solution represents a wave travelling in the +z direction withvelocity c. Similarly, f(z+vt) is a solution as well. This latter solution represents a wave travelling in the -z direction.So generally,

    ( ) ( )( )( )[ ]vtzvtyvtxftzEx =,In practice, we solve for either E or H and then obtain theother field using the appropriate curl equation.

    What about when sources are present? Looks difficult!

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 22

    Generalize for all components

    If we define the normal 3D position vector as:

    rkjz

    rkjy

    rkjx

    zyx

    CeE

    BeE

    AeE

    zkykxkrk

    zzyyxxr

    =

    =

    =

    ++=

    ++=

    similarly

    so

    then

    zCyBxAEeEE rkj ++== 00 where

    General expressionfor a plane wave

    +sign droppedhere

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 23

    Properties of plane waves

    For source free propagation we must have E = 0. If we satisfythis requirement we must have kE0= 0. This means that E0is perpendicular to k.

    The corresponding expression for H can be found by substitution of the solution for E into the E equation. The result is:

    EnkH

    = 0

    0

    Where n is a unit vector in the k direction.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 24

    Transverse Electromagnetic (TEM) wave

    Note that H is also perpendicular to k and also perpendicular toE. This can be established from the expression for H.

    E

    H

    Direction of propagationk,n

    Note that:

    knHE or =E and H lie on theplane of constantphase (kr = const)

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 25

    Plane waves at interfaces (normal incidence)

    Consider a linearly polarized (in x-direction) wave travelling inthe +z direction with magnitude Ei

    Ei

    Hi

    Er

    Hr

    Incident

    Reflected

    222111

    Transmitted

    z

    xArbitraryorientation!

    Et

    Ht

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 26

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 27

    Metallic Boundary

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 28

    Metallic Boundary

    Dielectric Metal

    H

    E

    Skin depth

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 29

    Boundary conditions

    We deal with a general dielectric interface and two specialcases. First the general case. For convenience we consider the boundary to be planar.

    Maxwells equations in differential form require known boundaryvalues in order to have a complete and unique solution. The so called boundary conditions (B/C) can be derived by consideringthe integral form of Maxwells equations.

    n111

    222

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 30

    General case

    Equivalent

    Et1 n111

    222 Et2Tangential E continuous

    n111

    222 Ht2

    Ht1

    n x (H1-H2)=Js

    n111

    222

    Bn1

    Bn2Normal B continuous

    n111

    222 D2n

    D1n

    n(D1-D2)=s

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 31

    Special case (a) Lossless dielectric

    n111=0

    222=0

    Bn1=Bn2 normal B fields continuous

    Ht1=Ht2 tangential H fields continuous (no current)

    Dn1=Dn2 normal D fields continuous (no charge)

    Et1=Et2 tangential E fields continuous)

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 32

    Special case (b) Perfect Conductor

    n111=0

    2 Perfect Electric Conductor Et2=Ht2=0

    Bn1= 0 Normal B(H) field is zero on conductor.

    Et1= 0 Tangential Electric field on conductor is zero.

    n H1=Js H field is discontinuous by the surface current

    n . D1= Normal D(E) field is discontinuous by surface charge

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 33

    Boundary conditions

    Continuity at the boundary for the tangential fields requires:

    (2) (1)

    tri

    tri

    HHHEEE

    =+

    =+ Fix signs whendefining impedance!

    211 :define Now ZHEZ

    HEZ

    HE

    t

    t

    r

    r

    i

    i ===

    Substituting into (1) and (2) and eliminating Er gives

    21

    22t coefficienon TransmissiZZ

    ZEE

    i

    t

    +==

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 34

    Plane Wave in Dispersive Media

    Recall the Maxwells equations:

    vDB

    JDjBBjE

    =

    =

    +=

    =

    0

    ( ) ( )

    ( )

    ( ) BjEBEj

    tBez,y,xE

    tBE

    ez,y,xEt;x,y,xE

    tj

    tj

    ==

    =

    =

    =

    1

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 35

    Plane Wave in Dispersive Media

    So far, for lossless media, we considered J=0, and v=0 but,there are actually two types of current and one of them shouldnot be ignored.

    Total current is a sum of the Source current and Conduction current.

    oo

    c

    oc

    JEjjJEEjH

    JDjHEJ

    JJJ

    +

    =++=

    +=

    =

    +=

    set

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 36

    Plane Wave in Dispersive Media

    Defining complex permittivity

    = j

    Maxwells equations in a conducting media (source free) can be written as

    00

    =

    =

    =

    =

    EH

    EjHHjE

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 37

    Plane Wave in Dispersive Media

    We have considered so far:

    Plane Wavesin Free space

    Plane Wavesin Isotropic Dielectric

    Plane Wavesin anisotropic Dielectric

    Plane Wavesin Dissipative Media

    00

    0

    0

    =

    =

    =

    =

    EH

    EjHHjE

    00

    0

    =

    =

    =

    =

    HE

    EjHHjE

    00

    =

    =

    =

    =

    BD

    EjHHjE

    00

    =

    =

    =

    =

    HE

    EjHHjE

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 38

    Plane Wave in Dispersive Media

    Wave equation for dissipative media becomes:

    ( )( ) ( )

    HH

    EE

    EjjEE

    HjE

    =

    =

    =

    =

    22

    22

    2

    The set of plane-wave solutions are:

    zj

    zj

    eEyH

    eExE

    =

    =

    0

    0

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 39

    Plane Wave in Dispersive Media

    Substituting into and

    yields the dispersion relationEE

    = 22 HH

    = 22

    =

    =and

    22

    Is the complex intrinsic impedance of the isotropic media.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 40

    Plane Wave in Dispersive Media

    Denoting the complex values:

    ( )

    ( ) ( )

    =

    =

    ====

    =

    =

    jzjjzjj

    xzzjzjjzj

    jIR

    eeEyeEyH

    ExeeExeExeExE

    e

    j

    IRIR

    IRIR

    00

    000

    then,

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 41

    Plane Wave in Dispersive Media

    Loss tangent is defined from

    tangent loss as defined is

    =

    ===

    j

    jj IR

    1

    =

    =

    =

    =

    tan

    jjj 1

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 42

    Plane Wave in Dispersive Media

    Slightly lossy case: 1

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 43

    Plane Wave in Dispersive Media

    Highly lossy case: 1>>

    =

    =2

    1 jj

    ( )j= 12

    = 2pd Skin depth

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 44

    Plane Wave in Dispersive Media

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 45

    Reflection & Transmission

    Similarly, substituting into (1) and (2) and eliminating Et

    12

    12t coefficien ReflectionZZZZ

    EE

    i

    r

    +

    ==

    We note that = 1+, and that the values of the reflection and transmission are the same as occur in a transmission linediscontinuity.

    Z1 Z2

    Not 1-

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 46

    Special case (1)

    (1) Medium 1: air; Medium 2: conductor

    iitt

    t

    iit

    m

    HEZ

    HZEH

    EZZEE

    jZZZ

    22 usethen

    2 So

    1 377

    12

    1

    2

    21

    ==

    =

    +==>>=

    This says that the transmitted magnetic field is almost doubledat the boundary before it decays according to the skin depth. On the reflection side Hi Hr implying that almost all theH-field is reflected forming a standing wave.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 47

    Special case (2)

    (2) Medium 1: conductor; Medium 2: air

    Reversing the situation, now where the wave is incidentfrom the conducting side, we can show that the wave isalmost totally reflected within the conductor, but that the standing wave is attenuated due to the conductivity.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 48

    Special Case (3)

    (2) Medium1: dielectric; Medium2: dielectric

    2

    02

    1

    01

    == ZZ ,1

    1

    2

    1

    2

    1

    +

    =

    This result says that the reflection can be controlled by varyingthe ratio of the dielectric constants. The transmission analogycan thus be used for a quarter-wave matching device.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 49

    /4 Matching Plate

    Z0 Zp Z2

    /4

    Air: r=1 Plate r'=? Dielectric r=4

    Transmission line theory tells us that for a match

    20ZZZ p =

    2 and 266 So

    1882

    7376 7376

    2

    0

    020

    ===

    ====

    ZZZ

    ZZZ

    rp

    r

    '

    .,.

    We will see TL lectures later

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 50

    Applications

    The principle of /4 matching is not only confined to transmissionline problems! In fact, the same principle is used to eliminate reflections in many optical devices using a /4 coating layer onlenses & prisms to improve light transmission efficiency.

    Similarly, a half-wave section can be used as a dielectric window.Ie. Full transparency. (Why?). In this case Z2=Z0 and thematching section is /2. Such devices are used to protect antennasfrom weather, ice snow, etc and are called radomes.

    Note that both applications are frequency sensitive and that thematching section is only /4 or /2 at one frequency.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 51

    Oblique Incidence

    The transmission line analogy only works for normal incidence.When we have oblique incidence of plane waves on a dielectric interface the reflection and transmission characteristics become polarization and angle of incidence dependent.

    We need to distinguish between the two different polarizations.We do this by first, explaining what a plane of incidence is, then we will point out the distinguishing features of each polarization. We are aiming for expressions for reflection coefficients.

    We note again that we are only dealing with plane waves

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 52

    Plane of Incidence

    Surfacenormal

    Dielectric interface in x-z plane

    x

    y

    z

    Plane of incidence containsboth direction of propagation vector and normal vector.

    Direction ofpropagation

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 53

    Parallel & Perpendicular Incidence

    HE

    x

    EH

    x

    y y

    Plane of incidence is the x-y plane

    E is Perpendicular to theplane of incidence

    E is Parallel to theplane of incidence

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 54

    Perpendicular incidence

    y

    Hi EiEr

    Hr

    x

    r i

    tHtEt

    22

    11

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 55

    Write math expression for fields!

    ( )[ ]

    ( ) ( )[ ]iiiii

    iii

    yxjZEyxH

    yxjEzE

    cossinexpsincos

    cossinexp

    11

    0

    10

    ++=

    +=

    ( )[ ]

    ( ) ( )[ ]rrrrr

    rrr

    yxjZEyxH

    yxjEzE

    cossinexpsincos

    cossinexp

    11

    0

    10

    +=

    =

    ( )[ ]

    ( ) ( )[ ]ttttt

    ttt

    yxjZEyxH

    yxjEzE

    cossinexpsincos

    cossinexp

    22

    0

    20

    ++=

    +=

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 56

    How did you get that?

    Within the exponential: This tells the direction of propagationOf the wave. E.g. for both the incident Ei and Hi

    ( )ii yxj cossin1 +A component in the x direction

    Another component in the y direction

    PropagatingIn medium 1

    Outside the exponential tells what vector components of the fieldAre present. E.g. for Hr

    ( )1

    0sincosZ

    Eyx rr +

    +x and +y components of Hr

    Perpendicular reflectioncoefficient

    E0/Z1 converts E to H

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 57

    Apply boundary conditions

    Tangential E fields (Ez) matches at y=0Tangential H fields (Hx) matches at y=0

    ( ) ( ) ( )tri xjxjxj sinexpsinexpsinexp 211 =+We know that =1+ , so then the arguments of the exponents must be equal. Sometimes called Phase matchingin optical context. It is the same as applying the boundaryconditions.

    tri jjj sinsinsin 211 ==

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 58

    Snells laws and Fresnel coefficients

    The first equation gives

    and from the second using it

    sinsin

    22

    11 2 ==

    ir =

    By matching the Hx components and utilizing Snell, we can obtain the Fresnel reflection coefficient for perpendicular incidence.

    ti

    ti

    ZZZZ

    coscoscoscos

    12

    12

    +

    =

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 59

    Alternative form

    Alternatively, we can use Snell to remove the t and write it interms of the incidence angle, at the same time assuming non-magnetic media (= 0 for both media).

    ii

    ii

    2

    1

    2

    2

    1

    2

    sincos

    sincos

    +

    =

    Note how both formsreduce to the transmissionline form when i=0

    This latter form is the one that is most often quoted in texts,the previous version is more general

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 60

    Some interesting observations

    If 2 > 1 Then the square root is positive, If 1> 2 i.e. the wave is incident from more dense to

    less denseAND

    Is real

    1

    22

    isin

    Then is complex and 1=This implies that the incident wave is totallyinternally reflected (TIR) into the more dense medium

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 61

    Critical angle

    When the equality is satisfied we have the so-called criticalangle. In other words, if the incident angle is greater than or equal to the critical angle AND the incidence is from more dense to less dense, we have TIR.

    1

    21

    = sinic

    For i> ic Then as noted previously.1=

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 62

    Strange results

    Now

    1A where

    imaginary! is 1

    ! 1 since so

    2

    2

    1

    2

    212

    1

    =

    ==

    >>=

    i

    ttt

    tit

    jA

    sin

    cossincos

    sinsinsin

    What is the physical interpretation of these results? To seewhat is happening we go back to the expression for thetransmitted field and substitute the above results.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 63

    Transmitted field

    ( )[ ]

    [ ] [ ]

    1 where 22

    1222

    20

    20

    ==

    =

    +=

    i

    t

    ttt

    A

    yxjEz

    yxjEzE

    sin

    expsinexp

    cossinexppreviously

    cos t=jA

    Physically, it is apparent that the transmitted field propagatesalong the surface (-x direction) but attenuates in the +y directionThis type of wave is a surface wave field

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 64

    Example

    Assume:r = 81 = 0r = 1

    Let i = 45evaluate TIRicic >==

    i1 so 386

    811 .sin

    y

    airwater

    Ei

    Hi

    21 x

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 65

    Example (ctd)

    =

    +

    +=

    +=

    ===

    +==

    ==

    6.4442.1

    5.0811707.0

    5.0811707.0

    1

    1

    /5.3928.6228.6145sin81cos

    38.645sin181sin Snell Using

    002

    2

    mNepA

    jjt

    t

    This means that ifthe field strength onthe surface is1Vm-1, then

    -1Vm421.== it EE

    Choose + signto allow forattenuationin +y direction

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 66

    Evaluate the field just above the surface

    Lets evaluate the transmitted E field at /4 above the surface.

    dB

    VmEt

    8854211027320

    2734

    4939421

    6

    10

    0

    ..

    .log

    ..exp.

    =

    =

    =

    =

    This means that the surface wave is very tightly bound to thesurface and the power flow in the direction normal to the surface is zero.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 67

    What about the factor ?0

    0

    k

    0

    0

    00000

    0 122

    ====

    ccfk

    This term has the dimensions of admittance, in fact

    0

    0

    000

    11

    ===

    ZY

    = 377space free of impedance Zwhere 0

    EnH

    = 0

    1

    And now

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 68

    Propagation in conducting media

    We have considered propagation in free space (perfect dielectricwith = 0). Now consider propagation in conducting media where can vary from a finite value to .

    +

    =

    ttJEE 2

    22Start with

    Assuming no free charge and the time harmonic form, gives

    22

    22

    22

    where0

    =

    =

    =+

    jEE

    EjEE

    Complex propagationcoefficient due tofinite conductivity

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 69

    Conduction current and displacement current

    In metals, the conduction current (E) is much larger than the displacement current (j0E). Only as frequencies increase tothe optical region do the two become comparable.

    E.g. = 5.8x107 for copper0 = 2x1010x 8.854x10-12 = 0.556

    So retain only the j term when considering highly conductivematerial at frequencies below light. The PDE becomes:

    002 = EjE

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 70

    Plane wave incident on a conductor

    Consider a plane wave entering a conductive medium at normalincidence.

    zx

    Ex

    Hy

    Free space Conducting medium

    Some transmittedMostly reflected

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 71

    Mathematical solution

    The equation for this is:002

    2

    =

    x

    x EjzE

    The solution is: zjx eEE

    00

    =

    We can simplify the exponent: ( )2

    1 00 jj +==

    So now has equal realand imaginary parts.

    2 with 00

    === zzx eeEE

    Alternatively write jzz

    x eeEE

    = 0

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 72

    Skin Depth

    The last equation

    gives us the notion of skin depth:

    112

    0

    ===

    On the surface at z=0 we have Ex=E0at one skin depth z= we have Ex=E0/e

    jzz

    x eeEE

    = 0

    field has decayed to 1/eor 36.8% of value on thesurface.

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 73

    Plot of field into conductor

    2 .

    z

    E0

    E0/e

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 74

    Examples of skin depth

    f

    2

    0

    106162 ==

    .

    at 60Hz =8.5x10-3 mat 1MHz =6.6x10-5 mat 30GHz =3.8x10-7 m

    = 5.8x107 S/mCopper

    Seawaterf

    210522 =

    .

    at 1 kHz =7.96m

    = 4 S/m

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 75

    Characteristic or Intrinsic Impedance Zm

    Define this via the materialas before:

    jZ

    cm

    == 00

    But again, the conduction current predominates, which means the second term in the denominator is large. With thisapproximation we can arrive at:

    ( )

    jjZm+

    =+=1

    21 0

    For copper at 10GHz Zm= 0.026(1+j)

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 76

    Reflection from a metal surface

    So a reflection coefficient at metal-air interface is

    00

    0 since 1 ZZZZZZ

    mm

    m

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 77

    Conductors and dielectrics

    Materials can behave as either a dielectric or a conductor depending on the frequency.

    EjEH += recall

    Conduction current density

    Displacement current density

    3 choices >> displacement current >> conductor current dielectric displacement current conductor current quasi conductor

  • Massachusetts Institute of Technology RF Cavity and Components for Accelerators 78

    A rule for determining whether dielectric or conductor