Electromanetic Formula Sp2011

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Transcript of Electromanetic Formula Sp2011

  • Formula Sheet for Comprehensive Exam, Electromagnetic SystemsSp 2011

    EE540 Microwave Devices and Systems

    1. Maxwells EquationsDifferential Form Integral Form

    t

    = BE SBE dt

    dLS

    .. =t

    += DJH SDH d

    tIdL

    S

    .. +=v= D. = dvd vvol

    S

    SD.

    0. = B 0. =S

    dSB

    (Notation and symbols follow the convention in Elements of Electromagnetics, 5th ed., M. N. O. Sadiku)

    2. Plane Waves

    ==o

    o

    HE

    1

    =v

    HES = je=

    +

    =12

    12 +==+

    = 12

    12

    2

    je

    3. For Lossy Media Permittivity )"'("'and rro jj

    ===

    Loss Tangent'

    tan

    =

    Propagation constant

    +2

    '811'

    '2

    and

    4. Propagation in Good Conductor

    Propagation Constant pi fjj )1( +=+= and Skin Depth pi f1

    =

    5. Transmission Line Theory

    Telegraphers Eqns. s

    s Vdz

    Vd 22

    2

    = ))(( CjGLjR ++=

    )()(

    0 CjGLjRZ

    +

    +=

    +

    +=

    lZZlZZ

    ZZL

    Lin

    tanhtanh

    0

    00

    +

    +=

    ljZZljZZ

    ZZL

    Lin

    tantan

    0

    00

    0

    0

    ZZZZ

    L

    LL

    +

    =

    6. Coaxial Line )/ln(2)/ln(

    '2io

    io

    rrLandrr

    Cpi

    pi ==

    1

  • 7. Open-Wire Line

    +==

    )2/(cosh41

    4)2/(cosh' 1

    1 rdLandrdC pi

    (Notation and symbols follow the convention in Microwave Engineering, 3rd ed., D. M. Pozar, John Wiley & Sons)

    8. Smith Chart Equations

    22

    2

    11

    1

    +=+

    +

    L

    i

    L

    Lr rr

    r and ( )

    222 111

    =

    +LL

    ir xx 9. Microstrip Line

    Wd

    rre /121

    12

    12

    1+

    ++

    =

    Z 0 =

    +

    dW

    Wd

    e 48ln60

    for W/d 1

    Z 0 = ( )[ ]444.1/ln667.0393.1/120

    +++ dWdWepi

    for W/d 1

    =dW

    282

    A

    A

    ee

    for W/d < 2

    =dW }

    +

    +rr

    r BBB

    pi

    61.039.0)1ln(2

    1)12ln(12 for W/d > 2

    where, A =

    ++

    ++

    rr

    rrZ

    11.023.011

    21

    600 and B =

    rZ pi

    02377

    ( )

    ( ) ,/12tan10 mNpk

    re

    erd

    =

    and mNp

    WZRs

    c /0

    = where, R s = 2/010. Waveguides

    2

  • 11. Z-Parameters

    NV

    VV

    .

    .

    .2

    1

    =

    NNNN

    N

    N

    ZZZ

    ZZZZZZ

    .............................

    ......................................................

    21

    22221

    11211

    NI

    II

    .

    .

    .2

    1

    or [ ]V = [ ]Z [ ]I

    12. Y-Parameters

    NI

    II

    .

    .

    .2

    1

    =

    NNN

    N

    YY

    YYYY

    ..............................

    .......................

    1

    21

    11211

    NV

    VV

    .

    .

    .2

    1

    or [ ]I = [ ]Y [ ]V

    3

  • 13. S-Parameters

    NV

    VV

    .

    .

    .2

    1

    =

    NNN

    N

    SS

    SSSS

    ......................1

    21

    11211

    +

    +

    +

    NV

    VV

    .

    .

    .2

    1

    or [ ]V = [ ]S [ ]+V

    forVVVS K

    j

    iij 0==

    ++

    k j

    14. ABCD-Parameters

    15. Impedance Matching with Lumped Elements

    4

  • X = L

    o

    L

    oL

    BRZ

    RZX

    B+

    1 X = ( )LoL RZR LX

    B = 22

    22/

    LL

    LoLLoLL

    XRRZXRZRX

    +

    + B =

    ( )o

    LLo

    ZRRZ /

    16. Single Shunt Stub Tuning Y = G + jB

    22

    2

    )()1(

    tZXRtR

    GoLL

    L

    ++

    +=

    22

    2

    )(())((

    tZXRZtZXtXZtRB

    oLLo

    oLLoL

    ++

    +=

    For RL Zo

    oL

    oLLoLL

    ZRZXRZRX

    t

    +=

    /])[( 22

    For RL = Zo o

    L

    ZXt

    2

    =

    For t 0 td 1tan21

    =

    pi For t < 0 )tan(

    21 1 td += pipi

    Open Circuited Stub: )(tan2

    1 1o

    o

    YBl

    =

    pi Short Circuited Stub: )(tan

    21 1

    BYl os

    =

    pi 17. Single Series Stub Tuning Z = R + jX

    22

    2

    )()1(

    tYBGtG

    RoLL

    L

    ++

    +=

    22

    2

    )(())((

    tYBGYtYBtBYtGX

    oLLo

    oLLoL

    ++

    +=

    For GL Yo

    oL

    oLLoLL

    YGYBGYGB

    t

    +=

    /])[( 22

    For GL = Yo

    o

    L

    YBt2

    =

    For t 0 td 1tan21

    =

    pi For t < 0 )tan(

    21 1 td += pipi

    Open Circuited Stub: )(tan21 1

    XZl oo

    =

    pi Short Circuited Stub: )(tan

    21 1

    o

    s

    ZXl

    =

    pi 18. Double Stub Tuning

    5

  • B 1 = B L + ( )

    ttGYGtY LL

    220

    20 1 +

    B 2 = ( )

    tGYGtGtGYY

    L

    LLL 0222

    00 1 ++

    The open circuited stub length is found as: 0l =

    pi21

    tan 1

    0YB

    The short circuited stub length is found as: sl =

    pi21

    tan 1

    BY0 where B= B 1 or B 2

    19. Quarter-Wave Matching Transformer

    Z = LZZ0 and

    = m

    pi2

    2

    pi

    mmmff

    fff

    ff 4222)(2

    00

    0

    0

    ==

    =

    = 2

    0

    0

    2

    1 2

    1cos4

    ZZZZ

    L

    L

    m

    m

    pi

    20. Three Port and Four Port Microwave Components

    Three Port Network (such as T-Junctions) [S] =

    333231

    332221

    131211

    SSSSSSSSS

    oZZZ111

    21

    =+

    N-way Wilkinson Power Dividers

    3

    2

    031

    KKZZ o

    +=

    )1( 20032

    02 KKZZKZ +==

    )1(0 KKZR +=

    Four Port Network (such as Directional Couplers) [S] =

    44434241

    34333231

    24232221

    14131211

    SSSSSSSSSSSSSSSS

    Coupling dBPPC

    3

    1log10= Directivity dBPPD

    4

    3log10= Isolation dBPPI

    4

    1log10=

    6

  • EE541 Electro-Optics

    Formula adopted from the textbook, Optoelectronics and Photonics, Principles and Practices, Prentice Hall, S. O. Kasap, (2001).

    E = hE(r, t) = Eo cos (t k r + o) v = = (r o o)- 0.5 n = c/ v vg = d/ dk I = v r o Eo2/ 2 tan p = n2 / n1 R = (n2 - n1)2 / (n2 + n1)2T = 4 n2 n1 / (n2 + n1)2 t = 1 sin = 1.22 / D d sin = m , m = 0, 1, 2, ---- (4n1a cosm) / m = m No sinM = (n12 n22)0.5V = 2a (n12 n22)0.5/ M = 1 + Int(2V / ) for small VM V2/2 for large V/ L (n1 n2) / c dB = 10 log (Pin/ Pout) / L external = Pout(optical) / IV int = [Po(int)/ h]/ (I / e) N2 / N1 = exp[- (E2 E1) / kBT] = 2 o (1.386 kBT / Mc2) 0.5 gth = - 0.5 ln(R1R2) / L L = mm / 2n, m = 1, 2, 3 ---- =(Iph / e)/ (Po/ h)R = Iph / Po = en (e + h) in = [2e(Id + Iph) B] 0.5 SNR = Signal Power / Noise Power Iph = eGoA (ln + W + Le) I = -Iph + Io [exp(eV/ nkBT) 1] FF = ImVm / IscVoc ne() -2 = cos2 ()/ no2 + sin2 ()/ ne2

    7

  • = 2L (ne no)/ I = Io sin2 (0.5 V/ V /2)

    EE 641: RF Wireless Communication SystemsList of Commonly Needed Expressions and Relationships

    I. Fundamentals of RF Wireless Communication Systems1. Spectral Efficiency spec = Data rate Rb (bits/sec) / Transmission bandwidth B (Hz)

    2. Power Efficiency pow = Radiated power Prad (Watts) / Power drawn from source PDC (watts)

    3. Shannons channel capacity

    +=

    NSBC 1log 2 bits/sec

    II. Electromagnetic Waves and Radiators

    1. Maxwells Equations( ) ( ), , . , . 0H EE H J E H

    t t

    = =+ = =

    2. Wave Equation2 2 2 20, 0E E H H + = + =

    3. Wave impedance of the medium/ 120 /rel rel pi = =

    4. Propagation constant of the medium = + j = j ( 2 ) ,if = r + j i, then = + j = j [ r(1 j i/ r ) ]

    5. Phase velocity of the electromagnetic waves in the medium (ohms) = ( /) = 120pi = 377

    6. Poynting VectorS = E H

    7. Radiation Intensity due to a source at the originU(r, , ) = r2 . S (r, , )

    8. Power radiated from a source at the origin

    8

  • = =

    =

    pi

    pi

    0

    2

    0

    sin),( ddUPrad9. Far Field condition

    Conditions of Far Field : R 2D2 / R >> DR >>

    10. Fields due to a Hertzian Dipole of length z and current I, placed at origin along z axis.E(r, , ) = Er(r, , ) ar + E (r, , ) a + 0 a

    2

    2 3

    cos cos( , , )2 ( ) ( )

    j rr

    I zE r e jr r

    pi

    =

    2

    2 3

    sin sin sin( , , )4 ( ) ( )

    j rj I z jE r er r r

    pi