17 Broadband Anti-Reflection Coating Design

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    Broadband Anti-reflectionCoating Design

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    UCF

    Quarter Wave Transformer (1)

    4g

    Z0 Z01

    0

    2

    01 ZR

    ZZ

    L

    in

    )(realRL

    001 ZRZ L

    Require the load is resistive.

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    UCF

    Quarter Wave Transformer (2)

    Narrow bandwidth

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    UCF

    Theory of Small Reflection (1)

    d

    T21T12

    Z1 Z2 1212

    1

    ZZ

    ZZ

    12

    21

    2

    121

    2

    1

    ZZ

    Z

    T

    21

    1

    212

    2

    1

    ZZ

    Z

    T

    je

    je

    je

    j

    e

    je

    L

    LZ

    312T

    3

    12T 3

    2

    11

    2

    23

    ZZ

    ZZ

    L

    L

    dkz

    21T

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    UCF

    Theory of Small Reflection (2)

    ...422

    32112

    2

    321121 jj eTTeTT

    n

    n

    jj

    eeTT

    0

    2

    32

    2

    321121

    j

    j

    eeTT

    2

    32

    2

    3211211

    1

    11212112 1,1,Since TT

    j

    j

    ee

    231

    2

    3111

    1

    111

    j

    j

    e

    e2

    31

    2

    31

    1

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    UCF

    Theory of Small Reflection (3)

    When the differences between Z1 & Z2, and Z2 & ZL are small,

    1|&|3| are small.

    If |13|

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    UCF

    Multi-section Transformer (1)

    ,01

    010ZZ

    ZZ

    Z0 Z1 Z2 ZN

    ZL

    0 1 2N

    ,1

    1

    nn

    nnn

    ZZ

    ZZ

    NL

    NLNZZ

    ZZ

    N

    n

    jnn

    jNN

    jj eeee0

    2242

    210 ...

    For small reflections:

    Note: at center frequency.2

    n

    nn

    n

    n

    nn

    nnnZ

    ZZxxxZ

    Z

    ZZ

    ZZ

    11

    1

    1 ,)1ln(since,ln2

    1

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    UCF

    Multi-section Transformer (2)

    ...2210 NjNjjNjNjN eeeee

    In designing wide-band transformers, we force the

    reflection coefficients to be symmetrical, i.e.

    Note, this does not imply that theZns are symmetrical.

    ,...,, 22110 NNN

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    UCF

    Multi-section Transformer (3)

    When N is odd,

    When N is even,

    Can synthesize any desired reflection coefficient response as a

    function of frequency by properly choosing ns and usingenough sections (N).

    2

    102

    1...2cos...2coscos2 Nn

    jN nNNNe

    cos...2cos...2coscos22

    110 Nn

    jN nNNNe

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    UCF

    Specifications on Bandwidth

    Design specs: center frequency f0 , bandwidth f ,maximum allowable reflection coefficient m

    Fractional bandwidth:

    mm

    f

    f 42

    2/

    2

    00

    0

    24 f

    f

    m

    or

    f

    m

    m

    0 2/0

    m 2

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    UCF

    Binomial Design (1)

    satisfies the above 2 conditions

    1-1,2,...Nnfor0

    2when0

    2

    n

    n

    d

    d

    NjeA 21

    Njjj eeeA

    NNA cos2||

    Require:

    We can let

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    UCF

    Binomial Design (2)

    Design specs: center frequency , bandwidth ,

    maximum allowable reflection coefficient

    Need to determine: (1) order N(2) all Zns.

    00

    20 ZZ

    ZZ

    A L

    LN

    mN

    L

    LmZZZZ cos

    0

    0

    mNN

    mm A cos2

    mf

    0f

    Letf=0 or

    At m

    0

    02ZZ

    ZZA

    L

    LN

    Step 1: determine N

    )ln(cos

    ln0

    0

    m

    L

    LmZZ

    ZZ

    N

    Round up to the next integer!

    (MatLab: ceil)

    Always overdesign a little bit.

    0

    24 f

    fm

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    UCF

    Binomial Design (3)

    !!

    !where,1

    0

    22

    nnN

    NCeCAeA Nn

    N

    n

    jnN

    n

    Nj

    ),1,:(Note 110 NCCCCCAC N

    N

    NNN

    nN

    N

    n

    N

    nn

    Step 2: determine Zn

    N

    n

    jn

    ne0

    2

    MatLab: nchoosek(N,n)

    The essence using this approximation can make the design to be consistent.We obtain N+1 n s for n = 0,1,N. But we need to find only N n s for n = 1,2,N.

    0

    )1(

    0

    0 ln22Z

    Z

    ZZ

    ZZA LN

    L

    LN

    n

    nn

    n

    n

    nn

    nnn

    ZZZxxx

    ZZ

    ZZZZ

    11

    1

    1 ,)1ln(since,ln21

    neZZ nn

    2

    1 use iteratively

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    UCF

    Binomial Design (5)

    00 ,

    x

    zik ki ak 1

    y

    rkr kak 1

    tkt k ak 2

    i

    r t10 , 20 , 10 , N N ,0

    10 , N

    1d 2d 1Nd Nd

    Design a multi-section binomial transformer for 45o TE incidence

    from air to a thermal plastic antenna radome material with r=3 atcenter frequency of 5 GHz. Require SWR = 1.01 from 4GHz to

    6GHz. All the materials are non-magnetic.

    Standing wave ratio SWR =m

    m

    1

    1

    1SWR

    1SWRor

    m

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    Binomial Design (6)

    0

    0

    02

    0

    2

    220

    2

    00

    2

    0

    22

    0

    2

    0

    22

    0

    sin

    2

    sin1

    sin

    1

    sin

    sinsin

    f

    c

    Zk

    Z

    ZZ

    kZ

    kkkkkk

    n

    irnzn

    gn

    i

    n

    rn

    irn

    nn

    irnzn

    n

    irnirnxrnzn

    Details in binomial.m

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    UCF

    Chebyshev Design (1)

    Ripple in passband!

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    UCF

    Chebyshev Design (2)

    Chebyshev function

    nTn cos)(cos

    )2cos(cos)1cos(cos2 nnnsince

    )(cos)(coscos2)(cos 21 nnn TTT

    Let cosx

    we have

    )()(2)( 21 xTxxTxT nnn

    xxxT

    xxTxxT

    xT

    34)(

    12)()(

    1)(

    3

    3

    2

    2

    1

    0

    If n is even, Tn (x) is even.

    If n is odd, Tn (x) is odd.

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    UCF

    Chebyshev Design (3)

    )coscos()( 1xnxTn

    )coshcosh()( 1xnxTn

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    UCF

    Chebyshev Design (4)

    )cos

    cos(

    ...2cos...2coscos2 10

    mN

    jN

    n

    jN

    TAe

    nNNNe

    mm

    0 2/0

    m 2

    |||)(| Amm

    obviously

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    UCF

    Chebyshev Design (5)

    Key: expansion of )cos(sec)cos

    cos(

    mN

    m

    N TT in terms of cos(n

    )()(2)( 21 xTxxTxT nnn

    )cos()]cos(cos2[

    )cos()cos(cos2)cos(

    21

    21

    pTpTp

    pTpTppT

    nn

    nnn

    Letmp sec

    from

    )1cos()1cos(coscos2 mmmsince

    )]cos(cos2)cos( 1 pTpS nn Let

    This means the coefficient of cosm ( ) inTn-1 will add to

    the coefficients of cos(m+1)and cos(m1) in Sn.

    MatLab program: Chebycoeff.m

    )2cos()1()cos(

    ,cos)cos(,1)cos(

    22

    2

    10

    pppT

    ppTpT

    for0m

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    UCF

    Chebyshev Design (6)

    Design specs: center frequency , bandwidth ,

    maximum allowable reflection coefficient

    Need to determine: (1) order N(2) all Zns.

    00

    )(sec0 ZZ

    ZZ

    AT L

    L

    mN

    mf

    0f

    Letf=0 or

    Step 1: determine N

    Round up to the next integer!

    Always overdesign a little bit.

    ||Am )coshcosh()( 1xnxTn

    0

    01)(secZZ

    ZZT

    L

    L

    m

    mN

    0

    01 1

    ))(seccoshcosh( ZZ

    ZZ

    NL

    L

    m

    m

    )(seccosh

    1cosh

    1

    0

    01

    m

    L

    L

    m ZZ

    ZZ

    N

    0

    24 f

    fm

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    UCF

    Chebyshev Design (7)

    )cos(sec

    ...2cos...2coscos2 10

    mN

    jN

    n

    jN

    TAe

    nNNNe

    ,...2,1,0

    2 )2cos()cos(secn

    nNmN nNDT Assume

    2

    2nNnNnAD

    ,...,, 22110 NNN

    Step 2: determine Zn

    If N is even 02

    ADN

    ,

    1

    1

    nn

    nnn

    ZZ

    ZZ

    From

    n

    nnn ZZ

    1

    11 use iteratively

    n

    nnn ZZ

    1

    11or

    0

    0

    1

    ))(seccoshcosh(

    1

    ZZ

    ZZ

    N

    A

    L

    L

    m

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    UCF

    Chebyshev Design (8)

    We obtain N+1 n s for n = 0,1,N. But we need to find only N n s for n = 1,2,N.There may always be a little error in the design. Since the whole methodology is

    developed from small reflection approximation, we need to check using rigorous

    impedance transformation formula or chain matrix approach and find whether thespecs are satisfied.

    (1) If N is odd:

    1

    1

    1 1

    1

    nn

    nn ZZUse for n = 1, , (N-1)/2

    n

    nnn ZZ

    1

    11Use for n = N, N-1, , (N+3)/2

    2

    3

    2

    1

    2

    1 NNN ZZZAlthough this will grantee

    2

    1

    2

    1 NN

    the value will deviate a little from the designed value.

    cos...2cos...2coscos22

    110 Nn

    jN nNNNe

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    UCF

    Chebyshev Design (9)

    (2) If N is even:

    1

    11

    1

    1

    n

    nnn ZZUse for n = 1, , N/2

    n

    nnn ZZ

    1

    11Use for n = N, N-1, , N/2+1

    This will never use2N whose actual value will deviate from the designed value.

    2

    102

    1...2cos...2coscos2 Nn

    jN nNNNe

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    UCF

    Chebyshev Design (10)

    m

    Find actual fractional bandwidth (due to overdesign)

    m

    f

    f 42

    0

    0

    01 1))(seccoshcosh(ZZ

    ZZN

    L

    L

    m

    m

    0

    01 1cosh1

    coshsecZZ

    ZZ

    N L

    L

    m

    m

    From the definition of Zn, we can find rn and then find kzn atf0.Then from kzndn = find dn.

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    UCF

    Tapered Line

    since

    dzZ

    Z

    dz

    deL dzzkj

    z

    z

    0

    0

    )(2

    ln2

    1)( 0