Adapted from notes by ECE 5317-6351 Prof. Jeffery Notes/Notes 5 5317... Prof. David R. Jackson Dept.

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Transcript of Adapted from notes by ECE 5317-6351 Prof. Jeffery Notes/Notes 5 5317... Prof. David R. Jackson Dept.

  • Prof. David R. Jackson Dept. of ECE

    ECE 5317-6351 Microwave Engineering

    Fall 2019

    1

    Notes 5 Smith Charts

    Adapted from notes by Prof. Jeffery T. Williams

  • Recall: ( ) ( ) ( )( )

    ( ) ( ) ( )( )

    ( ) ( )( ) ( ) ( )

    2 0 0

    20 0

    0 0

    2

    0 02

    1 1

    1 1

    11 1 1

    z z z L

    z z z L

    z L

    z L

    V z V e e V e z

    V VI z e e e z Z Z

    V z zeZ z Z Z I z e z

    γ γ γ

    γ γ γ

    γ

    γ

    + − + + −

    + + − + −

    +

    +

    = + Γ = + Γ

    = − Γ = − Γ

     + Γ + Γ = = =   − Γ − Γ   

    Generalized reflection Coefficient: ( ) 2 zLz e γ+Γ = Γ

    Generalized Reflection Coefficient

    2

    0

    0

    L L

    L

    Z Z Z Z

    − Γ =

    +

    0,Z β ( )V z

    ( )I z

    0z =z

    LZ +

  • ( )

    ( ) ( )

    2

    2L

    z L

    j z L

    R I

    z e

    e e

    z j z

    γ

    φ γ

    +

    +

    Γ = Γ

    = Γ

    = Γ + Γ

    Lossless transmission line (α = 0)

    ( ) ( )2Lj zLz e φ β+Γ = Γ

    Generalized Reflection Coefficient (cont.)

    { }Re 0 1L LZ ≥ ⇒ Γ ≤

    ( ) ( ) ( ) ( )

    0

    0

    0

    0

    L L L

    L L

    L L

    L L

    R jX Z R jX Z

    R Z jX R Z jX

    + − Γ =

    + +

    − + =

    + +

    ( ) ( )

    2 2 2 0

    2 2 0

    1L LL L L

    R Z X R Z X

    − + ⇒ Γ = ≤

    + +

    Proof:

    3

    Different forms for Γ(z) Magnitude property of Γ(z)

  • Complex Γ Plane ( )

    ( ) ( ) ( )

    ( )

    ( )

    ( )

    2

    2

    2

    2

    L

    L

    R I

    j z L

    j z L

    j d L

    j d L

    z

    z j z

    e

    e

    e

    e

    β

    φ β

    φ β

    β

    +

    +

    Γ = Γ

    = Γ + Γ

    = Γ

    = Γ

    = Γ

    = Γ

    Increasing d (moving towards

    generator)

    4

    ReΓ

    Im Γ

    2L dφ β−

    Γ

    Lossless line

    z d= − d = distance from load

    1

    Note: Going λ/2 on the line

    corresponds to going all the way around the Smith chart.

    2 dβ

    Clockwise movement!

  • ( ) ( )( )0 1 1

    z Z z Z

    z  + Γ

    =  − Γ 

    ( ) ( ) ( )( )0 1 1n

    Z z z Z z

    Z z  + Γ

    ≡ =  − Γ  Define

    n n nZ R jX= +

    Hence we have:

    Z Chart

    ( ) ( )

    1 1

    R I n n

    R I

    j R jX

    j  + Γ + Γ

    + =  − Γ + Γ 

    Next, multiply both sides by the RHS denominator term and equate real and imaginary parts. Then solve the resulting equations for ΓR and ΓI in terms of Rn or Xn. This gives two equations.

    5

    Note: The z dependence is being

    suppressed here.

    The Z chart is the “usual” Smith chart.

    Start with

  • 1) Equation #1: 2 2

    2 1 1 1

    n R I

    n n

    R R R

        Γ − + Γ =   + +   

    Equation for a circle in the Γ plane

    ,0 1 1

    1

    n

    n

    n

    R R

    R

      =  + 

    = +

    Center

    Radius

    Z Chart (cont.)

    6

    1

    ΓR

    ΓI

  • ( ) 2 2

    2 1 11R I n nX X

        Γ − + Γ − =   

       

    Equation for a circle in the Γ plane

    11,

    1 n

    n

    X

    X

      =  

     

    =

    Center

    Radius

    2) Equation #2:

    7

    1

    ΓR

    ΓI

    Z Chart (cont.)

  • Short-hand version

    8

    Γ plane Γ plane

    Rn = 1

    Xn = 1

    Xn = -1

    Z Chart (cont.)

  • 9

    Open ckt. (Γ=1)

    Imag. (reactive) impedance

    Match pt. (Γ=0)

    Real impedance

    Short ckt. (Γ= −1)

    Inductive (Xn > 0)

    Capacitive (Xn < 0)

    Γ plane Rn = 1

    Xn = 1

    Xn = -1

    Γ plane

    Z Chart (cont.)

  • Note: ( ) ( ) ( ) ( )0

    11 1 1

    z Y z

    Z z Z z  − Γ

    = =  + Γ 

    ( )( ) ( )( )0

    1 1

    z Y

    z  + −Γ

    =   − −Γ 

    ( ) ( ) ( )( )( )( ) ( ) ( )0 1 1n n n

    zY z Y z G z jB z

    Y z  + −Γ

    ⇒ ≡ = = +  − −Γ 

    ( )0 01 /Y Z≡

    Admittance Calculations with the Z Chart

    Define:

    ( ) ( )z z′Γ ≡ − Γ ( ) 1

    1n Y z

    ′+ Γ =  ′− Γ 

    Same mathematical form as for Zn:

    Conclusion: The same Smith chart can be

    used as an admittance calculator.

    10

    ( ) 1 1n

    Z z + Γ =  − Γ 

  • 11

    Short ckt. (Γ′ =1)

    Imag. (reactive) admittance

    Match pt. (Γ′ =0)

    Real admittance

    Open ckt. (Γ′ = −1)

    Capacitive (Bn > 0)

    Inductive (Bn < 0)

    Γ′ planeBn = 1

    Bn = -1

    Gn = 1

    Γ′ plane

    Admittance Calculations with the Z Chart (cont.)

  • Impedance or Admittance Calculations with the Z Chart

    12

    The Smith chart can be used for either impedance or

    admittance calculations, as long as we are consistent.

    The complex plane is either the Γ plane or the Γ′ plane.

    Normalized impedance or admittance coordinates

  • As an alternative way to do admittance calculations, we can continue to use the original Γ plane, and add admittance curves to the chart.

    ( ) ( )( )( )( ) ( ) ( ) 1 1n n n

    z Y z G z jB z

    z  + −Γ

    = = +  − −Γ 

    Y Chart

    ( ) ( )( )( )( ) ( ) ( ) 1 1n n n

    z Z z R z jX z

    z  + Γ

    = = +  − Γ 

    Compare with previous Smith chart derivation, which started with this equation:

    Side note: A 180o rotation on a Smith chart makes a normalized impedance become its reciprocal. 13

    Rn = 1 circle, rotated 180o, becomes Gn = 1 circle. Xn = 1 circle, rotated 180o, becomes Bn = 1 circle.

    ( ) ( ) ( ) ( )

    n n

    n n

    R z G z

    X z B z

    → ( ) ( )z zΓ → −Γ (rotation of 180o)

    Examples:

  • Y Chart (cont.)

    14

    Open ckt.

    Match pt.

    Gn = 0

    Short ckt.

    Inductive (Bn < 0)

    Capacitive (Bn > 0)

    Gn = 1

    Bn = +1

    Bn = -1

    Bn = 0

    Γ planeΓ plane

    The Y chart is the “mirror image” of the usual Smith chart.

  • Short-hand version

    Γ plane

    15

    Gn = 1

    Bn = -1

    Bn = 1

    Y Chart (cont.)

  • All Four Possibilities for Smith Charts

    16

    Z chart, used for admittance

    Γ′ plane

    Z chart, used for impedance

    Γ plane

    The first two are the most common.

    The third is sometimes convenient.

    The fourth is almost never used.

    1 2

    Γ′ plane

    4 Y chart, used for impedance

    Γ plane

    3 Y chart, used for admittance

  • ZY Chart

    17

    This is convenient for doing matching problems that involve both series and shunt elements (done later).

    Short-hand version

    Γ plane

    Gn = 1 Rn = 1

    Xn = 1

    Xn = -1

    Bn = -1

    Bn = 1

    Inductive

    Capacitive

  • The SWR is given by the value of Rn on the positive real axis of

    the Smith chart (Rnmax).

    Standing Wave Ratio

    Proof: 1

    SWR 1

    L

    L

    + Γ =

    − Γ

    ( ) ( )

    2

    2

    2

    2

    1 1

    1 1 1 1

    L

    L

    n

    j z L

    j z L

    j j z L

    j j z L

    z Z

    z

    e e e e e e

    β

    β

    φ β

    φ β

    +

    +

    +

    +

    + Γ =

    − Γ

    + Γ =

    − Γ

    + Γ =

    − Γ

    18

    max 12 0 1

    L L n

    L

    z Rφ β + Γ

    + = ⇒ = − Γ

    max n nR R=

    2 0L zφ β+ =2L zφ β+

    Γ plane

    ( )zΓ

  • At this link:

    http://www.sss-mag.com/topten5.html

    Download the following zip file: smith_v191.zip

    Extract the following files:

    smith.exe mith.hlp smith.pdf

    This is the application file

    Electronic Smith Chart

    19

    http://www.sss-mag.com/topten5.html

  • 0 50 100 50L

    Z Z j

    = Ω = + Ω

    , 0

    2 1LL n ZZ j Z

    = = +

    ( )

    / 4 0.4 0.2

    / 4 20 10

    g

    n

    g

    d Z j

    Z j

    λ

    λ

    =