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Transmission Line Theory KAU RFIC MMIC LAB

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Transmission Line Theory KAU RFIC MMIC LAB Powered By Docstoc
					      Chapter 2.
Transmission Line Theory



       Sept. 29th, 2008


                           1
2.1 Transmission Lines
• A transmission line is a distributed-parameter
  network, where voltages and currents can vary
  in magnitude and phase over the length of the
  line.
Lumped Element Model for a Transmission Line
• Transmission lines usually consist of 2 parallel
  conductors.
• A short segment Δz of transmission line can be
  modeled as a lumped-element circuit.

                                                 2
Figure 2.1
Voltage and current definitions and equivalent circuit for an
incremental length of transmission line. (a) Voltage and        3
current definitions. (b) Lumped-element equivalent circuit.
   • R = series resistance per unit length for both
     conductors
   • L = series inductance per unit length for both
     conductors
   • G = shunt conductance per unit length
   • C = shunt capacitance per unit length
   • Applying KVL and KCL,
                                 i( z, t )
 v( z, t )  Rzi( z, t )  Lz              v( z  z, t )  0 (2.1a)
                                    t
                                     v( z  z, t )
i( z, t )  Gzv( z  z, t )  C z                  i ( z  z , t )  0 (2.1b)
                                            t
                                                                             4
• Dividing (2.1) by Δz and Δz  0,
  v( z, t )                    i( z, t )
               Ri( z, t )  L            (2.2a)
    z                            t
  i( z, t )                    v( z, t )
              Gv( z, t )  C              (2.2b)
    z                             t
 Time-domain form of the transmission line,
  or telegrapher, equation.
• For the sinusoidal steady-state condition with
  cosine-based phasors,
  dV ( z )
             ( R  j L) I ( z ) (2.3a)
    dz
  dI ( z )
            (G  jC )V ( z ) (2.3b)
   dz                                                5
Wave Propagation on a Transmission Line
• By eliminating either I(z) or V(z):
   d 2V ( z )                      d 2 I ( z)
                2V ( z ) (2.4a)              2 I ( z ) (2.4b)
     dz 2                            dz 2

  where     j  (R  jL)(G  jC) the
  complex propagation constant. (α = attenuation
  constant, β = phase constant)
• Traveling wave solutions to (2.4):
  V ( z )  V0 e  z +V0 e   z , I ( z )  I 0 e  z  I 0 e   z (2.6)
                                                              


        Wave                      Wave
    propagation in           propagation in -
     +z directon                z directon
                                                                                 6
• Applying (2.3a) to the voltage of (2.6),
                   
  I ( z)             V0 e  z +V0 e   z 
             R  j L                         

• If a characteristic impedance, Z0, is defined as
         R  j L        R  j L                  V0         V0
  Z0                            , (2.7)            
                                                        Z0   
                        G  jC                   I0          I0

• (2.6) can be rewritten
           V0  z V0  z
  I ( z)     e       e     (2.8)
           Z0       Z0


                                                                     7
• Converting the phasor voltage of (2.6) to the
  time domain:
  v( z, t )  V0 cos(t   z   + )e z  V0 cos(t   z    )e z (2.9)

• The wavelength of the traveling waves:
       2
  =         (2.10)
         

• The phase velocity of the wave is defined as
  the speed at which a constant phase point
  travels down the line,
         dz 
  vp =     = = f      (2.11)
         dt 
                                                                           8
Lossless Transmission Lines
• R = G = 0 gives     j   j              LC   or
     LC ,   0 (2.12)
       L
  Z0 =             (2.13)
       C
• The general solutions for voltage and current
  on a lossless transmission line:
  V ( z )  V0 e  j  z +V0 e j  z ,
           I 0  j  z
  I ( z)      e         I 0 e j  z (2.14)
           Z0

                                                          9
                                     2         2
• The wavelength on the line:   =
                                      
                                           =          (2.15)
                                                LC

                                               1
• The phase velocity on the line:   vp =     =        (2.16)
                                              LC




                                                        10
 2.2 Field Analysis of Transmission Lines
 • Transmission Line Parameters




Figure 2.2 (p. 53)
                                                     11
Field lines on an arbitrary TEM transmission line.
• The time-average stored magnetic energy for 1
  m section of line:
         
         4 S
  Wm           H  H ds

• The circuit theory gives           Wm  L | I 0 |2 / 4

   L   2 S H  H ds
             | I0 |

                             
• Similarly, We  S E  E ds,       We  C | V0 |2 / 4
                 4
        
                      
                  
     C           2
                          E  E ds
             | V0 |   S




                                                           12
• Power loss per unit length due to the finite
  conductivity (from (1.130))
         Rs
  Pc 
         2    
              C1  C2
                        H  H dl

• Circuit theory                       Pc  R | I 0 |2 / 2
       Rs                                  (H || S)
  R
     | I 0 |2     
                  C1  C2
                            H  H dl


• Time-average power dissipated per unit length
  in a lossy dielectric (from (1.92))
          
  Pd 
          2       
                  S
                      E  E ds


                                                              13
• Circuit theory  Pd  G | V0 |2 / 2
         
   G
        | V0 |2   
                  C1  C2
                            E  E  ds


• Ex 2.1 Transmission line parameters of a
  coaxial line

• Table 2.1




                                             14
The Telegrapher Equations Derived form Field
Analysis of a Coaxial Line
• Eq. (2.3) can also be obtained from ME.
• A TEM wave on the coaxial line: Ez = Hz = 0.
• Due to the azimuthal symmetry, no φ-variation
   ə/əφ = 0
• The fields inside the coaxial line will satisfy
  ME.
   E   j H
   H  j E
  where      j 
                                                15
   E          ˆ E
                  1                               ˆ

 ˆ          z
               ˆ       (  E )   j (  H    H )
                                           ˆ
    z    z       
   H ˆ H       1                             ˆ

 ˆ          z ˆ       (  H )  j (  E   E )
                                          ˆ
    z     z       
Since the z-components must vanish,
       f ( z)            g ( z)
E             , H 
                         
From the B.C., Eφ = 0 at ρ = a, b  Eφ = 0 everywhere
H  0

E                      H                 E 
                                                    h( z )
         j H ,                j E           
 z                       z
                                                             16
h( z )                  g ( z )
          j g ( z ),            j h( z ),
 z                       z
The voltage between 2 conductors
             b                                     b   d            b
V ( z)            E (  , z ) d   h ( z )           h( z ) ln
              a                               a                 a
The total current on the inner conductor at ρ = a
             2
I ( z)          H  ( a, z ) ad  2 g ( z )
          0

V ( z )      ln b / a          I ( z )                      2V ( z )
         j             I ( z ),            j (   j )
 z             2                 z                           ln b / a
V ( z )                  I ( z )
           j LI ( z ),           (G  jC )V ( z )
 z                        z
                                                                         17
Propagation Constant, Impedance, and Power Flow
for the Lossless Coaxial Line
• From Eq. (2.24)
 2 E
             E  0
             2
                                     2   2 
 z   2



• For lossless media,       LC
• The wave impedance
            
           E
  Zw           /                             Ex 2.1
       H    

• The characteristic impedance of the coaxial
  line     V0 E ln b / a  ln b / a   ln b / a
            Z0                                   
                   I0       2 H           2              2   18
• Power flow ( in the z direction) on the coaxial
  line may be computed from the Poynting
  vector as
                                       
   1              1 2 b          V0 I 0               1
P   E  H  ds   
           
                                               d d  V0 I 0
                                                             

   2 s            2  0   a 2 2 ln b / a          2

• The flow of power in a transmission line takes
  place entirely via the E & H fields between
  the 2 conductors; power is not transmitted
  through the conductors themselves.


                                                          19
2.3 The Terminated Lossless Transmission Lines




 The total voltage and current on the line
 V ( z )  V0 e  j  z +V0 e j  z ,
          V0  j  z V0  j  z
 I ( z)     e           e       (2.34)
          Z0           Z0



                                                 20
• The total voltage and current at the load are
  related by the load impedance, so at z = 0
       V (0) V0  V0                       Z L  Z0 
  ZL =      =         Z0                 V           V0
                                              Z L  Z0
                                          0
       I (0) V0  V0

• The voltage reflection coefficient:
    V0 Z L  Z0
                          (2.35)
    V0   Z L  Z0

• The total voltage and current on the line:
  V ( z )  V0 e j  z +e j  z  ,
                                   
           V0  j  z
  I ( z)     e        e j  z  (2.36)
           Z0                    
                                                            21
• It is seen that the voltage and current on the
  line consist of a superposition of an incident
  and reflected wave.  standing waves
• When Γ= 0  matched.
• For the time-average power flow along the line
  at the point z:
                                       2

  Pavg    Re V ( z ) I ( z ) 
          1
          2
                                 1V 0

                                  2 Z0
                                               2 j  z
                                       Re 1   e          e 2 j z
                                                                      
                                                                         2
                                                                             
                 2

         
           1V
           2 Z0
               0
                      1   
                            2




                                                                         22
• When the load is mismatched, not all of the
  available power from the generator is delivered
  to the load. This “loss” is return loss (RL):
  RL = -20 log|Γ| dB
• If the load is matched to the line, Γ= 0 and
  |V(z)| = |V0+| (constant)  “flat”.
• When the load is mismatched,
 V ( z)  V0 1  e2 j z  V0 1  e2 j l  V0 1   e j ( 2  l ) (2.39)

 Vmax  V0 1    , Vmin  V0 1    (2.40)


                                                                             23
• A measure of the mismatch of a line, called the
  voltage standing wave ratio (VSWR)
        1          (1< VSWR<∞)
  SWR 
         1 

• From (2.39), the distance between 2 successive
  voltage maxima (or minima) is l = 2π/2β = λ/2
  (2βl = 2π), while the distance between a
  maximum and a minimum is l = π/2β = λ/4.
• From (2.34) with z = -l,
         V0 e j l
  (l )   j l  (0)e2 j l (2.42)
         V0 e
                                               24
• At a distance l = -z,
         V (l )     V0     e j  l  e  j  l       1  e 2 j  l
  Z in           Z0       j l          j l 
                                                      Z0                   (2.43)
         I (l )     V0       e  e                    1  e 2 j  l
          ( Z L  Z 0 )e j  l  ( Z L  Z 0 )e  j  l
      Z0
          ( Z L  Z 0 )e j  l  ( Z L  Z 0 )e  j  l
          Z L cos  l  jZ 0 sin  l
      Z0
          Z 0 cos  l  jZ L sin  l
          Z L  jZ 0 tan  l
      Z0                                                                   (2.44)
          Z 0  jZ L tan  l


 Transmission line impedance equation

                                                                                     25
Special Cases of Terminated Transmission Lines
• Short-circuited line
  ZL = 0  Γ= -1
  V ( z )  V0 e j  z  e j  z   2 jV0 sin  z ,
                                   
           V0  j  z                V0
  I ( z)     e
                       e j z   2
                                         cos  z
           Z0                         Z0

  Zin  jZ 0 tan  l       (2.45)




                                                            26
Figure 2.6
(a) Voltage, (b) current,
and (c) impedance (Rin
= 0 or ) variation
along a short-circuited
transmission line.




                            27
• Open-circuited line
  ZL = ∞  Γ= 1
  V ( z )  V0 e j  z  e j  z   2V0 cos  z ,
                                   
           V0  j z              2 jV0
  I ( z)     e
                      e j z  
                                          sin  z       (2.46)
           Z0                        Z0
  Zin   jZ 0 cot  l




                                                                  28
Figure 2.8
(a) Voltage, (b) current, and
(c) impedance (Rin = 0 or )
variation along an open-
circuited transmission line.




                                29
• Terminated transmission lines with special
  lengths.
• If l = λ/2, Zin = ZL.
• If the line is a quarter-wavelength long, or, l =
  λ/4+ nλ/2 (n = 1,2,3…), Zin = Z02/ZL. 
  quarter-wave transformer




                                                  30
Figure 2.9 (p. 63)
Reflection and transmission at the junction of two
transmission lines with different characteristic impedances.   31
2.4 The Smith Chart
• A graphical aid that is very useful for solving
  transmission line problems.
Derivation of the Smith Chart
• Essentially a polar plot of the Γ(= |Γ|ejθ).
• This can be used to convert from Γto
  normalized impedances (or admittances), and
  vice versa, using the impedance (or admittance)
  circles printed on the chart.


                                               32
Figure 2.10 (p. 65)
The Smith chart.      33
• If a lossless line of Z0 is terminated with ZL, zL
  = ZL/Z0 (normalized load impedance),
     zL  1                              1   e j
            e j             zL 
     zL  1                              1   e j

• Let Γ= Γr +jΓi, and zL = rL + jxL.
             (1   r )  j i               1   2  i2                   2 i
  rL  jxL                            rL          r           xL 
             (1   r )  ji               (1   r )2  i2          (1   r ) 2   i2

                 2                2
          rL            1 
   r           i  
                     2
                                   ,
        1  rL          1  rL 
                          2        2
                   1   1 
   r  1   i     
           2

                   xL   xL 
                                                                                   34
• The Smith chart can also be used to
  graphically solve the transmission line
  impedance equation of (2.44).
             1  e 2 j  l
  Z in  Z 0                   (2.57)
             1  e 2 j  l

• If we have plotted |Γ|ejθ at the load, Zin seen
  looking into a length l of transmission line
  terminates with zL can be found by rotating the
  point clockwise an amount of 2βl around the
  center of the chart.

                                                35
• Smith chart has scales around its periphery
  calibrated in electrical lengths, toward and
  away from the “generator”.
• The scales over a range of 0 to 0.5 λ.




                                                 36
 Ex 2.2 ZL = 40+j70, l = 0.3λ, find Γl, Γin and Zin




Figure 2.11 (p. 67)
Smith chart for Example 2.2.                          37
The Combined Impedance-Admittance Smith Chart
• Since a complete revolution around the Smith
  chart corresponds to a line length of λ/2, a λ/4
  transformation is equivalent to rotating the
  chart by 180°.
• Imaging a give impedance (or admittance)
  point across the center of the chart to obtain
  the corresponding admittance (or impedance)
  point.



                                                     38
 Ex 2.3 ZL = 100+j50, YL, Yin ? when l = 0.15λ




Figure 2.12 (p. 69)
ZY Smith chart with solution for Example 2.3.    39
The Slotted Line
• A transmission line allowing the sampling of E
  field amplitude of a standing wave on a
  terminated line.
• With this device the SWR and the distance of
  the first voltage minimum from the load can be
  measured, from this data ZL can be determined.
• ZL is complex  2 distinct quantities must be
  measured.
• Replaced by vector network analyzer.

                                              40
Figure 2.13 (p. 70)
An X-band waveguide slotted line.   41
• Assume for a certain terminated line, we have
  measured the SWR on the line and lmin , the
  distance from the load to the first voltage
  minimum on the line.
         SWR  1
  |  |
         SWR  1
• Minimum occurs when e j ( 2 l )  1
• The phase of Γ =     2 lmin
• Load impedance Z L  Z0 1  
                             1 




                                                  42
Ex 2.4
• With a short circuit load, voltage minima at z =
  0.2, 2.2, 4.2 cm
• With unknown load, voltage minima at z =
  0.72, 2.72, 4.72 cm
• λ = 4 cm,
• If the load is at 4.2 cm, lmin = 4.2 – 2.72 = 1.48
  cm = 0.37 λ
    ?,  ?, Z L  ?


                                                  43
Figure 2.14 (p. 71)
Voltage standing wave patterns for Example 2.4. (a) Standing
wave for short-circuit load. (b) Standing wave for unknown     44
load.
Figure 2.15 (p. 72)
Smith chart for Example 2.4.   45
2.5 The Quarterwave Transformer
Impedance Viewpoint
            RL  jZ1 tan  l
  Z in  Z1
            Z1  jRL tan  l
• For βl = (2π/λ)(λ/4) = π/2
         Z12
   Zin 
         RL
• In order for Γ = 0, Zin = Z0

   Z1  Z0 RL
                                  46
Figure 2.16 (p. 73)
The quarter-wave matching transformer.   47
Ex 2.5 Frequency Response of a Quarter-Wave
Transformer
• RL = 100, Z0 = 50
  Z1  Z0 RL  70.71
         Zin  Z 0
  |  |
         Zin  Z 0

        2   0      0      f 
  l            
           4   2      2   f0 



                                                48
Figure 2.17 (p. 74)
Reflection coefficient versus normalized frequency for the
quarter-wave transformer of Example 2.5.                     49
The Multiple Reflection Viewpoint

Figure 2.18 (p. 75)
Multiple reflection
analysis of the quarter-
wave transformer.




                                    50
     Z1  Z 0        Z 0  Z1              RL  Z1
1           , 2            1 , 3 
     Z1  Z 0        Z 0  Z1              RL  Z1
      2 Z1            2Z 0
T1           , T2 
     Z1  Z 0        Z1  Z 0
  1  T1T2 3  T1T2 3  T1T223 
                         2         2

                  
   1  T1T2 3  ( 2 3 ) n
                 n 0

          T1T2 3    1  1 2 3  T1T2 3
   1            
         1   23         1   23

                                                     51
• Numerator
1  1 23  T1T23  1  3 (1  T1T2 )
                                  2


          ( Z1  Z 0 )( RL  Z1 )  ( RL  Z1 )( Z1  Z 0 )
 1  3
                        ( Z1  Z 0 )( RL  Z1 )
     2( Z12  Z 0 RL )

  ( Z1  Z 0 )( RL  Z1 )




                                                              52
2.6 Generator and Load Mismatches
• Because both the generator and load are
  mismatched, multiple reflections can occur on
  the line.
• In the steady state, the net result is a single
  wave traveling toward the load, and a single
  reflected wave traveling toward the generator.
• In Fig. 2.19, where z = -l,
           1  l e2 j l        Zl  jZ0 tan  l
  Zin  Z0          2 j  l
                              Z0                    (2.67)
           1  l e               Z0  jZl tan  l
         Zl  Z0
  l              (2.68)
         Zl  Z0
                                                              53
Figure 2.19 (p. 77)
Transmission line circuit for mismatched load and generator.   54
• The voltage on the line:
                     Z in
  V (l )  Vg               V0 (e j  l  l e  j  l )
                 Z in  Z g
               Z in             1                                       By (2.67)
     V  Vg                                                   (2.70)
            Z in  Z g e j  l  l e  j  l
        0
                                                                         &
            Z0          e j l                                                   Z g  Z0
    
  V  Vg                                                      (2.71)     g 
   0
                         
         Z 0  Z g 1  l  g e2 j  l                                          Z g  Z0
                                                                         1  l
• Power delivered to the load:                                   SWR 
                                                                         1  l

                                               1                                      1 
                                                                                  2
                       1           2 1                                                 
  Pl  Re Vin I in   Re | Vin |    Re | Vg |2
      1                                                  Zin
                                                                                         
      2                2             Zin  2         Zin  Z g                      Zin 
                                                                                         
      1                      Rin
      | Vg |2
                                                (2.39)
      2        ( Rin  Rg )  ( X in  X g )
                           2                 2
                                                                                              55
• Case 1: the load is matched to the line, Zl = Z0,
  Γl = 0, SWR = 1, Zin = Z0,
         1                  Z0
  Pl      | Vg |2                        (2.40)
         2         ( Z 0  Rg ) 2  X g
                                      2




• Case 2: the generator is matched to the input
  impedance of a mismatched line, Zin = Zg
      1            Rg
  Pl  | Vg |2
                                     (2.41)
      2              
               4 Rg  X g
                  2     2
                                 
• If Zg is fixed, to maximize Pl,
  Pl                    1                            2 Rin ( Rin  Rg )
       0                                                                         0
  Rin     ( Rin  Rg )  ( X in  X g )
                       2                 2
                                             ( Rin  Rg ) 2  ( X in  X g ) 2 
                                                                                  2
                                                                               
                                                                                   56
  or        Rg  Rin  ( X in  X g )2  0
             2     2


    Pl             2 X in ( X in  X g )
         0                                   0
   X in                                   2 2
             ( Rin  Rg )  ( X in  X g ) 
                          2
                                           
  or X in ( X in  X g )  0
• Therefore, Rin = Rg and Xin = -Xg, or Zin = Zg*
• Under these conditions
          1           1
   Pl      | Vg |2        (2.44)
          2         4 Rg

• Finally, note that neither matching for zero reflection
  (Zl = Z0), nor conjugate matching (Zin = Zg*),
  necessary yields a system with the best efficiency.

                                                        57
2.7 Lossy Transmission Lines




                               58