Transmission Line Theory KAU RFIC MMIC LAB

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```					      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
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  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
• 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
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
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
• 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

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