Transmission lines:
wave propagation
v+(z)
z
Voltage wave propagating along the positive z direction (sinusoidal
time regime):
v  ( z )  (V0 e j 0 t )  e    z
The wave is represented thrugh a phasor (i.e. a rotating vector) both
in time and in space (coordinate z).
-1-
Meaning of propagation constant 
Given +j, it has:
v  ( z )   V0 e j t   e   z   e  j   z


Attenuation constant  it defines the rate of reduction (in space)
of the wave amplitude. It is measured in Neper/m or dB/m (1Np
= 8.686 dB)
Phase constant  it is the rate of variation of the phase of the
wave along the z coordinate (for t=cost). It is related to the
wavelength and to radian frequency:   represents
the phase velocity determined by the medium).  is measured in
rad/m (or °/m).
-2-
Physical parameters of the line
Sezione z
L z
R z
C z
G z
R, L, C, G, are the physical parameters of a transmission line. They depend
on the geometry of the transmission line and on the materials used for the
conductors and dielectric medium .
Through the physical parameters and the Kirckoff laws the Telegraphist
Equations can be derived, whose solution is the voltage (current) wave
propagating along the line
-3-
Secondary Parameters


Characteristic Impedance (Zc): it is defined as the input impedance of a
line with infinite length (i.e. there is only the wave propagating along
the positive z direction)
Propagation constant j
Formulas:
Zc 
L
,
C

1R 1
 G  Zc ,
2 Zc 2
    LC
These relations are valid for >> R/L, G/C; transmission lines used at
microwave frequencies always satisfy these conditions
-4-
Voltage and current on the line
I(z)
V(z)
v+(z)
v-(z)
ZL
V ( z)  v ( z)  v ( z)  V0+e j z  V0-e j z
V0+  j z V0-  j z
I ( z)  i ( z)  i ( z)  e  e
Zc
Zc

z
v+, i+: Incident waves

v-, i-: Reflected waves
The sign of the reflected wave of voltage is the same of the incident wave;
the sign of reflected wave of current is instead opposite to that of the
incident wave. In any case (incident or reflected), the ratio between the
wave of voltage and the wave of current is equal to Zc.
A reflected wave is produced by a discontinuity in the transmission line
structure (as for instance the termination with a load). The reflected wave is
eliminated by putting ZL equal to ZC.
-5-
Reflection Coefficient
Reflected Wave V0 e j  z V0  j 2  z
( z ) 
   j z   e
 0e j 2  z
Incident wave V0 e
V0
Properties of (z) with =0:
 The magnitude does not depend on z (it is constant along the line)
 The phase is periodic in z (perid equal to /2)
 The magnitude is always less than 1 with a passive load (the reflected
power must be less than the incident one).
-6-
|V(z)| as a function of z

V (z)  v  (z)  v  (z)  v  (z)  1   (z)
Vmax
|V|
V ( z )  V 0+ 1   0  e 
Vmin
V 0+
Vmin
z
j2 z


1  2 0  cos(2  z)  0
2
The magnitude of V is periodic (same period of ) with maxima and
minima given by:
V m ax  1  
,
V m in  1  
The Voltage Standing Wave Ratio (VSWR) is defined as the ratio of
these vlotages:
ROS 
V m ax
V m in
-7-

1
2

Impedance along the line
I(z)
Z(z)
ZL
V(z)
z
There is a biunivocal relationship between the reflection coefficient and
the impedance (normalized to Zc) seen along the line (in the load
direction):
Z (z)
1 V ( z) 1  ( z)


Zc
Z c I ( z) 1  ( z)
Relazione inversa:
Z (z) Zc  1
(z) 
Z (z) Zc  1
-8-
Impedance as a function of the
load
I(z)
V(z)
Zc , 
Vin
Z L  jZ c tan( L )
Zin 
 Zc
I in
Z c  jZ L tan( L )
ZL
L
Particular cases:
ZL= 0 (short circuit)
Z in 
Vin
L
 jZ c tan(   L )  jZ c tan(2 )
0
I in
ZL=  (open circuit)
Z in 
Vin
L
  jZ c cot(   L )   jZ c cot(2 )
0
I in
-9-
Graphic representation of 
 is a complex number in the polar
plane:

•

1
For the properties of , assuming the
line terminated with a passive load,
the point in the polar plane must be
within the circle with unit radius

-1
1
-1
- 10 -
Graphic representation of 
The curve in the polar plane representing
the reflection coefficient in a lossless line
terminated with a passive load is a circle
with radius equal to ||. The phase
changes of 2 with a variation of z equal to
/2
d
Characteristic points:
Matched line  =0 ( centre of the chart)
Open circuit  =1 (a)
Short circuit  =-1 (d)
Maximum of voltage (b) ( real and positive)
Minimum of voltage (c) ( real and negative)
- 11 -
c

b
a
Smith Chart

On the polar plane of  the curves representing the locus of points
where the real (or imaginary) part of zn=Z/Zc remain constant are
defined by:
 ( z)  1
ReZin   Re 
  cost (r)
 ( z)  1
 ( z)  1
ImZin   Im 
  cost ( x)
 ( z)  1

These curves are circles, whose center and radius depend on r (or
x). The Smith Chart is the graphic representation of such circles. It
allows to solve graphically several problems related to the
transmission lines used in microwave circuits.
- 12 -
Angle of  (it is measured in
degrees or in L/
Circle x = 1
Toward Source
Toward Load
Circle r = 1
Reference Axis
Admittance representation of the
Smith Chart
Impedance at :
zn 
1 
1 


’

Impedance at ’:
j    
1   e j 1   1
1   1   e
zn 




j
j    

1  1  e
1  e
1   zn
The point diametrically opposed presents the inverse impedance
(i.e.e the admittance) with respect the original point. As a
consequence, the Smith Chart can be used for representing either
the impedances or the admittances
- 14 -
Smith Chart
Moving at Γ=const
ZL
0
0
Asse d
  x     e j 2 x   e j0  e j 4 x  
0
0


 j 4 d 
j L
 j 2 d

 d    L  e
 L e
e
L

x
d
Displacement on a lossless line:
0
Asse x
Movement on a circle with constant radius
Displacement:
 from source  x increasing
  x    0  2 x  counterclockwise rotation
 from load  d increasing
  x    L  2 d  Clockwise rotation
NOTE: Γ (z) is a periodic function of z (d) with period λ/2
- 15 -
Smith Chart
Displacement at r (or x) constant
Displacement at x costant
xL
r
ΓL
zL = rL + jxL
Γin
rL
zin = (rL + r) + jxL
rin = r+rL
Displacement at r costant
xL
jx
ΓL
zL = rL + jxL
rL
zin = rL + j(xL + x)
- 16 -
Γin
xin = xL + x
Smith Chart
Displacement at g (or b) constant
Z Chart
ΓL
Γin
Γin
ΓI,in
yin = (gL + g) + jbL
gin =gL + g
ΓI,in
bL ΓI,L
- 17 -
gL
Γin
b
Γin
L+
yin = gL + j(bL + b)
ΓL
b
yL = gL + jbL
ΓL
bL
in =
jb
gL
b
in =
b
Displacement at g=const
L+
b
bL ΓI,L
b
yL = gL + jbL
bL
gL
ΓL
Displacement at b=const
g
Y Chart
Smith Chart
Example of graphical solution
in
jX A
d
B
Zc
α=0
L 
jB
ZL
Z L  Zc
 0.6259.0
Z L  Zc
Zc= 50 [Ω];
B= 0.05 [Ω-1]; (bn=2.5)
εr = [4];
X = -80 [Ω]; (xn=-1.6)
f0 = 3 [GHz];
d = 15 [mm]; (d=108°)
ZL= 20 + j40 [Ω];
xin
(Zn=0.4+j0.8)
=7.2 [°/mm]
ΓL
Γin ΓA
 B  0.75  116.6
2βL
2 d  40d  3.77 rad  216.0
ΓB
 A  0.7527.4
in  0.5131.1  Z in  Z c
1  in
 96  j 68 Ω 
1  in
- 18 -
bB