Wave Propagation in Linear Media | re-examined

3.6 Inhomogeneous transmission line
3Wave propagation in electromagnetic transmission lines
So far, the transmission lines we investigated were homogeneous, so that their parameters did
not depend on the position along the line. We now extend our analysis to inhomogeneous
transmission lines and brie y sketch the treatment of a so-called exponential line. Such
transmission lines are not only of academic interest | they have actually been used in the early
days of integrated circuit design for impedance transformation [95]. The equivalent circuit
of g. 3.1 and the respective di erential equations (3.2) are still valid, but the distributed
reactance and susceptance are now position-dependent,
X 0 = !L0 0 e x ; B 0 = !C0 0 e , x ; c =
1
p
L0 0 : (3.56)
0
C0
Inserting solutions of the form (3.3) into the di erential equations yields a propagation constant
0
k = @j
2
s
2!
c
1
2
, 1A
(3.57)
and a complex, position-dependent wave impedance
Z 0 =
0
@,j 0 +
2!C0
With the abbreviations and normalisations
= 2!
c
s L0 0
C0 0 , 2!C0 0
1
2
A e x : (3.58)
; X = x ; L = l ; (3.59)
the general solution in the pass band consists of a right- and left-going wave for the voltage
and a corresponding current
U = K e X 2 (1,j p 2 ,1)+j!t +(U0 ,K)e X 2(1+j p 2 ,1)+j!t
I = j2!C0 0 K=
1+j p e
2 ,1 ,X 2(1+j p 2 ,1)+j!t j2!C0
+ 0 (U0 , K)=
1 , j p 2 , 1
e , X 2 (1,j p 2 ,1)+j!t
(3.60)
(3.61)
with a complex constant K that must be determined to comply with the boundary conditions.
With the de nitions of the propagated energy,
and stored energy,
W = 1
4
P = 1
Re U I ; (3.62)
2
I I dX0
d!
46
+ U U dB0
d!
; (3.63)