12
S. Kar
With, J m (kr) and N m (kr) being the mth-order Bessel and Neumman functions,
respectively.
The wave impedance of the outward travelling wave and inward travelling wave
may be found by taking the ratio of E z and H ϕ in (2) and (3) with A and B set to
zero in respective cases, when we have:
For outward travelling wave:
Z
+
r = Z 0 (kr)e
j[ψ(kr)−θ(kr)]
(6)
For inward travelling wave:
Z
−
r = Z 0 (kr)e
− j[ψ(kr)−θ(kr)]
(7)
with
Z 0 (kr) = η 1
G 0 (kr)
G 1 (kr)
(8)
The input impedance Z i =
E z
H φ
i
, when load impedance Z L =
E z
H φ
L
is given by:
Z i = Z 0i
Z L (cos(θ i − ψ L ) + j Z 0L sin(θ i − θ L )
Z 0L cos(ψ i − θ L ) + j Z L sin(ψ i − ψ L )
(9)
Thus, the cap circuit impedance at the device plane (i.e. at r = r i ) has the real and
imaginary part expressions as [11]:
Re[Z i ] = Z 0 (r i )
Z L
Z 0 (r L )
⎡
⎢
⎣
1 + νζ
η 2 ζ 2 +
Z L
Z 0 (r L )
2
⎤
⎥
⎦
(10)
Im[Z i ] = Z 0 (r i )
⎡
⎢
⎣
η
2
ζ −
Z L
Z 0 (r L )
2 ν
η 2 ζ 2 +
Z L
Z 0 (r L )
2
⎤
⎥
⎦
(11)
where
Z 0 (r i,L ) = 377
h
2πr i,L
G 0 (kr i,L )
G 1 (kr i,L )
η =
sin(θ i − θ L )
sin(ψ i − ψ L )