End-Coupled, Half-Wavelength Resonator Filters - Design theory

Hairpin-Line Bandpass Filters
- Example
External quality factor for hairpin line: t
the hairpin line is 1.0 mm wide, which results in Z r = 68.3 ohm
on the substrate used. Recall that L = 20.4 mm, Z 0 = 50 ohm,
and the required Q e = 5.734. Substituting them into the eq.
yields a t = 6.03 mm, which is close to the t of 7.625 mm found
from the EM simulation above.
Yin
=
4
g
L λλλλ
{ } ( )
( ) ( ) ( ) ( )
Yin = jYr tan ⎡⎣ β L − t ⎤⎦ + jYr tan ⎡⎣ β L + t ⎤⎦ = jYr tan ⎡⎣ β L − t ⎤⎦ + tan ⎡⎣ β L + t ⎤⎦
= jB ω
ω ∂B ω ∂B ∂β
ω Y
2 ∂ω 2 ∂β ∂ω
2 u
ω= ω0 β = β0
2 2
{ ( ) sec ⎡β0 ( ) ⎤ ( ) sec ⎡β 0 ( ) ⎤}
0 0 0 r
b = = = L − t L − t + L + t L + t
p
⎣ ⎦ ⎣ ⎦
0L
=
2
π
condition:
β
ω0 Yr 2 2 ω0L 1 π Yr π Yr
= ⎡( L t) csc ( 0t) ( L t) csc ( 0t) Yr
2
2 u ⎣
− β + + β ⎤
⎦
= = =
p up sin ( β0t ) 2 ⎛ 2 2π ⎞ 2 2 ⎛ πt
⎞
sin ⎜ t sin ⎜ ⎟
π Y
⎜ ⎟ 2
r
λ ⎟ ⎝ L ⎠
⎝ g ⎠
2 2 ⎛ πt
⎞
sin
b
⎜ ⎟
2L
π Z0
Q
⎝ ⎠
2L −1
π Z0 Zr
e = = =
t = sin
G 1
2 ⎛ πt
⎞
π 2 Qe
2Z r sin
Z
⎜ ⎟
0
⎝ 2L
⎠
Example:
Interdigital Filter
- Concept
Prof. T. L. Wu
The filter configuration, as shown, consists of an array of n TEM-mode or quasi-TEM-mode
transmission line resonators, each of which has an electrical length of 90° at the midband frequency
and is short-circuited at one end and open-circuited at the other end with alternative orientation.
Parallel coupled-line BPF
Folding λ/2
resonator
Interdigital line filter with tapped line
Y1 = Yn denotes the single microstrip characteristic
impedance of the input/output resonator. Prof. T. L. Wu