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

Derivation concept (3/6)
Derivations on the image parameters
Y +
s
k, k 1
1.
2.
Ca
2
Jk, k+1
θθθθ
Y k, k+
1
s
θθθθ Y k, k+
1 θθθθ
=
2
π ω
θ
ω
0
Ca
2
A B
⎡ 1 1 ⎤ ⎢ ⎥ k , k+ 1 k , k+
1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ = ωC J k , k+
1 =
2
⎢ a
1 1
1 ⎢ ⎥ ωC
⎥ ⎢ a
⎣C D ⎦ j j 1⎥
⎢ ( ωC ) − ⎥
a ωC
⎣ 2 ⎦ ⎣⎢ jJ , + 1 0 ⎦⎥
a
k k ⎣ 2 ⎦ ⎢ jJ k , k+
1 − j ⎥
J k , k+ 1 2J
k , k+
1
C1D ⎛
1
ωC
⎞
a
Yimag1 = = J k , k + 1 1−
⎜ ⎟
A1B 1 ⎝ 2J
k, k + 1 ⎠
Y
⎡ −ωCa
1 ⎤
⎡ 1 0⎤ ⎡ 1 ⎤
0
⎡ 1 0⎤ ⎢
j
j
2J
J ⎥
C D
2 2
imag 2 = =
A2B 2
Derivation concept (4/6)
( ) 2
s
Y − Y + Y
2 2
k , k+ 1 k , k+ 1 k, k+
1
2
sinθ
cos θ
⎢⎣ ⎥⎦
⎡ j ⎤
⎡ A2 B2
⎤ ⎡ 1 0⎤ ⎢
cosθ sinθ
⎥ ⎡ 1 0⎤
⎢ ⎥ = ⎢ , + 1
2 2
, + 1 cot 1
⎥
Y
s k k
⎢ ⎥ ⎢ s
− − , + 1 cot 1
⎥
⎣C D ⎦ ⎣ jYk k θ ⎦ ⎢ , + 1 sin cos ⎥ ⎣ jYk
k θ ⎦
⎣ jYk
k θ θ ⎦
Design equations for interior sections
Lowpass Bandpass
' ( ω = 0)
= ( ω = ω )
Y Y
imag1 imag 2 0
' ' ( ω = ω ) = ( ω = ω )
Y Y
imag1 1 imag 2 1
2
=
1
J
⎡ Y j ⎤
s
k , k+
1
⎢ cosθ + cosθ sinθ
⎥
⎢
Yk , k+ 1 Yk
, k+
1 ⎥
⎢ 2
2 s
2
s ⎥
⎢ s cos θ ( Yk , k+
1 ) cos θ Yk
, k + 1
− 2 jYk , k + 1 + jYk , k+
1 sinθ − j cosθ
+ cosθ
⎥
⎢ sinθ Yk
, k+
1 sinθ
Y ⎥
k , k+
1
⎣ ⎦
k, k+
1
⎝ k , k+
1 ⎠
1.
2.
( ) 2
s
Prof. T. L. Wu
' ( ω = 0)
= ( ω = ω )
' ' ( ω = ω ) = ( ω = ω )
Y Y
2 2 2
Yk , k + 1 − Yk , k+ 1 + Yk
, k+
1 cos
⎛ ωC
⎞
a 1−
⎜ ⎟ =
2J sinθ
( ) 2
s
Y Y
imag1 imag2
0
imag1 1 imag 2 1
k, k+
1
2
' ⎛ ω1C
⎞ a 1−
⎜ ⎟ =
⎝ 2J k , k+
1 ⎠
2
Yk , k + 1 − Yk , k+ 1 + Yk
, k+
1
sinθ1
2
cos θ1
2
⎡ '
2 ⎛ ω1C
⎞ ⎤
a
2
J , 1 sin ⎢ 1 1 ⎥
k k+ θ − ⎜ ⎟ = Yk , k+ 1 −
⎢ 2J
k, k+
1 ⎥
s
2
2
Yk , k + 1 + Yk
, k+
1 cos θ1
J
( ) ( )
⎣ ⎝ ⎠ ⎦
'
s 2 2 2 ⎛ ω1Ca sinθ1
⎞
+ = − + ⎜ ⎟
⎝ 2 ⎠
2
( J k , k+ 1 Yk , k + 1) cos θ1 J k , k+
1 ( 1 sin θ1)
θ
2
J = Y
k, k+ 1 k , k+
1
Prof. T. L. Wu