Microwave Journal's sample issue

A
u
B

⎣ ⎢

C
u
u
⎤

D u

⎦
⎥=
⎡

Y

012

⎢
⎢
+ + cos( )
sin sin
θθ
θθ
ω 12

C m

⎢ ⎢ ⎢

0 12 sin( ) θ θ+ + jY j

⎣

Y02 1 2 sin sin θ θ
ωCm
jZ012 12 sin( ) cos cos θθ θ θ ω + −j Cm

⎡

A
C
l
B

⎢ ⎣

l
D
l

⎤

l

⎦
⎥=
⎡

0 1 2 cos cos θ θ ω

Y

⎢ + + cos( ) 1 2

θθ ⎢

Cm

⎢

2

⎢ ⎢ ⎣

1 2 sin( ) sin sin θθ θ θ ω

Y

++

jY j

0

C 0 12

m
jZ j
C 0 12

where Y0=1/Z0, and Cm is the coupling capacitance be-
tween the upper and the lower path. According to circuit
theory, the transmission matrix of the whole circuit can be
written as
AB
CD
A B AB
BB
A B AB
ul
ul lu

⎡
⎣
⎢
⎤
⎦
⎥=
+
+
+ ⎡⎣ ⎤⎦ + ⎡⎣ ⎤⎦− +
+

B D BD B B
B B BB
ul lu u l
u l ul
()

()

2

⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢

BB
BB
ul
u l+
+ B D u l BD
BB
lu
u l+
⎤
⎦
⎥
⎥
⎥
⎥
⎥
() 3

FERRITE PRODUCT RANGE

From Equations 1 and 2, it is found that Au + Al = Du
+ Dl, Bu = Bl and Cu = Cl. The transmission matrix can be
simplified as
AB
CD
AA
AA
B
u
u
u

B
AA
u
u+ 1
2

2

⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥

() 4

The transmission coefficient is
S
BZ
B AABZ AA Z
u u l uL u L
21 2

ul

where Zl is the system impedance. If S21 = 0, there are
attenuation poles existing. This means that the sufficient
and necessary conditions for the existence of the poles are
the denominator of S21 not equal to zero and Bu = 0. It is
found that
tan tan θ θ
ω 1 20

() 6

Because Cm is usually very small, 1/Z0ωCm is considered large enough at certain low frequency range. Generally, it is considered that θ1 θ2 for the external qualify factor. Equation 4 can be simplified as