Digital Signal Processing Reference
In-Depth Information
1. Incident wave:
v
i
(z)
=
v
i
e
−
jβ
1
z
1
Z
01
v
i
e
−
jβ
1
z
i
i
(z)
=
2. Reflected wave:
v
r
e
+
jβ
1
z
v
r
(z)
=
1
Z
01
v
r
e
+
jβ
1
z
i
r
(z)
=−
3. Transmitted wave:
v
t
(z)
=
v
t
e
−
jβ
2
z
1
Z
02
v
t
e
−
jβ
2
z
i
t
(z)
=
=
√
L
2
/C
2
are
the phase constants and characteristic impedances of transmission line
=
ω
√
L
1
C
1
,
β
2
=
ω
√
L
2
C
2
,
Z
01
=
√
L
1
/C
1
, and
Z
02
where
β
1
A
and
region
B
, respectively.
When the wave intersects a boundary between the transmission lines, the tan-
gential components of both the electric and magnetic fields across the interface
must remain continuous, as shown with equation (3-8). In other words, the tan-
gential component of the fields cannot change instantaneously. In our particular
scenario, the plane wave is propagating in the TEM mode in the
z
-direction, so
both the electric and magnetic fields are oriented parallel (i.e., tangent) to the
boundary of the dielectric interface. Subsequently, we can say that at the interface
(
z
=
0), the sum of the incident and reflected waves must equal the transmitted
wave.
v
t
(z
=
0
)
=
v
i
(z
=
0
)
+
v
r
(z
=
0
)
→
v
t
=
v
i
+
v
r
(3-98)
i
t
Z
02
i
i
Z
01
−
i
r
Z
01
i
t
(z
=
0
)
=
i
i
(z
=
0
)
+
i
r
(z
=
0
)
→
=
(3-99)
Since the incident waves are known, we can solve (3-98) and (3-99) simultane-
ously for the transmitted and reflected portions of the wave:
2
Z
02
Z
02
v
t
=
v
i
(3-100)
+
Z
01
v
i
Z
02
−
Z
01
v
r
=
(3-101)
Z
02
+
Z
01
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