28 Introduction to distributed circuits
•
Last lecture we learned that voltage and current variations on TL’s are
governed by telegrapher’s equations and their d’Alembert solutions —
the latter can be expressed explicitly as
+

Wire 2
Wire 1
+

0
f
(
t
)
R
g
R
L
I
(
z, t
)
V
(
z, t
)
z
l
Source
ckt
Transmission line
Load
Z
o
I
(
z, t
)
V
(
z, t
) =
V
+
(
t

z
v
) +
V

(
t
+
z
v
)
and
I
(
z, t
) =
V
+
(
t

z
v
)
Z
o

V

(
t
+
z
v
)
Z
o
in terms of
v
=
1
√
LC
and
Z
o
=
L
C
and
V
+
(
t
)
and
V

(
t
)
labeling signal waveforms propagated in
+
z
and

z
directions, respectively.
We will next solve a sequence of
distributed circuit problems
containing
TL segments
and
two terminal elements such as resistors and voltage (or
current) sources. In solving the problems, we will apply the usual rules of
lumped circuit analysis at element terminals and treat the TL’s in terms of
d’Alembert solutions above.
•
Consider a TL with a characteristic impedance
Z
o
extending from
z
= 0
to
z
=
l
, where a twoterminal
source
circuit (e.g., a receiving antenna)
modeled by a Thevenin equivalent with voltage
f
(
t
)
and resistance
1
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R
g
is connected between the TL terminals at
z
= 0
and a
load
(e.g.,
a receiver circuit) modeled by a resistance
R
L
terminates the line at
z
=
l
(see margin).
+

Wire 2
Wire 1
+

0
f
(
t
)
R
g
R
L
I
(
z, t
)
V
(
z, t
)
z
l
Source
ckt
Transmission line
Load
Z
o
I
(
z, t
)
–
We want to determine voltage and current signals
V
(
t
)
and
I
(
t
)
on the TL and the load in terms of source signal
f
(
t
)
.
•
Let us simplify the problem posed above by having
R
g
= 0
f
(
t
) =
u
(
t
)
R
L
=
Z
o
in which case the circuit takes the simplified form in the margin.
Circuit 1:
+

+

0
u
(
t
)
Z
o
I
(
z, t
)
V
(
z, t
)
z
l
Z
o
I
(
z, t
)
–
Now applying KVL at
z
= 0
we have
u
(
t
) =
V
+
(
t
) +
V

(
t
)
.
–
Next applying KVL and KCL at
z
=
l
we have
V
(
l, t
) =
I
(
l, t
)
Z
o
(since current
I
(
l, t
)
flows though the load
Z
o
to induce
V
(
l, t
)
),
which implies that
V
+
(
t

l
v
)+
V

(
t
+
l
v
) = (
V
+
(
t

l
v
)
Z
o

V

(
t
+
l
v
)
Z
o
)
Z
o
⇒
V

(
t
+
l
v
) =

V

(
t
+
l
v
)
,
2
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 Spring '08
 Kim
 Volt, LTI system theory, Electrical impedance, zo

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