Digital Signal Processing Reference
In-Depth Information
receiver. By equating the voltages and currents and applying Ohm's law, we
create a set of two simultaneous equations in two variables:
v(t <
0
)
=
i(<
0
)R
TX,lo
(11-6)
V
TT
−
v(t <
0
)
=
i(t <
0
)R
TT
(11-7)
The Bergeron diagram solves the equations graphically to give the steady-state
current and voltage.
To find the voltage and current values for the initial transition at the transmitter
end of the circuit, we recognize that the output of the transmitter is connected
to one end of the transmission line. By applying the circuit laws in the same
manner as above, we calculate the initial voltage and current wave magnitudes
at the transmitter. However, the Ohm's law expression for the transmission line
must comprehend the initial current and voltage flowing through the line:
v(
0
)
−
V
DD
=
i(
0
)R
TX
,
hi
(11-8)
v(t <
0
)
−
v(
0
)
=
[
i(
0
)
−
i(t <
0
)
]
Z
0
(11-9)
Note from equation (11-9) that the slope of the load line for the transmission
line is equal to
1
/Z
0
, just as we constructed it in our example. So the Berg-
eron diagram again solves the simultaneous equations while accounting for the
steady-state potential and current flow that existed on the line prior to the tran-
sition.
When the initial wave reaches the connection between transmission line and
receiver, we again apply the circuit laws and account for the current and voltage
of the incident wave:
−
V
TT
−
v(t
=
t
d
)
=
i(t
=
t
d
)R
TT
(11-10)
v(t
=
t
d
)
−
v(t
=
0
)
=
[
i(t
=
t
d
)
−
i(t
=
0
)
]
Z
0
(11-11)
The slope of this load line for the transmission line is 1
/Z
0
because it relates
the current and voltage for the reflected wave, and since we must account for the
current and voltage of the incident wave, it starts at the current and voltage,
i(t
0
)
, that we calculated in the preceding step.
The next step in the analysis returns back to the transmitter end of the system
using a load line with slope
=
0
)
and
v(t
=
1
/Z
0
starting from the receiver voltage and current
(i.e., the intersection between the previous transmission line load line and that
of the receiver). In this case the load-line slope is negative because we are
accounting for the reflected voltage and current waves at the transmitter. Since
the waves flow away from the transmitter and toward the receiver, the sign of
the current wave is negative.
The analysis continues in this fashion, alternating between transmitting and
receiving load lines using transmission load lines with alternating slopes of
−
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