2.14 The dreaded maths
119
( )
( )
( )
( )
( )
( )
In this case, where we assume that no losses are incurred along the line, the solution of (2.14-8) for
the voltage will be of the form
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shown in (2.14-9)
( )
[
]
The
and components of this equation represent the incident and reflected voltages
respectively. This is the equation which is used to model the incident and reflected waveforms which
we saw in section 2.8.
For the current we get a similar solution as shown by eq. (2.14-10)
( )
[
]
[ ]
Again the and component represent the incident and reflected currents respectively.
Remember that
⁄
√
, where represents the effective wavelength inside the line
(section 2.11, eq. (2.11-3))
As mentioned in section 2.3, eq. (2.3-4), in the lossless case
may also be expressed as
√
is related to the amplitudes of incident and reflected voltages and currents by equation (2.14-11).
The minus for the backward wave is significant and is indicative of the fact that the current is moving
in the negative direction, away from the load.
Now, to simplify our analysis, we may assume that we are in sinusoidal steady-state
26
at angular
frequency and hence omit the
term. This makes (2.14-8) a bit simpler as shown by (2.14-12).
( )
( ) ( )
( )
( )
( ) ( )
( )
25
We are using the complex exponential representation for our solution as we did in (2.14-1) however the
time domain solution will be similar to (2.14-5) i.e.
{ ( )} ( ) ( )
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Sinusoidal steady-state refers to the steady-state that gets established in a circuit when all the independent
sources are sinusoids of the same angular frequency
(2.14-8)
(2.14-9)
(2.14-10)
(2.14-11)
(2.14-12)
Conquer Radio Frequency
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