Question 6.13: A 12 V battery is connected via a switch to a transmission l...
A 12 V battery is connected via a switch to a transmission line that is 6 m long. The characteristic impedance of the transmission line is 50 Ω, the battery impedance is 25 Ω, and the terminated in a load impedance of 25 Ω. The velocity of propagation along this transmission line is 2 x 106 m/s. Find and sketch the voltage at the midpoint of this transmission line during the time interval 0 < t < 9 µs.
The amplitude of the wave that is launched on the transmission line is calculated from
V _{1}=\frac{Z_{ C }}{Z_{B}+Z_{C}} V _{ B }=\frac{50}{25+50} 12=8 V
The reflection coefficient at the load is equal to
\mathcal{R}_{ L }=\frac{Z_{ L }-Z_{ C }}{Z_{ L }+Z_{ C }}=\frac{25-50}{25+50}=-\frac{1}{3}
The reflection coefficient at the battery is equal to
\mathcal{R}_{B}=\frac{Z_{B}-Z_{C}}{Z_{B}+Z_{C}}=\frac{25-50}{25+50}=-\frac{1}{3}
In order to calculate the voltage at the midpoint of the transmission line, we make use of the bounce diagram. In this case, we clearly identify the amplitudes of the waves. The normalized time is \frac{ vt }{ L }=\frac{2}{6} t \quad The bounce diagram is first obtained.
Using the bounce diagram, the voltage at the midpoint of the transmission line is equal to 0 until the front of the wave arrives. The voltage increases to the amplitude of the wave, remains at that value until the wave that is reflected from the load impedance passes the midpoint. This reflected wave is reflected again at the battery and arrives at the midpoint. The picture on the right depicts the expected response of the oscilloscope that is located at the midpoint of the transmission line. The final value of this voltage is calculated from (6.59) to be
V=\frac{Z_{L}}{Z_{B}+Z_{L}} V_{B} (6.59)
=\frac{25}{25+25} 12=6 V
