A battery with zero internal impedance has an open circuit voltage of 100 volts. At a time t = 0, this battery is switched into a 50 Ω air-dielectric coaxial cable via a 150 Ω resistor. The cable is 300 m long and is terminated in a load of 33.3Ω. (a) Sketch a bounce diagram for the first 4 µs

Question

A battery with zero internal impedance has an open circuit voltage of 100 volts. At a time t = 0, this battery is switched into a 50 Ω air-dielectric coaxial cable via a 150 Ω resistor. The cable is 300 m long and is terminated in a load of 33.3Ω.

(a) Sketch a bounce diagram for the first 4 µs after the switch is closed.

(b) Draw a graph of the voltage that appears across the load impedance as a function of time.

(c) Find the asymptotic values of V_L and I_L [latex] t \rightarrow \infty . [/latex]

Accepted Answer

(a) The two reflection coefficients and the incident voltage step that propagates on the line are given by

[latex]\begin{gathered} \mathcal{R}_{B}=\frac{Z_{B}-Z_{C}}{Z_{B}+Z_{C}}=\frac{150-50}{150+50}=\frac{1}{2} \\ \mathcal{R}_{L}=\frac{Z_{L}-Z_{C}}{Z_{L}+Z_{C}}=\frac{33.3-50}{33.3+50}=-\frac{1}{5} \end{gathered} [/latex]

[latex]V_{1}=\frac{Z_{C}}{Z_{B}+Z_{C}} V_{B}=\frac{50}{150+50} 100=25 V[/latex]

In the (b) Since the coaxial cable is filled with air, the velocity of propagation equals [latex]3 imes 10^{8} m / s [/latex]. The voltage signal takes 1 µs s to travel from one end to the other.

(c) The asymptotic voltages and currents as computed from (6.59) and (6.60) are for [latex]V _{ t ightarrow \infty}=18.2 V[/latex] and [latex]I _{ t ightarrow \infty}=0.55 A[/latex].

[latex]V =\frac{Z_{ L }}{Z_{B}+Z_{L}} V_{B}[/latex] (6.59)

[latex]I =\frac{ V }{Z_{ L }}=\frac{1}{Z_{B}+Z_{ L }} V _{ B }[/latex] (6.60)

Question 6.14: A battery with zero internal impedance has an open circuit v...

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