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# 4.2 The Power Time Constant

Circuit theory supports the view of instantaneous power. When a 50-ohm load is placed across a 5-V voltage source, we expect that the power level to be 0.5 W immediately. What actually happens in the first few nanoseconds is different. Assume a fast switch and an ideal voltage source in series with a 50-ohm transmission line. When a 50-ohm load is placed on the far end of the transmission line, the voltage sags to 2.5 V. The power level at this time is 0.125 W. The power will stay at this level until a wave moving on the line reflects at a voltage source and returns to the load. Waves must make several round trips before the power level rises to near 0.5 W. If the line is 1 cm long, one round trip time on a typical circuit board is 0.13 ns.

Consider the case when the 50-ohm transmission line is connected to a 5-V source and the load is 5 ohms. The expected power level is 5 W. On a 50-ohm line the voltage sags to 0.45 V and the delivered power is only 0.0409 W. The sag in voltage means that a negative wave of 4.55 V is sent down the line toward the voltage source. A reflected wave at the voltage source brings energy back to the load. Waves continue to make round trips between the load and the voltage source until the voltage at the load rises to near 5 V. The time constant as given by Equation 2.7 depends on the parameters of the transmission line and the value of the load. For a 5-ohm load and a 50-ohm line that is 10 cm long, the time constant is 14.3 ...

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