### Section 4: Voltage Waves and Propagation Delay

The "resistance" of a resistor is defined as the relationship between the voltage and the current.  If you apply a DC voltage, a known current will flow through the resistor: .  This is the steady state response for a DC signal- the constant state given a constant input.

The "impedance" of a transmission line is defined as the relationship between the voltage wave and the current wave.  If you apply a voltage wave, a known current wave will flow through the resistor: V = i * Z.  The voltages and currents may vary over time even for a constant input.  This will be shown in the video below (wait for it).

One can refer to the "impedance" of a resistor, even though it has a "resistance".  A resistance may be referred to as an impedance, but in general an impedance is not a resistance.

Consider a transmission line, such as a coaxial line used to connect cable or satellite signals to a TV.  Assume the line is open (nothing is on the other end).  If a voltage is applied, in the steady state, no current will flow.  The resistance is too high.  However, it takes time for voltage waves to travel, so for the first few moments that the voltage has been applied, that side of the cable has no way of determining the steady state current.  The following video shows what happens.

#### Impedance versus Resistance

The 50Ώ impedance of the transmission line is the relationship between the propagating voltage and current waves. Any voltage wave traveling down the transmission line will have a corresponding current wave, maintaining a relationship between the potential and the actual electrons flowing down the line.

Figure 1.4.1: Impedance of a transmission line

Consider the transmission line shown in Figure 1.4.2: a switch connects a DC voltage source with a 50Ώ source impedance to a 1 ft transmission line terminated with a 50Ώ resistor. As long as the switch is open, the voltage between P1/P1' is 0V. At the moment the switch is closed, the voltage source (which includes the 50Ώ resistor) sees the 50Ώ transmission line, forming a voltage divider. Therefore, the voltage at P1 at the time the switch closes is 1V.

#### DC Voltage Wave Bounce, Matched

Since it takes time for energy to propagate, there is a delay between the switch closing and the voltage at P2 changing: at the exact moment the switch closes, the voltage at P2 is still 0V. In air, RF waves propagate at 299792458 m/s (approximately 1ft/ns). Therefore, the voltage at P2 will change approximately 1ns after the switch is closed.

Figure 1.4.2: Voltage wave traveling down a matched transmission line

Why is this a 1V pulse?  Because the 2V source has a source resistor of 50-ohm and is coupled to a 50-ohm transmission line.  From our voltage divider section:

At t = 0, the source resistor (50Ώ) and the transmission line (50Ώ) evenly splits the voltage, 1V across each.  The 1V voltage wave has a corresponding 20mA current wave traveling with it- this is the impedance relationship of the transmission line.

But what would happen if we expected the voltage wave to be the 2V from the source (it isn't).
1. A 2V voltage wave would have an associated 40mA current wave for the 50Ώ transmission line.
2. The current going into the transmission line must be equal to the current leaving the source resistor (Kirchoff's current law).
3. 40mA through the 50Ώ source resistor would represent a 2V drop... so there would be no voltage drop left to go across the transmission line.
4. This is a contradiction to our original assumption.  The only voltage wave that satisfies the impedance of the transmission line, the resistance of the source resistor, and Kirchoff's current law is for 1V to be across the source resistor, and 1V across the transmission line.

#### Main Point

The normal steady state analysis for a resistor divider (the two 50Ώ resistors connected by a foot of wires) ignores the propagation delay for voltages to set up along the line. After 1ns, the voltage wave analysis converges to the same answer as the simple steady state analysis.