### Section 5: Wave Bounce Diagram for Simple Structures

This analysis is more complicated if the resistances are not all equal, as they are in Figure 1.4.2. Consider the circuit shown in Figure 1.5.1. At the moment the switch is closed, the instantaneous voltage at P1 will be 1V (a 100Ώ source resistor into a 100Ώ transmission line). We know this because of the voltage divider relationship:

However, if you measure the DC resistance across points P1 and P1', you will measure a resistance of 33Ώ.  Over time, the steady state voltage at P1 will be 0.5V because:

How do we resolve this apparent conflict? The key is to understand the voltage waves.

As soon as the switch is closed, the voltage at P1 is 1V (first voltage divider formula).  When the voltage wave transitions at P2 from Z0 (100 ohms) to Z1 (50 ohms), the voltage reflection can be calculated with:

The voltage seen traveling left, back along Z0, is actually the sum of the incident wave (the original 1V) and the reflected wave (the -0.33V).

The voltage seen to continue on from P2 is the transmitted wave, and is

#### DC Voltage Wave Bounce with Mismatch

The amount of reflection (Γ, defined below) off of the resistors is a relationship between the impedances and depends on the direction of the wave: when the wave is traveling towards (and bounces off) the terminating resistor, Z2 = 70.7 and Z1 = 100; when the wave is traveling back to (and bouncing off) the source resistor, Z2 = 70.7 and Z1 = 50. Equation 1.5.1

Therefore, the amount of reflected voltage is the reflection coefficient times the voltage wave (so the voltage wave going back to the source is 0.1V), and the voltage at P2 is the sum of the incoming and outgoing voltage waves (0.586 in plus 0.1V out = 0.686V). This is still not 0.667V as expected for the steady state, but the voltage is starting to converge to the steady state value.

You can see the effects of the voltages bouncing off of these discontinuities if you look at the simulation in the video above.  Transitioning from high speed to low speed dielectric is just like going from a high impedance to a low impedance line- we see similar reflections.  Here, however, we don't have a short traveling pulse wave.  We have a constant DC voltage which is trying to figure out to what value it should settle.

#### Opens

An "open" (open transmission line) notionally has an infinite impedance.  In practice, an open is just a very large resistance, since even air has a resistivity and there is stray capacitance everywhere.  There is effectively no current flowing through an open.

Based on Equation 1.5.3, the voltage wave return off of the open is an equal magnitude reflection.  Since there is no resistance, there is no loss.  Since there is no loss, energy is conserved, and since the impedance on the return path is the same, the reflected wave has the same potential as the incident wave. Equation 1.5.2

The following video and figure explains how the incident wave and reflection bounce along the wave.

#### Shorts

A "short" (shorted transmission line) notionally has an impedance of zero.  In practice, a short is just a very small impedance, since except for superconductors, all metals have a small amount of resistance associated with them.  There is effectively no voltage across a short.

#### WaveMason, into a Short

Figure 1.5.3: Voltage bounce off a short (click to enlarge)

#### Optional: Wave Bounce Diagram

The Wave Bounce Diagram is a useful tool which helps track the voltages at P1 and P2 over time. It uses a vertical axis for time, and the voltage nodes (P1 and P2) go along the horizontal axis. The voltage at a point is equal to the sum of the existing voltage, plus the incoming and outgoing voltages. The middle voltages in the diagram are the actual voltage wave amplitude, not the actual voltage.  Therefore, for the 2nd bounce at P1, the voltage is equal to the existing voltage (5.86V) plus the incoming voltage wave (0.1V) plus the outgoing voltage wave (-0.0171V). One can see that after a few bounces, the voltage converges to the expected value.

Figure 1.5.4: Wave bounce diagram for two nodes (P1 and P2)

#### Main Point

The normal steady state analysis for a resistor divider (a 50Ώ resistor and a 100Ώ resistor connected by a foot of wires) ignores the propagation delay for voltages to set up along the line. After around 3ns (three bounces down the line), the voltage wave analysis converges to the same answer as the simple steady state analysis (within 0.1%).