2.9 Waves that Transition between Lines with Different Characteristic Impedances

When a wave traveling along a transmission line reaches a change in characteristic impedance, some of the energy is reflected and some of it enters the termination. The termination can be a resistor or a second transmission line. If a wave traveling in Z₀ reaches a termination impedance Z_L, the fraction that is reflected is given by

2.5

where ρ is called the reflection coefficient. If Z_L equals zero, the reflection coefficient is simply − 1. The reflected wave is then the original wave reversed in polarity. If Z_L = Z₀, there is no reflection. If Z₀ is large, then the reflection coefficient is near unity meaning that the reflected wave has the same polarity as the arriving wave. In this case, the two waves add together, doubling the voltage at the point of reflection.

The fraction of the wave that continues into the termination impedance is given by

2.6

where τ is called the transmission coefficient. If Z_L = Z₀ then τ is unity meaning that the transmitted wave is transferred without reflection to the new line or load. If Z_L is high compared to Z₀ then the voltage at the load is double the arriving voltage. This is a voltage doubler without active components.

If a transmission line is terminated in a resistor, the voltage at the resistor is the product of arriving voltage and transmission coefficient. When the termination resistor is higher than the characteristic impedance the voltage at the resistor is shown in Figure 2.7a. When the termination resistor is lower than the characteristic impedance the voltage at the termination resistor is shown in Figure 2.7b.

Figure 2.7 (a) The transmitted voltage and reflected wave for R > Z₀ and (b) the transmitted voltage and reflected wave for R < Z₀.

In a circuit board design there are many places where there are impedance mismatches. Examples might be at a terminating load, at a stub, at a branching transmission line, at an open circuit, or simply at a connector. The nature of the transmission and reflection at a transition in characteristic impedance is shown in Figure 2.8a and b.

Figure 2.8 (a) Reflection and transmission at a discontinuity where Z₂ > Z₁ and (b) reflection and transmission at a discontinuity where Z₂ < Z₁.

There are many transitions that are usually ignored. Examples include the IC die connections, connections to interposer boards, the rotation of field around a mounting pin, transitions through vias, etc.

In circuits where the leading edge takes time to transition, these reflections may not be important. For high clock rates (short rise and fall times), some of these transitions can pose a problem. Examples of problem reflections occur at transmission line branches and stubs. In general, for 50-ohm lines, tests have shown that changes in trace direction or via transitions through a single board layer have a minor effect on wave transmission. Via transitions through multiple layers can cause other problems (see Section 2.4 for a detailed explanation).