2.16 An Analysis of Cascaded Transmission Lines

The following analysis is for Case 2, as shown in Figure 2.11. The time of transit in line 1 is d₁ ε ₁^1/2/c, where c is the velocity of light. Similarly, in line 2, the time of transit is d₂ ε ₂^1/2/c. The ratio of round trip time is

2.24

When a load R_L is applied to line 2, a wave propagates on line 2 depending on the ratio of Z₂/R_L. The voltage at the load will stay constant until this wave reflects at line 1 and returns to the load. When the initial wave reaches line 1, a part of the wave is transmitted through to the voltage source. At the voltage source the wave is reversed in direction and amplitude. If line 1 is short then waves will make many round trips between the voltage source and line 2, while a wave makes one round trip on line 2. For each round trip on line 1 there will be some energy supplied to line 2. As these voltage increments reach the load, the load voltage will rise in an exponential manner.

The ratio of energy carried by waves on the two transmission lines is equal to the inverse ratio of characteristic impedances. The ratio of wave round trips is equal to the ratio of transit times as given in Equation 2.2. If we equate the ratio of characteristic impedances to the ratio of transit times, we have an approximate balance in the flow of energy in the two transmission lines. The resulting equation of balance is

2.25

where k is about 3. This factor accommodates the fact that the rising voltage at the load approaches its final value in an exponential manner.

It is interesting to use the above equation on an example. Assume d₁ is 1.0 cm ε ₁ = 4, Z₁ = 50, ε ₂ = 3600, and Z₂ = 1 then d₂ = 5 cm.

If d₂ is shorter than this value then the voltage at the load will sag to a lower value.

If Z₂ represents a decoupling capacitor then this capacitor must have a physical length of 5 cm to be effective. To reduce this length, the capacitor can have a lower characteristic impedance or a higher dielectric constant.

This build up of current flow in a short section of 50-ohm line illustrates the problem of pulling energy from the ground/power plane capacitance. The characteristic impedance of a typical connection is approximately 50 ohm. It takes nanoseconds before the reflection process can supply current from the planes through this high impedance. If there is a short section of connecting line, the delay is increased further. This is illustrated in curve D of Figure 2.9.