 Signal Propagation Model
 Hierarchy of Regions
 Necessary Mathematics: Input Impedance and Transfer Function
 LumpedElement Region
 RC Region
 LC Region (ConstantLoss Region)
 SkinEffect Region
 Dielectric Loss Region
 Waveguide Dispersion Region
 Summary of Breakpoints Between Regions
 Equivalence Principle for Transmission Media
 Scaling Copper Transmission Media
 Scaling Multimode FiberOptic Cables
 Linear Equalization: Long Backplane Trace Example
 Adaptive Equalization: Accelerant Networks Transceiver
3.3 Necessary Mathematics: Input Impedance and Transfer Function
The performance of the linear, timeinvariant transmission circuit shown in Figure 3.3 depends on four crucial factors: the characteristic impedance of the line Z_{C} [3.15], the raw oneway propagation function of the transmission line H [3.14], the source impedance Z_{S}, and the load impedance Z_{L}. All four of these complex phasor quantities vary with frequency. Although the figure is drawn representing a singleended coaxial configuration the same considerations apply to any form of conductive transmission circuit.
Figure 3.3. A transmission line complete with source and load impedances may be modeled as a cascade of three twoport circuits.
The input impedance v_{2}/i_{2} of a loaded transmission line is derived in Appendix C, “TwoPort Analysis.”
The gain G of the circuit of Figure 3.3, taking into account all the relevant loading effects and endtoend reflections, is (see Appendix C)
where 
G is the overall system gain from the opencircuited output of the driver v_{1} to the load v_{3}, 
Z_{S} is the source impedance of the driver (Ω), 

Z_{L} is the load impedance at the far end of the transmission structure (Ω), 

Z_{C} is the characteristic impedance of the transmission structure (Ω), and 

H is the oneway propagation function of the raw, unloaded transmission structure [3.14]. 
Formulation [3.17] is most useful for analyzing the behavior of lumpedelement and RCmode transmission structures. An alternate form applies best for the analysis of the lowloss structures (LC, skineffect, and dielectricloss regions).
If your objective is the undistorted conveyance of a signal from source to load, then you must ensure that the propagation function [3.17 or 3.18] remains flat over the band of frequencies covering the bulk of the spectral content of your data signal. The requirement for “undistorted conveyance” is equivalent to asking that each transition arrive intact, at full size, without significant dispersion of the rising or falling edge, and without any lingering aftereffects, like reflections or ringing. Such a waveform is useful for firstincidentwave switching, meaning that the first edge arrives with sufficient fidelity to be immediately and reliably used.
On the other hand, if you are willing to wait a few roundtrip times for your signal to settle, then you do not need your signals to arrive with firstincidentwave quality. You can in this case tolerate significant imperfections in the frequency response of your channel, all of which are cured by waiting for the steadystate condition to emerge, after which the signal may be reliably used. Waiting solves almost every signal integrity problem. If you can wait long enough, any transmission line will settle to a steady state. Of course, I assume the reason you are reading this book is that you cannot afford to wait! In that case, you have a direct interest in the gain flatness of the propagation function.
The bulk of the useful spectral content of a random data sequence spans a range from DC (zero frequency) up to a maximum upper bandwidth of
The knee frequency, f_{knee}, is a crude estimate of the highest frequency content within a particular digital signal. Presuming the propagation function G [3.17] remains flat to within x percent over the range 0 < f < f_{knee}, the expected distortion in the received waveform will be on the order of x percent.
The best assumption for the midpoint of the spectral content associated with the rising and falling edges of a digital signal is a little less than the maximum bandwidth:
A timedomain reflectometry (TDR) setup measures the gain from v_{1} to v_{2} (see Appendix C):
where 
TDR is the overall system gain from the opencircuited output of the driver v_{1} to the timedomain reflectometry observation point v_{2}, 
Z_{S} is the source impedance of the driver (Ω), 

Z_{L} is the load impedance at the far end of the transmission structure (Ω), 

Z_{C} is the characteristic impedance of the transmission structure (Ω), and 

H is the oneway propagation function of the raw, unloaded transmission structure [3.14]. 
The following sections detail the performance characteristics of each region and the means necessary to maintain acceptable flatness in [3.17] over the intended range of operation.
The stepresponse approximations for each region work best for transmission media with wide, welldefined regions. For example, the skineffect approximation presented below represents the performance of Belden 8237 beautifully over a range of four orders of magnitude, from 10^{5} to 10^{9} Hz. As you approach the edge of each region, however, the step response begins to mutate into a new shape characteristic of the next region.
The modeling of complete systems, including arbitrary source and load impedances and combinations of operation in all regions, is considered at the end of this chapter.

The undistorted conveyance of a signal from source to load requires a propagation function that remains flat over a band of frequencies covering the bulk of the spectral content of the data signal.