Performance Regions
 Sep 12, 2003
 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
Figure 3.1 displays the propagation function of six distinct types of coaxial cabling, plus one typical pcb trace. The horizontal axis shows the operating frequency in units of Hz. The vertical axis shows cable attenuation in units of dB. Both axes use logarithmic scales.^{23}
Figure 3.1. The attenuation curve for all copper transmission media is divided into distinct regions, with a characteristic relation in each region between the attenuation a and frequency.
Each curve may be divided into distinct regions, with a characteristic shape to the loss function in each region. The hierarchy of regions, in order of increasing frequency, proceeds generally in the same order for all copper media:

RC region

LC region

Skineffect region

Dielectric loss region

Waveguide dispersion region
Within each region the requirements for termination differ, as do the tradeoffs between length and speed. Remarkably, a common signal propagation model accurately describes almost any type of metallic transmission media across all four regions.
3.1 Signal Propagation Model
This model computes the transfer function and impedance of any forms of metallic cabling. It is appropriate for use with cables made in multiwire, ribbon, UTP, STP, or coaxial format. It also works for pcb traces, both striplines and microstrips, up to a frequency of approximately 10 GHz.
Table 3.1 describes the six parameters to the model, and further explanation is provided in the notes that follow.
Table 3.1. Parameters for Metallic Transmission Model
Name 
Meaning 
Units 
Ref 

R_{DC} 
DC resistance of conductors 
Ω/m 
[2.37] 
ω_{0} 
Frequency at which AC line parameters are specified 
rad/s 
see notes 
R_{0} 
Real part of AC resistance at frequency ω_{0} (real part of z_{i}) 
Ω/m 
[2.43] 
θ_{0} 
Angle formed by the real and imaginary parts of complex electric permittivity (arctg ε′′/ε′) at frequency ω_{0}, rad; for small angles, θ_{0} ≈ tan θ_{0} 
rad 
[2.73], [2.85] 
Z_{0} 
Characteristic impedance at frequency ω_{0} 
Ω 
[2.11] 
v_{0} 
Velocity of propagation (inverse of group delay) at frequency ω_{0} 
m/s 
[2.29] 
This model was developed over the course of many years, combining the best ideas from many versions of the international cabling standards with my own research.
This model is available in computerexecutable form. It has been coded in MathCad 2000 syntax. A working version of the model along with a set of example spreadsheets and cable parameters is available through http://www.sigcon.com.

Frequency ω_{0} is assumed sufficiently high that the skin effect has substantially eliminated the internal inductance of the signal conductors, and the proximity effect has caused the distribution of current on the surface of the conductors to assume its highfrequency, magnetically dominated form, but not so high that you must concern yourself with nonTEM waveguide modes. In some geometrical arrangements such a setting is impossible. For example, a gigantic 3mm 50ohm microstrip as typically used in microwave circuits might exhibit significant nonTEM behavior at frequencies as low as 10 GHz. Typical pcb geometries for digital circuits, being much smaller, enjoy the benefits of a much higher onset frequency for nonTEM behavior.

For simulations of data transmission systems operating above the onset of the skin effect, ω_{0} would ideally be located at or near onehalf the transmission symbol frequency.

For simulations of data transmission systems operating entirely within the RC dispersion region a value for R_{DC} and line capacitance C are required, but the other parameters may be ignored or set to default values:
These substitutions are equivalent to accepting C as a constant value independent of frequency and then setting the minimum possible value of L consistent with the speed of light.

If you have set ω_{0} commensurate with the rise/fall bandwidth of your digital signals, then parameter Z_{0} represents the average impedance over a section of transmission line with a delay comparable to the duration of one rising (or falling) edge. Parameter Z_{0} is assumed to be purely real—the model supplies the imaginary parts of the overall characteristic impedance Z_{C}(ω).

Parameter R_{DC} includes the total resistance of both outgoing and return conductors. For ribbon cables the total DC resistance sums the resistance of the outbound wire plus 1/N times the resistance of the N return wires. For twistedpair cabling the total DC resistance is twice the resistance of either wire alone. For coaxial cables the total DC resistance is the sum of center conductor and shield resistances.

Parameter R_{0} similarly includes the total resistance of both outgoing and return conductors, with the path of current flow determined by the proximity effect.

Sometimes R_{DC} is specified directly on the data sheet, in which case you should use that value instead of calculating it yourself from the wire sizes. Datasheet values take into account the resistivity of various alloys and coatings used in the construction of the conductors. Always use datasheet values when modeling coppercoated steel wire or tinned or silvercoated conductors.

Parameter θ_{0} models the effective dielectric loss for the mixture of insulating materials and air surrounding the conductors.

The model assumes uniform values for all parameters along the length of the transmission line.

The model does not take into account temperature variations. All parameters must be specified at their worstcase values.

The transmission line attenuation is maximized when you combine low Z_{0}, low v_{0}, large R_{DC}, large R_{0}, and large θ_{0}.
Here are the model equations:
First use the specification of the real part of the skineffect resistance, R_{0}, which is specified only at a single frequency, to produce a resistance model covering a range of frequencies.
where 
R_{0} is the AC resistance of the wire at frequency ω_{0}, in ohms, and 
R_{AC} is the complexvalued impedance due to the combination of skin effect and proximity effect. 
The factorindicates that the skineffect resistance grows proportional to the square root of frequency.
Another way to write the same equation is to substitute , which leads to
The factor (1 + j) indicates that for all positive frequencies the real and imaginary parts of the complex skineffect impedance are equal. You may also check that the real part of the complex skineffect impedance equals R_{0} when ω = ω_{0}.
Next we must model the crossover from DC resistance to AC resistance. I use a squarerootofsumofsquares type mixing function. This mixing function matches the measured skineffect data presented in ^{[28]} and also produces the correct lowfrequency value for the internal inductance of a round wire, 50 nH/m, given in ^{[26]} and ^{[27]}.
The transmission line is characterized by a nominal external inductance L_{0} per meter, which I define as a constant, calculated from the characteristic impedance and velocity. This is the value of inductance measured at frequency ω_{0}. Variations in inductance with frequency (which occur as part of the internal inductance) are incorporated into the imaginary part of R.
where 
Z_{0} is the nominal characteristic impedance at frequency ω_{0}, and 
v_{0} is the velocity of propagation at frequency ω_{0}. 
The transmission line is characterized by a nominal value of shunt capacitance C_{0} per meter, which is calculated from the characteristic impedance and velocity.
Formulas [3.5] and [3.6] are mathematical inverses of the telegrapher’s relations [2.30] and [2.33].
Value [3.6] is next expanded into a frequencyvarying model of the complex capacitance C(ω) of the transmission line. The imaginary part of jωC(ω) represents the capacitive reactance. The real part of jωC(ω) represents dielectric losses within the transmission line. The ratio of the real part of jωC(ω) to the imaginary part of jωC(ω) equals the dielectric loss tangent. A nonzero loss tangent induces a slow degradation in effective capacitance with increasing frequency.
The transmissionline propagation coefficient, per meter, is defined in numerous texts ^{[29]}:
The transfer function for a transmission line of length l meters is computed from the propagation coefficient,
The attenuation in dB is defined as the negative of the transfer gain in dB.
The characteristic impedance modeled as a function of frequency is
These equations correctly model the transfer function and impedance at low frequencies (in the dispersionlimited mode), midfrequencies (in the skineffectlimited mode), and at extremely high frequencies (in the dielectriclosslimited mode).
In the midtohigh frequency range the parameter R_{DC} provides an amount of loss that is flat with frequency. The parameter R_{0} provides an amount of loss that grows (in dB) in proportion to the square root of frequency. The parameter θ_{0} provides an amount of loss that grows (in dB) in direct proportion to frequency. At all frequencies the magnitude and phase responses match to produce a causal, minimumphase response.

The signal propagation model computes the transfer function and impedance of cables made in multiwire, ribbon, UTP, STP, or coaxial format. It also works for pcb traces, both striplines and microstrips, up to a frequency of approximately 10 GHz.

The parameter R_{DC} provides an amount of loss that is flat with frequency.

The parameter R_{0} provides an amount of loss that grows (in dB) in proportion to the square root of frequency.

The parameter θ_{0} provides an amount of loss that grows (in dB) in direct proportion to frequency.

At all frequencies the magnitude and phase responses match to produce a causal, minimumphase response.
3.1.1 Extracting Parameters for RLGC Simulators
Transmissionline parameters R, L, G, and C may be extracted from the expressions in the previous section.
Some simulators require a discrete table of parameters. Each row of the table represents the four values R, L, G, and C sampled at one particular frequency. The number of rows (frequency sample points) may be unlimited, with the frequencies spaced either linearly or exponentially on some kind of dense grid. The simulator interpolates between the listed points to do its work. Assuming you have the list of sample frequencies ω_{n} in units of rad/s, Table 3.2 shows how to prepare the vectors R, L, G, and C from which you may prepare your table. The expressions properly account for the influence of the inductive component of skin effect (internal inductance) on the total inductance L and the influence of the imaginary part of C(ω) on G.
Other simulators expect you to provide values of only six parameters from which they extrapolate the full range of R, L, G, and C at all frequencies. I’m a little suspicious of these types of simulators because there is no consistent, standard way to perform the extrapolation. I have heard reports of some simulators generating noncausal waveforms. You should check carefully the step response of your simulator (especially under a condition with lots of dielectric loss but very little resistive loss) to make sure it gives you the correct causal dielectric response, as shown in Chapter 4, “FrequencyDomain Modeling.” Some don’t. One popular combination of parameters from which other values may be extrapolated appears in Table 3.3.
There is no consistent standard as to whether the value L_{0} includes the internal inductance or not. You’ll have to check the documentation of your simulator. The equations above assume it is not included in L_{0}; therefore L_{0} in Table 3.3 just equals L_{0}, which in my system of definitions is the external inductance. If you wish to include the internal inductance, then you should evaluate the total line inductance at a frequency well below the skineffect onset (e.g., 1 rad/s) as shown in [3.12].
Table 3.2. R, L, G, and C Vectors Sampled at Frequencies ω_{n}
Parameter 
Value 
Units 

Series resistance 
Ω/m 

Series inductance 
H/m 

Shunt conductance 
S/m 

Shunt capacitance 
F/m 
Table 3.3. Six R, L, G, and C Parameters from Which Other Values May Be Extrapolated
Parameter 
Value 
Units 

Series resistance 
Ω/m 

Series inductance 
H/m 

Shunt conductance 
S/m 

Shunt capacitance 
F/m 

Series skineffect resistance 
Ω/(mHz^{1/2}) 

Shunt conductance (dielectric loss) 
S/(mHz) 