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 
W /m 
[2.37] 
w 
Frequency at which AC line parameters are specified 
rad/s 
see notes 
R 
Real part of AC resistance at frequency w (real part of z i ) 
W /m 
[2.43] 
q 
Angle formed by the real and imaginary parts of complex electric permittivity (arctg  e ''/ e ') at frequency w , rad; for small angles, q tan q 
rad 
[2.73], [2.85] 
Z 
Characteristic impedance at frequency w 
W 
[2.11] 
v 
Velocity of propagation (inverse of group delay) at frequency w 
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 .
NOTES:
Equation 3.1
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.
Here are the model equations:
First use the specification of the real part of the skineffect resistance, R , which is specified only at a single frequency, to produce a resistance model covering a range of frequencies.
Equation 3.2
where 
R is the AC resistance of the wire at frequency w , in ohms, and 
R AC is the complexvalued impedance due to the combination of skin effect and proximity effect. 
The factor indicates 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
Equation 3.3
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 when w = w .
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] .
Equation 3.4
The transmission line is characterized by a nominal external inductance L per meter, which I define as a constant, calculated from the characteristic impedance and velocity. This is the value of inductance measured at frequency w . Variations in inductance with frequency (which occur as part of the internal inductance) are incorporated into the imaginary part of R .
Equation 3.5
where 
Z is the nominal characteristic impedance at frequency w , and 
v is the velocity of propagation at frequency w . 
The transmission line is characterized by a nominal value of shunt capacitance C per meter, which is calculated from the characteristic impedance and velocity.
Equation 3.6
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 ( w ) of the transmission line. The imaginary part of j w C ( w ) represents the capacitive reactance. The real part of j w C ( w ) represents dielectric losses within the transmission line. The ratio of the real part of j w C ( w ) to the imaginary part of j w C ( w ) equals the dielectric loss tangent. A nonzero loss tangent induces a slow degradation in effective capacitance with increasing frequency.
Equation 3.7
The transmissionline propagation coefficient, per meter, is defined in numerous texts [29] :
Equation 3.8
The transfer function for a transmission line of length l meters is computed from the propagation coefficient,
Equation 3.9
The attenuation in dB is defined as the negative of the transfer gain in dB.
Equation 3.10
The characteristic impedance modeled as a function of frequency is
Equation 3.11
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 provides an amount of loss that grows (in dB) in proportion to the square root of frequency. The parameter q 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.
POINTS TO REMEMBER
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 w 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 ( w ) 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 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 ; therefore L in Table 3.3 just equals L , 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 w n
Parameter 
Value 
Units 

Series resistance 
W /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 
W /m 

Series inductance 
H/m 

Shunt conductance 
S/m 

Shunt capacitance 
F/m 

Series skineffect resistance 
W /(mHz 1/2 ) 

Shunt conductance (dielectric loss) 
S/(mHz) 
Equation 3.12
Fundamentals
Transmission Line Parameters
Performance Regions
FrequencyDomain Modeling
Pcb (printedcircuit board) Traces
Differential Signaling
Generic BuildingCabling Standards
100Ohm Balanced TwistedPair Cabling
150Ohm STPA Cabling
Coaxial Cabling
FiberOptic Cabling
Clock Distribution
TimeDomain Simulation Tools and Methods
Points to Remember
Appendix A. Building a Signal Integrity Department
Appendix B. Calculation of Loss Slope
Appendix C. TwoPort Analysis
Appendix D. Accuracy of Pi Model
Appendix E. erf( )
Notes