The total perunitlength series impedance z of any transmission line may be partitioned into external and internal series impedances: z = z e + z i .
The external series impedance z e represents mostly the inductance L e formed by magnetic flux occupying the spaces between the signal conductors (i.e., in dielectric insulating medium).
Equation 2.46
where 
L e is the external series inductance (H/m). 
The external series impedance z e also incorporates a small real part representing resistive losses encountered in the space surrounding the conductors (like radiation resistance). The resistive component of z e , I shall ignore under the assumption that conductors normally used for digital applications are designed not to significantly radiate.
Practical conductors posses two additional contributions to their series impedance: L i and R i . The internal inductance L i represents the magnetic flux that penetrates the walls of the conductor (see Figure 2.14). The term R i represents the resistance of the conductor. The internal impedance z i captures both resistive and inductive terms:
Figure 2.14. In this coaxial configuration a noticeable proportion of the lowfrequency magnetic flux lies within the signal conductor and shield.
Equation 2.47
where 
R i is the internal series resistance ( W /m), and 
L i is the internal series inductance (H/m). 
As you move to higher and higher frequencies the magnetic fields (and associated currents) penetrate the walls of the conductors less and less significantly, thus shrinking the internal inductance and raising the internal resistance. Figure 2.15 captures the changes to internal inductance and resistance that happen within an RG58/U coaxial cable.
Figure 2.15. Below the onset of the skin effect both L i and R i hold constant.
Below w d (the skineffect onset frequency), both resistance and inductance hold constant. The internal resistance is determined by the conductivity and crosssectional area of the conductors, while the internal inductance depends on the conductor geometry. For round conductors, the low frequency value of internal inductance is m /8 p . For other conductor shapes (like square or rectangular conductors), the internal inductance is undoubtedly less; however, the exact value of internal inductance for an arbitrary shape is not easily calculated.
Above w d , the internal resistance R i grows in proportion to the square root of frequency, while the internal inductance L i shrinks at the same rate. This result holds for all good conductors , meaning materials for which ( s >> w e ), a property shared by most metals up to extremely high (i.e., optical) frequencies. In all but the lowest impedance transmission configurations L e exceeds L i by a substantial factor even at DC, with the difference becoming even more pronounced about w d . Not far above w d , the internal inductance shrinks to insignificance.
Table 2.4 summarizes the asymptotic values of R i and L i for a round conductor , assuming it is located sufficiently far from its return path so k p = 1.
A complete derivation of the distribution of current within a round conductor appears in [19] and [15] . Unfortunately, the derivation may be obtained only at the high mental cost of invoking Bessel functions. These derivations apply to the case of one round conductor exposed to a uniform electric field oriented along the axis of the conductor. The conductor must be made of a "good conductor" material for which s greatly exceeds w e at all frequencies of interest (a good assumption for almost all metals). Under these assumptions, the value of L i due to redistribution of current within the center conductor equals precisely m /8 p . The complete form of the internal impedance z i for a round conductor is
Equation 2.48
where 
h is the intrinsic impedance of a good conducting material, 
I and I 1 are modified Bessel functions of order zero and one respectively, 

w is the frequency of operation (rad/s), 

m is the magnetic permeability of the conducting material (H/m), 

s is the conductivity of the conducting material (S/m), and 

r is the radius of the wire (m). 

The value of m for nonmagnetic materials is 4 p ·10 “7 (H/m). 

For annealed copper at room temperature, s = 5.800 ·10 7 (S/m) . 
In general, the intrinsic impedance of a material is given by . In a good conductor, however, the term s greatly exceeds j w at all frequencies, so the intrinsic impedance reduces to . The intrinsic impedance of a good conductor therefore takes on the phase angle of , which precisely equals p /4.
When evaluating [2.48], you need to know that many software implementations of the modified Bessel functions I and I 1 exist for realvalued arguments, but few accept complex arguments. That problem may be solved by resorting to two specially tabulated Bessel functions, ber n ( v ) and bei n ( v ), defined in the following way:
Table 2.4. RoundWire Values for R i and L i , Assuming k p = 1 and k a = 1
w << w d 
w >> w d 


Internal resistance R i ( W /m) 

Internal inductance L i (H/m) 

Internal impedance z i = R i + j w L i ( W /m) 

NOTE ”Where a is the crosssectional area (m 2 ), p is the perimeter of a crosssection of the signal conductor (m), s is the conductivity (S/m), and m is the permeability (H/m) of the conductor. The operating frequency is w (rad/s). The conductor is assumed smooth and well separated from a large, lowresistance return path. 
Equation 2.49
The functions ber n ( v ) and bei n ( v ) are available in MathCad, where they go by the name of the Bessel Kelvin functions. These same functions are tabulated elsewhere [6] , [7] .
Figure 2.16 plots the internal impedance predicted by [2.48]. Please keep in mind that equation [2.48] works only for round wires. Figure 2.16 depicts the real and imaginary parts of the complex internal impedance, R i and j w L i , slightly different from the parameters R i and L i (without the j w ) shown in Figure 2.15. One fascinating aspect of this figure is how, at high frequencies, the real and imaginary parts of complex internal impedance converge to a common value. This convergence reveals a deep connection between the real and imaginary parts of any network function. Any causal , minimumphase impedance function growing at a flat rate of +10 dB/ decade must have a phase angle of exactly p /4. The phase angle of p /4 bestows upon the function equal real and imaginary parts.
Figure 2.16. Equation [2.48] for round wires predicts both real and imaginary parts of the internal impedance.
2.8.1 Practical Modeling of Internal Impedance
Although the closedform model [2.48] is mathematically quite beautiful, it isn't widely used for two reasons:
For most signalintegrity simulations, an approximation will suffice. I'll present two possibilities here. The basic approximation produces the correct asymptotic behavior at DC and at high frequencies, but lacks accuracy in the transition region. The better approximation provides improved accuracy in the transition region. Both approximations use the same parameters:
w 
Frequency of operation (rad/s). 
R DC 
The total DC series resistance per meter of the transmission line ( W /m), 
w 
A particular frequency (rad/s), chosen well above the onset of the skin effect but below the onset of surface roughness and nonTEM modes, and 
R 
The real part of the skineffect impedance at the particular frequency w . ( W /m). R should be set to take into account the proximity effect, if present, although in the following discussion for roundwire problems not in proximity to other conductors, you may assume k p 1. This value may be computed using [2.43] or [2.44]. 
Both approximations use the same expression for the highfrequency (AC) series impedance of the wire. Note that this expression comprises equal real and imaginary parts, meaning a phase angle of 45 degrees, which is correct for a causal, minimumphase network function with a slope of +10 dB/decade.
Equation 2.50
In both cases the real part of z i will be identified as the internal resistance, while the imaginary part of z i will be interpreted as the internal inductance: .
Here are the two approximations for z i .
Simple approximation:
Equation 2.51
Better approximation:
Equation 2.52
Figure 2.17 compares both approximations to the Besselfunction solution for a round wire. The asymptotic behavior of both functions is good. Both functions produce the correct impedance at very high frequencies, with the correct upward tilt of +10 dB/decade and a phase angle of +45 degrees. They also both produce the correct DC resistance at low frequencies. The difference between them lies in their behavior near the skineffect onset frequency and in the predicted value of internal inductance.
Figure 2.17. Two decades to either side of the skineffect onset frequency, the real part of z SIMPLE still errs by 10%.
In particular, at w d the real part of z SIMPLE looms far too large, exceeding R DC by 6.02 dB. At that same frequency the real part of z BETTER is only 2.09 dB larger than R DC , much closer to the correct value (according to the Besselfunction solution) of 2.04 dB.
The second problem with z SIMPLE concerns its predicted value of L i at low frequencies. The imaginary part of z SIMPLE should have a positive slope of +20 dB/decade at low frequencies. Instead, it has a slope of only +10 dB/decade, an artifact that effectively predicts infinite inductance at DC.
These shortcomings combine to produce fractional errors in z SIMPLE on the order of 50% at w d , falling off to approximately 10% at frequencies two decades on either side of w d . Don't use z SIMPLE near the transition region. If your simulation operates exclusively at frequencies above w d , then approximation z SIMPLE may be somewhat improved by assuming R DC = 0.
If you require a more accurate approximation, try z BETTER . In the simulation of RG58/U coaxial cable the peak error in z i induced by the use of z SIMPLE is 56.7%, where the error is expressed as a percentage of z i . The error in z BETTER under the same conditions is only 6.7%. Because the internal impedance z i represents only a part of the overall series impedance z s , calculating the same errors as a fraction of the overall series impedance z s results in values of 25.4% and 1.4% for z SIMPLE and z BETTER , respectively. If you need even better accuracy in the transition region than provided by z BETTER , use a twodimensional fullwave electromagnetic field solver.
2.8.2 Special Issues Concerning Rectangular Conductors
For rectangular conductors there exist no closedform solutions for internal inductance. Nor can the internal inductance be evaluated using ordinary quasistatic 2D calculations, because the phase of the current deep inside the conductor differs substantially from the phase on the surface. This difference in phase introduces new complications into the mathematics of the problem ”complications not handled by commonly available 2D field simulation software.
Fortunately, the exact value of L i matters little in the computation of many signalintegrity results, because the external inductance L e always exceeds the internal inductance L i for all but the very lowestimpedance configurations. As the frequency rises above w d , the difference becomes even more pronounced so that any error in L i is quickly swamped. The largest simulation errors lie near the skineffect onset frequency, falling off on either side of the onset. For signalintegrity simulations of transmission lines at reasonable impedances (i.e., L e >> L i ) or at frequencies well above w d , this author uses approximation z BETTER for rectangular conductors.
Below the skineffect onset, the magnitude of the internal inductance L i plays a role in determining the phase delay of the transmission line. For nonround conductors, approximation z BETTER makes some particular assumptions about the lowfrequency value of L i . The values of L i implied by the use of approximation z BETTER are found by evaluating Im( z BETTER )/ w in the limit as w 0.
Equation 2.53
The evaluation proceeds by first factoring R DC from under the radical .
Equation 2.54
Take the first two terms in the Taylor's series approximation for
Equation 2.55
Take the imaginary part and simplify the expression.
Equation 2.56
Substitute [2.37] for R DC . Substitute [2.43] for real part of the skineffect impedance, R , evaluated at frequency w . Assume k a = k r = 1.
Equation 2.57
Plug in definition of the skin depth.
Equation 2.58
The final result is
Equation 2.59
Table 2.5 lists the lowfrequency internal inductance values asserted by approximation z BETTER for various conductor geometries. Please don't make the mistake of assuming these are authoritative values of L i (0) for the listed geometries (except for the round wire, which is exact). These are merely reasonable assumptions made by the approximation Z BETTER .
If you need more accurate models in the vicinity of the skineffect onset, you should use a 3D fullwave electromagnetic field solver.
POINTS TO REMEMBER
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