High-Speed Signal Propagation[c] Advanced Black Magic
Authors: Johnson H., Graham M.
Published year: 2005
2.8 Modeling Internal Impedance
The total per-unit-length 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).
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 low-frequency magnetic flux lies within the signal conductor and shield.
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 RG-58/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 skin-effect onset frequency), both resistance and inductance hold constant. The internal resistance is determined by the conductivity and cross-sectional 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  and  . 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
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 real-valued 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. Round-Wire Values for R i and L i , Assuming k p = 1 and k a = 1
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  ,  .
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 , minimum-phase 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 closed-form model [2.48] is mathematically quite beautiful, it isn't widely used for two reasons:
For most signal-integrity 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:
Both approximations use the same expression for the high-frequency (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, minimum-phase network function with a slope of +10 dB/decade.
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 .
Figure 2.17 compares both approximations to the Bessel-function 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 skin-effect onset frequency and in the predicted value of internal inductance.
Figure 2.17. Two decades to either side of the skin-effect 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 Bessel-function 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 RG-58/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 two-dimensional full-wave electromagnetic field solver.
2.8.2 Special Issues Concerning Rectangular Conductors
For rectangular conductors there exist no closed-form solutions for internal inductance. Nor can the internal inductance be evaluated using ordinary quasi-static 2-D 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 2-D field simulation software.
Fortunately, the exact value of L i matters little in the computation of many signal-integrity results, because the external inductance L e always exceeds the internal inductance L i for all but the very lowest-impedance 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 skin-effect onset frequency, falling off on either side of the onset. For signal-integrity 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 skin-effect 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 low-frequency 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.
The evaluation proceeds by first factoring R DC from under the radical .
Take the first two terms in the Taylor's series approximation for
Take the imaginary part and simplify the expression.
Substitute [2.37] for R DC . Substitute [2.43] for real part of the skin-effect impedance, R , evaluated at frequency w . Assume k a = k r = 1.
Plug in definition of the skin depth.
The final result is
Table 2.5 lists the low-frequency 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 skin-effect onset, you should use a 3-D full-wave electromagnetic field solver.
POINTS TO REMEMBER
High-Speed Signal Propagation[c] Advanced Black Magic
Authors: Johnson H., Graham M.
Published year: 2005