## Characteristic impedance of lossy line

For a hypothetical, 150-micron (6-mil), 50Ω FR-4 stripline, Figure 1 illustrates the relative influence of skin-effect and dielectric losses on the characteristic impedance of a lossy transmission line. The chart depicts the characteristic impedance of a trace with only skin-effect and dc-resistive losses (assuming a perfect dielectric), a trace with only dielectric losses (assuming zero resistance), and a combination of both.

The topmost three curves in each case show the real part of impedance, and
the bottom three curves show the imaginary part. Compared with a base case that
includes no losses of any type, skin-effect losses *increase* the real part
of the impedance curve in the vicinity of the skin-effect onset at *ω*_{δ}, and the dielectric losses *decrease* the real part
of impedance in the same area. The two effects almost cancel each other in the
band above *ω*_{δ}, causing the characteristic impedance in this
region to approach the asymptotic value marked *Z*_{1} more
rapidly, and more completely, than when either effect is present alone. The
cancellation is nothing more than a grand coincidence. Do not depend on the
cancellation effect to stabilize the impedance of a practical design, as the
values of dielectric loss in most materials vary substantially with temperature
and water content.

At frequencies above the onset of the dielectric-loss-limited mode, *ω*_{δ} dielectric losses have the effect of increasing the real
impedance.

Dielectric losses progressively diminish the available line capacitance as you move to higher and higher frequencies, thereby causing an upward tilt to a plot of characteristic impedance versus frequency. In the time domain, the tilt suggests that the effective impedance measured over short scales of time should exceed that measured at larger scales of time. Figure 2 illustrates precisely that effect.

The detailed blowup of the first plateau in the TDR (time-domain-reflectometry) response for a lossy transmission line shows three waveforms: the response of a trace with only skin-effect and dc-resistance losses (assuming a perfect dielectric), a trace with only dielectric losses (assuming zero resistance), and a combination of both.

The dielectric losses produce a negative slope in the first plateau. The resistive losses create a positive slope. Working together, the two effects almost cancel, in this example, creating a slope less steep than when either effect is present alone-the same peculiar coincidence mentioned above.

Dielectric loss distorts the slope in the first plateau, obliterating your ability to accurately infer the resistance of the line from a single TDR plot.

*EDN* takes this excerpt from the forthcoming Prentice Hall
publication,* High-Speed Signal Propagation: More Black
Magic *, by Howard Johnson, ISBN
013084408X, February 2003. Adapted by permission of Pearson Education Inc, Upper
Saddle River, NJ.