Several publications report the analytical expressions of unit step voltage responses of lossy lines. These theoretical formulas are typically derived applying the Inverse Laplace Transform to the expressions of line voltages and currents in the s plane. Coaxial cable made by solid copper cylindrical conductors is the most studied case.

1. Copyright Piero Belforte 2015
1
UNIT STEP RESPONSES OF LOSSY LINES
Several publications report the analytical expressions of unit step voltage responses
of lossy lines. These theoretical formulas are typically derived applying the Inverse
Laplace Transform to the expressions of line voltages and currents in the s plane.
Coaxial cable made by solid copper cylindrical conductors is the most studied case
[1,2]. Theoretical impulse response of microstrip and stripline structures are reported
in [3] and the related step response can be obtained by integration.
These formulations are very important because can be used to model lossy
interconnects in time domain using the BTM modeling technique and the related pwl
behavior fitting (PWLFIT) supported by DWS [4]. They can be also used as a first
guess of responses in the de-embedding loop to get measurement-derived BTM
models [4].
Skin effect losses on coaxial cables.
According to reference [1] pag.1, the unit step response of a coaxial cable due to skin
effect only is given by :
(1)
where :

2. Copyright Piero Belforte 2015
2
A similar formula, related to coaxial cable is found on ref.[2], Chapt.5 and shown in
Fig.1:
Fig.1: Unit step voltage and current of a coaxial cable due to skin effect losses [2]
Generic lossy lines
From the integrals of impulse responses reported in reference [3]:
Skin effect losses
(2)
Dielectric losses
(3)
where
(4)

3. Copyright Piero Belforte 2015
3
Application examples
A RG58 coaxial cable 1.83m long with a .5dB/m attenuation at 1Ghz, shows the
following theoretical step response (Fig. 2), according to (1).
Fig. 2: Skin effect only theoretical step response of a 1.83m long RG58 coaxial cable
Fig.3 shows the differences among measurement-derived response, theoretical skin
effect only and Vector Fitting (VF) -derived including dielectric losses.
Fig. 3: Step response of a 1.83m long RG58 coaxial cable: comparison among theoretical (skin
effect only, dotted), VF derived including dielectric losses (tandelta=8e-4, dashed) and
measurement-derived BTM model (solid)

4. Copyright Piero Belforte 2015
4
Fig.4 shows the step response of a 1m long RG223 coaxial cable [5]. The skin effect only and
dielectric only responses, calculated according to (1) and (3) respectively,are compared to the VF
derived and measurement derived BTM models [5].
Fig. 4: Step response of a 1m long RG223 coaxial cable: comparison among theoretical (skin effect
only, dotted and dielectric only, dash-dotted ), VF derived including dielectric losses (tandelta=8e-
4, dashed) and measurement-derived BTM model (solid)
References
[1] Lawrence Livermore Radiation Laboratory, UC Berkeley, 1964 "Pulse response
of coaxial cables" Pulse response coaxial cables
[2] Philip C. Magnusson,Andreas Weisshaar,Vijai K. Tripathi,Gerald C. Alexander
"Transmission Lines and Wave Propagation" Fourth Edition
Transmission Lines and Wave Propagation
[3] R.B Wu "Fast Eye-Diagram Analysis", National University of Taiwan, EDAP
2013 : FAST EYE DIAGRAM
[4] P.Belforte "Vector vs Piecewise linear fitting for Signal and Power Integrity
simulation" Research Gate July 2013 VECTFIT VS PWLFIT
[5] P.Belforte "Physical implementantion effects on Multi-gigabit/sec coaxial cable
performance" Research Gate July 2013 Physical effects on coaxial cable
performance