Coaxial cables are often neglected during the characterization of high-speed digital components, even though such transmission lines can have a major electrical influence on a test setup. Coaxial cables are sometimes treated as lossy elements, and assigned a lumped capacitance and/or inductance, although the electrical effects of a coaxial cable can be much more complex than a single capacitance value.

What follows is an examination of how the electrical performance of a coaxial cable, notably loss and distributed capacitance/inductance, can affect the integrity of a high-speed digital pulse. The article will review the creation of a high-speed digital pulse, how to launch the pulse from an ideal output impedance, follow its transmission down an imperfect coaxial cable, and show how to make measurements at a perfect load.

To apply transmission-line theory to this analysis, the pulse will be created in the time domain and then converted to the frequency domain. To use the analysis tools available, the pulse is created as a discrete time pulse. The Fourier Transform, and inverse Fourier Transform for the pulse are¹

where:

M = 17,
N = 2^M,
K = 1 + 2^{M − 1}
t_n = Δt(− 2^{M − 1}),
Δt = 0.25 ps,
n = 0...(N − 1)
f_k = kf₀,
f₀ = 1/(2^MΔt), and
k = 0...(K − 1). Note that the values for M and Δt have been chosen as required for optimal resolution and bandwidth.

At the input, this pulse is very narrow and has a finite rise time. It can be expressed mathematically as:

SEE EQ. 2

where:

A = amplitude,
π = the pulse width, and
t_r = the 100-percent rise time, and

SEE EQ. 3

Transmission-line theory is well established and this analysis should be straightforward. Still, it may provide some reassurance by running a simple analysis that can be confirmed by means of time-domain-analysis techniques. In this case, a pulse will be launched from a nonideal source and transmitted a short distance to a nonideal load. The source and the load are connected by means of a short, lossless coaxial cable. The source is chosen to be 60 Ω, the cable is 50 Ω, and the load 40 Ω (Fig. 1). These impedance mismatches will generate multiple reflections, which can be calculated using time-domain-to-frequency-domain-to-time-domain theory, and then verified using time-domain transmission-line theory.

The analysis will be presented as a time-domain graph of the source voltage, the input voltage, and the output voltage. The first step in the analysis is to convert the source voltage into the frequency domain by means of the Fourier Transform. In order to calculate the input voltage, the input impedance, which can be determined by the scattering parameter (S parameter) S₁₁, is required. The S₁₁ term can be calculated from the circuit's ABCD parameters²:

SEE EQ. 4

where:

βl = the electrical length of the cable [ = 2πf(0.5 ns)].

From the ABCD parameters, S₁₁ is³

SEE EQ. 5

The input impedance is:

The input voltage and current become:

The output voltage and current can be calculated using the ABCD parameters and the input voltage and current: