Improve and simplify cable simulation

Bernard Hyland, Maxim Integrated Products -July 30, 2012

This article covers the two main loss effects related to cables (the skin effect and dielectric losses) and presents a simple cable modeling method for use in standard SPICE simulators.

Cables are used in many high-frequency board designs and can become a critical element in the signal path. This is especially true for signals that exceed 500MHz. If not modeled as part of that system, cables can lead to unexpected system performance degradation and to costly time delays in debugging and corrections. Even with this understood, cables are notoriously difficult to model correctly. Using a simple transmission line model may not effectively model this element because it is difficult to model a cable both in the frequency and the time domains.

Cable nonideal dispersive effects can affect system performance. These cable effects are seen on drivers, buffers, and comparators [1]. In the driver or buffer, the low-frequency dribble up (Figure 1) primarily degrades propagation delay versus pulse-width dispersion. It also degrades minimum pulse time and rise time. In the comparator, the low-frequency dribble up primarily degrades delay versus pulse width and propagation delay versus overdrive. However, it also degrades the minimum pulse width. This article discusses the two main loss effects related to cables (the skin effect and dielectric losses) and presents a simple method for modeling the cable for use in standard SPICE simulators.

Figure 1. Example of cable loss effects (simulated results from Figure 3 below).

Why model cable?
As frequencies start to exceed 500MHz, the cable starts to noticeably impact the bandwidth of the signal path and begins to degrade this path in many ways. To understand the effects of cables at all frequencies, the cable needs to be modeled. Based on that model, more intelligent decisions can be made about the type of cable to be used. In addition, the parameters of interest that are being degraded in the signal path can then be understood.

Cable losses
There are two main loss mechanisms with cables: skin-effect and dielectric losses.
At high frequencies, the signal tends to propagate along the surface of the conductor. This is known as skin effect. (Note that this article will not rigorously discuss skin-effect losses.) The skin depth δ, is defined as:

Where ω is the frequency in radian per second; μ is the conductor’s permeability in H (Henries per meter); and σ is the conductor’s conductivity in S (Siemens per meter).

Equation 1 shows that the skin depth decreases with the square root of the frequency. Alternatively, since the lower the skin depth the higher the resistance per unit length, then the resistance per unit length, RL, increases with the square root of frequency.

The dielectric constant of insulators has an imaginary component. The dielectric constant,  ε, is defined as ε  = ε’ + jε’’ = ε’(1 + jtanδ), where ε’ is the real component of the dielectric constant; and tan&;delta; is the loss tangent or dissipation factor of the dielectric. The dielectric constant affects the capacitance and the CI (capacitance per unit length) will change to CI(1 + jtanδ).

Total cable loss
By including the skin-effect and dielectric losses in the model, an ideal cable model (per unit length) can be now modified to include those losses (Figure 2). Here Ll, Rl, and Cl are the per unit length for inductance, resistance, and capacitance, respectively.

Figure 2. Ideal cable representation.

The propagation constant is defined as jk = √ZY, where Z is the distributed series impedance and Y is the distributed parallel admittance.

From Figure 2 we see:

Where Zo is the cable’s characteristic impedance; εr is the relative dielectric constant; and c is the speed of light.

We are actually seeking the cable gain, H(??). Therefore, H(??) = e-jkl, where l is the length of the line.
Using the findings above:

Therefore, skin-effect losses (α1) dominate at lower frequencies and dielectric losses (α2) dominate at higher frequencies.

Note that in real-world cables H(??) varies somewhat from the approximations given above. However, this model is accurate enough for most automated-test-equipment (ATE) work, where the attenuation increases to 6dB at the most.

We will be using Equations 6, 7, and 8 later to model the cable.
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