Introduction
I have been doing some work that involves computing the characteristic impedances of cables. The work has involved creating some tables in Mathcad for comparison with tables from a government specification. Since I am always looking for real-life computations to use as Mathcad training examples for my staff, I thought I would blog about this work. This blog post shows a couple of different ways that I computed a table of characteristic impedances for wire pairs of different gauges and at different frequencies. I used Mathcad range variables to create my table. I then compared my computed results with the results listed by the US government in a table put out by the Rural Elecrification Administration (REA), which is now known as the Rural Utility Service (RUS).
My plan is to use this post as an example for a class that I plan to teach in a few months.
Background
REA has many specifications that control how phone lines are connected in the United States. Figure 1 is an excerpt from one of their specifications that lists the characteristic impedances for insulated wires of various gauges and at various frequencies. This particular wire is referred to as polyolefin-insulated cable (PIC). This insulation is similar in performance to polyethylene.
Figure 1: Characteristic Impedance Table Excerpt for Polefin Insulated Conductors (PIC).
This table is from an old paper specification that was filled with markups. It is the only copy I have.
Analysis
Derivation of Key Relationships
Figure 2 shows my derivation of a couple of equations that relate Z0 (characteristic impedance) and vSignal (signal speed) to L0 (unit inductance) and C0 (unit capacitance). We need L0 and C0 to compute the characteristic impedance of the cable.
Figure 2: Derivation of Unit Capacitance and Inductance in Terms of Signal Velocity (vSpeed) and Impedance (Z0)
Figure 3 shows how we can compute L0 and C0 using the physical dimensions of the cable (s: conductor separation, and d: conductor diameter) and the dielectric constant of the insulation (ϵR: relative permittivity)(Source).
Figure 3: Formulas for the Impedance and Signal Velocity on a Wire Pair.
Calculation When Given Unit Capacitance
Figure 4 is a rather dense illustration of how I used the formulas of Figures 2 and 3 to compute the characteristic impedances for the same cable type as is documented in the REA document. The REA document specifies the unit capacitance of the cable and I have to assume a signal velocity. Most cables that I know of have a signal velocity of ~66% of the speed of light.
Figure 4: Calculation When Given Capacitance Per Unit Length.
Calculation When Given Wire Construction
Figure 5 is another dense illustration that estimates the characteristic impedance versus frequency and wire gauge using the cable dimensions and the relative permittivity of the insulation. To vary things a bit, I presented the output using an Excel component.
Figure 5: Characteristic Impedance Calculation Using Wire Geometry Specifications.
Conclusion
The impedances I obtained by both methods agree fairly well with the values listed by the REA. The largest differences are at low frequency. The differences are small enough not be significant for my work. I want to be able to use computed values rather than tables because they are more convenient for me to work with in Mathcad and Excel.