Quote of the Day

The great aim of education is not knowledge but action.

— Herbert Spencer

## Introduction

The popularity of Power Over Ethernet (PoE) has proven that customers find value in using a single cable for both data and power distribution. Unfortunately, copper-based Cat 5e/6 cable is limited to a 100 meters because of data transmission issues. To circumvent this limit, some equipment vendors are using composite fiber/copper cables – a single cable that contains fiber for data and large gauge copper wire for power distribution (Figure 1).

Some composite cables distribute power using a single pair of 22 AWG wire (power and ground). Other composite cables use Cat 6 cables with four pairs of 23 AWG wire to distribute power. I have also seen some proposal that have completely separate power and data cables.

To develop an intuitive feel for how these proposals work, I decided to work through a simple example from Belden (Figure 2) – the choice of Belden was completely arbitrary. Belden is recommending the use of Cat 6-based composite cables over a 22 AWG-based composite cable.

The analysis is simple and I normally would not post it – it consists of determining the resistance of copper wire at different temperature, gauges, and lengths. I decided to post this analysis because I used Mathcad's minimize function to optimize my model, which I do not recall demonstrating before. This will be an example that I will include in my set of reference worksheets for my staff.

## Background

### Definitions

- American Wire Gauge (AWG)
- This is an archaic way of specifying the diameter of wire. Also known as the Brown & Sharpe wire gauge, AWG is a standardized wire gauge system used since 1857 predominantly in North America for the diameters of round, solid, nonferrous, electrically conducting wire. (Source)
- Contact Resistance (
*R*)_{C} - The term contact resistance refers to the contribution to the total resistance of a system which can be attributed to the contacting interfaces of electrical leads and connections as opposed to the intrinsic resistance, which is an inherent property, independent of the measurement method. (Source)
- Reach
- I use the term reach for the maximum useful distance that a cable can service. This assumes a completely straight cable run, which rarely occurs in real life.

### Copper Wire Resistance Modeling

See this web page for background on modeling the resistance of copper wire.

### Objective

Belden gives the following reaches (Figure 3) for a one-pair 22 AWG cable and a four-pair Category 6 (i.e. 23 AWG) cable in this document.

## Analysis

### Utility Functions

Many of my posts use a pair of utility functions that compute the resistance of annealed copper wire as a function of temperature, wire gauge, length. I have discussed these utility functions (Figure 4) in a previous post – I will include them here for easy reference.

### Calculation Setup

Figure 5 shows how I setup my calculations for a commonly used temperature (50 °C). It also shows the Belden table that I will use for comparison with my model.

### Basic Circuit and Reach Formula Equation

Figure 6 shows the reference circuits and the formula I used to model the voltage drop for a given load current. In Figure 6, I define a parameter called λ that represent the two-way resistance in the cable per meter. This means that the total resistance in the cable is given by .

### Case Modeling

Figure 7 shows the how I modeled:

- Copper resistivity versus temperature for a 100 m length and specific wire gauges
- PoE source voltage, load voltage, and load power specifications – these values are set by the type of PoE (i.e. Type 1, Type 2, Type 3 – not codified).
- Wire configuration using an array called pairs
- Contact resistance

### Table Generation

We are now ready to determine the contact resistance values that will minimize the least-square error between my model and the Belden specification (Figure 8). I do not expect perfect agreement because Belden provides no details on their analysis assumptions.

I consider these values reasonably close considering that I am trying to guess their analysis assumptions.

## Conclusion

While simple, this analysis provided a nice example for how to use Mathcad's minimize function.