# RF Math Made Easy

*At some point, you’ll probably need to perform “RF mathematics” when installing or supporting a wireless LAN. In most cases, this will involve dealing with logarithmic relationships, which is a bit beyond what most of our brains can handle directly. So let’s take a look at what might make this pesky math a bit easier.*

RF signals have amplitudes with units of watts, which represent the amount of power in the signal. As the power (i.e., number of watts) increases, the signal will travel farther.

A watt is a linear value that follows mathematical relationships that we’re very all familiar with. For example, the result of doubling 10 milliwatts (mW) is 20mW. We certainly don’t need to do any serious number crunching to get that result.

As an alternative, RF signal power has the unit of dBm, which is a *logarithm* value. It’s the amount of power referenced to 1mW, which simply means that 0dBm equals 1mW. By the way, the little “m” in dBm is a good reminder of the 1mW reference. The dBm values are positive above 1mW and negative below 1mW. Beyond that, math with dBm values gets a bit harder.

A problem is that literature refers to both linear (watts) and logarithmic (dBm) units. The output power of an access point, for example, is generally given in mW. Most analyzers, though, display output power in dBm. Often, you must perform conversions so that all values are in the same units in order to determine path loss, calculate EIRP (equivalent isotropically radiated power), and so on.

The following are relationships between mW and dBm:

dBm = (10Log_{10}(mW))

and

mW = 10^{(dBm/10)}

This is a good time to point out that dBm values don’t fit into the linear world. For example, doubling the equivalent power in watts of 20dBm is not 40dBm. You can see this for yourself by using a calculator and finding the value in watts for 20dBm. The answer is 100mW. Now perform the calculation for 40dBm, and you’ll get 10,000mW. That’s much more than doubling the power.

If you have a calculator with a logarithm (LOG) key, then you can punch in the numbers and get the results. For example, plug 100mW into the equation for dBm, and you should find that the answer is 20dBm. Try another one by converting 26dBm to mW. You should get 400mW.

If you want to impress your friends, you can remember some simple relationships and easily do RF math in your head, probably faster than with a calculator. A simple relationship is that if you multiply a value in mW by 10, the equivalent value in dBm increases by 10 (dividing mW by 10 decreases dBm by 10). Here’s a quick list that provides some typical equivalent values worth memorizing based on this rule:

- 0dBm = 1mW
- 10dBm = 10mW
- 20dBm = 100mW
- 30dBm = 1,000mW (1Watt)

Memorize this table, and you’ll be off to a very good start. Also, if you multiply the value in mW by two, the equivalent value in dBm increases by 3dBm (divide mW by two and subtract 3dBm). Something to understand is that every 3dB of antenna gain adds 3dBm to the value of dBm and doubles the number of mW.

As a practical example, let’s assume that you want to know the overall effective power in dBm of an access point that is set to 100mW transmit power with an antenna having 6dB gain. You might find yourself going through this exercise to know if your system is within regulatory limitations. You could use a calculator and convert 100mW to dBm using the formula from above.

Or, if you remember that 20dBm equals 100mW, then all you have to do is add 6dBm to 20dBm and end up with the correct answer of 26dBm. The antenna gain of 6dB adds 3dBm to the value twice (it doubles the power twice).

Now, let’s make life a little more difficult. What is the value in dBm of a 200mW signal? (Do this one in your head without a calculator.) Recall that 20dBm equals 100mW, and doubling 100mW to 200mW adds 3dBm. So, the answer is 23dBm. That’s much faster than using a calculator!

With a little practice, you’ll be a wiz at this type of math. You might want to have a calculator nearby at first to check your work, but you’ll soon be running through these types of problems in your head completely on your own!

*Jim Geier is the principal of Wireless-Nets, Ltd, a consulting firm focusing on the implementation of wireless mobile solutions and **training*.* *