Fundamentals: Working out ERP

Image: Marco Verch CC BY 2.0

I got asked this question on Facebook the other day and the answer was too long to type in the comments.

The question was, how does one calculate the radiated power (ERP) at the antenna if the transmit power of the repeater is say 25W.

The Effective Radiated Power (ERP) of a repeater is determined by:

1. The transmit power of the Repeater as measured at the antenna socket
2. Combined losses of all cables between the Repeater and Antenna
3. Insertion loss of any filtering equipment
4. The gain of the antenna

Since TX power and ERP are both measured in Watts and because items 2-4 are measured in dB, it makes sense to first convert everything into the same measurement scale. Practically, this means converting Watts to dBm (or more accurately dBmW) - and then back again.

Watts to dBm

The easiest way to do the conversion in your head, is to remember that 0dBm (0dBmW) is 1mW (one milliwatt or 0,001W). So 10mW will be 10dBm; 1W will be 30dBm and 100W will be 50dBm.

This is done using the following formula:

Where: Pd is the Power expressed in dBm and Pw is the same in Watts.

If you're using a calculator like mine, the keystrokes are {Pw} [÷] 0.001 [=] [LOG] [×] 10 [=]. If you're using a HP; you'll more than likely know what to do.

An example

Now that all the parameters are in dB, it's a simple task to add all the values. Remember that anything that has a loss, has a negative value as far as gain goes.

The best way to illustrate this, is to use a practical example.

Here is a very simple repeater site: repeater; duplexer; co-ax and antenna. The transmit power of the repeater is set to 25W or 44dBm (actually 43,979dBm).

The internal cables and feedline will almost certainly be different. Typically, internal cables (the cables that connect the repeater to the duplexer could be something like RG223. On VHF, this cable has a loss of 0,187dB/m. Your cable supplier will have a table showing the losses for all cables they sell. Lets say that the length of the cable between the TX output and Duplexer, is 3m in which case the loss will be 0,561dB.

The duplexer will have a certain insertion loss. For example, the TPRD-1544 from Telewave has an insertion loss of 1,5dB (or a gain of -1,5dB).

Depending on the application, the feedline will usually be a larger-diameter low-loss cable. On the other hand, the internal cables between duplexer and repeater will usually be a thinner cable which has a smaller bending radius.
RG214, for example, has a diameter of 10,8mm and 0,160dB/m loss. If the length of the cable is 35m then the loss will be 5,6dB (or gain of -5,6dB).

The antenna will have a certain gain. The gain depends on what type of antenna it is. For example, a single folded dipole will have 0dBd gain. Be careful, some manufacturers will state the antenna gain in dBi. An antenna with 2,15dBi gain will actually have 0dBd gain.
In this case, let's say that here we are using a 2-stack dipole from Procom. This antenna has a gain 5dBd.

So in summary:

TX Power: 44dB
Internal cables: -0,561dB
Duplexer: -1,5dB
Feedline: -5,6dB
Antenna: 5dB

Since these values are all in dB, one just needs to add them all up. So the result is 41,339dB.

The last step is to convert back to Watts. To do this, use this formula:

Again, Pw is power is Watts and Pd is the same in dBm.
On a calculator, the keystrokes would be: {Pd} [÷] 10 [=] [2ndF] [Log] [÷] 1000 [=].
So in this case 13,61W.

The above is only true if the VSWR is 1:1 (i.e. no return loss).

I suck as Maths!

There are lots of tool on the internet which can do some of this for you. Here is what I found:

I hope this helps a few people. If you would like see more stuff like this, please let me know...

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