How to calculate passive repeater gain?

Question

An active club in my area moved its repeater to a new location behind a big hill, and we can't join the regular social nets anymore. I'm not holding my breath that anyone in the gated HOA at the top of said hill would let me install any sort of active equipment. But there is some undeveloped area up there where I might be able to Accidentally Misplace some scrap metal and a little bit of coax.

How much gain could I expect from a Passive repeater formed by simply connecting a receive antenna pointing in one direction to a nearby transmit antenna pointing in another?

When calculating a link budget directly between two radios, I can simply add the gains of the "transmitter" and "receiver" antennas [understanding the antennas may reverse roles of course], and that combined gain offsets any path/other loss. But it seems pretty apparent that I can't just account for the passive repeater antennas in the same way — dropping two 12 dBi yagis connected together somewhere in the far field along the original path isn't going to suddenly improve my signal at the repeater by 24dB, is it!?

But yet isn't that what the link budget equation would imply, even through two hops? Basically the path loss gets split between the two paths, and then the "received power" at the passive antennas becomes the "transmitted power" to the original destination? Everything in the link budget just adds and subtracts, so it seems like nesting one link inside another should would just add up!

Answer 1

As I suspected, interposing a passive repeater does not somehow magically just add in a bunch of additional gain. The key reason why not stems from this error in my initial question:

Basically the path loss gets split between the two paths

My thinking being that if path loss from A to C was e.g. 100 dB then if you put B in between, then the path loss from A to B would be 50 dB and the path loss from B to C would be 50 dB and so the total would still be just 100 dB. Maybe you can already see the problem here?

Unfortunately when you halve the path length, you can't just "halve" the path loss in decibels. Well actually, it's better than that! Halving the path length quarters the path loss. But what that means is that the free space path loss (FSPL) amount goes down by 6 dB, regardless of its starting value. (Logarithm math! Halving [-3dB] the radius of a sphere quarters [-6dB] its surface area, and FSPL is all about the surface area of an isotropic sphere.)

So if you divide a free space path with loss of 100 dB into two equal paths, each half has a quarter of the original loss — but you now end up with a total path loss of 94 dB + 94 dB = 188 dB to overcome!*

(The "exactly in half" division is actually the worst case of the options that don't add any additional distance of their own. The sky — or should I say THE MOON! — is the limit for worst case when your "passive repeater" ends up significantly increasing the total distance covered.)

Back to the closer-to-direct layover options, dividing the route into one short hop and one long hop still adds a lot of FSPL, but shows where you might start seeing wins. For a concrete example, at 148.24 MHz the free space path loss over 10 miles would be 100 dB. If you move the "split" closer and closer to one end the total (combined) path loss does go down. With a 9.95 mile path (99.957 dB) plus a 0.05 mile path (53.979 dB loss) you've only got to offset around 54 dB of additional loss instead of the 88 dB of the halfway-between placement.

That's example still leaves a lot of loss to try make up with antenna gain alone, but consider that in the real world where the direct path is far worse than a "free space" path of the same distance. In a canyon or a tunnel where the direct line of sight is essentially totally blocked, you might do well using 10–20 dB of antenna gain to offset a 99.957 dB + 53.979 dB loss, compared to the loss of a "direct" path through bedrock!

To summarize:

How much gain could I expect from a Passive repeater formed by simply connecting a receive antenna pointing in one direction to a nearby transmit antenna pointing in another?

I don't think there's much of a shortcut to calculating "passive repeater gain" than by:

1. Determining the baseline actual [not "free space"] loss of the direct A to C path
2. Proposing a specific location B for the passive repeater, and determining what the actual path losses of both A to B and B to C would be. Add those losses together.
3. The "passive repeater gain" is then the difference between the baseline and via-repeater path losses, plus the antenna gain that could be provided at B
4. […plus or minus any multipath effects that would also be introduced…?!]

Overall it's a lot more like the dealing with the radar equation than like calculating coax loss per foot!

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* Thinking in terms of "one big sphere" vs. "two small spheres" and their relative surface area threw me off a bit at first. In the "big sphere" case the antenna captures a tiny fraction of the signal. In the "two small spheres" case you get a 4x bigger fraction of the signal… but then that fraction becomes the signal you have to capture next. Compare a 1% chance of winning the lottery yourself, to a 5% chance of your friend giving you the winnings of a lottery they have a 5% chance of winning!