## What’s the deal with S-parameters?

As a relative newcomer to microwave and RF, there are certain things that the industry takes for granted that I find very weird when I first encounter them. It took me a long time to really understand the concepts of isolation and directivity, for example, and why do 2.4 mm connectors look just like SMA connectors when they don’t fit together? Couldn’t we color code them or something?

S-parameters are one of these things. I know from working with our datasheets and test data that the S parameters represent power loss. For example, and insertion loss of 3 dB corresponds to a power loss of ~50%. This is because converting 10 dB to linear units, you use the formula 10^(-3dB/10) = .50118723362727, the point is that 3 dB is *approximately* a 50% power loss, though not exact. There is no fundamental reason this is so, it just conveniently happens to be about the same.

According to our microwave bible, “Microwave Engineering” by David Pozar, the scattering matrix [*S*] is defined according to the forward and reverse voltage waves as [*V-*] = [*S*][*V+*]. A specific element of the matrix is defined as a formula, but the formula states roughly that “*Sij* is found by driving port *j* with an incident wave of voltage *V+j*, and measuring the reflected wave amplitude, *V-i*, coming out of port *i*. This is the definition that is used in all of the books and papers that I have seen using scattering parameters.

Here is the tricky part, that everyone in the microwave industry takes for granted, and has been hazy for me for a while. If the S-parameters are defined in terms of voltage, than a 3 dB reduction should mean a 50% reduction in voltage, equivalent to a 75% reduction in power. A 3 dB loss (or gain) however never means a 75% reduction in power, or a 50% reduction in voltage. This is because:

*S parameters in linear units always refer to the amplitude (voltage or current), while S parameters in logarithmic (dB) units always refer to power*.

There, I said it. It is one of those things that is usually gets lost or taken for granted going from the classroom to the lab. It’s like amplitude balance: even though it is called amplitude balance, it refers to the the difference in power between two outputs, and it is quoted in dB. So when calculating between linear and logarithmic S parameters, you have to use 10^(X dB/20), while for power it is 10^(X dB/10). It’s weird, and you just have to get used to it.

I hope this clears things up for some poor students out there. Good luck on your finals and happy late spring break!

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