[Radio Wrangler]

By looking at TOI and SFDR, you're looking at a processed result of some measurements, often a good number of measurements.

Set up an intermodulation test and progressively step up the levels of the pair of tones. At each step, measure the levels of the various products. Also measure the level of the wanted output.

You can now plot them all on an X-axis of the applied tone level. Do all levels in dBm for convenience.

For a well-behaved device you will see that the third order products go up 3dB for every 1dB increase in the applied tones. Fifth order products 5dB, fourth order 4dB and you get the pattern. It's terribly convenient! Go high and you'll see the intermod products start to level off and eventually saturate, or something gets too hot and burns out. But look at lower levels and you get excellent straight lines. Project the straight section of the third order intermod beyond the levelling and saturation, and it intersects the straight line of the wanted output (projected also as necessary) Where they cross is your true TOI point. The linear section of the wanted output has a 1:1 slope (dB/dB slope) while you'll see the third order linear section has a 3dB/1dB slope. So the gap between the wanted and unwanted 3rd order product closes at 2dB per 1dB rise in applied tones.

The spurious free dynamic range is defined as the range of variation of input level between the noise floor, and the input level which makes the intermod products just come up to equality with the noise floor. So it's twice the difference between noise floor and TOI point. IN THEORY. You will only get this if the upper level is still within the nice linear region of the intermod level plot.

There are also badly behaved devices whose intermod plot lines are not linear, curving smoothly into saturation. Some can be decidedly kinky and even jumpy towards the top end. Things like crystal filters, some very high level mixers and some feedback amplifiers do these sorts of things. But, the lower slopes are usually well behaved and you can extrapolate to a TOI crossing from them. The results are valid and will predict low level behaviour. Doing the plot and seeing where things go silly will tell you what levels not to approach, and where the cosy TOI modelling breaks down.

So, why the convenient dB/dB slopes? They seem almost too good to be true and are reliable at low enough levels, and the factors are accurate, not just close.

Take a device and model its transfer function as a mathematical power series. The zeroeth power gives you DC output if you're bothered. The first power gives your wanted output, the square power gives you second order distortions, the cube power gives you third order, the fourth and so on

Differentiate this series to get the slopes and you see that as per normal calculus, the power of a term is multiplied into its coefficient. And there is the basis of those convenient scales for each order of product.

The bad behaviour at higher levels would be modelled properly if you took the time to make a complicated enough series to model high level behaviour in detail. No-one does, it's easy to take the simple view because we only want to use devices in their well-behaved region and aren't bothered just how bad the badly behaved region is.

So there you have it.

You need to do a full plot of the intermod product power so that you can see this in action and feel comfortable about it.

You need to do a full plot of a new device so you can check where it goes badly behaved and the small-signal assumptions and models fail.

Jeremy's mentioned the handy estimate of TOI versus 1dB compression point. It's pretty good on all well-behaved devices. On the bad guys it's an assumption that will have you.

There are other rough rules of thumb. Howard Swain was responsible for one relating relative levels to relative amounts of compression. Handy for spectrum analyser designers deciding where to pitch maximum operating levels and designing to meet an overall compression figure which is actually the budget for the total of several contributing stages. Noise figure instruments are touchier in this area than spectrum analysers so it's one reason why the purpose-designed NF instruments out perform general analysers with a software personality.

David