Rules of Thumb
Kirt's Cogitations™ #212

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Rules of Thumb

Rules of thumb are a great tool to have available as long as you have confidence in the general accuracy of the rule. Depending on which source you consult, the term “rule of thumb” has many possible origins, but most refer to some part of the of the thumb (probably one belonging to some king) being used to approximate length, like the distance from the tip of the thumb to the first joint being about an inch. From there, just about any sort of mnemonic for approximating a quantity has been called a rule of thumb.

Many common rules of thumb exist, like the “Rule of 72,” whereby for exponential growth at a constant rate is obtained by dividing 72 by the percent growth rate so as to arrive at the period for doubling the original amount. For example, if a population grows 10% every year, then it doubles in 72/10 = 7.2 years. The “real” number in this case is (log 2)/(log 1.1) = 7.273, but it is close to 7.2 (a mere 1% error), so at least for this example, the rule of thumb holds. Let us assume it holds for any case since it has persisted for a long time.

Another common rule of thumb is the Tailor's Rule of Thumb (possibly where the rule of "thumb" originated). Tailors used to measure the circumference of a client's thumb to approximate the circumference of the wrist (2x), the neck (4x), and the waist (8x). For myself, the multiplication factors are  2.4x, 5.7x, and 11x, respectively (2.8" thumb, 6.8" wrist, 16" neck, 32" waist). Hmmm, I would hate to wear that suit, because according to the rule of thumb, my shirt sleeves would be only 5.6" and the neck would be 11.2," the pants waistline would be a mere 22.4." Either my thumb is too thin or the rest of me is way too fat. Maybe I measured my thumb incorrectly.

Electromagnetic energy travels about one foot in one nanosecond in free space (actually 1.01670336 ns), and in one nanosecond, it travels about one foot (actually 0.98357106 ft). Yet another useful rule of thumb.

OK, so what's the point? Here's the point. Recently, the subject surfaced again regarding what value to use for the relationship between the 3rd-order intercept point (IP3) and the 1 dB compression point (P1dB). Most people will say it is 10 to 12 dB. Many software packages allow the user to enter a fixed level for the P1dB to be below the IP3 when the actual P1dB value is not known. For instance, if a fixed level of 12 dB below IP3 is used and the IP3 for the device is +30 dBm, then the P1dB would be +18 dBm. I was tempted to simply propagate that rule of thumb, but decided to actually test it empirically.

In order to check the theory, IP3 and P1dB values from 53 randomly chosen amplifiers and mixers were entered into an Excel spreadsheet (see here). The components represent a cross-section of silicon and GaAs; FETs, BJTs, and diodes; connectorized and surface mount devices. A mean average and standard deviation was calculated for the sample, and everything was plotted on a graph (see here).

As it turns out, the mean is 11.7 dB with a standard deviation of 2.9 dB, so about 68% of the sample has P1dB values that fall between 8.8 dB and 14.6 dB below the IP3 values. What that means is that the long-lived rule of thumb is a pretty good one. A more useful exercise might be to separate the samples into silicon and GaAs to obtain unique (or maybe not) means and standard deviations for each.

An interesting sidebar is that where available, the IP2 values were also noted. As can be seen in the chart, the relationship between IP2 and P1dB is not nearly as consistent.

Of equal motivation for the investigation was the desire to confirm or discredit the use of the noise figure and IP3 type of cascade formula for use in cascading component P1dB values. As discussed elsewhere, the equation for tracking a component from its linear operating region into its nonlinear region is highly dependent on the entire circuit structure, and one model is not sufficient to cover all instances. Indeed, the more sophisticated (pronounced “very expensive”) system simulators provide the ability to describe a polynomial equation that fits the curve of the measured device. Carrying the calculation through many stages is calculation intensive. Some simulators exploit the rule of thumb of IP3 versus P1dB tracking and simply apply the IP3 cascade equation to P1dB. As with other shortcuts, as long as the user is aware of the approximation and can live with it, it's a beautiful thing.

The RF Cascade Workbook series of spreadsheets has assiduously avoided attempting a P1dB cascade calculation for the reason noted above. Instead, a saturated power (Psat) value was provided and the program simply flagged a condition where the linear power gains would cause a stage output power that was greater than the entered Psat value. Future version of RF Cascade Workbook will incorporate the P1dB cascade and use the rule of thumb method for calculations.

While on the subject of rules of thumb, it would be very useful to have you go to the RF Cafe Forum and add any that you know, whether they apply to engineering, science, woodworking or anything else. If enough good rules of thumb are posted, I will create a dedicated page for them, and give credit to you for each if desired. Thanks for your help!