Accurate prediction of harmonic levels from non-linear gain expansions, Part 1

Relationship between powers of voltage and harmonics of frequency
Figures of merit for a signal path, such as HD3 and IM3, can be expressed in terms of the coefficients of a polynomial expansion of the function that relates the output signal to the input signal. (See Pengfei Zhang's article for a useful background). Perhaps in an attempt to simplify the printed expressions, the power series expansion of the (assumed weakly) non-linear gain function is typically terminated after the 3rd power of the voltage, the higher order terms relegated to the land of the ellipsis The 3rd harmonic (HD3) is nearly always the most significant harmonic spurious component and the temptation to associate the generation of HD3 exclusively with the 3rd power of the voltage is strong " but wrong, as we'll see.

This is because the mapping between powers of voltage and harmonics of frequency is not one-to-one but many-to-many. Each of the odd powers of the voltage produces a spray of odd harmonics of the input fundamental. A significant amount of HD3 is contributed by these higher order terms, and neglecting them will certainly lead to an inaccurate prediction.

To quantify this, the harmonic spectrum of cosⁿ(ωt) can be derived straightforwardly by recognizing that

where we use cos(ωt) instead of sin(ωt) simply because the expressions are cleaner (less minus signs and js to keep track of); obviously a cosine wave is essentially 'the same thing' as a sine wave.

This allows us to write down coefficients of cosⁿ(ωt) from a binomial expansion:

so if we put e^jωt = a and e^-jωt = b, manage powers of 2 in the denominator and use the symmetry of the binomial coefficients

to pair up terms of the form a^rb^n-r and a^n-rb^r, we end up with:

for k≤ n/2. In the commonly used case where n=3 we immediately get the usual expression:

But we can also see that higher powers of cos(ωt) also generate significant amounts of HD3:

and so on. Note that the coefficients of cos(3ωt) get higher as the power of cos(ωt) increases. So we can't simply discard the higher order terms before we check what effect they have on the final answer.

Incidentally, we're looking at odd power examples here primarily because modern circuit techniques for push-pull and differential operation generally make even order components less of an issue (though this isn't always so, for instance, in very broadband applications such as broadcast TV and cable modem systems). The expressions do work just as well for even powers. Here's a tip: when building an Excel spreadsheet to do these sums, the COMBIN(number,number_chosen) worksheet function gives you the value of the binomial coefficient. Let's just check this, using the example of cos⁷(ωt) above. Here's a simulation of a voltage source set to give a waveform of 1*cos⁷(2pi*1*t):

We can clearly see in figure 1b that the signal has a fundamental at 1Hz and clear 3rd, 5th and 7th harmonics, and nothing else. Are the levels correct? Allowing for the conversion of a 1Vpk sinusoid to -3.01dBV, we find completely insignificant discrepancies (table 1.1):