How can I quantify a device’s nonlinearity? part 2

A figure of merit called the third-order intercept point indicates how well an RF device minimizes third-order intermodulation products.

In part 1, we looked at the 1-dB compression point, which serves as a figure of merit for devices such as RF power amplifiers. An additional specification, the third-order intercept point, abbreviated IP3 or TOI, is particularly relevant for circuits that have multiple input frequencies. For example, assume that an amplifier sums two cosine waves having frequencies of f₁ = 2 GHz and f₂ = 2.5 GHz. Because of nonlinearities, the circuit generates an output spectrum with a variety of interfering frequencies as shown in Figure 1. In a previous post on intermodulation distortion, we noted that low- and high-side third-order intermodulation products at 2f_1­ – f­₂ and 2f₂ – f­₁, respectively, can be particularly troublesome because they are close to the fundamentals and difficult to filter. Choosing an amplifier or other device with a high IP3 rating can minimize the levels of these products.

How do we know that the 2f₁ – f­₂ and 2f₂ – f­₁ signals represent third-order products?
Good question. In general, for a nonlinear system, the output power P_O as a function of an input signal s can be expressed as a power series:

Here, c₁s  represents our desired signal. The term c₀ is an offset that we can compensate for, and all the higher- order terms represent interference. So, let’s begin our search for 2f_1­ – f­₂ and 2f₂ – f­₁ with the second-order term. To generate second-order intermodulation products, the nonlinear device squares the input signals:where w₁ = 2pf₁ and w₂ = 2pf₂. 

Figure 2 shows the expanded and annotated form of s²t. Here, the cos² terms represent the second harmonics of the fundamental input signals. The middle term—the intermodulation product in red — can be expressed in the form of a trigonometric identity as shown in the figure, where we can see that the intermodulation product represents the sum and difference of the input frequencies.

Neither the second-order harmonics nor the sum and difference frequencies appear to be significant in Figure 1, so we will have to look elsewhere for the culprit contributing to the interference, and the third-order term of our power series is next up. Recall from high school algebra that (a+b)³=a³+3a²b+3ab²+b³, so we can calculate s³t, the cube of our input signal, as shown at the bottom of Figure 2. The cos³ terms represent third harmonics, which are far away from the fundamentals and therefore are not of concern, so we’re left with the two third-order intermodulation products, highlighted in red and blue.

It’s not obvious that the s³(t) red and blue terms contain 2f₁ – f₂ and 2f₂ – f₁,  or 2w₁ – w₂ and 2w₂ – w₁.
Right. We could do some algebra to present a proof involving more trigonometric identities — this application note has details. Alternatively, we can just take the fast Fourier transform, as discussed in an earlier series, of the red and blue terms to demonstrate that the components at 1.5 and 3 GHz do result from third-order processes. As shown in Figure 3, the red term contributes to the low-side product at 1.5 GHz, and the blue term contributes to the high-side product at 3 GHz. Note that Figure 3 shows additional third-order products at 6.5 GHz (2f₁ + f₂) and 7 GHz (f₁ + 2f₂); these would be easy to filter.

Great, so how do we find IP3?
Note that as input power increases, our desired output increases linearly, while the unwanted third-order products increase cubically. At some hypothetical point (not reachable in real life), the power levels of the third-order products equal the power levels of the desired output—that point is IP3. In part 3, we’ll show how to measure IP3, and we’ll plot it on top of our 1-dB compression-point diagram from part 1.