*Nonlinearities in active circuits and passive components can add harmonic and nonharmonic distortion.*

This series has investigated intermodulation, the process by which the application of two frequencies to a nonlinear system results in the system generating frequencies that equal the sum and difference of the input frequencies. In Part 1 and Part 2, we looked at useful applications of intermodulation — for example, the modulation of a carrier and the upconversion and downconversion of the modulated carrier. We concluded part 3 by looking at intermodulation distortion (IMD) — the appearance of unwanted frequency components in what should be a linear system. Specifically, we asked the Microsoft Excel charting function to draw a fundamental cosine wave, and it introduced nonharmonic frequency components.

**What would be a real-world example of unwanted IMD?
**Consider first the linear amplifier in

**Figure 1**. If the three resistors are equal, the amplifier’s output voltage is the negative of the sum of the input voltages (we’ll address the passive component later).

As shown in **Figure 2a**, if f* _{1}(t)* is 1.5 kHz (orange trace) and

*f*is 2 kHz (blue trace), the circuit generates the red trace. A fast Fourier transform of that trace shows only the original frequencies, with no harmonic or nonharmonic components (

_{2}(t)**Figure 2b**).

**How can we introduce nonlinearities?
**We can have the amplifier clip the output voltage, as shown by the blue trace in

**Figure 3a**. That waveform’s FFT (

**Figure 3b**) shows the resulting frequency content. From what we have learned so far about intermodulation, we won’t be surprised to see the difference frequency at 0.5 kHz and the sum frequency at 3.5 kHz. In general, given the fundamental low-side tone

*f*and fundamental high-side tone

_{1}*f*, we can expect intermodulation products at these frequencies:

_{2}m*f _{1}* + n

*f*, where m and n equal 0, ±1, ±2, ±3, and so on.

_{2}Figure 3b labels several examples in red. Of particular note are the high- and low-side third-order products. They are of relatively high magnitude and close to the fundamental frequencies, so they are difficult to filter out.

**Is IMD a problem only with active circuits?
**Components that are supposed to be passive, such as connectors, can exhibit nonlinear behavior that introduces what’s called passive intermodulation (PIM) distortion. As an example, let’s apply frequencies

*f*= 2 kHz and

_{1}*f*= 3 kHz to the inputs of the Figure 1 circuit. The mysterious “passive” component between the amplifier and the output voltage

_{2}*v*creates an open-circuit condition of 0.29-ms duration every 2.5 ms, corresponding to a frequency

_{0UT}(t)*f*equaling 0.4 kHz, giving us the waveform shown in

_{OPEN}**Figure 4a**.

**Isn’t that an odd failure mode?
**Indeed, such repetitive failure modes aren’t unheard of. Consider a faulty slip ring or loose connector on constant-speed rotating or reciprocating machinery, for example.

**OK, I’ll buy that. What’s the effect?
**Figure 4b shows the FFT of the Figure 4a waveform. Because of the specific frequency values chosen for this example, the intermodulation products related only to the fundamental frequencies

*f*and

_{1}*f*appear on the vertical gridlines associated with integer frequency values. I’ve labeled these in red. The many additional intermodulation products are associated with

_{2}*f*. I’ve labeled several of these in blue, including the ones between the fundamental frequencies’ high- and low-side third-order products. Clearly, there is so much IMD that filtering will not be an option here—replacement of the defective passive component is the solution.

_{OPEN}**Is PIM the same as insertion loss, which we looked at in a ****previous article****?
**PIM and insertion loss are different phenomena. Insertion loss becomes an issue when signal wavelengths are short relative to conductor wavelengths — for example, at microwave frequencies. Indeed, installations such as cellular base stations can have PIM problems but not insertion-loss problems, or vice versa.

But PIM and other IMD issues extend down to the audio range. In the examples here, I’ve been using kilohertz values. The wavelength for a 1-kHz signal is 300,000 m, so unless we are transmitting over a very long distance, insertion loss and return loss won’t be an issue.

**Where can I learn more about IMD?
**See these papers:

- Passive Intermodulation (PIM) | Anritsu America
- Keysight Amplifier Distortion Measurements using MW PNA.pdf
- Application Note Intermodulation Distortion (rohde-schwarz.com)