Third-Order Intermodulation Products in Non-Linear Circuits

In radio frequency (RF) engineering, understanding the distinction between linear and non-linear circuits is fundamental to effective system design. Linear circuits preserve signal integrity by preventing distortion and unwanted interactions between multiple inputs. In contrast, non-linear circuits produce harmonics and intermodulation products, which can degrade performance and interfere with adjacent channels.

The third-order intercept point (IP3) is a key parameter used to quantify non-linearity and predict distortion levels. This article examines the characteristics of linear and non-linear circuits, the relevance of IP3 in performance evaluation, and the challenges posed by Passive Intermodulation (PIM) in practical RF systems.

What is a Linear Circuit?

A linear circuit has a transfer function H(x) such that given two inputs A and B:

Where c is an arbitrary constant.

A transfer function that exhibits this property is a linear function:

Where x is an arbitrary function.

Then:

and

And the two are different only by a constant a₀.

This is important because it means the circuit faithfully reproduces multiple signals without creating interactions between them. This is usually a desirable feature for an amplifier

A linear passive circuit that combines two RF signals in a 50 Ω environment might look like Figure 1:

Figure 1 - Linear Combiner Circuit

What is a Non-Linear Circuit?

A non-linear circuit has a transfer function with higher-order terms. It can be modeled with a Taylor series:

Amplifiers are always slightly non-linear, particularly as the operating point approaches compression. The a₂, a₃, and higher coefficients are small, but non-zero. Diode detectors are intentionally biased to a point on the I-V curve where a2 is maximized (usually around 30 μA).

Let’s plug in a pair of sine waves Asin(x) and Bsin(y) into the second order term and arbitrarily set a₂ to 1.

After applying two trig identities:

and  

We get:

This gives us second harmonics of sin(x) and sin(y), the sum and difference of the two frequencies, and a constant DC term. Note that if x is greater than y one can recast ABcos(y-x) as ABcos(x-y) since cos(-a) = cos(a).

The sum and difference terms are the properties of a mixer. The constant, DC term gives us the A and B envelopes of the two sine waves. An RF detector relies on this property. It’s interesting to note that the two original tones have vanished. The a1 and other odd-order terms are needed to reproduce them.

If we plug the two sine waves into the third-order term we get:

The identity,

is also used here.

We get the two original tones, their third harmonics, and the third-order intermodulation terms 2x+y, 2y+x, 2y-x, and 2x-y. There is no constant detection term. Note that the two most troublesome terms, 2y-x and 2x-y, which are close by in frequency to the two input frequencies, increase as AB² and A²B, respectively, essentially third order if A and B are equal.

The sin(x) and sin(y) also appear but add very little to the original input tones since it is assumed that coefficient a3 is tiny compared to a₁.

What is IP3?

If two tones of identical amplitude are passed through a non-linear device, the two tones will create the 2x-y and 2y-x intermodulation products as shown in Figure 2.

Figure 2 - Intermodulation Products

If the two tones increase in amplitude, the intermodulation (intermod) products will increase by three times as much. Specifically, if the tones are increased by 5 dB, the intermodulation products will increase by 15 dB, at least for small values of the products. It is possible to extrapolate the third-order intercept point: the level where all four tones will have equal amplitude. This is the third-order intercept value shown in Figure 3. If it is referenced to a device's output, it is called the Output IP3 or OIP3; if referenced to the input, it is called the Input IP3 or IIP3. An amplifier usually has its OIP3 specified, while a mixer typically has its IIP3 specified. The two numbers are interchangeable as they differ only by the device's gain or loss.

Figure 3 - IP3 Chart

It is best to measure the intermod products at low levels, perhaps 60 dB down from the main tone amplitude. This ensures that the tones are well away from compression and the effects of 5th and higher-order non-linearities are negligible. 

To calculate IP3, measure the level of the two equal tones (T) and the level of the highest intermod product (IM). Then:

For example, if the two tones are at +10 dBm and the highest of the two IMs is at -60 dBm, then the IP3 is (10 – (-60))/2 or +35 dBm. The two intermods may differ in amplitude somewhat due to higher-order products. These higher-order intermodulation products can be at the same frequency as the third-order ones and can significantly reduce them, but cannot greatly increase them.

What is Passive Intermodulation?

Passive Intermodulation, or PIM, is a nonlinear response caused by passive materials. For instance, a chain-link fence surrounding a transmitting site might have a combination of aluminum links and metal clamps of some other alloy, and some corrosion may also be present. The galvanic interaction of two metals, plus the interaction with corrosion products, sets the stage for PIM. Strong RF signals at two or more frequencies can pass through a galvanic metal junction and create third-order mixing products. These products may interfere with sensitive receivers, which might also be on site. Tracking down PIM in radio site infrastructure can be very challenging, to say the least.

RF connectors may contain different alloys and must undergo PIM testing before being used in high-power RF applications.

Why does IP3 matter?

Suppose a communications system has equally spaced channels, and two high-power RF signals are transmitted on adjacent channels. In that case, the third-order intermodulation products will land on either side of the two signals, directly in the middle of the channels above and below, interfering with existing traffic. A receiver will be listening on those frequencies somewhere, and it does no good to design a low noise-figure receiver system if spurious signals jam the receiver frequency.

How is an IP3 Measurement Made?

IP3 is measured by injecting two frequency tones into a Device Under Test (DUT). If the device is to be used in a communications system with 25 kHz channel spacing, the tones are set to that spacing. The DUT's output is connected to a spectrum analyzer or a Vector Network Analyzer receiver to view the two tones and any third-order intermodulation products.

A test setup like the one in Figure 4 might be used. Several things must be considered to ensure an accurate measurement. What happens when the output of one signal source leaks across the combiner and enters the output of the second source? Will the two signals mix and create intermod products? Also, what is the IIP3 of the spectrum analyzer? Are the intermod products being generated by the DUT or in the analyzer? 

It’s easy to test for a receiver IIP3 problem in the spectrum analyzer. Step the input attenuator of the spectrum analyzer down by 10 dB and see if the intermod product levels change. If they do, the analyzer was adding products. Step the attenuator down another 10 dB and ensure there is no change. When this occurs, the intermods are not due to the spectrum analyzer.

By adding attenuators to the output stages of the signal sources, you can reduce the mixing of signals. Circulators might also be used. Using a signal combiner with high input port-to-port isolation is also desirable.

Figure 4 - IP3 Test Setup

If the spectrum analyzer or vector network analyzer receiver's IIP3 is too low to make a proper measurement and dialing up the input attenuation is insufficient, it may be necessary to add attenuation to the DUT's output. Of course, this additional attenuation must be factored into the IP3 calculation.

Using a Vector Network Analyzer to Measure OIP3

To measure OIP3 directly, a 4-port vector network analyzer with two sources may be used. The two sources must be set to the same amplitude and with the desired frequency separation. The output of the DUT is attached to a third port, and the receiver measures the intermodulation products. A free VNA software plug-in is available for Copper Mountain Technologies VNAs to automate the measurement and perform OIP3 measurement over a chosen frequency range.