Cascaded 2-Tone, 3rd-Order Compression Point (IP3)

When two or more tones are present in a nonlinear device, intermodulation products are created as a result. A power series describes all of the possible combinations of generated frequencies. 3rd-order products lie near in frequency to the two input tones and are therefore very likely to fall inband at the output. As a device is driven farther into its nonlinear region, the amplitudes of the third order products increase while the powers of the input tones decrease. If the device was not limited in output power, then the powers of the intermodulation products would increase in power until they were eventually equal in power with the input tones at the output.

Assuming a gain of 1 (0 dB) the slope of the fundamental gain line would be 1:1; the slope of the 3rd-order gain line would be 3:1. The slope of the 3rd-order product line is 2:1 relative to the fundamental gain line. Accordingly, the 3rd-order products increase in power at twice the rate of the input tones and are always three times farther away from the IP3 than the input tones when not near saturation.

The power of the 3rd-order products can be predicted when the IP3 is known, or the IP3 can be predicted when the relative amplitudes of the 3rd-order tones and the input tones are known.
 

Equal Input Powers
(see below for unequal powers)

P3rd-order products = Pinput tones@output - 2 · (IP3 - Pinput tones@output)  {dBm}

P3rd-order products = 3 · Pinput tones@output - 2 · IP3  {dBm}

IP3 =3/2 · Pinput tones@output - 1/2 P3rd-order products {dBm}

Intercept point IP2 IP3 P1dB drawing graph
Here's a great link to a paper on intermod frequency prediction for any number of tones.

Cascading IP3 Values in a Chain of Components

Calculating the cascaded values for IP3 for the system budget requires the following operation based upon ratios for gain and IP3 (do not use decibel values). The standard format for indicating decibel values is to use upper case letters, i.e., OIP3 for units of dBm. The standard format for indicating watt values is to use lower case letters, i.e., oip3 for units of mW or W.

Cascaded components for calculating IP2 and IP3
A Typical Chain of Cascaded Components

This equation gives the method for calculating cascaded output IP3 (oip3) values based on the oip3 and gain of each stage. When using the formula in a software program or in a spreadsheet, it is more convenience and efficient to calculate each successive cascaded stage with the one preceding it using the following format:

Conversion of the IP3 and IP2 reference may be accomplished with the following equations:

IP3OUTPUT = (IP3INPUT + Gain)  {dBm}

or

IP3INPUT = (IP3OUTPUT - Gain)  {dBm}

The following equation is a series expansion of the mixing (multiplying) of two pure tones:

Multi-tone mixing equation

IP3 spectral content unequal tone powersUnequal Input Powers

PL = P2 - 2*(IP3 - P1)

PU = P1 - 2*(IP3 - P2)

Where power units are kept constant in dBm or dBW