Cascaded 2-Tone, 3rd-Order Intercept 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. 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.

Cascading IP3 Values in a Chain of Components

Calculating the cascaded values for 3rd-order intercept point (IP3) for the system budget requires use of ratios for gain and power levels for IP3 (do not use dB and dBm values, respectively). The standard format for indicating decibel values is to use upper case letters; i.e., IP3 for units of dBm. The standard format for indicating power values is to use lower case letters; i.e., ip3 for units of mW.

Conversions:   ip3 = 10^IP3/10  ↔  IP3 = 10 * log₁₀ (ip3)

where ip3 has units of mW and IP3 has units of dBm

A Typical Chain of Cascaded Components

Combining 2 Stages at a Time for Calculations

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 convenient and efficient to calculate each successive cascaded stage with the one preceding it using the following format, per the drawing (above-right). Note that the ip3_N-1 term in the denominator is the same as the cumulative ip3 up through the previous (N-1) stage.

These formulas are used to convert back and forth between input- and output-referenced IP3 values:

IP3_Output = (IP3_Input + Gain)  {dBm}

IP3_Input = (IP3_Output - Gain)  {dBm}

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

Equal Input Powers

(see below for unequal powers)

P_{3rd-order products} = P_{input tones@output} - 2 · (IP3 - P_{input tones@output})  {dBm}

P_{3rd-order products} = 3 · P_{input tones@output} - 2 · IP3  {dBm}

IP3 =3/2 · P_{input tones@output} - 1/2 P_{3rd-order products} {dBm}

Unequal Input Powers

P_L = P₂ - 2*(IP3 - P₁)

P_U = P₁ - 2*(IP3 - P₂)

where power units are kept constant in dBm or dBW.