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Cascaded 2-Tone, 2nd-Order Intercept Point (IP2)

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. 2nd-order products lie far in frequency to the two input tones and are therefore not very likely to fall inband at the output in a narrow band system. As a device is driven farther into its nonlinear region, the amplitudes of the 2nd-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 2nd-order gain line would be 2:1. The slope of the 2nd-order product line is 1:1 relative to the fundamental gain line. Accordingly, the 2nd-order products increase in power at the same rate as the input tones and are always twice times farther away from the IP2 than the input tones when not near saturation.

The power of the 2nd-order products can be predicted when the IP2 is known, or the IP2 can be predicted when the relative amplitudes of the 2nd-order tones and the input tones are known.
 

P2nd-order products = 2 · Pinput tones@output - IP2  {dBm}

IP2 =2 · Pinput tones@output - P2nd-order products {dBm}

Intercept point IP2 IP3 P1dB drawing graph - RF Cafe

Cascading IP2 Values in a Chain of Components

Calculating the cascaded values for IP2 for the system budget requires the following operation based upon ratios for gain and IP2 (do not use decibel values). The standard format for indicating decibel values is to use upper case letters, i.e., OIP2 for units of dBm. The standard format for indicating watt values is to use lower case letters, i.e., oip2 for units of mW or W.

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

This equation gives the method for calculating cascaded output ip2 (oip2) values based on the oip2 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:

Cascaded IP equation - RF Cafe

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

IP2OUTPUT = (IP2INPUT + Gain)  {dBm}

or

IP2INPUT = (IP2OUTPUT - Gain)  {dBm}

Thanks to Bernard R. for pointing out the missing square in the denominators.

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