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RF Cascade Workbook

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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.

See cascade calculations for NF, IP2, IP3, and P1dB.

Click here to view an example of a cascaded system. |

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 * log_{10}
(ip3)

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

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).

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}
= P_{2} - 2*(IP3 - P_{1})

P_{U} = P_{1} - 2*(IP3 - P_{2})

where power units are kept constant in dBm or dBW.