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The main-line loss of a coupler is defined as the loss between the input and output ports. Theoretical loss is defined as that of a perfect (lossless) coupler, for which the input power is the sum of the (main-line) output power and the power at the coupled port. Therefore, the theoretical main-line loss is dependent upon the fraction of the input power that is outputted at the coupled port. The conversion between them can be expressed mathematically. This application note derives the formula and provides a conversion table.
A lossless coupler transfers all input power to the output and coupled ports: Pin = Pout + Pcoupled (1).
In practice, positive decibel values are given for main-line loss and coupling (the fraction of the input power that is outputted at the coupled port):
ML (main-line loss in dB) = 10 log10(Pin / Pout) (2) and
CPL (coupling in dB) = 10 log10 (Pin /Pcoupled) (3)
With these three equations, the CPL to ML conversion can be obtained.
From equation (1), Pout/Pin = 1 – Pcoupled/Pin (4)
From equation (3), Pcoupled /Pin = 10^–CPL/10 (5)
Substitute (4), and then (5), into (2):
ML = 10 log10 (Pin / Pout) = 10 log10 [1/(1 – Pcoupled/Pin)]
= 10 log10 [1/(1 – 10 ^–CPL/10 )] (6)
