How to calculate Return loss of cascaded system?

Question

If system 1 has 10 dB return loss and system 2 has 20 dB return loss then what is the overall return loss of cascaded system of 1&2 ?

Answer 1

Assuming the systems have no internal loss.

Reflected power ratio for system 1 is 0.1, for system 2 is 0.01. $ = \Gamma^2 = 10^{-RL/10}$

Power transmitted ratio for system 1 is 0.9, for system 2 is 0.99 $ = 1-\Gamma^2$

The first reflection back from each system will be the largest and subsequent reflections will be heavily attenuated, so we approximate by only considering the first reflections.

If we take A to be the power incident on system 1:

• first reflection from system 1 will be $0.1*A$
• incident power on system 2 will be $(1-0.1)*A = 0.9*A$
• reflected power from system 2 will be $(0.01)*(0.9)*A$

• reverse-transmission of system 2 reflected power through system 1 will be $(0.9)*(0.01)*(0.9)*A$

• total power reflected (B) from both systems (first reflection only) $B = 0.1*A + (0.9)*(0.01)*(0.9)*A = 0.1081*A$

• return loss of total system $RL = 10*log_{10}(0.1081) = -9.66dB$

Answer 2

Some time as passed since the original posting and I suppose the student has lost interest in this problem. But for the sake of completeness, I offer the following.

It must be said (as other commentators mentioned) that the problem as stated lacks sufficient information to give an exact result. Given more scattering parameters of each system, a conversion to T parameters to facilitate multiplying and then converting back to S parameters could be utilized. When only the combined S11 result is desired, this conversion process can deploy shortcuts to save on conversions.

Given the lack of required information , a statistical approach to the problem will yield the best available estimate. If we assume that the two stated Return Losses are unrelated, then applying RSS (Root of Sum of Squares) will yield a reasonable estimate (this is sometimes called "addition in quadrature").

First convert the RL figures to their respective reflection coefficient (Γ) or S11 scattering parameter:

$$ \Gamma_1 = 10^{10/-20} = 0.3162 $$

$$ \Gamma_2 = 10^{20/-20} = 0.1 $$

Then compute the RSS:

$$ \text{RSS} = \sqrt{\Gamma_1^2 + \Gamma_2^2} = 0.3316 $$

And finally, convert the RSS back to a combined RL:

$$ \text{RL} = -20 \log(0.3316) = 9.6 $$

It must again be stated that this results in a statistical estimate. If in fact there is significant gain, attenuation, or impedance transformation in these systems, this result could be substantially in error. In such a case, the order of the cascading could also have an effect on the results. But given the lack of information, it serves as a reasonable first order estimate. Meanwhile, I hope the good professor has removed or appropriately amended the question for future test pools.

It should be noted that this method can be used successfully when attempting to account for distributed reflections from connectors, etc. subject to the limitations noted above.