# How to calculate Return loss of cascaded system?

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 ?

• Return loss is just 1 of the 4 S-parameters. You need them all for systems 1 & 2, and also the return loss for whatever you put after system 2. – Juancho Jun 11 '16 at 11:34
• @Juancho : I was asked this question in exam and only this information was given. – aparna Jun 12 '16 at 6:10
• If that was the only information you were given, you would first find the return loss looking at the load, then use system 2's S matrix to find the RL looking into it, then do the same with system 1 until you finally have the RL looking into the entire network. – Andrew W. Jun 13 '16 at 17:26
• @Andrew W. I tried your method n got answer as 9.5dB but actual answer is 9.63dB. can I approximate it to 9.63dB? – aparna Jun 15 '16 at 6:34
• Hum. In most practical calculation you probably could since that only about a 3% power difference.To get a more specific answer and account for all reflections you would need the actual S matrix, not just the insertion losses. (I prefer converting to the ABCD matrix which makes the cascading calculation simpler (although it can be annoying to convert back and forth). The link below has good examples of both methods and the pros/cons of each. ABCD method on page 40, S method using signal graphs on 54. .... ee.sc.edu/classes/Fall12/elct865/Class_Notes/… – Andrew W. Jun 20 '16 at 18:23

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:

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

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

$$\text{RSS} = \sqrt{\Gamma_1^2 + \Gamma_2^2} = 0.3316$$
$$\text{RL} = -20 \log(0.3316) = 9.6$$