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20th May 2019, 10:55 #1
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Unitary property of scattering matrix
From sparameter lossless network definition , The sum of the incident powers at all ports is equal to the sum of the reflected powers at all ports.
But how do we derive the above unitary property expression involving S22 and S21 ?

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20th May 2019, 12:01 #2

22nd May 2019, 15:23 #3
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Re: Unitary property of scattering matrix
Hi promach,
Mathematically speaking, a unitary matrix is one which satisfies the property [S]^* = [S]^{1}. Rearranging, we see that [S]^* [S] = [I], where [I] is the identity matrix.
Inserting the [S] matrix into this equation, we can then see that any column dotted with itself is equal to unity. Conversely, if any column is dotted with any other column, the product is equal to 0.
So if we take your 2x2 scattering matrix and look at some examples, we would find that S12^2 + S22^2 = 1. If the network is reciprocal, then S12 = S21. We would also have that S11^2 + S21^2 = 1 and (S11*)(S12) + (S21*)(S22) = 0.Last edited by PlanarMetamaterials; 22nd May 2019 at 15:28.

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23rd May 2019, 01:42 #4

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24th May 2019, 13:41 #5
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Re: Unitary property of scattering matrix
This could be explained by law of conservation of energy. The injected power must be equal to the reflected power + the transmitted power for a lossless network.

26th May 2019, 15:45 #6
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26th May 2019, 17:26 #7

5th June 2019, 02:57 #8
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Re: Unitary property of scattering matrix
@pancho_hideboo
For ideal lossless case:
transmitted power + reflected power = source power
So, divide the whole expression just above by source power, we have the following:
S21^2 + S22^2 = 1

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5th June 2019, 13:22 #9
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Re: Unitary property of scattering matrix
You can not understand things correctly.
Wrong.
We can not get this directly from "transmitted power + reflected power = source power".
We have followings from "transmitted power + reflected power = source power".
S11^2 + S21^2 = 1
S12^2 + S22^2 = 1
However from matrix nature of kindergarten level, S21=S12 is satisfied.
So S21^2 + S22^2 = 1 is satisfied.
This does not require reciprocity at all.
Learn basic things surely.
Attached is an excerption from kindergarten level book.
Here "~" means conjugate complex and transpose.Last edited by pancho_hideboo; 5th June 2019 at 13:35.
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