MICROWAVE ENGINEERING SCATTERING MATRIX AND IT’S PROPERTIES 1

Scattering matrix:

Scattering matrix DEFINATION: The process by which the incoming particles are transformed (through their interaction) into the outgoing particles is called Scattering. A physical theory of these processes must be able to compute the probability for different outgoing particles when we collide different incoming particles with different energies. In physics, the scattering matrix (or S-matrix ) relates the initial state and the final state of a physical system undergoing a scattering process. 2

History:

History The S-matrix was first introduced by John Archibald Wheeler in the 1937 paper "'On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure” In the 1940s Werner Heisenberg developed, independently, the idea of the S-matrix. Due to the problematic divergences present in quantum field theory at that time Heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. In doing so he was led to introduce a unitary "characteristic" S-matrix 3

The Scattering Matrix:

The Scattering Matrix The scattering matrix relates the voltage waves incident on the ports to those reflected from the ports. The scattering parameters can be calculated using network analysis technique. Otherwise, they can be measured directly with a vector network analyzer. Once the scattering matrix is known, conversion to other matrices can be performed. Consider the N-port network in Fig. 4

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or S ii the reflection coefficient seen looking into port i when all other ports are terminated in matched loads, S ij the transmission coefficient from port j to port i when all other ports are terminated in matched loads. 5

Figure A photograph of the Hewlett-Packard HP8510B Network Analyzer. This test instrument is used to measure the scattering parameters (magnitude and phase) of a one- or two-port microwave network from 0.05 GHz to 26.5 GHz. Built-in microprocessors provide error correction, a high degree of accuracy, and a wide choice of display formats. This analyzer can also perform a fast Fourier transform of the frequency domain data to provide a time domain response of the network under test. Courtesy of Agilent Technologies.:

Figure A photograph of the Hewlett-Packard HP8510B Network Analyzer. This test instrument is used to measure the scattering parameters (magnitude and phase) of a one- or two-port microwave network from 0.05 GHz to 26.5 GHz. Built-in microprocessors provide error correction, a high degree of accuracy, and a wide choice of display formats. This analyzer can also perform a fast Fourier transform of the frequency domain data to provide a time domain response of the network under test. Courtesy of Agilent Technologies. 6

A matched 3B attenuator with a 50 Ω Characteristic impedance :

A matched 3B attenuator with a 50 Ω Characteristic impedance 7 Evaluation of Scattering Parameters

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Show how [S] [Z] or [Y]. Assume Z 0 n are all identical, for convenience Z 0 n = 1. where Therefore , For a one-port network, 8

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To find [Z], Reciprocal Networks and Lossless Networks As in Sec. 4.2, the [Z] and [Y] are symmetric for reciprocal networks, and purely imaginary for lossless networks. From 9

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If the network is reciprocal, [ Z ] t = [ Z ]. If the network is lossless, no real power delivers to the network. 10

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For nonzero [ V + ], [ S ] t [ S ] * =[ U ], or [ S ] * ={[ S ] t } -1 . Unitary matrix If i = j , If i ≠ j , Application of Scattering Parameters The S parameters of a network are properties only of the network itself (assuming the network in linear), and are defined under the condition that all ports are matched. 11

A Shift in Reference Planes:

A Shift in Reference Planes 12 Figure Shifting reference planes for an N -port network.

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[S]: the scattering matrix at z n = 0 plane. [S ' ]: the scattering matrix at z n = l n plane. 13

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Generalized Scattering Parameters:

Generalized Scattering Parameters 15 Figure An N -port network with different characteristic impedances.

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The generalized scattering matrix can be used to relate the incident and reflected waves, 17

Figure :

Figure 18

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