Single-Ended and Differential S-Parameters
Differential circuits have been important incommunication systems for many years. In the past,differential communication circuits operated at lowfrequencies, where they could be designed andanalyzed using lumped-element models andtechniques. With the frequency of operationincreasing beyond 1GHz, and above 1Gbps fordigital communications, this lumped-elementapproach is no longer valid, because the physicalsize of the circuit approaches the size of awavelength.Distributed models and analysis techniques are nowused instead of lumped-element techniques.Scattering parameters, or S-parameters, have beendeveloped for this purpose [1]. These S-parametersare defined for single-ended networks. S-parameterscan be used to describe differential networks, but astrict definition was not developed until Bockelmanand others addressed this issue [2]. Bockelman’swork also included a study on how to adapt single-ended S-parameters for use with differential circuits[2]. This adaptation, called “mixed-mode S-parameters,” addresses differential and common-mode operation, as well as the conversion betweenthe two modes of operation.This application note will explain the use of single-ended and mixed-mode S-parameters, and the basicconcepts of microwave measurement calibration.
/*
* EULER S ALGORITHM 5.1
*
* TO APPROXIMATE THE SOLUTION OF THE INITIAL VALUE PROBLEM:
* Y = F(T,Y), A<=T<=B, Y(A) = ALPHA,
* AT N+1 EQUALLY SPACED POINTS IN THE INTERVAL [A,B].
*
* INPUT: ENDPOINTS A,B INITIAL CONDITION ALPHA INTEGER N.
*
* OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T.
*/
