Microwave Engineering Europe - December 2008 - (Page 17) X-PARAMETERS 17 from approximating the entire dependence on every B-wave as linear in each spectral component of each incident A-wave. At the other extreme (no approximation) is the general case where each B-wave depends nonlinearly on each component of every incident A-wave. In practice, it is usually only one or two levels beyond S-parameters that are required to give an excellent approximation to the nonlinear B(A) relations, even under very strongly nonlinear conditions. For much of this article, we consider the simplest case beyond S-parameters where we retain the full nonlinear dependence for the B-waves on at least one spectral component of one incident signal, specifically A11, the fundamental component incident at the input port. In this case, we say the DC and A11 tone together determine the large-signal operating point (LSOP). All the other spectral components are assumed to enter the constitutive relations linearly. This situation applies to PAs approximately matched to 50 ohms and is expedient for many commercial off-the-shelf low to medium power high frequency amplifiers (e.g. for handset applications) even when they are fully saturated or operated at large degrees of compression. The formula for this “spectral linearization” of Equation 1 is given by Equation 2 [2]. The sums are over all port and harmonic indices, except the pair (1,1). The pure phase, P, is given by: P = ejPhase(A11) Equation 2: There are three types of terms in Equation 2. The first term is the response of a perfectly terminated (at all harmonics) nonlinear system with DC bias stimulated by a single largeamplitude sinusoidal input at port one. The additional X(S) functions of bias and large-tone amplitude are the coefficients that multiply the complex phasors, Agh, for all harmonics and also the fundamental component incident on port 2. These are analogous to “hot” Sparameters, parameterized by the large-signal drive. The summation indicates there is a contribution to B at each harmonic index f from every harmonic component incident from each port. Thus, frequency translation properties are included — unlike linear Sparameters. The final set of terms in Equation 2 multiply the phasors Agh*, the conjugate of the complex phasors Agh. There is no analogy in S-parameter theory for these terms. Their origin can be understood in terms of mixer analysis, and a graphical representation is given in Figure 2. The conjugate terms are related to image frequencies in the mixing of the large incident tone, A11, and the small tones, Agh, simultaneously incident on the DUT. The phase dependence on A11 in Equation 1 can be shown to be outside of the argument of the X-parameters by application of the principle of time-invariance [2, 3]. (Note: references [2,3, 6] use a different convention than the present article for the X-parameter indices and subsume the X(F) terms into the X(S) terms compared to Equation 2). Equation 2 is valid as long as the presence of all spectral components Agh are sufficiently small that they alter neither the DC conditions nor produce intermodulation products beyond first UMTS Power Amplifier WCDMA PA orld C Mobile W - 19th February 16th na Barcelo 2B97 nd all 2 , Sta H at: Visit usngress 2009 o Mitsubishi Electric is offering a complete set of high efficiency power amplifiers for UMTS mobile applications. Mitsubishi's PA's are well recognized in the cellular market for highest efficiency and excellent performance. They are available for several interfaces and many UMTS bands. www.mitsubishichips.com · semis.info@meg.mee.com MITSUBISHI ELECTRIC EUROPE B.V. (Semiconductor European Business Group) Gothaer Straße 8 · D-40880 Ratingen · Phone: +49 (0) 2102/486 0 · Fax: +49 (0) 2102/486 4140 Microwave Engineering ● December 2008 ● www.mwee.com http://www.mitsubishichips.com http://www.mitsubishichips.com http://www.mwee.com
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