Microwave Engineering Europe - December 2008 - (Page 18) 18 X-PARAMETERS order in their amplitude. For some high power amplifiers operated under highly mismatched conditions the incident wave at port 2 at the fundamental frequency may be large enough that it must be included in the LSOP rather than treated linearly as in Equation 2. An example of this is presented later. While more complicated than linear Sparameters, Equation 2 nevertheless represents a dramatic simplification of the general constitutive relations of Equation 1, while retaining great utility for nonlinear applications. Equation 2 can be identified from far fewer measurements, reducing the size of the resulting data set, the model complexity, and the data acquisition and model simulation time. Nevertheless, this “classic 1-tone X-parameter” approximation allows the generation of harmonics and the effects of (moderate) mismatch at the fundamental and harmonic frequencies, to be included in the model. It includes all AM-AM and AM-PM affects, all even and odd harmonics generated by the nonlinear device in response to the large tone, and includes the DC effects of RF amplitude variation and therefore PAE and bias currents versus power. All these effects also have a sensitivity to source harmonics at each port. For a two-port amplifier, even without considering harmonics at all, there are six X-parameters relating port-to-port scattering at the fundamental frequency in the presence of a large input drive. This is two more than the four traditional linear two-port S-parameters. These six X-parameters are written in Figure 3 where the three associated with the scattered wave at the output port are also plotted, as a function of incident power (magnitude of A11) for a power amplifier. The first two X-parameters reduce to the familiar S21 and S22 in the limit of small input drive. The new term, X(T)21,21, vanishes as the input power decreases. Its origin is a third-order intermodulation interaction between A11 and A21. At large input drive, the X(T)21,21 term becomes significant, and becomes even larger than the conventional “hot” S-parameter term X(S)21,21. Large X(T)21,21 is characteristic of fully saturated amplifiers such as are commonly used in GSM applications [4]. It therefore must be extracted and included in order to properly take mismatch into account. The linear dependence on A21 and A21* through multiplication by X-parameters X(S)21,21 and X(T)21,21, respectively, leads to an elliptical dependence on the match versus the phase of A21, consistent with data, unlike “hot” S-parameters that fail to come close even to the gross features of the DUT mismatch response while being driven with a large signal [4]. X-parameter extraction For each probe tone, there are three distinct terms that land at the same frequency (Figure 2). The extraction methodology uses three (or more) independent measurements to separate the terms. The ideal approach is presented in Figure 4. The idea is to directly identify the X(F) term by excitation only with a single large tone, and then to do two more experiments per port per harmonic frequency, with different (approximately orthogonal) phases, to obtain sufficient measurements from which to extract the set of three X-parameters. In practice, the source produces harmonics and the harmonic mismatches of the measurement system require a more sophisticated approach. In the Agilent NVNA X-parameter application, a measurement and extraction procedure is implemented that Figure 2: This diagram illustrates the origin of the X(T) terms based on mixing products. Three contributions fall on each harmonic due to mixing products that are first order or below in the small tone. At the third harmonic, the on-frequency component is 3f0 + 0f1, and is captured by the X(F) term. The upper sideband is f0 + f1, and is captured by the X(S) term. The lower sideband is 5f0 – f1, and is captured by the X(T) term. Because this mixing term involves –f1, the phase dependence on Aj2 is also inverted so the X(T) term multiplies Aj2* instead of Aj2. Figure 3: Equations at the top show how the six (fundamental frequency) terms of Xparameters replace the typical four terms of S-parameters for a two-port. The equations at the right show how these terms reduce to S-parameters for small signals. The plot on the left shows the importance of the X(T) term increasing as a function of input power, becoming even more important than the X(S) term (which corresponds to “Hot S22”) at some power levels. Microwave Engineering Europe ● December 2008 ● www.mwee.com http://www.mwee.com
For optimal viewing of this digital publication, please enable JavaScript and then refresh the page. If you would like to try to load the digital publication without using Flash Player detection, please click here.