# Berman and Mahle Phase Model

In [12], Berman and Mahle proposed a model suited to TWTAs for multiple access communications satellite applications. This model only represents the AM/PM distortions that are given by the following form:

where *A* refers to the magnitude of the input signal and *ф _{с}(А)* is phase shift, relative to the output phase in small signal conditions, introduced to the output signal by the AM/PM distortion. The values for model coefficients

*k*and

_{b}k_{2},*k*

*are found through optimization. Figure 4.12 presents the relative phase shift versus the normalized input power characteristic calculated with Equation 4.40 for a TWTA. The values of the coefficients used to generate Figure 4.12 are k*

_{3}_{1}= 0.372, k

_{2}= 5.14, k

_{3}= 0.27 [12].

# Thomas-Weidner-Durrani Amplitude Model

Thomas, Weidner, and Durrani presented a method in [13] to model the normalized amplitude’s nonlinearity in TWTAs. The mathematical expression for the model can be given as:

Figure 4.12 Phase shift transfer function characteristic of the Berman and Mahle model

where, here also *A* refers to the magnitude of the input signal, *M(A)* is the level of the small-signal normalized output signal, *A _{s}* is the input saturation level, and

*A*is the input signal’s level where the compression starts. Model coefficients

_{c}*a*and

*ft*are normally found by optimization to fit experimental data. In this model, the phase distortion is modeled using the Berman and Mahle model given by Equation 4.40.

Figure 4.13 shows the AM/AM and AM/PM characteristics for TWTA, where AM/AM characteristic is obtained from Thomas-Weidnar-Durrani model and the AM/PM characteristic is approximated using the Berman-Mahle model.