Mathematical Problems in Engineering

Volume 2017, Article ID 1390295, 19 pages

https://doi.org/10.1155/2017/1390295

## On Optimal Truncated Biharmonic Current Waveforms for Class-F and Inverse Class-F Power Amplifiers

^{1}Department of Power, Electronics and Communication Engineering, Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21000 Novi Sad, Serbia^{2}Computer Science Department, Indiana University Bloomington, 150 S. Woodlawn Ave., Bloomington, IN 47405, USA

Correspondence should be addressed to Ladislav A. Novak; ude.anaidni@kavondal

Received 4 October 2016; Revised 25 January 2017; Accepted 1 February 2017; Published 16 March 2017

Academic Editor: Alessandro Lo Schiavo

Copyright © 2017 Anamarija Juhas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, two-parameter families of periodic current waveforms for class-F and inverse class-F power amplifiers (PAs) are considered. These waveforms are obtained by truncating cosine waveforms composed of dc component and fundamental and either second or third harmonic. In each period, waveforms are truncated to become zero outside of a prescribed interval (so-called conduction angle). The considered families of waveforms include both discontinuous and continuous waveforms. Fourier series expansion of truncated waveform contains an infinite number of harmonics, although a number of harmonics may be missing. Taking into account common assumptions that for class-F PA the third harmonic is missing in current waveform and for inverse class-F PA the second harmonic is missing in current waveform, we consider the following four cases: (i) (ii) , (iii) and (iv) , We show that, in each of these cases, current waveform enabling maximal efficiency (optimal waveform) of class-F and inverse class-F PA is continuous for all conduction angles of practical interest. Furthermore, we provide closed-form expressions for parameters of optimal current waveforms and maximal efficiency of class-F (inverse class-F) PA in terms of conduction angle only. Two case studies of practical interest for PA design, involving suboptimal current waveforms, along with the results of nonlinear simulation of inverse class-F PA, are also presented.

#### 1. Introduction

There is continuous interest in class-F PA and its dual inverse class-F PA (see, e.g., [1–8]). Finding optimal and suboptimal nonnegative waveforms for PAs also attracted substantial interest within the research community (see, e.g., [5, 9–16]) and can be regarded as a part of the so-called waveform engineering [15, 16].

In this paper, we consider a two-parameter model of periodic current waveform, defined within fundamental period aswhere stands for is conduction angle, and are parameters, is constant, and or This family includes both continuous and discontinuous waveforms. Waveform of type is a truncated biharmonic current waveform.

The current waveform of type (1) is an even function and therefore its Fourier series expansion contains dc component and cosine terms. Coefficients of the Fourier series expansion can be expressed aswhere and denotes unnormalized sinc function Coefficients of fundamental harmonic and harmonic can be also obtained from (3) by using that Notice that change of parameter and/or causes the change of the whole harmonic content of waveform of type (1).

Corresponding voltage waveform of PA is assumed to be of the formwhere , and or

A number of existing models of current waveforms (both continuous and discontinuous) can be embedded in model (1), including the most used continuous model of current waveform for classical PA operation (see, e.g., [5]):Model (5) can be obtained directly from (1) by setting and

Another widely considered continuous model of current waveform is one-parameter model of type (see, e.g., [5])which can be also obtained from (1) by setting the value of parameter toTypically, continuous current waveform of type (6) is paired with voltage waveform of type (4) with for biharmonic mode of PA operation [5, 17]. In [17], the value of parameter (for and ) is obtained via optimization of efficiency, subject to the constraint that harmonic coefficient is nonpositive. A special case of biharmonic mode with is also analyzed in [5, 17].

The two-parameter model of continuous current waveform for PA with has been considered in [18]. This model can be obtained from (1) by setting , and

The one-parameter model of discontinuous current waveform is used in [3]. The authors of [3] considered case in the context of class-F PA and case in the context of inverse class-F PA. Current waveform proposed in [3] for class-F can be obtained from (1) by setting , and Furthermore, parameter is determined from the condition that the third harmonic is missing in current waveform. The same model of current waveform is also used in [19] in the context of continuous class-F PA.

On the other hand, for inverse class-F, the one-parameter model of discontinuous current waveform proposed in [3] can be obtained from (1) by setting , and In this context, parameter is obtained from the condition that the second harmonic is missing in current waveform.

For class-F PA, we use the common assumption that the third harmonic in current waveform is missing, (see, e.g., [2, 3, 7]). In this case, the corresponding voltage waveform is of type (4) for Also, for inverse class-F PA, we use the common assumption that the second harmonic in current waveform is missing, (see, e.g., [2, 3]). The corresponding voltage waveform is of type (4) for . In what follows, current waveforms of type (1) satisfying condition are denoted by In Sections 2–4, we consider current waveform of type (1) for and To be more specific, we consider the following four cases:(i),(ii) and ,(iii),(iv) and Cases (i) and (ii) correspond to class-F mode of PA operation, whereas cases (iii) and (iv) correspond to inverse class-F mode. To the best of our knowledge, cases (ii) and (iv) are not widely explored in waveform modeling for PA design.

The efficiency of PA can be expressed via basic waveform parameters and as , provided that parameter () is equal to the quotient of fundamental harmonic amplitude and dc component of the current (voltage) waveform (see, e.g., [1]). In class-F and inverse class-F PA, a higher harmonic component could appear in at most one of the waveforms in current-voltage pair (see, e.g., [1]). This fact implies that current and voltage waveforms can be optimized independently for class-F and inverse class-F PA. Therefore, nonnegative current waveform of type that ensures maximal efficiency (optimal waveform) of class-F PA or inverse class-F PA is a waveform of type with maximal parameter

For all cases (i)–(iv), in Section 2, we prove that current waveforms of type (1) enabling maximal efficiency are continuous for all conduction angles of practical interest for class-F and inverse class-F PA. Closed-form expressions for parameters of such waveforms as functions of conduction angle are also provided in Section 2. In Section 3, an independent numerical verification of the values of parameters of optimal current waveform is described. In Section 4, we consider efficiency of class-F and inverse class-F PA, using the results obtained in Section 2. In Section 4.1, maximal efficiency of class-F PA with optimal current waveforms for or is provided. Section 4.2 is devoted to inverse class-F PA with optimal current waveforms for or In Section 5, we consider continuous current waveforms of type (6) satisfying relaxed condition for and , instead of used in the context of class-F and inverse class-F PA. Two case studies involving suboptimal current waveforms for class-F and inverse class-F are also presented in this section. As a practical validation of the proposed approach, comparisons of the results derived in this paper with the result of nonlinear simulation of inverse class-F PA with CGH40010F HEMT are provided in Section 6.

#### 2. Optimal Current Waveform of Class-F and Inverse Class-F PA

In this section, current waveforms of type (1) that satisfy condition for and are considered. In what follows, we prove that current waveform , which in pair with voltage waveform of type (4) enables maximal efficiency, is continuous for all conduction angles of practical interest for class-F and inverse class-F PA (i.e., for those conduction angles listed in Table 1). In this section, we also provide closed-form expressions for parameters of such waveforms.