Incremental Behavioral Modeling of Envelope-dependent Nonlinear Distortion in Dynamic Supply Power Amplifiers

This paper presents behavioral modeling of an envelope tracking (ET) power amplifier (PA) excited by an slewrate limited envelope. In order to relax the slew-rate requirements of current efficient envelope amplifiers (EA), we can drive the ET PA with a slower version of the signal’s envelope. However, the resulting amplified signal suffers from nonlinear distortion due to the time-variant gain effects that arise when using the slower version of the signal’s envelope. In this paper this time-variant effects are presented and characterized. A new envelope-dependent (ED) behavioral model is presented and compared with a simpler static AM-AM and AM-PM modeling. Experimental results showing the performance of the proposed ED model were obtained with an ET PA based on a GaN transistor. I. INTRODUCCIÓN Dynamic supply architectures such as the envelope tracking (ET) power amplifier (PA) can be combined with linearization techniques such as digital predistortion (DPD) to enhance power efficiency and preserve linearity within the transmitter. ET PAs (see Fig. 1) result very attractive from the implementation point of view, because ET can be applied in conventional transmitters by only substituting the static supply for a dynamic one. Dynamic supply can be carried out by means of an envelope amplifier (EA) that has to efficiently supply the required voltages and currents to the RF transistor drain at the speed imposed by the changes of the RF envelope [1]. In OFDM-based modulations the bandwidth of the signal’s envelope is several times the bandwidth of the baseband complex modulated signal. Therefore, the EA has to efficiently supply envelopes with high peak-to-average power ratios (PAPR) and bandwidths ranging from DC up to several MHz. In order to relax the slew-rate requirements of the EA, recent works have proposed methods to iteratively reducing the envelope bandwidth [2] or reducing its PAPR [3]. The method recently published in [4], consist in processing the real signal’s envelope E(n) to obtain, in realtime, a slew-rate limited version Es(n). Fig. 2 shows both the real and slew-rate limited envelopes. ET PA was performed in [5] using the slew-rate limited version of the envelope. Results showed that by using the slower envelope to supply the PA drain the efficiency improvement is slightly reduced with respect to supplying with the real envelope. However, the slew-rate requirements are also relaxed, which permits considering the use of highly efficient commercial EAs. Unfortunately, by using the slower version of the signal’s envelope, the resulting amplified signal suffers from nonlinear distortion due to the time-variant gain effects (envelope dependent) that arise. The scope of this paper is to present and characterize these time-variant (envelope dependent) gain effects. Therefore, in Section II we present these nonlinear timevariant gain effects that depend on the supply. Then, in Section III an envelope-dependent (ED) behavioral model to characterize these effects is presented. Experimental results showing the performance of the proposed ED model in comparison to a static one are presented in Section IV. Finally, conclusions are given. Fig. 1. Block diagram of an envelope tracking PA architecture. Fig. 2. a) RF signal, b) real envelope, c) slew-rate limited envelope. II. ET PAS SUPPLIED BY SLEW-RATE LIMITED ENVELOPES Fig. 3 shows the normalized AM-AM characteristic considering an ET PA supplied with the slow version of the envelope, Es (the maximum Es voltage is 28V), while Fig. 4 shows the AM-AM characteristic considering only 3 selected Es values (in fact, it is considered a range of voltages around each Es). As it can be derived from Fig. 3 and Fig. 4, due to the fact that the supply and the input signals are not univocally related, as a consequence time-variant nonlinear gain effects arise. For a given input it is possible to have a range of different outputs. Therefore, the ET PA presents an envelope-dependent (ED) behavior. Fig. 3. Normalized AM-AM characteristic for all Es values. Fig. 4. Normalized AM-AM characteristic for some Es values. III. MODELING THE SLOW-ENVELOPE DEPENDENT TIMEVARIANT NONLINEAR GAIN EFFECTS As it is well-known, the input-output relationship of a PA operating under constant supply voltage can be formulated as A A A A x x g y ⋅ = ) ( (1) with xA and yA being the PA complex baseband input and output signals respectively. The complex function gA(·) is only dependent of the input envelope. A x , and constitutes the AM-AM and AM-PM characteristics ( ) ( ) ( ) PM A A j g x AM A A A A g x g x e = ⋅ (2) The static nonlinear modulus () AM A g ⋅ and phase () PM A g ⋅ terms of the gain can be expressed as polynomials