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SECTION I

## INTRODUCTION

Fig. 1. The power smith tube. The vertical axis represents the input power, while the horizontal cross section of the tube is a conventional smith chart [18].

The variation of PAE and ACPR with both load reflection coefficient $\Gamma_{L}$ and input power $P_{in}$ is well documented. Snider describes how the load impedance providing optimal efficiency varies with input power level and demonstrates that the output power is significantly dependent on both the load impedance and the input power level [13], and Nemati et al. show variation of the optimum PAE load impedance with changing input power [14]. Fu and Mortazawi show the design of a reconfigurable power amplifier using tunable varactors and also show that the load impedance in the Smith chart providing optimum efficiency is different for different output power levels [15]. Hajji et al. show that a change in input power can be useful in obtaining constant intermodulation rejection [16], related to the adjacent-channel spreading of amplifiers under broadband signal excitation. Load modulation to provide efficiency in Doherty amplifiers is discussed by Nam et al. [17].

Our previous work introduces the input power Smith tube (Fig. 1) to visualize how a criterion (such as PAE or ACPR) varies with both load reflection coefficient and input power [18]. Previous Smith chart extensions include a spherical extension of the Smith chart to consider both positive and negative resistances in circuit design [19], [20], a Smith chart generalization allowing fractional circuit elements to be considered [21], and an adjusted Smith chart for design with lossy transmission lines [22].

The purpose of the present paper is to show how an optimum combination of $\Gamma_{L}$ and $P_{in}$ can be obtained quickly, using a small number of measurements, through a fast search algorithm in the power Smith tube. The desired optimization is an example of biobjective optimization [23], [24] and is applied to the well-known trade-off of linearity and efficiency [25], [26]. A previous paper by our group demonstrates optimization for linearity and efficiency using only the real and imaginary parts of $\Gamma_{L}$ as the input parameters [27], and the present paper extends the solution of this problem to a third dimension, allowing simultaneous optimization of input power with $\Gamma_{L}$ to provide the highest PAE possible under ACPR constraints.

Section II presents the basics of the search algorithm to be applied in the Smith tube. Section III describes simulation results of the algorithm's application. Section IV provides results from measurement testing of the algorithm. Finally, Section V provides conclusions based on the presented results.

SECTION II

## ALGORITHM DETAILS

The search algorithm is based on estimation of the PAE and ACPR gradients within the three-dimensional input power Smith tube. First, the input power, on the vertical axis of the Smith tube, is normalized so that the user-defined maximum input power is assigned the value 1 and the user-defined minimum input power is assigned the value −1. This scales the vertical limitations of the cylindrical search space (the Smith tube) to the same dimensions as the planar values of $\Gamma_{L}$ in the horizontal dimension. The normalized input power $p_{in}$ (with a lowercase $p^{\prime\prime}$) is defined as follows (all power values are in dBm, decibels with respect to 1 milliwatt, for this paper). TeX Source

This equation normalizes the range of $P_{in}$ to set the minimum to −1 and the maximum to 1. The normalized input power $p_{in}$ represents the input power for purposes of calculating and evaluating gradients.

Although the three-dimensional search is limited by the power Smith tube's cylindrical space, the search vectors can be laid out in Cartesian dimensions, because the horizontal coordinates represent the real and imaginary parts of $\Gamma_{L}$. The vertical coordinate $p_{in}$ provides the third of the three Cartesian dimensions. Using Cartesian coordinates allows direct extension of the search space used in the two-dimensional algorithm for the same objectives [27].

A search vector $\overline{v}$ can be added to a candidate in the Smith tube to find the subsequent candidate in the search. When the candidate from which the search is operating does not possess an acceptable ACPR value, this search vector is defined as follows: TeX Source where TeX Source and TeX Source $D_{s}$ is the search distance parameter and is provided by the user. It provides scaling for both components of the search vector [(3) and (4)]. $ACPR_{meas}$ is the measured value of ACPR at the present candidate point. $ACPR_{target}$ is the maximum acceptable value of ACPR. $ACPR_{worst}$ is the largest value of ACPR measured to this point in the search. $\theta$ is the angle between vectors $\hat{a}$ and $\hat{b}$. In (2), the vectors $\hat{a}$ and $\hat{b}$ represent vectors in the direction of ACPR steepest descent and the bisector between the ACPR steepest descent and PAE steepest ascent vectors. On the Pareto optimum locus, which is the curve of constrained solutions, the gradients will be oppositely directed [6], and the value of $\theta$ (the half-angle between the gradients) will be 90°. Thus, $\theta_{target}=90^{\circ}$ in (4). $\theta_{meas}$ is the measured value of $\theta$ at the candidate under consideration. Fig. 2 shows that the gradients can be evaluated in the three-dimensional space. If PAE is represented by the variable $p$, the gradient for the PAE is given in terms of the unit coordinate vectors $\hat{\Gamma}_{r},\hat{\Gamma}_{i}$, and $\hat{p}_{in}$ by TeX Source

The partial derivatives to calculate this gradient are estimated from a measurement separated from the candidate in the Smith tube by a neighboring-point distance $D_{n}$ in each coordinate direction, as shown in Fig. 2. From a measurement of the change in power $\Delta p$ at a change in the real part of the load reflection coefficient, $\Gamma_{r}$, the partial-derivative estimation follows as TeX Source

From a measurement of the change in power for a change in the imaginary part of the load reflection coefficient, $\Gamma_{i}$, the partial-derivative estimation follows as TeX Source

Finally, from a measurement of the change in power $\Delta p$ at a change in $p_{in}$, the calculation follows as TeX Source

Because normalized power $p_{in}$ (lowercase) is being used, as defined by (1), the same step $D_{n}$ is used in the vertical direction for the partial derivative estimation as for the horizontal directions. This completes the estimation of the gradient vector given by (5). The unit vector in the direction of increasing PAE can be taken by dividing the gradient by its magnitude: TeX Source

Fig. 2. Gradient evaluation in three dimensions.
Fig. 3. Search vectors in three dimensions in the cases in which (a) ACPR is not within constraints and (b) ACPR is within constraints.

An approach similar to (5) through (8) can be used to estimate the gradient for ACPR. Because minimization is desired for the ACPR, the direction of optimal travel is given by a unit vector $\hat{a}$ oriented oppositely to the ACPR gradient: TeX Source

The three-dimensional bisector of $\hat{a}$ and $\hat{p}$ is defined as the arithmetic mean of the vectors $\hat{p}$ and $\hat{a}$. Because $\hat{a}$ and $\hat{p}$ are both unit vectors, the mean (bisector) given by TeX Source is also a unit vector.

Fig. 3 shows the resultant search vectors for the cases when ACPR at the present candidate is out of compliance (Fig. 3a) and when ACPR at the present candidate is within compliance (Fig. 3b). When the ACPR is in compliance, the search vector (2) is changed to have a component in the $\hat{p}$ direction instead of the $\hat{a}$ direction: TeX Source

A noticeable result of using (2) when the search is out of ACPR compliance and (12) when the search is in compliance is that the search firsts attempts to try to reach the region of ACPR compliance. When the initiating candidate is out of compliance, (2) is used, meaning that the search vector has a component of $\hat{a}$, which causes a component of travel toward the ACPR minimum. An $\hat{a}$ component will be present in each search vector until the region of ACPR compliance is reached. When an initiating candidate is inside the region of ACPR compliance, (12) is used, meaning that the $\hat{a}$ component of (2) is replaced by a $\hat{p}$ component of the same magnitude. This means that the search replaces its initial quest for the acceptable region by going toward the PAE optimum. It progresses with this component directed toward the optimum PAE point, as long as it remains in compliance. In the results shown in the following section, note that the first effort of each search is to reach the region of ACPR compliance. If the starting point of the search is outside the ACPR compliance region, this can result in an initial downward trajectory of the search for several candidates, followed by a subsequent upward trajectory once the search enters the ACPR compliance region.

The speed and resolution of the algorithm's convergence are dependent on the starting point and the parameter values selected for the search distance $D_{s}$ and neighboring-point distance $D_{n}$. Although further analysis could be performed to find the optimal values of these parameters given domain expertise and desired outcomes, this topic is saved for a later paper. In general, larger values of $D_{s}$ are suggested for use if the vicinity of the end point is unknown; if the search is believed to start near the optimum, then a smaller value of $D_{s}$ may reduce the number of measurements.

If the search reaches the region of ACPR compliance and then tries to leave, the search distance parameter $D_{s}$ is divided by two, and the search returns to the last candidate in the acceptable ACPR region and calculates a new search vector using the modified $D_{s}$. In addition to this penalization for leaving the acceptable region, the search vector tends to get smaller as it approaches the optimum, because $ACPR_{meas}$ tends to approach $ACPR_{target}$ and $\theta_{meas}$ tends to approach $\theta_{target}$, decreasing the search vector components according to (3) and (4), respectively. When the size of the search vector $\vert \overline{v}\vert$ decreases below a prespecified value (often the same as $D_{n}$), the search ends, and the measured point with the highest PAE that meets ACPR requirements is declared the constrained optimum.

SECTION III

## SIMULATION RESULTS

The algorithm was first tested in simulations using a nonlinear transistor model in the Advanced Design System (ADS) simulator from Keysight Technologies. A cosimulation was constructed using MATLAB to perform algorithm-controlled ADS simulations. In the cosimulation, MATLAB is used to control an ADS nonlinear circuit simulation when supplied a netlist by ADS, including the values of $P_{in}$ and $\Gamma_{L}$, and then MATLAB reads the results from ADS and performs algorithmic decisions, recalling ADS for each required circuit simulation in the algorithm's execution. For the design, a constraint of $\mathrm{ACPR}\leq-40\text{dBc}$ was imposed on the design. The goal was to obtain the highest PAE, while meeting this ACPR constraint. For comparison with the algorithm results, load-pull simulations spanning the Smith chart were performed at multiple input power levels and are shown in Fig. 4, as presented in [6]. This exhaustive measurement shows that the ACPR-constrained optimum point is at $P_{in}=21.75\text{dBm}$ and $\Gamma_{L}=0.80/\underline{180^{\circ}}$. The maximum PAE providing $\mathrm{ACPR}\leq-40\text{dBc}$ is 40.18% at this point.

Fig. 4. Constant ACPR surface from simulation data for $\mathrm{ACPR}=-40\ \text{dBc}$ and PAE surface representing the maximum PAE from simulated load-pull data. The ACPR-constrained optimum solution occurs where the two surfaces intersect; the surfaces are collinear at this point. The ACPR-constrained optimum occurs at location $\Gamma_{L}=0.80/\underline{180^{\circ}}$ at a power level of 21.75 dBm, where PAE $=40.18\%$ and $\mathrm{ACPR}=-40.23\text{dBc}$.
Fig. 5. Simulated search algorithm trajectory through the power smith tube using $D_{s}=1$, input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$, and starting location $\Gamma_{L}=0.50/\underline{90^{\circ}}, P_{in}=5\text{dBm}$. The search required 29 measured points and converged to the end point $\Gamma_{L}=0.79/\underline{-178.6^{\circ}}$ and $P_{in}=21.58\text{dBm}$, where $\mathrm{PAE}=41.13\%$ and $\mathrm{ACPR}=-40.08\text{dBc}$.

Simulation tests of the algorithm were performed using a stepsize $D_{s}=1.5$, neighboring-point distance $D_{n}=0.05$, and input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$. Because $D_{n}$ and $D_{s}$ have units consistent with the dimensions of the Smith chart, they possess the same units as load reflection coefficient $\Gamma_{L}$ and are unitless. Figure 5 shows the trajectory of a search starting from $\Gamma_{L}=0.5/\underline{90^{\circ}}, P_{in}=5\text{dBm}$. A total of 33 measured points was required for the search to converge to the end parameter values of $\Gamma_{L}=0.79/\underline{-178.6^{\circ}}$ and $P_{in}=21.58\text{dBm}$. At this end point, $\mathrm{PAE}=41.13\%$ and $\mathrm{ACPR}=-40.08\text{dBc}$ were obtained. Fig. 6 shows the search trajectory for a starting point of $\Gamma_{L}=0.80/\underline{0^{\circ}}, \ P_{in}=0\text{dBm}$. Notice that while the starting conditions were very different in both cases, the end point values of $\Gamma_{L}, P_{in}$, PAE, and ACPR are very similar. This indicates that both searches converge to approximately the same location in the Smith tube.

Fig. 6. Simulated search algorithm trajectory through the power smith tube using $D_{s}=1$, input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$, and starting location $\Gamma_{L}=0.90/\underline{0^{\circ}}, P_{in}=0\text{dBm}$. The search required 42 measured points and converged to the end point $\Gamma_{L}=0.77/\underline{176.4^{\circ}}$ and $P_{in}=21.21\text{dBm}$, where $\mathrm{PAE}=41.44\%$ and $\mathrm{ACPR}=-40.05\text{dBc}$
Fig. 7. Simulated search algorithm trajectory through the power smith tube using $D_{s}=1.50$, input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$, and starting location $\Gamma_{L}=0.75/\underline{-90^{\circ}}, P_{in}=25\text{dBm}$. The search required 26 measured points and converged to the end point $\Gamma_{L}=0.75/\underline{-179.6^{\circ}}$ and $P_{in}=21.07\text{dBm}$, where $\mathrm{PAE}=40.76\%$ and $\mathrm{ACPR}=-40.76\text{dBc}$.

Figs. 7 through 9 show the search results for additional different starting combinations of $\Gamma_{L}$ and $P_{in}$. Table I summarizes the search results for the different starting combinations. The results show that the final values of PAE vary less than 1%. The ending ACPR values are all within 0.2 dB of the −40-dBc limiting value. The end point $\Gamma_{L}$ coordinates are also similar, and the variation across end point $P_{in}$ values is less than 1 dB. The results all compare well with the optimum value obtained from the exhaustive load-pull search. Interestingly, the PAE values obtained from the algorithm are all slightly better than the optimum obtained through the exhaustive measurement, because the points measured in the Smith tube for the exhaustive measurement are spaced in a way that the resolution of the optimum's determination is less than the shorter, algorithm-based search. This shows that excellent resolution can be obtained by the fast search, even if no initial information is known about the location of the optimum.

Fig. 8. Simulated search algorithm trajectory through the power smith tube using $D_{s}=1$, input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$, and starting location $\Gamma_{L}=0.0/\underline{0^{\circ}}, P_{in}=18\text{dBm}$. The search required 28 measured points and converged to the end point $\Gamma_{L}=0.76/\underline{175.7^{\circ}}$ and $P_{in}=20.89\text{dBm}$, where $\mathrm{PAE}=40.77\%$ and $\mathrm{ACPR}=-40.14\text{dBc}$.
Fig. 9. Simulated search algorithm trajectory through the power smith tube using $D_{s}=1$, input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$, and starting location $\Gamma_{L}= 0.85/\underline{180^{\circ}}, P_{in}=27\text{dBm}$. The search required 25 measured points and converged to the end point $\Gamma_{L}= 0.76/\underline{179.3^{\circ}}$ and $P_{in}= 21.23\text{dBm}$, where $\mathrm{PAE}=40.89\%$ and $\mathrm{ACPR}=-40.18\text{dBc}$.
Fig. 10. Measurement setup.
Table I Simulation results for different starting reflection coefficients
SECTION IV

## MEASUREMENT RESULTS

The algorithm was measurement tested using the nonlinear load-pull test bench in the laboratory of the authors (Fig. 10). The input waveform is supplied by the signal generator with variable input power. Measurements of PAE are performed with the power meter, and a spectrum analyzer is used to measure ACPR. The algorithm was tested on a Skyworks SKY5017-70LF InGaP packaged amplifier. This device is different from the modeled device used for the simulation test of the algorithm, providing unrelated behavior for examining algorithm performance. Measurement tests of the algorithm were performed using a stepsize $D_{s}=1.5$, neighboring-point distance $D_{n}=0.05$, and input power range $-5\text{dBm}\leq P_{in}\leq 2\text{dBm}$. An ACPR limit of −27.5 dBc was specified for this search.

For comparison with the algorithm results, Fig. 11 shows a plot of the constant ACPR surface for the −27.5-dBc limiting value, extracted from exhaustive, traditional load-pull measurements performed at multiple $P_{in}$ values. The maximum PAE value, as assessed by the traditional load pull, is 7.39%, obtained for $\Gamma_{L}=0.47/\underline{-41.5^{\circ}}$ and $P_{in}=1.5\text{dBm}$.

Figs. 12 through 16 show the search trajectory for different starting combinations of $\Gamma_{L}$ and $P_{in}$. The results of the searches show excellent correspondence to each other, as summarized in Table II. Variation in end point $P_{in}$ across the five searches is less than 0.5 dB. All ACPR end point values are beneath the limit and within 0.4 dB of the limiting value. End point PAE varies by less than 0.5% across all of the searches. The number of measurements ranges from 23 to 39. As in the simulation case, the number of measurements required appears to be dependent on factors, including the search starting point location relative to the optimum point.

Fig. 11. Constant ACPR surface from measurement data for $\mathrm{ACPR}=\ -27.5\text{dBc}$, with the constrained optimum point indicated.
Fig. 12. Measured search algorithm trajectory through the power smith tube using $D_{s}=1, D_{n}=0.05$, input power range $-5\text{dBm}\leq P_{in}\leq 2\ \text{dBm}$, and starting location $\Gamma_{L}=0.6/\underline{-135^{\circ}}, P_{in}=2\text{dBm}$. The search required 31 measured points and converged to the end point $\Gamma_{L}=\ 0.42/\underline{-44.32}$ and $P_{in}=1.60\text{dBm}$, where $\mathrm{PAE}=7.32\%$ and $\mathrm{ACPR}=\ -27.55\text{dBc}$.
Fig. 13. Measured search algorithm trajectory through the power smith tube using $D_{s}=1, D_{n}=0.05$, input power range $-5\text{dBm}\leq P_{in}\leq 2\ \text{dBm}$, and starting location $\Gamma_{L}=0, P_{in}=0\text{dBm}$. The search required 19 measured points and converged to the end point $\Gamma_{L}=0.45/\underline{-46.61^{\circ}}$ and $P_{in}=1.58\text{dBm}$, where $\mathrm{PAE}=7.05\%$ and $\mathrm{ACPR}=-27.86\text{dBc}$.
Fig. 14. Measured search algorithm trajectory through the power smith tube using $D_{s}=1, D_{n}=0.05$, input power range $-5\text{dBm}\leq P_{in}\leq 2\ \text{dBm}$, and starting location $\Gamma_{L}=0.8/\underline{0^{\circ}}, P_{in}=-3\text{dBm}$. The search required 35 measured points and converged to the end point $\Gamma_{L}=\ 0.43/\underline{-24.12^{\circ}}$ and $P_{in}=1.57\text{dBm}$, where $\mathrm{PAE}=7.49\%$ and $\mathrm{ACPR}=\ -27.51\text{dBc}$.
Table II Measurement results for different starting reflection coefficients
SECTION V

## CONCLUSIONS

A fast search algorithm has been presented and validated for simultaneous optimization of power amplifier load reflection coefficient and input power for maximum PAE, while meeting ACPR requirements. The search has been demonstrated with simulations and measurements using the search space of the power Smith tube. In both simulation and measurement, the results of searches taken from multiple starting points show excellent convergence as compared with traditional load-pull evaluations performed at multiple values of input power. Furthermore, the results show the ability to improve the resolution of the solution. This results in the ability to obtain improved PAE, while meeting ACPR requirements without having to perform load-pull measurements over very closely spaced input power values. Excellent repeatability of the algorithm from multiple starting points is observed in both the simulation and measurement results.

Fig. 15. Measured search algorithm trajectory through the power smith tube using $D_{s}=1, D_{n}=0.05$, input power range $-5\text{dBm}\leq P_{in}\leq 2\ \text{dBm}$, and starting location $\Gamma_{L}=0.9/\underline{180^{\circ}}, P_{in}=-4\text{dBm}$. The search required 39 measured points and converged to the end point $\Gamma_{L}=\ 0.42/\underline{-40.33}$ and $P_{in}=1.67\text{dBm}$, where $\mathrm{PAE}=7.23\%$ and $\mathrm{ACPR}=\ -27.80\text{dBc}$.
Fig. 16. Measured search algorithm trajectory through the power smith tube using $D_{s}=1, D_{n}=0.05$, input power range $-5\text{dBm}\leq P_{in}\leq 2\ \text{dBm}$, and starting location $\Gamma_{L}=0.75/\underline{45^{\circ}}, P_{in}=1.5\text{dBm}$. The search required 24 measured points and converged to the end point $\Gamma_{L}=\ 0.39/-38.37^{\circ}$ and $P_{in}=1.21\text{dBm}$, where $\mathrm{PAE}=7.29\%$ and $\mathrm{ACPR}=\ -27.51\text{dBc}$.

### Acknowledgement

The authors thank Keysight Technologies for cost-free loan of the Advanced Design System simulation software.

## Footnotes

Refereeing of this contribution was handled by S. Blunt.

This work has been funded under a grant from the National ScienceFoundation (Award ECCS-1343316).

Corresponding author is C. Baylis, E-mail: (charles_baylis@baylor.edu).

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None

## Keywords

### INSPEC: Controlled Indexing

optimisation, power amplifiers, search problems, vectors

## Corrections

None

This paper appears in:
IEEE Transactions on Aerospace and Electronic Systems
Issue Date:
April 2016
On page(s):
928 - 937
ISSN:
0018-9251
INSPEC Accession Number:
16051359
Digital Object Identifier:
110.1109/TAES.2015.150335
Date of Current Version:
2016-05-26
Date of Original Publication:
No Data Available