## Introduction

Increasing demands on radar systems, including the need for participating in dynamic spectrum allocation and ability to quickly adapt to changing detection needs, are causing new radar design approaches to be considered for future systems. Cognitive radar has been proposed as a solution [1], [2]. Dynamic spectrum allocation has already become prevalent in many of the radar bands, including the upper part of the radar S-band allocation. Discussions of codesigning the circuit and waveform to meet the increasing demands of a future radar system have been ongoing [3]. Many radar system designers have focused on optimizing the radar waveform, in many situations, with the nonlinear transmitter amplifier “in the loop.” Patton and Rigling have created waveform optimization techniques for situations where the autocorrelation and waveform amplitude are constrained [4], where spectrum constraints are present [5], and where the waveform and the matched filter are jointly optimized [6]. Ryan *et al.* have demonstrated hardware-in-the-loop optimization for polyphase coded frequency modulation waveforms in the linear amplification with nonlinear components amplifier topology [7]–[9]. Cognitive radio technology has also been applied to radar applications [10], providing a framework for radar systems to be able to optimize components in “real time,” that is, to tune their components during operation for optimum performance, according to their surroundings and needs.

While waveform optimization has been an intense matter of focus, recent attention has been given to design of radar circuitry to support the reconfiguration in next-generation radar systems. Recent work by Guerci and Kingsley has investigated issues of circuit design for cognitive and adaptive radar [11], [12]. One particular roadblock to the implementation of reconfigurable radar transmitters is the power-handling capability of traditional reconfigurable circuit components. While this is beyond the focus of the present paper, parallel developments in high-power tunable components promise that the techniques we present will soon be implementable at power levels needed to accommodate radar transmission [13].

The power amplifier is the largest consumer of energy in the transmitter chain. As a result, its power-added efficiency (PAE) is crucial to the overall power efficiency of the transmitter. As more wireless systems begin to participate in dynamic spectrum allocation, where the operating frequency and bandwidth are allocated to spectrum users in real time, reconfigurable circuitry will be needed to allow the system to operate with excellent efficiency and performance at the frequency assigned for its system's use. A second important characteristic of transmitter devices is to meet standards of adjacent-channel power. This can be assessed by measuring the adjacent-channel power ratio (ACPR) of the device. In this work, we consider the application of tunable matching networks for adaptive power amplifiers, but other networks requiring tunable matching impedances could also use these techniques.

A recent paper by our group shows the design and implementation of a varactor tuning network for an adaptive amplifier [14]. A downfall reported by this work is that the matching network begins to perform nonlinearly at higher values of input power. As a result, the characterization, which determines the load reflection coefficient presented to the amplifier preceding the matching network, is not valid, creating difficulty in optimizing the system clearly, and in assessing the gain and PAE accurately. While a result may still be achieved, nonconvexities in the output power characteristic may cause the algorithm to reach a result that is less than optimal.

Varactor networks have been used for some time in tunable matching networks, and nonlinearities have been a significant problem in the use of tunable matching networks. However, most of the literature focuses on examining the linearity limits of the networks rather than characterizing to allow the networks to operate in their nonlinear regions. Entesari provides a comparison between matching network and filter designs using radio-frequency (RF) micro-electrical mechanical systems (MEMS), barium strontium titanate (BST), and GaAs varactors for a WCDMA front-end, and shows that the RF MEMS design has the highest third-order intercept point (IP3), followed by the BST and GaAs varactor designs [15]. Nemati demonstrates the design of tunable varactor networks, and states that nonlinearities in the network are caused by variation of the capacitance values due to the large-signal modulation of the varactor bias voltages [16]. Shen uses two-tone intermodulation measurements in efforts to assess nonlinearity [17], and Hoarau additionally employs 1-dB compression measurements of the matching network [18]. Modeling of the distortion in variable-capacitance diodes is explored by Meyer [19], and Buisman shows that placing varactors in antiseries and antiparallel combinations can mitigate distortion, validating this with improvements in simulated and measured third-order intermodulation results [20]. Andersson shows how a varactor's behavior can be understood as it is pressed into nonlinearity, and demonstrates a load-pull optimization of the varactor for second-harmonic performance [21]. Park shows that the third-order intermodulation products are a significant function of varactor bias voltage, and states that some varactors can vary by nearly 50% in their capacitance value based on typical bias voltages and large-signal levels [22]. In a 2013 patent, Spears describes the use of a look-up table for reconfigurable impedance matching of an antenna, including the scenarios of precharacterization of matching states over a range of antenna impedances and input power values, among other parameters [23].

Fig. 1 shows the topology and layout of the tunable-varactor matching network, designed for use at 1.3 GHz based on the approach of Fu [24]. Fig. 2 shows the characterized states of the matching network for the load reflection coefficient

This paper presents a fast search algorithm that can be used for real-time optimization of a tunable-varactor matching network in reconfigurable power amplifiers, including an approach for overcoming power-dependent nonlinearities in the matching network. The results will be applicable in real-time impedance matching with matching networks containing nonlinear devices.

In Section II, load-pull contours taken using the tunable varactor matching network with and without power-dependent characterization at nonlinear power levels are compared to each other and to measurements taken using a traditional Maury Microwave mechanical tuner. The use of the power-dependent characterization in fast load-pull search algorithms designed for real-time constrained optimization is demonstrated in Section III. Some conclusions and suggestions for future work are presented in Section IV.

## Load-Pull Results Comparison With and Without Power-Dependent Characterization

Comparing load-pull results measured with the tunable-varactor matching network shows improvement by using characterizations performed close to the value of output power expected from the transistor. Measurement comparisons were performed for a Microwave Technologies MWT-173 field-effect transistor (FET) under two different conditions. Fig. 4 shows traditionally measured load-pull results for PAE and ACPR at a bias of

For comparison, Fig. 5 shows PAE and ACPR load-pull results measured at the same bias and input-power conditions using the tunable-varactor matching network. The load-pull in Fig. 5 was performed using a standard small-signal characterization of the varactor network. The maximum PAE location predicted by the varactor tuner appears significantly different than the maximum PAE location predicted from measurements with the Maury system. In addition, the maximum PAE value (approximately 35%) obtained from the varactor-tuner load pull is higher than those obtained from the mechanical tuner (approximately 30%).

Because the varactor network is expected to perform nonlinearly at this input power level, a power-dependent characterization of the varactor network was performed. A look-up table was developed based on the power input to the varactor network (output power of the MWT-173 during the load pull). To perform the power-dependent characterization, a gain estimation measurement is performed near each

Fig. 6 shows the PAE and output power contours obtained by a load-pull measurement using the power-dependent characterization. The PAE optimum location is much closer to the optimum location found with the mechanical-tuner measurement, and the optimum PAE value is also closer to the optimum PAE found in the mechanical-tuner measurement (approximately 31%).

An investigation as to why the PAE value is overstated when a small-signal characterization is used reveals that as the input power increases, the value of

Next, a second bias and input-power condition is considered:

## Fast Load-Impedance Optimization Algorithm Using the Varactor Matching Network

A fast search for the value of

The power-dependent characterization is used in the algorithm by using an iterative process that selects the appropriate power level and characterization. An initial guess is made to estimate the power input to the varactor network from the output of the device. The characterization that corresponds to the guessed power value is then selected. A measurement of the power is then performed, and using the S-parameters from the characterization associated with the guessed power value, an input power to the varactor network is calculated. If the difference between the initial guess and the measured value are within a certain limit then the process is stopped and that characterization is selected. Otherwise, the guess is set to the measured value and the process is repeated until it converges. Fig. 12 shows a flowchart that summarizes this process. The power-dependent characterization can result in some additional measurements if inaccurate guesses are made.

The algorithm was applied to tune the MWT-173 FET with bias condition

Fig. 13 shows the results of a modified gradient search using a small-signal characterization starting at __/−45°__. Results from multiple starting points using the small-signal characterization are shown in Table I. The average end PAE value found is 34.4%, with 3.2% PAE standard deviation. The average number of measurements is 13, with an average time per measurement of 4.35 s. It can be seen that the search takes multiple turns; this is consistent with the fluctuating shapes and nonconvexities shown in the Fig. 5 contours. Table I summarizes small-signal characterization search results for multiple

Fig. 14 shows the search trajectory on the Smith Chart using the power-dependent characterization with the same starting __/−45°__) as the small-signal characterization based search shown in Fig. 12. The search path is much more direct than when using the small-signal characterization. Results from multiple starting points for the power-dependent search are shown in Table II. The average end PAE value found is 32.06%, with .33% PAE standard deviation. The average number of measurements is 13.31, with average time per measurement of 4.4 s. The power-dependent characterization is, in general, expected to require more measurements, due to the need to estimate the output power value at new candidate points for purpose of applying the characterization. When using the small-signal characterization, no estimation of the output power is needed. Another notable difference in comparing the end PAE values is the significantly smaller standard deviation obtained for the power-dependent characterization. This indicates that the convergence of the algorithm is more consistent for the power-dependent characterization than for the small-signal characterization.

Additionally, it can be noted that the PAE values obtained using the small-signal characterization are higher than the PAE values obtained using the power-dependent characterization. This makes sense considering the concept that

It is also useful to compare the results of the tunable-varactor load-pull search with results taken by applying the same search algorithm with the Maury Microwave mechanical tuner. Table III shows the results of this load-pull search from multiple starting values of

The results of the searches are compared in Table IV. The average PAE reported using the small-signal characterization is higher than the PAE values reported using the other measurements due to the de-embedding through the varactor network using an incorrect

A similar comparison was performed for a second input-power and bias condition: __/−45°__ (see Fig. 9). From a full load-pull measurement using a small-signal characterization, the best PAE providing ACPR ≤−27 dBc is 24.47% at __/−45°__ (see Fig. 8). For this bias and input power, the small-signal characterization provides a higher constrained optimum value of PAE. This is due to the interacting shapes of the PAE and ACPR contours as measured in the Smith Chart for the different load-pull results.

Fig. 15 shows the trajectory of the search algorithm in the Smith Chart as performed with the varactor matching network for starting point __/0°__ using the small-signal characterization, and Fig. 16 shows the trajectory for starting point __/45°__ using the power-dependent characterization with the varactor matching network. Table V shows the results of fast searching from multiple starting

The common conclusion that can be drawn by examining the search summary comparisons in Table VIII is that the varactor tuning network converges to consistent PAE values with a higher degree of repeatability for the large-signal characterization than for a small-signal characterization. Similar to the first set of operating conditions used, the end PAE standard deviation is much smaller for the large-signal characterization than for the small-signal characterization. Due to the inaccuracy of the small-signal characterization to predict the value of

While it appears that the power-dependent characterization helps to hone the results for higher accuracy, it also appears that that a useful result for real-time reconfigurable radar is reached with the varactor matching network even when a small-signal characterization is used. A significant time savings is accomplished by using the varactor tuner, regardless of the characterization. Requirement of 3–4 s per measurement is primarily due to equipment overhead, which will be significantly reduced in a real-time optimization from a cognitive radio platform in an adaptive radar.

## Conclusion

A constrained optimization for the PAE under ACPR constraints has been demonstrated using a tunable-varactor matching network. The varactor matching network can tune to provide repeatable results for the optimization from multiple starting reflection coefficient values, requiring approximately one-fifth of the time per measurement required by a traditional bench-top load-pull tuner. This matching network serves as a first-level prototype of a reconfigurable matching network to be used in an adaptive radar transmitter. Further, the results show that the use of a power-dependent characterization can account for nonlinearities in the tuner itself, providing more consistent and accurate search results. In many cases, more measurements are required for the power-dependent characterization due to needed repeated measurements when an error in power estimation is incurred. The fast reconfiguration time and accuracy of this tuning algorithm will allow the radar to adapt in a congested environment to coexist with communication and other radar systems. Parallel developments in high-power tunable passive components are expected to allow this algorithm to be useful in real-time radar transmitter optimization.

### ACKNOWLEDGMENT

The authors would like to thank J. Clark of the Army Research Laboratory for his helpful comments and assistance.