spectrum is becoming increasingly congested, and new techniques
continue to emerge for dynamic radar spectrum sharing. Much of the
S-band radar allocation in the United States continues to be
reallocated, and the 3.45 to 3.7 GHz range is now shared between radar
and wireless communications . Cognitive radar systems –
can assess and respond to their environment, and many cognitive radar
systems can sense spectrum use and then adjust operating frequency,
bandwidth, and the transmitted waveform to coexist with other wireless
operations , –.
An important part of cognitive radar transmission in spectrum sharing
scenarios is the ability to optimize transmission range and efficiency
after changing to a new operating frequency. If a tunable matching
network is present, such adjustments can be performed in real-time.
has been shown that power-amplifier output power and power-added
efficiency (PAE) can be significantly enhanced by strategic design of
load-impedance terminations at harmonic frequencies. Stancliff describes
an approach to perform loadpull measurements using harmonic
terminations, varying fundamental and harmonic load impedances to
maximize the output power or efficiency of a nonlinear power amplifier .
Benedikt describes harmonic tuning using a 30 W active loadpull system,
where a voltage wave is injected back toward the amplifier to emulate
the waveform reflected from a given load impedance. The voltage and
current waveforms can be viewed and engineered using an approach called
“waveform engineering” to maximize power or efficiency .
Vadala describes the use of low-frequency current-voltage measurements
to perform characterizations that allow harmonic terminations to be
selected . A back-and-forth method of fundamental and harmonic load- and source-pull measurements is demonstrated by Colantonio .
harmonic tuning techniques are useful because narrow-band, tunable
matching networks are used rather than wideband, fixed matching
networks. Bode  and Fano  have demonstrated that requiring higher bandwidth theoretically results in degrading the quality of the match .
If a network can be designed over a narrow band and adjusted in
real-time when the operating frequency is changed to re-optimize
performance, then better output power or efficiency at each frequency is
likely to be obtained, leading to increased range of the radar and more
efficient use of transmitter power supplies.
Harmonic Tuning Algorithm
simultaneous tuning of the fundamental and harmonic impedances (related
to reflection coefficients) can be performed using the algorithm
described in Fig. 1. Fig. 1 shows an illustration of the gradient search in three dimensions for simultaneous fundamental (ΓL,1) and second-harmonic (ΓL,2)
reflection-coefficient optimization, a concept easily extended to
additional dimensions as more harmonics can be tuned. A completely
reflective termination is desired for the second-harmonic load impedance
because it will prevent power from being delivered to the load at the
second-harmonic, reflecting it back into the device and forcing it to be
eventually delivered to the load at the fundamental frequency. As such,
the optimum second-harmonic reflection coefficient is expected to have a
magnitude of 1, and that only phase θ2 will need to be adjusted, providing a second-harmonic reflection coefficient ΓL,2=1/θ2–––––.
Visualization of gradient estimation in three-dimensional search space.
Neighboring points are used to assess the change in PAE in each
dimensional direction, allowing estimation of the gradient. (b) Search
step visualization in three-dimensional search. The search proceeds in
the direction of the PAE gradient to the next candidate.
At the beginning of the gradient search, neighboring points are measured by varying each by the neighboring-point distance Dn
from the initial candidate. The direction of steepest ascent can then
be ascertained through a gradient calculation. The search then proceeds
in the direction of steepest ascent a predefined search distance, Ds, to the next candidate, as shown in Fig. 1(b). The value of the criterion (i.e.
PAE) is evaluated at Candidate 2, and if the PAE at Candidate 2 is
higher than at Candidate 1, the search process is repeated at Candidate
2. If instead the PAE at Candidate 2 is less than the PAE at Candidate
1, the search distance is divided by two and the shorter search vector
is used from Candidate 1 to find a closer candidate in the same
direction, and the PAE at this candidate is evaluated. The process
carries forward until the search distance is less than the resolution
When this happens, the search is concluded and the point with the
highest PAE measured is chosen as the optimum. The progression of the
search has been adapted from the steepest-ascent search described by
Wilde , with some minor changes.
The Smith Tube  is a useful tool for visualizing the search progression, and is a three-dimensional extension of the ΓL,1 Smith Chart. The third (vertical) dimension is used for an additional parameter, as shown in Fig. 2. This search is performed using four parameters: Re(ΓL,1),Im(ΓL,1),θ2, and θ3.
This four-dimensional search can be visualized with two
three-dimensional trajectory plots, each plotted in a Smith Tube where
the horizontal dimensions are Re(ΓL,1) and Im(ΓL,1). For the first Smith Tube, the vertical dimension is θ2, and for the second Smith Tube, the vertical dimension is θ3.
The Smith tube 
is a three-dimensional, cylindrical extension of the Smith chart, with
the third axis representing an additional parameter. Here the third axis
represents θ2, the phase of ΓL,2.
Optimization Search Results
algorithm was tested experimentally in simulations using Keysight
Technologies' Advanced Design System (ADS) microwave circuit simulation
software. A Modelithics model for the On Semi MMBFU310LT1 junction
field-effect transistor model with VGS=−0.8 V, VDS=5.75 V was used in a simulation setup allowing independent control of the fundamental, second, and third harmonic impedances.
A. Traditional Harmonic Load-Pull Pre-Design
comparison with search algorithm results, a traditional harmonic-tuning
design approach was used, where fundamental load-pull measurements were
first used to select ΓL,1 to provide the highest PAE with ΓL,2=ΓL,3=1 (open-circuit harmonic terminations with θ2=0∘,θ3=0∘). Following this selection, this value of ΓL,1 was fixed and a second-harmonic load-pull was performed to select ΓL,2 to further increase PAE to the highest available value. Finally, the selected values of ΓL,1 and ΓL,2 were fixed at the determined optimum values and a third-harmonic load-pull simulation was performed to select the value of ΓL,3 maximizing PAE.
Fig. 3 shows the fundamental load-pull contours that describe the PAE variation over ΓL,1 with fixed ΓL,2=ΓL,3=1.
The optimum PAE for fundamental impedance variation is 22.7% at ΓL,1=0.63/40.6∘––––––.
500 MHz simulated fundamental load-pull contours with fixed
open-circuit second- and third-harmonic terminations, showing optimum PAE=22.7% at ΓL,1=0.63/40.6∘––––––,1.136% PAE step between contours.
shows the second-harmonic load-pull contours with the fundamental
reflection coefficient fixed to the previously determined fundamental
optimum (ΓL,1=0.63/40.6∘––––––) and ΓL,3=1. The optimum PAE=40.1% at ΓL,2=1/73.0∘––––––. This termination was then fixed for the second-harmonic, and a third-harmonic load-pull simulation, as shown in Fig. 5, was performed.
Traditional 500 MHz simulated second-harmonic load-pull contours with fixed fundamental termination ΓL,1=0.63/40.6∘–––––– and fixed open-circuit third-harmonic termination ΓL,3=1, showing optimum PAE=40.1% at ΓL,2=1/73.0∘––––––,1.337% PAE step between contours.
The optimum third-harmonic termination was found to be ΓL,3=1/79.3∘––––––.
The contours indicate that the best point is slightly inside the Smith
Chart but using that point in further calculations provides worse
performance than rounding the magnitude to 1. However, the third
harmonic termination only increases the PAE by approximately 0.7%, and
the second-harmonic termination must have much greater influence than
the third harmonic upon device performance for this device and operating
conditions. After fixing the second- and third-harmonic terminations to
the optima, the fundamental load-pull was re-performed, as shown in Fig. 6,
allowing an additional 2% PAE increase. A PAE increase from 22.7% to
45.2% is obtainable by performing harmonic tuning and then re-tuning the
fundamental. As such, second-harmonic tuning doubles the efficiency of
the transmitter amplifier.
Traditional 500 MHz simulated third-harmonic load-pull contours with fixed fundamental termination ΓL,1=0.630/40.6∘––––– and second-harmonic termination ΓL,2=1/73.0–––––∘, showing optimum PAE=40.8% at ΓL,3=0.97/79.3––––∘,0.136% PAE step between contours.
Traditional 500 MHz simulated fundamental load-pull contours with fixed ΓL,2=1/73.0–––––––∘ and ΓL,3=1/79.3–––––––∘, showing optimum PAE=45.2% at ΓL,1=0.73/43.8∘––––––,2.259% PAE step between contours.
B. New Fast Gradient Search for Real-Time Reconfiguration in Frequency-Agile Radar Transmitters
Using the fast gradient search technique of Fig. 1, a search was performed in the four-dimensional space consisting of the four real search parameters Re(ΓL,1),Im(ΓL,1),θ2, and θ3.
A magnitude of 1 was fixed for both the second- and third-harmonic
reflection coefficients. Fundamental and harmonic terminations are
adjusted simultaneously in the fast search, a feat not accomplishable
using the traditional methods.
The search was performed to maximize PAE, and its trajectory is shown in Fig. 7. The search trajectory, as evidenced by Fig. 7(a) and Fig. 7(b),
first shows a significant adjustment in the fundamental reflection
coefficient (the plane of both Smith Tubes) and then shows a significant
upward trajectory in the θ2
direction, indicating that the second-harmonic phase is adjusted. The
third harmonic does not show significant adjustment for this example.
For this device, an optimum of 44.5% PAE is obtained at ΓL,1=0.74/42.6∘––––––,θ2=73.7∘, and θ3=−0.66∘ with a total of 66 points assessed (termed “measurements”). Table I
compares the fundamental harmonic impedance terminations and
performance results of a traditional design versus the fast optimization
search. Additionally, the values for ΓL,1 and θ2 are very similar to the values found through the traditional sequential design. It appears that θ3
has little impact on the PAE of the device based on the small
additional PAE obtainable from tuning the third harmonic impedance as
shown in Fig. 5 (compared to Fig. 4 for second-harmonic termination). As such, it seems the search does not focus much on the optimization of θ3 since the sensitivity of PAE to θ3 is low.
tubes describing the 500 MHz PAE optimization trajectory magnitude of
fundamental load reflection coefficient with (a) Second-harmonic
reflection-coefficient phase θ2 and (b) Third harmonic reflection-coefficient phase θ3. Optimum PAE=44.5% was obtained at ΓL,1=0.74/42.6∘––––––,θ2=73.7∘, and θ3=−0.66∘ with a total of 66 measurements.
Fig. 8 shows the progression of PAE and output power during the search of Fig. 7. Fig. 8(a)
shows that PAE plateaus at approximately 23%, corresponding with the
end of its trajectory parallel to the horizontal plane of the Smith
Tube. This part of the search seems to result in a large change in the
fundamental termination but negligible change to the second- and
third-harmonic phase values. The green triangles in Fig. 8(a)
indicate the measured points that represent the highest PAE values
measured at or before their measurement numbers. Significant change in
the trajectory of the search happens between measurements 25 and 30, and
the second-harmonic termination phase begins to be adjusted, providing
approximately 45% PAE, an increase of over 20% PAE from the results
obtained in the first part of the search, where the primary adjustments
were performed on the fundamental termination. In this search, very
little adjustment of the third-harmonic reflection coefficient ended up
occurring. Notably, the locations of the reflection-coefficient
terminations resulting from the fast optimization search are similar to
the traditional design results shown in Fig. 6, both in terms of Smith Chart location and maximum PAE achievable, as summarized in Table I. In addition, similar benefits were seen from performing second-harmonic tuning in addition to fundamental tuning.
(a) PAE and (b) Amplifier output power Pout versus simulated measurement number during the 500 MHz PAE optimization search of fig. 8
To demonstrate the need for reconfiguration in changing
frequencies, a second search was performed at a frequency of 300 MHz
beginning at the fundamental and harmonic reflection coefficient values
selected as the end point of the first search. Changing frequency while
presenting the same reflection coefficients reduces the PAE by 30.0% to
17.3%. The search trajectory is displayed in Fig. 9,
starting at the previous frequency's optimum, and shows that the
optimum location moves significantly when changing frequency. The
algorithm allows reconfiguration to obtain PAE=47.3% with 31 measurements required for the search. Fig. 10 shows the PAE and output power versus measurement number, and Table I compares the results of this fast search with traditional load-pull design and the fast optimization results at 500 MHz.
tubes describing the 300 MHz PAE optimization after frequency change
trajectory magnitude of fundamental load reflection coefficient with (a)
Second-harmonic reflection-coefficient phase θ2 and (b) Third harmonic reflection-coefficient phase θ3. Optimum PAE=47.3% was obtained at ΓL,1=0.72/25.6∘––––––,θ2=36.7∘, and θ3=−0.95∘ with a total of 31 measurements.
Table I: Comparison of results for traditional design and fast PAE searches at 500 and 300 MHz
An additional search was performed at 500 MHz using the same device and bias settings. For this search, however, output power Pout was maximized instead of PAE, which can be directly optimized to maximize radar range. Fig. 11 shows the results of this search, and Fig. 12 shows the PAE and output power during the output-power optimization search.
(a) PAE and (b) Amplifier output power Pout versus simulated measurement number during the 300 MHz PAE optimization search after the frequency change shown in fig. 9.
tubes describing the 500 MHz output-power optimization trajectory
magnitude of fundamental load reflection coefficient with (a)
Second-harmonic reflection-coefficient phase θ2 and (b) Third harmonic reflection-coefficient phase θ3. Optimum Pout=19.48 dBm was obtained at ΓL,1=0.635/38.3∘––––––,θ2=62.63∘, and θ3=0.09∘ with a total of 61 measurements.
(a) PAE and (b) Amplifier output power Pout versus simulated measurement number during the 500 MHz Pout optimization search of fig. 11.
fast real-time search allowing simultaneous optimization of the
fundamental and harmonic impedances in a radar transmitter power
amplifier has been demonstrated in simulations. Gradient searches to
optimize PAE or output power, which is related to radar range, have been
demonstrated in the four-dimensional Smith Tube search space. The
search results show significant improvement over only optimizing the
fundamental reflection coefficient. Next steps include the development
of a high-power harmonic tuning network and implementation of the
algorithm in tuning this device.
The authors are grateful to Modelithics for
the donation of model libraries to Baylor University under the
Modelithics University Program.