## Introduction

Much effort has been expended into the development of real-time configurable and optimizable RF transmitter and receiver systems. One requirement for such a system is the ability to alter the impedances presented to the system's amplifiers. Traditional load pull impedance tuners are too slow for real-time operation and too large for integration into a final deployable system. In their place, various reconfigurable impedance tuners have been designed that are able to adapt faster and are housed in a much smaller package.

Although such newer technologies may not require the use of a characterization when used to optimize an amplifier system [1], it is nevertheless helpful to have a characterization during development and testing as amplifier behavior is most easily understood and analyzed when performance is mapped to a Smith Chart.

Generating
a tuner characterization can be a time-consuming effort since every
impedance point within the characterization table must be individually
tuned to and then measured. Because the precise relationship between the
tuner's fundamental parameters and the resulting reflection coefficient
(denoted in this paper using *P* → Γ) may not be understood at the
start of the characterization process, the selection of a small set of
fundamental parameter settings that provide a given coverage of the
Smith Chart is nontrivial. In situations where the behavior of the
relationship *P* → Γ cannot be easily modelled, manual trial and
error can lead to a good distribution of parameter settings that results
in the desired characterization. However, an automated approach that is
able to determine a sufficient distribution greatly reduces the time
required and results in a more consistent result between users.

## Relationship of Tuner Parameters to Impedance

Traditional
load-pull tuners implemented using a slide screw assembly (such as the
Maury Microwave Automated Tuner System (ATS) tuner shown in Fig. 1) exhibit a relatively straightforward relationship between fundamental tuner parameters and reflection coefficient [2]. In general, the distance of a mismatch probe from a transmission line (denoted as *h*
in this paper) is able to control the magnitude of the tuner's
reflection coefficient |Γ| while the position of that probe along the
transmission line (denoted as *x*) is able to control the phase of the reflection coefficient ∠Γ. That is, the relationship (*h*, *x*) → Γ can be described as:

An approximation of this relationship is illustrated in Fig. 2.

If using a tuner with a more complicated relationship than the previous (*h, x*) → Γ, these methods are not applicable. In particular, the tuner for which we demonstrate our method, designed by Semnani [4] and later adapted to employ commercial-off-the-shelf actuators [5],
utilizes two adjustable evanescent mode resonant cavities. We will
refer to this tuner as the evanescent mode cavity tuner (EMCT). This
tuner is also shown in Fig. 1.
The height of each cavity is adjusted by extending plates attached to
linear actuators, whose extension lengths are denoted in 0.5 μm
increments as *n*_{1} and *n*_{2}. For this
device, the input resonant cavity height selects a tuning circle on the
Smith Chart with center and radius dependent on the value of *n*_{1}, and the output resonant cavity selects a point on this circle dependent on the value of *n*_{2}, as illustrated in Fig. 3. As with the slide screw tuner, these relationships are nonlinear, with larger values of (*n*_{l}, *n*_{2}) resulting in more significant impedance variations. In this situation, the relationship (*n*_{l}, *n*_{2}) r cannot be simplified into two independent relationships as was done for (*h*, *x*) → Γ.

While a model can be determined by calculating the center and radius of each tuning circle associated with *n*_{1}, each circle model would require sweeping *n*_{2}, which is complicated by its nonlinear behavior. The approach presented by Spirito [3] could be applied for a fixed value of *n*_{1}, but the resulting mapping of *n*_{2} would not be transferrable to other values of *n*_{1}.
This would allow for the creation of a characterization consisting of
points uniformly distributed across the Smith Chart, at the cost of two
passes over the fundamental parameter domain.

If uniformity can be
sacrificed in favor of ensuring the spacing between points in the final
characterization is less than a desired maximum, then a more generic
algorithm built upon an interval-halving approach can be used, with
fewer restrictions on the nature of the relationship *P* → Γ. The following section illustrates this algorithm using a tuner with arbitrary parameters *P* = (*n*_{1}, *n*_{2}).

## Recursive Interval-Halving Maximum Separation Algorithm

Let the sorted set of allowed values for the parameters (*n*_{1}, *n*_{2}) be represented as *N*_{1} and *N*_{2}, the Euclidian distance between two values of Γ be represented as *δ*Γ, and the desired maximum point separation be represented as ΔΓ. For this discussion, sweeps of *n*_{2} will be performed for each selected value of *n*_{1}.

For a given *n*_{2} sweep, start by measuring Γ for the first and last points of the set *N*_{2} and *δ*Γ
between these two points. If this distance is greater than desired
maximum point separation ΔΓ, then select the next chosen value for *n*_{2} as the midpoint of the set *N*_{2}. Note that this has divided the set *N*_{2} into two different intervals: one that runs from the first value of *N*_{2} to the midpoint, and a second from the midpoint to the final value of *N*_{2}. Calculate *δ*Γ for the endpoints of each of these two intervals and divide each interval in half if *δ*Γ > ΔΓ for each interval. If *δ*Γ
≤ ΔΓ for either of the intervals, then this interval does not need to
be halved further. This process can be repeated recursively for each of
the new intervals until *δ*Γ ≤ ΔΓ for all intervals, indicating the desired point separation is reached. This process is illustrated in Fig. 4.

Note
that this process is not guaranteed to achieve a uniform spacing of
points in the resulting characterization, or even within a single
parameter sweep. For example, consider the simple example of
characterizing the arbitrary one-dimensional function

*x*≤ 100 with a desired point separation of no more than 25, as shown in Fig. 5. While a uniformly spaced characterization would contain around 7 points evenly spaced on the y-axis from 0 to 150, the resulting characterization contains 10 points with some intervals sampled beyond the desired density, such as the three points within the range 0 ≤

*f*(

*x*) ≤ 25.

A similar process can be used for the higher-level parameter *n*_{1}. Complete the described n_{2} sweep process of Fig. 4 for both the first and last points of the set *N*_{l} to create the first *n*_{1} interval. To determine if this interval should be halved, compute *δ*Γ between the two *n*_{2} sweeps from the *N*_{1} endpoints. This provides a measure of *δ*Γ between the *N*_{1} endpoints as a function of *n*_{2}. For each continuous interval of the *n*_{2} sweeps where the requirement *δ*Γ ≤ ΔΓ is violated, halve the *n _{L}* interval and perform additional sweeps of

*n*

_{2}, with

*n*

_{2}constrained to the violating interval. This process is illustrated in Fig. 6, and an example subdivision with multiple violating n

_{2}intervals is illustrated in Fig. 7.

## Limitations on Tuning Parameter Relationship

While this characterization algorithm functions without generating a model of the underlying relationship between the fundamental parameters and the resulting impedance, there are some constraints that must be maintained to ensure good performance.

First, the relationship *P* → Γ is assumed to be smooth—that is, continuous changes in *P*
will result in continuous changes in Γ. Violating this assumption could
prevent the interval-halving process from eventually converging. We are
unaware of any physically realizable tuner designs that are capable of
violating this assumption.

Secondly, it is assumed that the
behavior of a given fundamental parameter is such that all resulting
values of r within a radius ΔΓ correspond to a continuous set of
fundamental parameter values. In other words, the parameter sweep should
not revisit regions of the Smith Chart that were reached during earlier
portions of the sweep. If this assumption is violated, it is possible
that the interval-halving process converges sooner than desired. This
assumption is actually violated by the EMCT of [5] for the end points of a given *n*_{2}
sweep for extreme values of Compensation for this is accomplished by
requiring that a minimum set of points per sweep is visited. If the
process converges before this amount, the sweep interval is divided
according to a predetermined spacing which ensures the full sweep
behavior is observed, and then the interval-halving process is executed
with this initial seed of intervals.

Finally, this process is not well suited for a tuner that is controlled using fundamental parameters with a binary functionality, such as the tuner presented by Calabrese [6]. Such a tuner is incompatible with the interval-halving approach as the interval for each parameter is indivisible by nature (each parameter only has two states—no interval halving is possible).

## Results

An example characterization generated using the approach of Section III with ΔΓ = 0.1 and the EMCT of [5] is shown in Fig. 8. The regions of *N*_{1} and *N*_{2} that were sampled more densely during the characterization process are shown in Fig. 9. Note that the spacing in Fig. 9 needed to obtain the characterization of Fig. 8 is extremely irregular and cannot be obtained by any simple linear or exponential sweep.

To
achieve the specified ΔΓ, the algorithm visited 971 impedances in 276
seconds, achieving a maximum point separation of 0.08. While this is
close to the desired maximum point separation, the average point
separation of 0.03 is much lower. This is because the tuning circles
associated with each *n*_{1} value overlap in many instances even for distant values of *n*_{1}.
As the characterization algorithm only checks for point spacing within a
circle and between adjacently selected circles, oversampling can occur
along overlapping circles.

To
compare this method with the state of the art, characterizations were
performed of the Maury tuner using the same method. Unlike the EMCT,
each adjustable component of the Maury tuner requires different amounts
of time to set to new positions. As such, it is recommended to use the
slower component as the higher-level fundamental parameter *n*_{1}.
For the Maury tuner, the faster parameter is the probe position, and
the slower parameter is the carriage position. To demonstrate,
characterizations using either option for *n*_{1} and ΔΓ =
0.2 are shown in Figs. 10 and 11 below. These characterizations both
provide comparable coverage to that achieved by the Maury software
characterization with a target Γ separation of 0.2 shown in Fig 2.

Results for each characterization are summarized in Table I. As expected, characterizing the Maury tuner is much faster when the probe position is used as *n*_{2},
completing in 67% of the time despite using 46 more measurements.
Additionally, our algorithm outperforms the approach built into the
Maury tuner's software, evaluating more than twice as many points in 30
fewer seconds.

## Conclusion

An efficient approach for choosing tuning parameter values required to achieve a minimum Smith Chart separation in load-pull tuner characterizations has been presented. This characterization approach is expected to be especially useful and efficient in situations where the relationship between fundamental tuning parameters and reflection coefficient is not easily able to be modeled or understood.

### ACKNOWLEDGEMENT

This research was funded by the Army Research Laboratory (Grant No. W911NF-16-2-0054). The views and opinions expressed do not necessarily represent the opinions of the U.S. Government.