A Fast Two-Tone Active Load-Pull Algorithm for Assessing the Non-linearity of RF Devices

I.   Introduction

The rapid evolution of modern wireless communication system is asking for stringent regulatory emission in spectrum to prevent the generation of co-channel and adjacent channel interference due to the growing nonlinear behaviour of the radio frequency (RF) circuits^[1-3]. Hence the nonlinear characterization of the RF device is essential to optimize its performance in terms of output power,efficiency and linearity, as well as to mimic its behavior over different operating regimes and impedance terminations at system-level simulations^[4,5].

Since modern wireless communication systems utilize wide-band digitally modulated signals to maximize channel throughput over available channel capacity, traditional single-tone based continuous wave (CW) excitation testing is insufficient to characterize the non-linearity behaviour of the device^[6], which is because CW signal doesn’t has envelope shape like digitally modulated signals. On the other hand, although two-tone stimulus signal is also arguably unable to fully represent the nonlinear behaviour of the device under test (DUT) due to lacking of time-varying envelop shapes^[7], it is still the mostly used method in the RF analysis techniques because of the advantages of easier to understood and more intuitive to be applied in the RF analysis techniques^[8]. Actually, many figure of merits of nonlinear response, such as inter-modulation distortion, cross-modulation, memory effects, and gain compression/expansion, can be clearly illustrated using two-tone measurement^[9].

It is a well established fact that device’s non-linearity performance, as well as power and efficiency performance, is significantly dependent upon the load impedance^[10,11]. Hence the designer often relies on load-pull measurement, which is the technique of monitoring the nonlinear performance of DUT while driving it with different load impedance, to design the impedance matching network of the circuit. Based on the way of impedance tuning, load-pull systems can be classified as passive or active systems, or a combination of both^[12,13].

In passive technique, the desired impedance is synthesized by controlling the position and length of stubs in a stepper-motor driven tuning mechanism. Although the passive load-pull technique has the advantages of relatively higher power handling capability, low implementation cost and convenient, its limitation in synthesizing the low impedance and slow measurement speed has restricted its applications^[14]. These drawbacks can however be overcome by using the active load-pull technique, in which the reflection coefficient is synthesized by controlling the amplitude and phase of the injected signal with a phase-synchronized signal source. Such systems can emulate the load impedance near the boundary of the smith chart with much faster speed than passive technique, and can be implemented at high-accuracy real-time load-pull systems^[15].

However, the existing load-pull systems are mostly CW signal based though two-tone test has been extremely popular for nonlinear characterization. The newly developed harmonic passive tuners are rather expensive and only have limited tuning capability for two-tone test^[16]. Previous researchers have also reported an envelope active load pull architecture for assessing the nonlinear behaviour of DUT^[17], but this solution utilizes the closed loop feedback which has the potential issue of instability. Furthermore, the envelope active load-pull system is typically expensive and complicated because powerful embedded signal processing chips and high-precise dedicated samplers are necessary in such systems.

In this paper, a fast and intelligent active two-tone load-pull algorithm is introduced. This algorithm extends the classic nonlinear poly-harmonic distortion (PHD) model to two-tone stimulus format, based on which the required injection signals in the real time active load-pull system are predicted. This paper presents a comprehensive description and analysis of the proposed two-tone load-pull algorithm, along with the experimental evaluation and validation using a gallium nitride high electron mobility transistor (GaN HEMT) in a real two-tone load-pull measurement bench. This paper is, to our knowledge, the first report of an optimized two-tone real-time active load-pull algorithm, and can be further used in the nonlinear communication circuit analysis applications.

II.   Analysis of Two-Tone Active Load-Pull Algorithm

It is well known that for obtaining the optimum performance from a RF device the best matching condition at the device output is required. As it is difficult to physically design and realize various matching networks to identify the best output matching, an approach is adopted whereby various loading conditions are synthesized and device performances are measured under those load impedance. This process of synthesizing loading conditions has been termed as “load-pull”.

Active load-pull technique is a process in which the desired load reflection coefficient is achieved by actively injecting signals at the DUT output port rather than using a physical tuner. In the active open-loop load-pull technique, as can be shown in Fig.1, the injected signal $A_2$ is from external signal generator and is independent of the transmitted signal, $B_2$. $\Gamma_{load}$, the impedance presented to the load reference plane, is synthesized by controlling the magnitude and phase of the externally injected travelling wave $A_{2,set}$. The synthesized reflection coefficient depends on the delivered power of the RF generator and the effective gain of the drive amplifier. Under this setup, any impedance can be synthesised, including the impedance that lay outside the Smith Chart, i.e. $\left| \Gamma_{load}\right|>1$.

Figure  1.  A generic description of active open-loop load-pull technique

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The open-loop active load-pull setup however requires custom synthesizing algorithms for iterative convergence to desired reflection coefficients because the output of DUT, which is the transmitted traveling wave $B_2$, is dependent on device operating conditions. This makes the technique effectively too slow for high throughput measurement applications. Moreover, improper active load-pull algorithm will set an extensive $A_{2,set}$ value, which could present short impedance to the DUT, and further draw excessive current and blow up the transistors. This situation will get severer as most power amplifiers and signal generators are not perfectly 50Ohm matched, hence $\Gamma_{S}$ is introduced to account for the load mismatch of the measurement system introduced by the drive amplifier and the signal source, which is dependent on $A_{2,set}$.

To develop the active load-pull algorithm, an active load-pull error model is shown at Fig.2. In this error model, $\Gamma_L$ represents the load reflection coefficient applied to the DUT for the specified port; $A_{2,set}$ is the signal generated by the generator; $T_S$ refers to the transmission coefficient accounting for the loss or gain in the specified port network between the signal generator and the DUT, and $\Gamma_S$ represents the system reflection coefficient. Hence Eq.(1) can be concluded using the mason rule:

Figure  2.  The error model of open-loop active load-pull

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In the above equation, $\Gamma_L$ can be calculated by $B_2$ and $A_2$ after calibration of the system; $T_S$ is continuously measured by taking account for the nonlinear behaviour of the pre-amplifiers in the system using Eq.(1) from the measured data and the RF signal of the generator. $A_{2,set}$ hence is continuously computed using Eq.(1) via numerical technique through iterative convergence process.

For the two-tone load-pull system, the load emulation process is more complicated. Suppose the two tone frequencies are $w_1$ and $w_2$ separately, $p$ is the tone number for the specified frequency, and $h$ is the harmonic number of the specified impedance, Eq.(1) can be rewritten as Eq.(2) in two-tone format:

Therefore, if prior knowledge of $B_2$ is unavailable, the iteration process to calculate Eq.(2) will be very time consuming. In the scenario of harmonic load-pull where two or more frequencies are introduced, the iteration process will be further extended due to the change in the fundamental tone causing distortions in harmonics. This problem is further compounded by the fact that the utilized numerical techniques have disadvantages such as multiple roots and numerical oscillations^[18], which severely limit the application of active load-pull technique.

Recent advancements in device behavioral modeling have seen the introduction of the PHD model framework^[19-21], which provides an alternative potential solution to the active load-pull issue. As well as containing magnitude and phase relating to the spectral components of the input signal, PHD model is a black box model framework which uses the traveling waves to describe the cross-frequency components due to nonlinear response and can be further extended to two-tone signal stimulus scenario. Therefore a local derived model can assist the calculation of required injection signals. This will be further discussed in the following section.

III.   Two-Tone Load-Pull Algorithm Design from Nonlinear Behavior Model

The PHD model is a black-box frequency domain modeling technique that has been presented as a natural extension of S-parameters under large-signal conditions. Black-box definition indicates that no prior knowledge about the internal structure of the device is required and thus, PHD model is quite fit for the load-pull applications where devices characteristic are typically unknown. PHD model uses the concept of harmonic superposition to represent the nonlinear response as:

In the above equation, the $p$ and $q$ are the port index, and $m$ and $n$ are the corresponding harmonic index. Also $P$ refers to the normalized phase of injection signals. Eq.(3) is a generalised equation that shows the phase normalized output $B_{ph}$ waves being the linear summation of the input $A_{qn}$ waves and their conjugates.

For the simplicity of analysis, the discussion is started from the single tone scenario to conclude the formulation of active load-pull algorithm. As discussed in Eq.(1), where $B_{2h}$ is the waves directly related to the injection signal, a third order extension of Eq.(3) can be rewritten in a simple manner as Eq.(4).

In the above equations, P and Q are the input and output a-wave harmonic phase operators. A similar formulation of Eq.(4) has also been reported^[22] before. Adapt this generalized formulation and assume that the magnitude of the input signal $|A_{11}|$ during this process is held constant, it can be simplified to a X-parameter alike formulation as Eq.(6) by considering only the linear third order mixing terms.

As can be seen from Eq.(6), $B_{2h}$ is related to 3 parameters as $S_{2,h,0},S_{2,h,1}$ and $S_{2,h,-1}$. In the load-pull measurement scenario, $S_{2,h,0}$ can be deduced from the linear output response of $A_{11}$. $S_{2,h,1}$ and $S_{2,h,-1}$ can be extracted by applying three different perturbation signals to the incident $A_{21}$ wave, then simultaneous the three equations to solve these the rest of parameters. Once the computed value of $B_{2h}$ is acquired, the desired injection signal $A_{2,set}$ in Eq.(1) to achieve the desired load emulation can now be analytically computed. If the resulting load accuracy is not sufficient, the process can be repeated.

In the two-tone scenario, the incident wave and reflected waves can be written as $a_{(i,h1,h2)}$ and $b_{(i,h1,h2)}$, where $i$ is the port number, $j$ and $k$ are the non-linear orders of tone $w_1$ and tone $w_2$, respectively. The nonlinear frequency mix product of the two-tone stimulus signal can be represented by $m = |h1|+|h2|$, which is an integer bigger than 0, and all the frequency mix product can be represented in this way, as Fig.3 illustrates.

Figure  3.  Terms generated up to 4th order by two-tone stimulus

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Extension Eq.(4) to two-tone stimulus is achieved by expanding the PHD model formulation to account for the additional varying stimuli at the relevant mixed product harmonics. The behavior $b_{i,h1,h2}$ must be described now by a set of functions on respective to phase vectors $P_1$ and $P_2$ which are the incident wave phase operators of two-tone stimulus, and $Q_1$ and $Q_2$ are the reflected wave phase operators.

This general formulation, unfortunately, implies a very complicated model with many coefficients. To make the analysis easier, it can be generalised as:

where $K_{h1,h2,m,n} = f(|A_{1,1}|,|A_{1,2}|,|A_{2,h1}|,|A_{2,h2}|)$, $m$ and $n$ are the nonlinear orders of tone 1 and tone 2 respectively. As $\Gamma_{h1,h2}$ can be considered as the target reflection coefficients, it is hence possible to extract those parameters by varying the amplitude and phase of $A_{2,h1}$ and $A_{2,h2}$ while the driving two-tone stimulus signal held at a constant power level. This parameter extraction process involves the constructing of multiple measurement results matrix and using Least Square method to solve. If the resulting load accuracy is not sufficient, the newly measured $B_{2,h1,h2}$ of the achieved load impedance can be reintroduced to Eq.(8) to extract the new series of model parameters and update the local model to achieve more precised predictions.

IV.   Implementing Two-Tone Active Load-Pull Algorithm

Eq.(6) and Eq.(8) only suggest the way of calculating injection signals in the ideal scenarios. However in reality the insertion gain of the load drive amplifiers and the loss of couplers and cables, as described by the error model shown in Fig.2 as $T_{s}$, have to be taken into account. On the other hand, $\Gamma_{S}$ accounts for the load match of the measurement system. As discussed in Section II, the calculated signal $A_{2,set}$ has to compensate the above two variables to include the physical state of the system.

In Eq.(1), only $T_{S}$ and $\Gamma_{S}$ are unknown, because $A_{2}$ and $B_{2}$ can be directly measured and $A_{2,set}$ is pre-known. Therefore, these two parameters in Eq.(1) can be calculated with two distinct measurements.

As quite a few distinct measurements are required to compute the local model, an optimization is necessary to maximize the use of existing set of model parameters. If the emulated load impedance is within the acceptable tolerance of the target load, the existing model would have converged without requiring an update. The overall two-tone load-pull process can be described as the follows:

1) Calculate the system variables $T_{S}$ and $\Gamma_{S}$ at both the two tones frequencies $w_{1}$ and $w_{2}$. To achieve this goal, firstly drive the DUT to the large signal operation point (LSOP), then vary the $A_{2,set}$ vector value and do multiple measurements. With the raw measurement data collected $T_{S}$ and $\Gamma_{S}$ can be calculated using Eq.(1).

2) Synthesize the desired impedance at the first frequency $w_{1}$. In this step the phase-varied perturbation $A_{2}$ signal needs to be presented in the system to calculate the two parameters in Eq.(6). If the desired impedance didn’t meet the target precision, the current measurement would be used to extract the new series of parameters, until the target is meet. Obviously the more measurements are taken, the more accurate the established behaviour models can be, and hence more precise would be the synthesized impedance.

3) Once the desired impedance at $w_{1}$ is met, the impedance at the second frequency $w_{2}$ can be synthesized using the Eq.(8). At this step, the $A_1$ signals at both frequencies are held constant. Again $A_2$ signals with three varying phases will be injected to calculate the local model which is used for predicting the $A_{2,set}$. If the desired impedance at the second frequency is not met, new measurement will be added until the required precision achieved. The flow chart in Fig.4 summarizes the implementation of the algorithm.

Figure  4.  Flow graph of the algorithm implementation

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V.   Experiment Setup and Verification

To examine the validity of our proposed two-tone load-pull algorithm, a test bench is built as shown at Fig.5. A dedicated measurement software programmed using Python is developed to control the measurement instruments and run the introduced load-pull algorithm. The measurements were carried out on a 10 μm×60 μm GaN HEMT transistors operating at $w_{1}$ = 3 GHz and $w_{2}$ = 3.1 GHz.

Figure  5.  The large signal measurement bench used in this paper

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The detailed topology graph of the proposed two-tone active real time load-pull system is illustrated at Fig.6. A Ceyear 3672D vector network analyzer (VNA) from RI.41 is used as the main test instrument, whilst the travelling waves are acquired via the dual direction couplers and feeding into the internal receivers of the VNA. The two independent internal sources of the VNA, as they are phase coherent, are used as $a_{1,w1}$, the excitation of frequency $w_{1}$, and the $a_{2,w1}$, the load injected signal of this frequency, separately. For frequency $w_{2}$, two Agilent 8267D signal generators are used as the excitation and load emulation signal sources for this frequency. This setup is a typical open loop active load-pull system with real-time data acquisition capability^[23]. The Keysight 6700 DC supply is used to provide the gate and drain DC bias for the DUT. At the load side, a drive PA with 45 dBm gain and output power $P_{sat}$ of 45 dB is used to amplify the $a_2$ signals of both tones, and a source tuner is used to pre-match the DUT. A cascade 11000 probe station is employed for the on-wafer measurement.

Figure  6.  The proposed two-tone load-pull topology graph

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Prior to the load-pull measurement, the calibration is performed in two steps. First, a relative vector calibration is performed to determine complex error terms between the VNA receivers and the probe tips. At this step, a thru, reflect, match (TRM) on-wafer calibration as Ref.[24] introduced is carried out. Secondly, to further measure the absolute power at device plane, a power calibration as introduced in Ref.[25] is performed using a coaxial power meter.

To extract the system error terms, an initial measurement is performed at the system impedance, which is only close to 50 Ohm due to the mismatch of cables and couplers. Afterwards three more measurements are taken with the load power $A_{2,set}$ 10 dB lower than $A_{1,set}$ but with 3 different phases to $A_{1,set}$. The $T_{S}$ and $\Gamma_{S}$ are therefore calculated using Eq.(1).

Fig.7 illustrates the process taken by the algorithm to achieve the target impedance. The starting points are the system impedance with reflection coefficient as – 0.05 + 0.06i. To synthesize the target impedance at $w_{1}$ frequency, three perturbations point are set as the $a_{2,set}$ injection signals. These create three offset points (indicated by red circle-shaped markers) which allow the calculation of a local behaviour model. This local model is then used to compute a new value of $a_{2,set}$, which moves the load impedance $\Gamma_{L}$ to a new position (0.28+0.56i), which in this case is not within the tolerance range (set at 5%) of the algorithm. The algorithm therefore requires the model to update itself at this stage, hence add this measurement data to the process of calculating local model. Repeat the algorithm, and a new position of the load (0.29+0.56i) is obtained within 1 % of the target (0.29+0.55i), which implies that the algorithm has now converged to the desired impedance. Once the impedance at the first frequency is converged, the converging process will be continued at the second frequency with the same process.

Figure  7.  The load impedance convergence process for the first tone

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To further test the robustness and the speed of proposed load-pull algorithm, we set offset circle grids consisting of 31 points for both frequencies. Noticed that the optimum load impedance at these two frequencies are different, which could be caused by both the frequency difference and the modulation effect taken by the two-tone stimulus, the grids set at these two frequencies are slightly offset. Fig.8 shows the actual converged points with the target points at the first frequency. With 5% convergence error tolerance limit is set, the load-pull convergence accuracy is assured. Fig.9 further shows the output power contours at this frequency.

Figure  8.  The emulated points for single-tone stimulus

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Figure  9.  The maximum output power at the first frequency

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Once the optimum impedance of the first frequency is found, The two-tone load-pull algorithm is started by setting the target load of the first frequency at the optimum impedance, and use a different grid for the target points of the second frequency. Because of the modulation caused by the two-tone stimulus, the converge error at this time is bigger, but still maintained within 5% tolerance limit. Fig.10 shows the converged points and Fig.11 shows the output power contours at two-tone stimulus.

Figure  10.  The emulated points for two-tone stimulus

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Figure  11.  The maximum output power at two-tone stimulus

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To evaluate the overall performance,the ratio of total number of measurements to the number of target points is employed as the figure of merit to quantify the efficiency of the algorithm. From this definition, an ideal algorithm would have 100% efficiency if only one measurement is required per load impedance point. Obviously this efficiency is also related to the convergence error limit set during the measurement. For most application scenarios, the load-pull measurement is only a guidance for the power amplifier design, hence usually the tolerance limit is defined as 5%^[26]. The typical value for previously reported single-tone active load-pull system is from 5% to 15% using numerical method^[27], and by the author’s knowledge, there is no reported load-pull algorithm efficiency for two-tone stimulus possibly due to the exponential increased complexity.

Fig.12 firstly shows the number of measurements required in the convergence of each load point in single-tone stimulus, as well as the convergence errors between the target and emulated values for both conventional method and our proposed method. The convergence error limit is set as 5%. As can be seen, using our proposed algorithm, the convergence error is consistently less than the conventional method even though the set error tolerance limit is the same. For convergence efficiency the proposed algorithm is 50.8% (averaging 1.96 measurements per load impedance) compared to 12% (averaging 8.29 measurements per load impedance) using the conventional method. Fig.13 further shows the results by applying the proposed algorithm to two-tone stimulus. As can be seen, the convergence errors are still moderately lower than conventional method, and the efficiency is now much higher than conventional method with 38.2% (averaging 2.61 measurements per load impedance) compared to 6% (averaging 15.51 measurements per load impedance). These results clearly suggest the efficiency of the proposed method.

Figure  12.  The total number of measurements and convergence for single-tone stimulus

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Figure  13.  The total number of measurements and convergence for two-tone stimulus

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The mechanism behind the proposed efficient load-pull algorithm is unlike traditional load-pull algorithm which has to take redundant measurements to achieve load convergence, all the measurements data have been employed for the load converge calculating process by continuously updating the constructed behaviour model. Obviously, with a lower convergence limit set, the number of measurements will increase accordingly. Although setting extremely small convergence error tolerance for load-pull measurement is usually regarded as unnecessary since the parasitic components and self-heating of the devices would anyway lead to certain amount of offset during the further power amplifier design process, the ability to achieve convergence in extreme precision is still valuable since it provides the options for users to achieve trade-off between measurement speed and accuracy. To further test the capability of the proposed method, another series of two-tone load-pull experiments were carried out by setting the convergence error limit from 10% to 1% and comparing the total number measurement needed. As can be seen in Fig.14, for the 31-points grid, our proposed method always achieve convergence even at 1% error limit. However with the decreasing of error tolerance, the number of total measurement needed using conventional method increase dramatically with more than 800 measurements for 4% error limit. For error limit set below 4%, the conventional method can not converge.

Figure  14.  The total number of measurements at different convergence error limit

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VI.   Conclusions

We have proposed a fast, intelligent, and robust approach of two-tone active load-pull algorithm for the purpose of assessing the nonlinear performance of RF devices. Based on the two-tone extension format of tradition PHD behaviour models, this method use real time measurement data to build local behaviour model, which is further being used to predict the value of injection signal for emulating the desire load.

This algorithm is implemented using off-the-shelf measurement instruments and verified with actual GaN transistor. It is shown that the load impedance convergence speed is greatly improved without trade-off accuracy. Hence the realized two-tone on-wafer load-pull measurement system is opening more possibilities of characterizing the nonlinear performance of RF devices under 5G and 6G communication signals.