Introduction
The Doherty power amplifier (DPA) is, nowadays, the wireless base-station workhorse in what RF signal amplification is concerned [1]–[4].
Accurate nonlinear models for the state-of-the-art high-power transistors are very difficult to obtain mostly because of the thermal issues and the distributed nature of these devices [5]. Therefore, the conventional Doherty design process is normally based on load-pull and S-parameter measurements. From these measurements, the optimal power load (
There are several ways to incorporate measurements in CAD simulators, which can be divided into two groups: look-up tables (LUTs) and equation-based behavioral models.
LUT-based models are perhaps the most direct way to incorporate measurements in CAD simulators. However, there are some known difficulties in data interpolation and extrapolation capabilities of such approaches. On one hand, they require a considerable amount of measurements to be accurate within a given extraction area and those measurements must be stored in memory during simulation. On the other hand, they also reveal poor extrapolation capabilities, as demonstrated in [9]. These drawbacks promote the usage of equation-based approaches that can accurately interpolate the data and reduce the required measurements.
The most well-known equation-based behavioral models in the literature can be divided into poly-harmonic distortion (PHD) models [10], Padé-approximation-based formulations [11], and artificial neural networks (ANNs) [12]. The PHD model is a black-box, frequency-domain, modeling technique based on the idea of extending S-parameters for large-signal conditions [9]. Over time, several behavioral formulations based on this approach have been proposed, such as the X-parameters [10] and the Cardiff model [13], [14]. Both formulations have been successfully used in power amplifier design. In [10] and [13], the X-parameters proved to be suitable for Doherty PA design, as long as they are made load-impedance-dependent, which requires an internal LUT for the model parameters for each load condition. In [14], an active device behavioral model is created through the Cardiff formulation. In this model, the core principle is to take advantage of the periodicity of the phase to create a multidimensional Fourier series to approximate a PHD function. Other options such as Padé-approximation-based models [11], [15] and Bayesian inference-based models [16], [17] have been proposed, but all of them need a large number of coefficients or have to internally incorporate an LUT to mimic the load dependence.
The previously mentioned model implementations rely on polynomial forms in which the number of parameters grows exponentially with the number of inputs and with the polynomial order, a problem known as the Curse of Dimensionality [18]. In the ANN case, the number of parameters only grows either linearly or quadratically [5], reducing the model complexity since it is able to represent complex functions with a low number of coefficients, [5], [9], [19]. In the active device modeling framework, the sigmoid is one good example of an activation function normally used for the neurons, as its graphic shape resembles the typical transistor current source or the PA’s gain compression characteristics. For all these reasons, ANNs are conquering their space in the microwave circuits modeling world [19], [20].
Most of these models are very accurate in mimicking the single-ended PA responses, where we are mostly interested in the high-power region and so small signal is not critical except for stability analysis. However, the Doherty PA introduces an additional challenge: the small-signal behavior of the peaking PA becomes relevant near the first efficiency peak, since it will determine the impedance presented to the carrier device. Unfortunately, erroneous extrapolation can occur when a behavioral model is considered, especially when it is extracted from passive load-pull data. This issue will be further explored in Sections II and III.
In this article, we intend to solve the problem in ANN-based models by creating a new behavioral model formulation that incorporates the small-signal S-parameters of the transistor. With the improved extrapolation for very low power levels, it will be possible to accurately predict the DPA’s AM/AM and AM/PM characteristics and the load modulation trajectories for both the carrier and the peaking PAs.
This article is organized as it follows. In Section II, we take the conventional modeling approach as a starting point to analyze and demonstrate the problem and then describe in detail the proposed formulation. The validation results are presented in Section III where we demonstrate the validity of the proposed formulation by comparing its predictions of a 700-W DPA performance with those of a conventional ANN behavioral model.
Since the main goal is to evaluate the accuracy of the model formulation, we chose a transistor with an available equivalent-circuit model. Even though this model is not very accurate, as it is normal for high-power devices, it allows us to assess the viability of the proposed formulation without any measurement or calibration errors. Thus, if we compare the available circuit model with the ANN models, any differences are solely due to the ANN model formulation.
ANN Formulations Based on Passive Load-Pull Data
In cellular base stations, it is necessary to use devices with large gate peripheries to achieve the desired high-power levels. This requires the use of transistors with built-in prematching circuitry allowing the transformation of the very low impedances at the intrinsic reference plane, to manageable impedance values, at the package plane [5], [21]. Along with the package, this prematching circuitry sets the harmonic impedances at the intrinsic reference plane, regardless of the harmonic terminations at the output [5]. Therefore, extracting behavioral models for these high-power devices becomes simplified, since only the fundamental and baseband data are needed [23]. Besides this modeling simplification, in this work, the modeled data are restricted to the zone that contains the efficiency and power termination of the transistor, not the entire Smith chart.
Hence, the swept power load-pull data, which PA designers have already been using to characterize field-effect transistors, can be used to develop a behavioral model.
A. Conventional Formulation
The direct approach to behavioral modeling using power waves is to represent the reflected waves, \begin{equation*} [b_{p1},b_{20}]=F_{\mathrm {ANN}}\left ({a_{11},a_{21} }\right)\tag{1}\end{equation*}
The
Taking all this into consideration, the ANN model outputs are, then, the real and imaginary parts of the scattered power waves,
Note that although the ANN formulation is formulated as
The ANN training step is performed until the load-pull measurements are predicted with an acceptable degree of accuracy. The selection process of the ANN architecture will be discussed in Section III. However, for a robust practical implementation in CAD environment, additional care must be taken to ensure well-behaved interpolation and extrapolation capabilities for correct simulations and power amplifier design. Thus, different formulations can lead to different predictions when used in a simulator, although the obtained error during the extraction can be almost the same.
Improper interpolation within the range of measurement conditions usually occurs due to overfitting, which can be minimized through division of the original measured data into three sets for training, validation, and testing.
The extrapolation capability, however, depends greatly on the formulation. While large-signal extrapolation is usually important during simulation convergence, if the underlying formulation provides a smooth and limited response, the simulator is able to converge in most situations. However, the small-signal extrapolation can be problematic since there are PA architectures that use a mix of class AB and class C devices whose large-signal and small-signal performance is interdependent.
Although with this formulation the large signal response of the device under CW excitation can be well-represented, as long as the complexity of the model is enough to represent the measurements, it can be demonstrated that this behavioral model approach is not suitable for DPA design. The model may not only incorrectly extrapolate
To explain the problem of wrong small-signal gain prediction, let us analyze how the error in the output reflected wave, \begin{align*} \Delta G_{T}=&\hat {G}_{T}-G_{T}=\frac {\left |{ \hat {b}_{21} }\right |^{2}-\left |{ a_{21} }\right |^{2}}{\left |{ a_{11} }\right |^{2}}-\frac {\left |{ b_{21} }\right |^{2}-\left |{ a_{21} }\right |^{2}}{\left |{ a_{11} }\right |^{2}} \\=&2\frac {\left |{ b_{21} }\right |\cos \left ({\Delta \theta }\right)}{\left |{ a_{11} }\right |^{2}}\left |{ \Delta b_{21} }\right |+\frac {\left |{ \Delta b_{21} }\right |^{2}}{\left |{ a_{11} }\right |^{2}}\tag{2}\end{align*}
\begin{equation*} \Delta G_{T,\mathrm {max}}=2\frac {\left |{ b_{21} }\right |}{\left |{ a_{11} }\right |^{2}}\left |{ \Delta b_{21} }\right |+\frac {\left |{ \mathrm {\Delta }b_{21} }\right |^{2}}{\left |{ a_{11} }\right |^{2}}.\tag{3}\end{equation*}
As seen in (3), the error
A possible alternative that partially solves these low-power extrapolation problems could be to use a gain formulation. This would force \begin{equation*} [b_{p1}/a_{11},b_{20}]=F_{\mathrm {ANN}}\left ({a_{11},a_{21} }\right)\tag{4}\end{equation*}
Thus, for small values of \begin{equation*} \Delta G_{T\mathrm {,max}}=2\frac {\vert b_{21}\vert }{\left |{ a_{11} }\right |}\left |{ \hat {e} }\right |+\left |{ \hat {e} }\right |^{2}.\tag{5}\end{equation*}
As seen from (5), this formulation does not completely solve the problem since one term of the error is still inversely proportional to
As previously explained, characterization of high-power devices is normally done at their package reference planes. Because of that, OMNs are usually optimized at this plane, from where the optimum efficiency and output power loads were extracted. When these previously characterized high-power active devices are used in Doherty PA arrangements, for small power levels, the peaking amplifier should be seen as an open circuit, one of the reasons why some offset lines are needed at the peaking PA output [22]. Unfortunately, the OFF-state output admittance of the device changes with frequency due to the parasitic elements of the package, and so the impedance presented by the peaking PA can be different from the desired open circuit within the PA bandwidth. This is represented in Fig. 3 by
Illustration of how
Although gain formulation may solve the single-ended performance at small signal, the simulation of a DPA configuration will certainly have problems both in small-signal and peaking impedance prediction, since \begin{equation*} S_{22}=\left.{ \frac {b_{21}}{a_{21}} }\right |_{a_{11}=0}\left.{ \frac {F_{\text {ANN}}}{a_{21}}a_{11} }\right |_{a_{11}=0}=0.\tag{6}\end{equation*}
This means that a different formulation must be considered, where
To solve this problem, active load pulling could be a solution to capture the behavior of the peaking PA in those conditions. Unfortunately, there are cases where these systems cannot be used: either the required nonlinear network analysis instrumentation is too expensive [19]; [23]–[25] or the DUT is a very large device, which requires a very high power and linear excitations to synthesize the loads. The alternative is to use a passive load-pull system, though the capability to synthetize active loads is lost.
B. Proposed Formulation
Since the behavioral model will be extracted from passive load-pull characterization data collected at the transistor package reference plane, we can embed the peaking PA \begin{align*} \begin{cases} \displaystyle b_{p1}=\left [{F_{\mathrm {ANN}}\left ({a_{11},a_{21} }\right)+s_{p1} }\right]a_{11}+s_{p2}a_{21} \\ \displaystyle b_{20}=F_{\mathrm {ANN}}\left ({a_{11},a_{21} }\right) \end{cases}\tag{7}\end{align*}
Note that now, even for the case where
Although, in this work, we have focused on modeling a prematched device for a high-power Doherty PA, where the prematching reactances act as harmonic traps, and thus the harmonic excitation at the package plane does not have any significant impact on the behavior of the device, the proposed formulation can also be expanded to include this dependence. That formulation, along with a brief analysis, can be found in Appendix A.
Application to a Practical Doherty PA
This section presents the results of the conventional and the proposed formulation in a Doherty PA. The objective is to demonstrate the problems derived from the conventional formulation and to validate our proposed formulation.
The ANN models were implemented in the Advanced Design System (ADS) circuit simulator through an FDD block, which enables the creation of nonlinear components based on user-defined equations, in the frequency domain. This ADS model is then compared with the existent commercial circuit model for performance evaluation. Although the main goal is to evaluate the behavioral model in a DPA simulation, the single-ended class B and C amplifiers are also used in the analysis.
A. Behavioral Model Extraction
The carrier and peaking PA behavioral models were extracted via ADS simulations using the passive load-pull approach.
In the single-ended simulations, the models were cross-validated, that is, different sets of loads were used for extraction and validation. These loads are represented in Fig. 5 alongside with the two loads selected for the gain and efficiency power sweep plots (black crosses). The extraction was done by sweeping the power from 23 to 48 dBm, that is, 25 dB of back-off, while the validation was done from 13 to 48 dBm, that is, 35 dB of back-off. In this way, it is possible to observe the model extrapolation in very low-power regions.
Using the corresponding MATLAB toolbox, the ANN model was extracted for several layer sizes as shown in Fig. 6. An ANN with three layers of 15 neurons each ([
Performance of several ANN architectures for this implementation (sum of squared errors (sse)/complexity/extraction time).
ANN structure for the conventional formulation. For the proposed formulation, the structure is the same, but with the respective outputs.
The Bayesian regularization algorithm from MATLAB was used to train the ANNs used in this work. This algorithm not only seeks to optimize the cost function but also seeks to reduce the number of effective parameters and lower their values. This has proven to be beneficial to improve the interpolation capabilities of the model and reduce the overfitting problem.
Note that the behavioral model is parametrized in frequency and in the
B. Load-Pull Contours
As previously discussed, the conventional formulation can accurately represent the active device as long as the transistor is not driven outside the extraction region. So, if we only analyze the model considering the power or efficiency contours, the model will seem to be accurate enough for PA design, as can be seen in Fig. 8 for class B biasing. In fact, if we compare with our proposed formulation, we will not detect any significant differences.
(a) Efficiency and (b) output power contours for class B biasing (
The models must be tested in different conditions to be properly evaluated. Thus, we started by evaluating the single-ended PA case. Since we are interested in designing a Doherty PA, both class B and class C will be analyzed.
C. Single-Ended Amplifier
By taking the conventional and proposed model formulations and simulating the gain and efficiency over a large power sweep for class B biasing, we obtain the results presented in Fig. 9(a). Note that these results show us two different load terminations and test the lower power extrapolation by 5 dB.
Gain and efficiency plots for (a) class B and (b) class C biasing for two different load conditions.
By observing these class B results, the differences between the formulations are already clear in the low-power extrapolation region. However, the problem aggravates for the class C amplifier. At low input power, this amplifier operates in the cutoff region, so small-signal problems will become much more noticeable, as can be seen in Fig. 9(b). Here, we can observe that in the conventional formulation, the gain prediction in the extrapolation region does not behave in a controlled manner.
By including the S-parameters in the proposed model, it remains very accurate in the low-power region. This demonstrates that the small-signal extrapolation problems can indeed be solved using the proposed formulation.
D. Doherty Power Amplifier
To validate the proposed behavioral model approach for DPA simulations, we designed and implemented a 700-W, two-way asymmetric DPA at 1840 MHz. The DPA is composed of three 250-W LDMOS devices, one for the carrier PA and two for the peaking PAs that will provide the appropriated load modulation. The circuit and behavioral models were used to simulate the DPA in the ADS simulator. The complete DPA schematic is presented in Fig. 10.
Fig. 11 depicts the efficiency, gain, and AM/PM performance obtained for both formulations (conventional and proposed) using the designed Doherty PA at 1.84 GHz.
Comparison between the circuit model and ADS behavioral models (conventional and proposed) of gain, efficiency, and AM/PM curves.
It is possible to observe that the conventional formulation is inaccurate in the low-power region. In fact, in Fig. 9 we can already observe this issue. Naturally, the same problem happens in Doherty operation and is further aggravated by the wrong prediction of the peaking impedance trajectory at back-off, which results in a wrong impedance presented to the carrier. In Fig. 12, this problem is highlighted in a red circle. Since this error cannot be controlled during the model extraction due to improper formulation, the model will extrapolate differently for different extractions, resulting in a high variance on the performance prediction.
Comparison between the circuit model and ADS behavioral models (conventional and proposed) of main and peaking: (a) impedances trajectory at the fundamental and (b)
On the contrary, our proposed formulation presents a good agreement with the circuit results, solving the aforementioned extrapolation problems and thus proving to be much more suitable to predict the Doherty behavior.
To further validate the proposed model formulation,
Comparison of dc and fundamental drain current curves between the circuit and ANN models, for both formulations (conventional and proposed), for the carrier PA.
In addition, in Figs. 13 and 14 we show the comparison of dc and fundamental drain current curves between the circuit and ANN models for both formulations. As observed in Fig. 14, the error in the prediction of the ANN model with the conventional formulation is greater than that of the proposed formulation.
Comparison of dc and fundamental drain current curves between the circuit and ANN models, for both formulations (conventional and proposed), for the peaking PA.
To further illustrate the advantages of the proposed model, regarding the small-signal extrapolation issue, another example is presented in Appendix B, where the behavioral model is used to predict the circuit simulation performance of a two-stage amplifier.
E. Model Validation and Experimental Results
It is now important to evaluate the model in several frequencies to correctly verify its validity in Doherty operation. The Doherty PA was implemented and measured at 1.805, 1.84, and 1.88 GHz.
In Fig. 15, AM/AM, AM/PM, and efficiency curves are presented, where we can verify that the behavioral model successfully emulates the circuit model. This means that the model can be effectively used as a substitute to circuit models in cases where those are not possible to obtain. In Fig. 16, the AM/AM, AM/PM, and efficiency measurements are shown.
Comparison of AM/AM, AM/PM, and efficiency obtained with the DPA circuit simulation and with the proposed behavioral model formulation.
Comparing the simulation results with the measurements, it is possible to observe that except for the frequency of 1880 MHz, the output power predicted by the simulations is higher. Furthermore, the high-power measured efficiency is lower than the simulated one, and some differences start to be noticeable in the AM/PM curves from 50 dBm onward of the output power.
These results may indicate that the transistor large-signal nonlinear model is not completely accurate, as is common for high-power devices, in addition to some imprecision in the Doherty implementation and measurement process.
To increase the accuracy of the proposed ANN behavioral models, they should be extracted directly from the swept power passive load-pull measurements, which most of the time is easier than to extract a more accurate circuit-level model for high-power transistors, this being one of the ANN behavioral model advantages.
Conclusion
In this work, we demonstrated that incorporating the active device small-signal S-parameters in the behavioral model formulation solves small-signal and active load prediction problems in a Doherty PA configuration. These problems occur due to improper extrapolation of the conventional model formulation and cannot always be solved simply by performing more measurements. For example, if a passive load-pull system is used, the active loads cannot be tested.
Based on an ANN, a behavioral model was formulated, implemented in a commercial simulator, and validated for an implementation of a DPA. The final model formulation allows the use of passive load systems without compromising its accuracy, by including the description of the S-parameters, a typically simple, commonly available, added measurement. As demonstrated, the model can predict the active loads produced by the Doherty peaking PA, so that there is no need to use an active load-pull system. This advantage is particularly useful for very high-power transistors, commonly used in wireless communications.
Appendix A
Appendix A
The model formulation proposed in this work can be expanded to include the output second-harmonic dependence according to (8). Note that in this case, the ANN function can be understood as the conversion gain from fundamental to second harmonic. In that sense, similar to what we have done for the fundamental formulation when we included the S-parameters, here we are including a conversion gain parameter, \begin{align*}&\begin{cases} \displaystyle b_{p1}=\left [{F_{\mathrm {AN}\mathrm {N}_{\mathrm {p1}}}\left ({a_{11},a_{21},a_{22} }\right)+s_{p1} }\right]a_{11}+s_{p2}a_{21} \\ \displaystyle b_{20}=F_{\mathrm {AN}\mathrm {N}_{20}}\left ({a_{11},a_{21},a_{22} }\right) \\ \displaystyle b_{22}=\left [{F_{\mathrm {AN}\mathrm {N}_{22}}\left ({a_{11},a_{21},a_{22} }\right)+G_{21,2} }\right]a_{11}^{2}+s_{22}a_{22} \end{cases} \tag{8}\\&\left.{ G_{p_{2}p_{1},h_{p_{2}}}=\frac {b_{p_{2}h_{p2}}}{a_{p_{1}1}^{h_{p2}}} }\right |_{a_{22}\mathrm {=0, }a_{12}=0.}\tag{9}\end{align*}
This G-parameter may be extracted from the load-pull information by the least-squares method. The block diagram of this implementation can be found in Fig. 17, where the mathematical manipulation that is required to extract the ANN is presented.
As previously mentioned, the transistor used in this work is a 220-W prematched LDMOS device whose prematching reactances behave as harmonic traps, making it insensitive to the input and output harmonic terminations. Thus, to validate this expanded model, we have now selected the CGH40010F packaged transistor, a 10-W GaN HEMT device from Wolfspeed. The load terminations used in this part are shown in Fig. 18. The power level was swept from −5 to 30 dBm, the center frequency was 2 GHz, and the device was biased at
Extraction and validation loads for the second-harmonic implementation (
Fig. 19 shows the simulated efficiency and output power load-pull contours for a second harmonic where the efficiency is maximized, in (a), and minimized, in (b). As shown, the error is acceptable for both cases and the model can correctly predict the behavior of the transistor.
To demonstrate the prediction capability of the model for a wider range of second-harmonic terminations, Fig. 20 presents the efficiency at 3 dB of gain compression for the same fundamental load. The second-harmonic loads are always kept in the edge of the Smith chart, that is,
Efficiency at 3 dB of gain compression for a constant fundamental frequency load and several second-harmonic loads.
Appendix B
Appendix B
The small-signal problem treated in this article is relevant for many other applications beyond the Doherty operation. Any configuration where several stages of amplifiers are used may not be possible to simulate if the S-parameters cannot be accurately predicted. For instance, in a multistage monolithic microwave integrate circuit (MMIC) PA, where several devices are cascaded to boost the gain, it is very important to adapt the input of one PA to the output of a previous one. Thus, correctly predicting the input impedance,
A simple two-stage cascaded PA, represented in Fig. 21, was built in the simulator, to demonstrate the impacts of poor small-signal prediction on these architectures. The represented S-parameter block guarantees the correct load and source terminations of the first- and second-stage amplifiers, respectively. For these simulations, the CGH4010F packaged transistor model from Cree was used.
The devices used in the first and second PA stages were biased above and below the threshold voltage,
The two-stage PA gain and efficiency profiles are shown in Fig. 22(a). This shows that the conventional formulation starts to fail as soon as it starts to extrapolate, leading to a wrong small-signal prediction. On the contrary, the formulation presented in this work is able to mimic the reference circuit model. This is also evidenced by the input loads of both PAs, represented in Fig. 22(b). The conventional formulation fails to predict both impedances for all power levels. However, the prediction of the input impedance of the second-stage PA is better. This is expected, since the power level on the second stage is higher, and so the small-signal extrapolation problems are less notable. The proposed formulation proves to be highly accurate, demonstrating that unlike the conventional approach, this formulation is not only suitable for Doherty operation but also it can be used in many other circuits such as the herein shown multistage PA.
(a) Gain and efficiency power sweeps and (b) input impedance for the two-stage cascaded power amplifier (