Basic theory of the new model

Support vector regression

SVR is a regression technique based on the popular support vector machine (SVM) machine learning classification technique, developed by Vapnik et al. in 1996 [Reference Drucker, Burges, Kaufman, Smola, Vapnik, Mozer, Jordan and Petsche28]. It is a nonparametric technique (i.e. it makes very few assumptions about the form of the modeling functions used) based on supervised learning (i.e. training data are required to determine the model parameters).

Suppose that n sets of multi-input single-output training data are given as $\{ {{\bf x}_i, \;{\rm \;}y_i} \} _{i = 1}^n$, where ${\bf x}_i\in {\bf R}^d$ denotes the d-dimensional input variables, with $y_i\;\;{\bf R}$ denoting the scalar output value. The unknown function relating inputs and output can be expressed as

(1)$$f( {\bf x} ) = {\bf w}^{\rm T}{\bf x} + b, \;$$

where ${\bf w}$ is a weighting vector and b is a scalar bias term. The SVR technique leads to a convex optimization problem that attempts to find the simplest model for the observed data while permitting an acceptable error (± ɛ), when constructing the model $f( {\bf x} )$, as seen in Fig. 1. This is known as an ɛ-insensitive error function and is introduced to improve the sparsity of the solution.

Fig. 1. Support vector regression.

The error function for SVR is expressed as

(2)$$\min \,\displaystyle{1 \over 2}\vert {\vert {\bf w} \vert } \vert ^2 + C\displaystyle{1 \over n}\mathop \sum \limits_{i = 1}^n ( {{\rm \xi }_i^ + { + } {\rm \xi }_i^- } ), $$

where C is the control for the model complexity. Slack variables, ${\rm \xi }_i^ \pm$, are introduced to penalize points that lie outside the ɛ-insensitive region (usually in a linear fashion, as here) and to ensure the optimization problem is feasible (this is similar to the soft margin approach in the standard SVM technique for classification). This constrained optimization problem can be solved using the method of Lagrange multipliers [Reference Shen, Heikman, Moran, Coffie, Zhang, Buttari, Smorchkova, Keller, DenBaars and Mishra5].

For applications to nonlinear transistor modeling, where typically complex relations exist between input and output harmonic components, the linear model in (1) should be replaced by the nonlinear model $\mathop \sum \limits_{i = 1}^n \beta _iK( {{\bf x}_i, \;{\bf x}} ) + b, \;$ where the β _i are model coefficients and K is the kernel function.

Proposed modeling method

As described in previous work [Reference Cai, King, Yu, Liu and Sun27], under certain conditions, the relationship between the incident and scattered waves at all ports and harmonics can be represented by

(3)$$B_{\,pm} = f_{\,pm}( {\overbrace{{A_{qn}}}^{{}}} ) = \mathop \sum \limits_{i = 0}^k \beta _{\,pm, i}K( \overbrace{{A_{qn, i}}}^{{}}, \;\overbrace{{A_{qn}}}^{{}}) + b_{\,pm}$$

where $\overbrace{{A_{qn}}}^{{}}$ are the incident waves, A ₁₁, A ₁₂, …A ₂₁, A ₂₂, …. The describing function f _pm captures how the complex amplitude of the scattered waves, B _pm, depends on the complex amplitude of (potentially all) the A _qn, where p and q are in the output and input port indices, respectively, and m and n are the output and input harmonic indices, respectively. The quantities β _pm,i and b _pm, are the respective model coefficients and bias terms. The topology of the model is described in Fig. 2.

Fig. 2. Block diagram of existing SVR based behavioral model (RF).

From the figure, it can be seen that the model presented in [Reference Cai, King, Yu, Liu and Sun27] employs two SVR machines, one each to represent the real and imaginary parts of the output scattered wave. The real and the imaginary parts of the incident waves, A _qn, are fed as separate inputs to both machines.

This existing model, however, only can be used to predict the behavior at a fixed frequency; it cannot accurately predict across a frequency band, as the model allows for no input frequency terms.

In this work, we extend the model (3) to include, explicitly, the frequency of operation. The model topology, now including this frequency input parameter, is shown in Fig. 3. The topology of the broadband SVR model for dc is given in Fig. 4. Similar to the RF topology, the frequency information is provided to the model as an input.

Fig. 3. Block diagram of the broadband SVR based behavioral model (RF).

Fig. 4. Block diagram of the broadband SVR based behavioral model (DC).

The model topology in Fig. 3 implies that the following adjustment must be made to (3), giving

(4)$$B_{\,pm} = f_{\,pm}( {\overbrace{{A_{qn}}}^{{}}\semicolon \;f} ) = \mathop \sum \limits_{i = 0}^k \beta _{\,pm, i}K( \overbrace{{A_{qn, i}}}^{{}}, \;\overbrace{{A_{qn}}}^{{}}\semicolon \;f) + b_{\,pm}, \;$$

where it can be seen that frequency f is now included implicitly as a parameter. The proposed broadband SVR model is obtained after it has been trained on a set of training data $\left\{ {{f_t},\overbrace {{A_{q{n_t}}}}^{},{\rm{}}{B_{pm}}_t} \right\}$. After that, the model can be used for behavioral prediction.

Model extraction

The four different kernel functions considered for the SVR algorithm are a linear function, a RBF, a sigmoidal function, and a polynomial function.

The details of the nature and implementation of these kernel functions are given in Table 1. As can be seen from the table, the kernel functions contain hyperparameters that must be optimized to obtain the optimal model. More information on this procedure is found in [Reference Cai, Yu, Sun, Chen and King24].

Table 1. Four different kernels.

Explanation, γ: gamma; l: bias coefficient; d: degree.

Model validation

In this section, the proposed model is verified against experimental data to demonstrate the model ability to predict, accurately, the response across a range of frequencies and input power levels. The test bench consists of a Keysight PNA-X combined with a Focus Microwaves active load-pull system. A block diagram of the test bench is shown in Fig. 5 with the corresponding setup picture shown in Fig. 6.

Fig. 5. Block diagram of test bench.

Fig. 6. Experimental test bench setup.

In the first experimental validation, the model ability to predict the response across a range of frequencies is demonstrated. The Wolfspeed CGH40010 device is used in the validation. The device is biased at −3 and 28 V, at gate and drain, respectively. The fundamental scattered wave at port-2, B ₂₁, is taken as the comparison quantity, and 300 load points are used for training, with higher-order impedances set to the matched condition. The input power fixed at +20 dBm. The broadband SVR model is extracted for fundamental frequencies at 1.5, 2, and 2.5 GHz. The prediction performance can be seen from Fig. 7. The model can provide accurate prediction across all three frequencies, with +20 dBm input power. In addition, the corresponding dc behavioral prediction is also examined. The performance is given in Fig. 8. Similar to the RF results, the proposed model provides excellent performance.

Fig. 7. Measured and modelled results for (a) 1.5 GHz, (b) 2.0 GHz, and (c) 2.5 GHz cases with +20 dBm input power.

Fig. 8. Measured and modelled corresponding I_DS results, (a) 1.5 GHz, (b) 2.0 GHz, and (c) 2.5 GHz cases with +20 dBm input power.

Figures 9 and 10 show measured and modelled versions of both the fundamental scattered wave, B ₂₁, and the drain dc current, I_ds, as the load impedance varies, for excitations of +24 and +29 dBm at 2.5 GHz. It is seen that the measured data are in strong agreement with the model.

Fig. 9. Measured and modelled results for P_in,av of (a) +24 dBm, and (b) +29 dBm, with a 2.5 GHz input signal.

Fig. 10. Measured and modelled corresponding I_DS results, for P_in,av of (a) +24 dBm, and (b) +29 dBm, with a 2.5 GHz input signal.

The measured and modelled second harmonic scattered wave, B ₂₂, for excitation of +29 dBm at 2.5 GHz is also provided in Fig. 11. As can be seen from the results, the proposed model shows high harmonic fidelity, a key requirement for modern waveform-based PA design.

Fig. 11. Measured and modelled second harmonic scattered wave, B ₂₂, results for P_in,av =  +29 dBm, with a 2.5 GHz input signal.

Finally, a one-tone power sweep of the DUT is also shown in Fig. 12. In this test, the specific power levels used for model extraction/training, are: −20, −15, 0, +8, +15, +22, +25, and +28 dBm. Once the model is obtained, it is used to predict the behavioral of the DUT with the input power varying from −20 to +30 dBm. Shown are the gain and the output power measurements and corresponding modelled results across an input available power sweep ranging from −20 to +30 dBm at 2 GHz. The proposed model can accurately predict the behavior of the DUT across different input power levels.

Fig. 12. One-tone power sweeps on an unmatched 10-W device placed in a test fixture. Shown are transducer gain, and output power measurements and model results over a sweep ranging from −20 dBm to +30 dBm, with 2 GHz.

Class J PA theory

This section applies the proposed model to a practical engineering design problem. Using the method described in this paper, a model is extracted from a 10 W GaN device, and used for the design of a modern harmonically-tuned class-J PA. In the remainder of this section, we provide a brief overview of the theory of this mode of PA operation.

The class-J mode of PA, as introduced in [Reference Cripps6], relies on a drain voltage waveform that is “engineered” through appropriate load terminations. The bias point of the class-J mode is set according to class-B or “deep” class-AB modes. The ideal voltage and current waveforms of the transistor can be seen from Fig. 13 [Reference Cripps6].

Fig. 13. Ideal class-J normalized voltage and current waveforms.

As described in Table 2 [Reference Wright, Lees, Benedikt, Tasker and Cripps29], the ideal normalized output fundamental and second harmonic components are given, along with the corresponding fundamental and second-harmonic impedances in (5) and (6), where R _L is the load impedance. The third-harmonic component is assumed short in ideal class-J theory.

Table 2. Ideal normalized half-wave voltage component

In order to ensure broadband operation at high efficiency with sufficient output power, compromises are required for setting the design values for the load resistance [Reference Roff, Benedikt, Tasker, Wallis, Hilton, Maclean, Hayes, Uren and Martin30]. Using the proposed broadband SVR model, a matching network is synthesized across the full fundamental and harmonic bands.

(5)$$Z_{\,f_0} = R_L( {1 + j} ), $$

(6)$$Z_{2f_0} = R_L\left({0-j\cdot \displaystyle{{3\pi } \over 8}} \right). $$

Practical class J PA design

A class-J PA circuit was designed and fabricated in this section to validate the proposed model under real-world conditions. To begin the design stage, the proposed model is firstly extracted using load-pull-based data. The same 10 W GaN packaged transistor from the previous section is used, with load-pull data measured at several frequencies between 2.5 and 3.5 GHz, with an input available power of +29 dBm.

The general design procedure for the PA is given in previous work [Reference Wright, Lees, Benedikt, Tasker and Cripps29]. Through analysis of the device IV-characteristics (with due regard to the knee region), the optimum fundamental impedance can be fixed. After tuning the harmonic impedance, a class-J mode PA design can be obtained. For practical implementation of the output matching/waveform engineering network, the simplified real frequency technique is employed to synthesize the broadband design [Reference Yarman31].

Based on a load-pull simulation using the proposed broadband SVR model across the desired frequency band, a target package-plane load impedance trajectory across frequency is determined, as shown in Fig. 14. This impedance is chosen such that it provides a drain efficiency greater than 65% and output power of over 10 W. As can be seen from the results, once the maximum power delivery area and maximum drain efficiency area have been found by the proposed SVM model, a compromise has to be made since these regions are not coincident, in general. As shown in Fig. 14, one such suboptimal overlapping area is identified within which the output power is greater than 40 dBm and the drain efficiency is over 65 %. A matching circuit is designed, targeting this “area” of the load Smith chart, providing a predetermined compromise between PA output power and drain efficiency. The resulting current-generator-plane load impedance trajectory across frequency can also be obtained via de-embedding, as shown in Fig. 15. This allows the designer to verify that the PA operates in the desired mode – class-J, this is the case. The corresponding fundamental and second harmonic load impedance at the external drain package-plane, for which a matching circuit must be synthesized, is given in Fig. 16, along with the fundamental optimal source matching impedance trajectory.

Fig. 14. Compromise area (DE>65% & Pout>10 W) used by designers to choose fundamental load trajectory, from 2.5 GHz to 3.5 GHz, looking at external drain package plane.

Fig. 15. Compromise (Drain Efficiency > 65% & Pout >10 W) target S-parameters line used for designers to choose fundamental and second harmonic load impedances i.e. looking at current generator plane.

Fig. 16. Compromise (drain efficiency>65% and Pout>10 W) target S-parameters trajectory used for designers to choose fundamental and second harmonic load impedances i.e. looking at external drain package plane.

After the matching networks are determined, the PA is fabricated on Rogers 4350B microstrip board. This board has a substrate thickness equal to 0.762 mm and a nominal relative permittivity equal to 3.48. The layout of the PA is shown in Fig. 17. A photograph of the fabricated PA is shown in Fig. 18.

Fig. 17. Actual size diagram of fabricated class-J PA.

Fig. 18. Photograph of fabricated class-J PA.

The performance of the fabricated PA across the targeted frequency band is shown in Fig. 19, along with the simulation results from the proposed broadband SVR model. An excellent match of the output power, efficiency, and gain, between the model simulation results and measurements of the realized class-J PA is seen in the figure. The realized PA provides an average power added efficiency of 66.34%, an across-the-band-average output power of 41.01 dBm, and gain of 12.01 dB.

Fig. 19. Class-J PA measured and modelled from proposed model results.