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A general neurospace mapping technique for microwave device modeling
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 37 (2018)
Abstract
Accurate modeling of nonlinear microwave devices is critical for reliable design of microwave circuit and system. In this paper, a more general neurospace mapping (NeuroSM) method is proposed to fulfill the needs of the increased modeling complexity. The proposed technique retains the capability of the existing dynamic NeuroSM in modifying the dynamic voltage relationship between the coarse model and the desired model. The proposed NeuroSM also considers dynamic current mapping besides voltage mappings. In this way, the proposed NeuroSM generalizes the previously published NeuroSM methods and has the potential to produce a more accurate model of microwave devices with more dynamics and nonlinearity. A new formulation and new sensitivity analysis technique are derived to train the general NeuroSM with dc, small, and largesignal data. A new gradientbased training algorithm is also proposed to speed up the training. The validity and efficiency of the general NeuroSM method are demonstrated through a real 2 × 50 μm GaAs pseudomorphic highelectron mobility transistor (pHEMT) modeling example. The proposed general NeuroSM model can be implemented into circuit simulators conveniently.
Introduction
Microwave transistors are key components in the next generation wireless communication systems [1,2,3,4], such as cognitive multipleinput multipleoutput (MIMO) systems [5,6,7], and cognitive relay network [8, 9]. With the increasing complexity of communication circuit and system structure, designers rely more heavily on computeraided design (CAD) software to achieve efficient design. Microwave device models are essential to CAD software. The accuracy of these models can even decide whether the communication circuit and system design is successful or not. Due to rapid technology development in semiconductor industry, new microwave devices constantly arrive. Models suitable for previous devices may not fit new devices well. There is an ongoing need for new accurate models.
In recent years, neurospace mapping (NeuroSM) technique [10] combining artificial neural networks [11] with space mapping [12] has been recognized in microwave device modeling with the advantages of good efficiency and accuracy. In NeuroSM, neural networks are used to automatically map and modify an existing equivalent circuit model also called coarse model to a desired/accurate model through a process named training. In order to fulfill the needs of the increased modeling complexity and the industry’s increasing need for tighter accuracy, several improvements on the basis of [10] were subsequently studied to enhance the modeling accuracy and efficiency, such as NeuroSM with the output mapping [13], dynamic NeuroSM [14], and analytical NeuroSM with sensitivity analysis [15]. NeuroSM with the output mapping [13] was introduced, through incorporation of a new output/current mapping, for modeling of microwave devices. Compared to the NeuroSM presented in [10], NeuroSM with the output mapping is more suitable for modeling nonlinear devices with more nonlinearity due to the additional and useful degrees of freedom from the output mapping neural network. In order to accurately model nonlinear devices which have higher order dynamic effects (e.g., capacitive effect or nonquasistatic effect) than that of the coarse model, dynamic NeuroSM was introduced [14]. However, when the modeling devices have both more nonlinearity and high order dynamics, in such case, even though existing NeuroSM [13, 14] is used to map the coarse model towards the device data, the match between the trained NeuroSM models and the device data may be still not good enough. More effective NeuroSM methods need to be investigated to overcome the accuracy limit of the NeuroSM presented in [13, 14].
In this paper, we propose a more generalized NeuroSM approach including not only static mapping but also dynamic mapping, and considering both voltage mapping and current mapping for the first time. This paper is a further expansion of the work in [13, 14]. Compared to [13] where only static mapping is used, the proposed technique is more suitable for modeling nonlinear devices with higher order dynamic effects and nonquasistatic effect that may be missing in the coarse model due to inclusion of dynamic mapping. Compared to [14], the general NeuroSM considers not only input voltage mapping, but also output current mapping, further refining the existing coarse model. In this way, well trained general NeuroSM model can represent the dynamic behavior and largesignal nonlinearity of the microwave devices more accurately than the coarse model, NeuroSM model with the output mapping [13], as well as dynamic NeuroSM model [14]. The modeling results of a real 2 × 50 μm GaAs pseudomorphic highelectron mobility transistor (pHEMT) demonstrate the correctness and validity of the proposed general NeuroSM technique.
Concept of the general NeuroSM model
Suppose the existing equivalent circuit model is a rough approximation of the behavior of the microwave device. We name this existing model as the coarse model. Let the desired model that accurately matches the device data be called the fine model. Just take field effect transistor (FET) modeling as an example, let the gate and drain voltages and currents of the coarse model be defined as v_{ c } = [v_{c1}, v_{c2}]^{T} and i_{ c } = [i_{c1}, i_{c2}]^{T}, respectively. Let the terminal voltages and currents of the fine model as v_{ f } = [v_{f1}, v_{f2}]^{T} and i_{ f } = [i_{f1}, i_{f2}]^{T}, respectively.
Suppose the total number of voltage delay buffers at gate and drain be the same and both equal to N_{ v }. Let τ be the time delay parameter. To represent timedomain behavior, the time parameter t is introduced. Figure 1 illustrates the signal flow of the general NeuroSM model. At first, the present voltages of the fine model v_{ f }(t) as well as their historyv_{ f }(t − τ), v_{ f }(t − 2τ), …, and v_{ f }(t − N_{ v }τ) are mapped into the coarse model voltages v_{ c }(t). Because the formula of the mapping is unknown and usually nonlinear, a neural network is used to learn and represent the mapping. While the NeuroSM presented in [10] uses a static neural network such as multilayer perceptron (MLP), we propose to use a time delay neural network (TDNN) to map the coarse model to fine model. In functional form, v_{ c }(t) can be described as
where f_{ANN} represents the input/voltage mapping neural network, and w_{ 1 } is a vector containing all the weights of the input mapping neural network. As seen from Eq. (1), voltages at gate and drain of the coarse model depend on not only the present voltages of the fine model, but also their history signals making the proposed technique more suitable for modeling the dynamic behavior of the nonlinear devices. Then, after the coarse model computation, the coarse model currents i_{ c }(t) can be obtained. Suppose the total number of current delay buffers at gate and drain be the same and both equal to N_{c}. At last, i_{ c }(t) and their history i_{ c }(t − τ), …, i_{ c }(t − N_{ c }τ) as well as the present voltages of the fine model v_{ f }(t) are mapped by another TDNN to the external currents as
where h_{ANN} represents the output/current neural network, and vector w_{2} contains all the output mapping neural network weights. Compared to [14], the new output neural network mapping further refines the coarse model current signals to produce the fine model outputs. The combined dynamic voltage mapping neural network, coarse model, and dynamic current mapping neural network is called the general NeuroSM model.
The proposed general NeuroSM is more general than NeuroSM technique presented in [10, 13, 14]. While N_{ v } = 0, then the general NeuroSM model without the output mapping is static NeuroSM model [10]. While N_{ v } = 0 and N_{ c } = 0, then the general NeuroSM model belongs to the NeuroSM model with the output mapping [13]. While N_{ v } > 0, then the general NeuroSM model without the output mapping is the dynamic NeuroSM model [14]. In this way, the proposed general NeuroSM generalizes the previously published NeuroSM technique. Furthermore, while N_{ v }> 0 and N_{ c }> 0, a new NeuroSM technique is presented for the first time. Compared to the NeuroSM introduced in [10, 13, 14], the new NeuroSM is more suitable for modeling the microwave devices with high order dynamics and nonlinearity due to inclusion of dynamic mapping as well as current mapping.
Proposed analytical formulation of the general NeuroSM model for training
The general NeuroSM model will not be accurate unless the dynamic voltage and dynamic current mapping neural networks are trained suitable. In order to train the general NeuroSM efficiently with typical types of transistor modeling data, the relationship between the dynamic voltage and current mapping neural networks with typical types of transistor data, such as DC, biasdependent S parameter, and largesignal harmonic data need to be derived.
In the DC case, present voltage signals of the fine model v_{ f }(t) as well as its history, i.e.,v_{ f }(t − τ), …, and v_{ f }(t − N_{ v }τ) are all equal and defined as V_{f, DC}. Similarly, present current signals of the coarse model i_{ c }(t) as well as its history, i.e.,i_{ c }(t − τ), …, and i_{ c }(t − N_{ c }τ) are all equal and defined as I_{c, DC}. The response of the general NeuroSM model at V_{f, DC}can be generally described as
where
The smallsignal S parameter of the general NeuroSM model can be calculated by transforming its Y parameters Y_{ f }, which can be obtained by mapping Y parameters of the coarse model Y_{ c }. In functional form, Y_{ f } can be described as
where
where the firstorder derivatives of f_{ANN} and h_{ANN} can be obtained at the bias V_{f, Bias} using adjoint neural network method [15]. Superscript k and l represent the index of voltage and current delay buffers, respectively. Equation (5) includes two parts. The first part is in the form of multiplications of three matrices, which are defined as the output/current Ymapping matrix, i.e., the sum of products of e^{−jωlτ} and ∂h_{ANN}/∂i_{ c }, Y parameter matrix of the coarse model Y_{ c }, as well as the input/voltage Ymapping matrix, i.e., the sum of products of e^{−jωkτ} and ∂f_{ANN}/∂v_{ f }. The other part is the sensitivity matrix of h_{ ANN }. Equation (5) is more general than formulas of smallsignal Y parameter of the NeuroSM models in [10, 13, 14] due to the consideration of the new effects of current mappings and dynamic mappings. For largesignal case, we need to derive the relationship between HB computation and dynamic voltage and current mapping neural networks so that model training can be performed with harmonic data. Let the harmonic current of the general NeuroSM model and coarse model at a generic harmonic frequency ω_{ k } be I_{ f }(ω_{ k }) and I_{ c }(ω_{ k }), respectively. The I_{ f }(ω_{ k }) can be evaluated as
where
where the subscript k represents the index of the harmonic frequency, k = 0, 1, 2, …, N_{ H }, where N_{ H } is the number of harmonics considered in HB simulation. N_{ T } is the number of time sampling points, W_{ N }(n, k) is the Fourier coefficient for the nth time sample and the kth harmonic, superscript * denotes complex conjugate, and m represents the index of voltage delay buffers, m = 0, 1,…, N_{ v, }. As seen from (7)~(9), apart from changing the nonlinearity of the coarse model, dynamic voltage and current neural network mappings can also change the dynamic order so that the proposed general NeuroSM has the potential to model the microwave devices with high order dynamics and nonlinearity.
Sensitivity analysis of the general NeuroSM model with respect to mapping neural network weights
Let the number of hidden neurons of the dynamic voltage and current mapping neural networks be N_{hv} and N_{hc}, respectively. Let generic symbols w_{1, i} (i = 1, 2, …, N_{hv}) and w_{2, i} (i = 1, 2, …, N_{hc}) be internal weights of the voltage and current mapping neural network, respectively. w_{1, i} and w_{2, i} are the ith component of vectors w_{1} and w_{2}, respectively. In order to train the general NeuroSM efficiently, gradient information provided by sensitivities of the model with respect to w_{1, i} and w_{2, i} is needed [16].
(1) DC sensitivity: let the DC output at gate and drain of the general NeuroSM model be I_{f, DC}. The sensitivities of I_{f, DC} with respect to w_{1, i} and w_{2, i} are described in functional form as
where \( {\boldsymbol{G}}_c={\left(\partial {\boldsymbol{I}}_{c,\mathrm{DC}}^T/\partial {\boldsymbol{V}}_{c,\mathrm{DC}}\right)}^T \) is the DC conductance matrix of the existing coarse model, and the firstorder derivatives ∂f_{ANN}/∂w_{1, i} and ∂h_{ANN}/∂w_{2, i} can be calculated by neural network backpropagation [17].
(2) S parameter sensitivity: S parameter sensitivity can be obtained by converting its Y parameter sensitivity. The smallsignal Y parameter sensitivities of the general NeuroSM model with respect to w_{1, i} and w_{2, i} are shown in Eqs. (12) and (13), respectively. These two equations can be obtain by differentiating (5) with respect to w_{1, i} and w_{2, i}, respectively.
where the secondorder derivative of the dynamic voltage and current mapping neural networks f_{ANN} and h_{ANN}, which are the differentiation of the Jacobian matrix \( \partial {\boldsymbol{f}}_{\mathrm{ANN}}^T/\partial {\boldsymbol{i}}_c\left(t l\tau \right) \) and \( \partial {\boldsymbol{f}}_{\mathrm{ANN}}^T/\partial {\boldsymbol{v}}_f\left(t k\tau \right) \) with respect to w_{1, i} and w_{2, i}, can be obtained by the adjoint neural network backpropagation [17], respectively.
(3) HB sensitivity: the sensitivities of the largesignal harmonic current of the general NeuroSM model with respect to w_{1, i} and w_{2, i} at a generic harmonic frequency ω_{ k }, k = 0, 1, 2, …, N_{ H } can be described in functional form as
where G_{ c }(t_{ n }) at the mapped voltage of coarse model v_{c}(t_{ n }) is the nonlinear conductance matrix of the existing coarse model at time point t_{n}.
Sensitivity analysis of the general NeuroSM model with respect to coarse model parameters
Let x be a generic variable in the coarse model. In case the coarse model parameter needs to be treated as a variable in circuit optimization, it is useful to obtain the sensitivity for DC, biasdependent S parameter, and largesignal HB responses of the general NeuroSM model due to changes in the generic optimization variable x.
(1) DC sensitivity: the sensitivity of I_{f, DC} with respect to x is derived as
where \( \partial {\boldsymbol{I}}_{c,\mathrm{DC}}^T/\partial x \) is the DC current response due to changes in coarse model variable x evaluated at the mapped bias V_{c, DC}.
(2) S parameter sensitivity: S parameter sensitivity with respect to coarse model variable x can also be calculated by converting its Y parameter sensitivity. The Y parameter sensitivity is shown as
where ∂Y_{ c }/∂x is the sensitivity for Y parameter of the coarse model due to changes in x. ∂i_{ cr }/∂x, r = 1, 2 is the derivative of coarse model current with respect to x, which can be calculated by coarse model sensitivity analysis.
(3) HB sensitivity: the sensitivity of the harmonic current of the general NeuroSM model with respect to x at a generic harmonic frequency ω_{ k }, k = 0, 1, …, N_{ H } is shown in Eq. (18), where ∂i_{ c }(t_{ n })/∂x is the sensitivity of the nonlinear current of the coarse model with respect to x at time sample t_{ n }.
Proposed training algorithm for the general NeuroSM model
Training is the key step to determine the general NeuroSM model. The model development process needs two phases: initial training and formal training.

A.
Initial training
Before the nonlinear device data from simulation or measurement is used for formal training, the general NeuroSM model is first initialized to be equal to the original coarse model. In such case, the dynamic voltage and current neural networks are initialized to learn unit mappings, i.e., to learn the relationships v_{c1}(t) = v_{f1}(t), v_{c2}(t) = v_{f2}(t), i_{c1}(t) = i_{f1}(t), and i_{c2}(t) = i_{f2}(t) in the entire operation range of the nonlinear device.

B.
Formal training
In this phase, the weights of dynamic voltage and current mapping neural networks, i.e., w_{ 1 } and w_{ 2 }, are trained such that the overall training error of the general NeuroSM model can be reduced to satisfy the specifications. The overall training error for combined DC, smallsignal S parameter, and largesignal HB training is defined as the total difference between all nonlinear device data and the general NeuroSM model as:
where I(.), S(.), and HB(.) are the DC, biasdependent S parameter, and HB responses of the general NeuroSM model, respectively. Take FET modeling as an example, vector I(.) contains gate and drain current I_{f1} and I_{f2}, which can be computed by Eq. (3). Vector S(.) is achieved from the Y matrix defined by Eq. (5). HB responses of the general NeuroSM model, i.e., HB(.) can be calculated by Eq. (7). I_{ D }, S_{ D }, and HB_{ D } represent the DC current, smallsignal S parameter, and largesignal HB responses of the modeling device, respectively. The subscript k \( \left(k=1,2,\dots, {N}_{V_{f2}}\right) \), l \( \left(l=1,2,\dots, {N}_{V_{f1}}\right) \), j (j = 1, 2, …, N_{freq}), m (m = 1, 2, …, N_{ H }), and n (n = 1, 2, …, N_{ P }) denote the indices of V_{f2}, V_{f1}, frequency, harmonic frequency, and input power level, respectively. \( {N}_{V_{f1}} \), \( {N}_{V_{f2}} \), N_{freq}, N_{ H }, and N_{ P } are the total number of V_{f1}, V_{f2},frequency, harmonic frequency, and input power level, respectively. Diagonal matrices A, B, and C contain all the scaling factors, which are defined as the inverse of the minimumtomaximum range of the I_{ D } data, S_{ D } data, and HB_{ D } data, respectively. The training error calculation of the general NeuroSM model for combined DC and S parameter training as well as HB training further illustrates in Fig. 2. Figure 2a, b is error calculation for combined dc and smallsignal S parameter training as well as largesignal HB training, respectively.
The objective of the model training is to minimize the error E defined in (19) by optimizing w_{ 1 } and w_{ 2 }. In general, gradientbased training algorithm is used. After training, the general NeuroSM model with appropriate hidden neurons and delay buffers can accurately represent the nonlinear behavior of the modeling device.
Discussion
The proposed NeuroSM model, after being trained for a specific range, is very good at representing the nonlinear behavior of the microwave device within the training region. However, when we use model in a wider range than the training range, inappropriate derivative information of the model outside the training range may mislead the iterative process into slow convergence or even divergence during largesignal simulation. One possible way to solve the divergence problem is to use appropriate extrapolation technique. For general NeuroSM technique, a simple and effective extrapolation technique is used to improve the convergence of the model [18].
For simplification, the proposed general NeuroSM technique is formulated for 2port fieldeffect transistor (FET) modeling. This approach can be further extended to nport network, where all the notations and equations are extended accordingly. After the generalization, the proposed general NeuroSM technique has the potential to be used for developing models of microwave devices with trapping effect.
The format of the general NeuroSM model presented so far is to map the voltage input signals between the coarse and fine models. Hence, our approach presented so far is applicable to modeling voltage controlled devices, such as FET and HEMT. It is possible to extend the method to a mixed input mapping case, where the dynamic input mappings are for a mixture of port voltage and current signals. In that way, our approach can be extended to modeling current controlled devices, such as HBT.
The frequency limit of the proposed general NeuroSM model depends on the frequency limit of training data. For example, if the frequency in the training data extends to millimeter wave bands, the proposed general mapping will be even more important because of the need of capacitive effects, nonquasistatic effects, and nonlinear effects in the model. In this case, more hidden neurons and time delay buffers maybe needed to guarantee the accuracy of the proposed general NeuroSM model.
A pHEMT modeling using the proposed general NeuroSM method
This example illustrates the use of the general NeuroSM for modeling of a real 2 × 50 μm GaAs pHEMT device. The training and test data is obtained from measurement. An enhanced Angelov model including a thermal subcircuit to model the selfheating effect of the device proposed in [19] is used as the existing coarse model. Even though parameters in enhanced Angelov model are extracted as much as possible, there are still distinct differences between the model and measured data. Thus, NeuroSM is used to bridge the gap between the coarse model and measured data. We then apply the previously published NeuroSM technique such as NeuroSM with the output mapping [13] and dynamic NeuroSM [14] to get more accurate models. After training, the accuracy of the two NeuroSM models is clearly improved compared to that of the coarse model, as shown in Fig. 3. However, the previous NeuroSM techniques at their best are still insufficient to achieve the desired accuracy. Then, our proposed general NeuroSM is used to get a more accurate model.
Training was firstly done in NeuroModelerPlus [20] using DC and biasdependent S parameter data for 400 iterations. Then, training refinement was done using combined DC, biasdependent S parameter, and HB data at 189 different biases for 3600 iterations. Harmonic data used for HB training was measured at 7.5 GHz fundamental frequency and different input power levels (− 10~ 3 dBm). Time delay parameters are both 0.008 ns. The number of hidden neurons for both voltage and current mapping neural networks is 30.
Results
After training, we compared the DC, biasdependent S parameter, and largesignal HB responses of the pHEMT device with those computed from the coarse model, NeuroSM with the output mapping [13], dynamic NeuroSM [14] with 5 delay buffers and 30 hidden neurons, and the proposed general NeuroSM model with 5 delay buffers and 30 hidden neurons both for dynamic voltage and current mapping neural networks as shown in Fig. 3. In Fig. 3a, b–e, f represent the comparisons of dc, S parameter at two test bias points (0.7 V, 2.4 V) and (0.3 V, 5.2 V), as well as HB responses at different input power levels (− 10~ 3 dBm), respectively. As observed from Fig. 3, the responses computed from the proposed general NeuroSM are closest to the data among all the four models in this comparison. We obtain further improvement in model accuracy using general NeuroSM technique because additional and useful degrees of freedom provided by the new dynamic current mappings at the gate and the drain in the general model. The increased accuracy of the general NeuroSM model helps to improve the accuracy of circuit and system simulation, such as simulation to predict power performance and linearity of highfrequency PA designs.
There are two important factors that impact the accuracy of the dynamic NeuroSM model and the proposed general NeuroSM model, i.e., number of hidden neurons and delay buffers. To show the results further, we compared the training and test error of the dynamic NeuroSM and general NeuroSM with different delay buffers and hidden neurons as shown in Table 1. As seen in Table 1, general NeuroSM with 30 hidden neurons and 5 delay buffers both for dynamic voltage and current mapping neural networks are suitable for this example.
The proposed general NeuroSM model can be conveniently implemented into the existing circuit simulators such as Keysight ADS for highlevel circuit and system design. Figure 4 shows the proposed general NeuroSM model structure in ADS. The time delay parameter is 0.08 ns. In this figure, the dynamic voltage mapping neural networks are embedded as the functions in two 7port symbolically defined devices (SDDs), i.e., SDD7P1, and SDD7P2. Similarly, the dynamic current mapping neural networks are embedded as the functions in two 9port SDDs, i.e., SDD9P1 and SDD9P2. Time delay voltage and current signals can be obtained using voltage controlled voltage sources with delay parameters, i.e., SRC1~SRC8. After implementing the general NeuroSM model into ADS, we have also compared simulation speed between coarse model, dynamic NeuroSM, and the proposed general NeuroSM model on an Intel i53230M 2.6 GHz computer as shown in Table 2. The simulation was performed by Monte Carlo analysis of 200 HB simulations. As seen in Table 2, the simulation time is 48.32 s using coarse model, compared to 57.17 s using general NeuroSM, showing that the simulation speed of the proposed general NeuroSM is acceptable in view of its good accuracy.
Conclusions
This paper has presented a general NeuroSM technique for nonlinear device modeling. By modifying the dynamic current and dynamic voltage relationships in the existing coarse model, the proposed general NeuroSM model can exceed the accuracy limit over the coarse model, the NeuroSM model with the output mapping, and the dynamic NeuroSM model. Compared to previously published NeuroSM, the proposed general NeuroSM has demonstrated much improved performance in terms of accuracy by a pHEMT modeling example. The general NeuroSM model can be applied to microwave circuit and system design.
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Acknowledgements
The authors would like to thank Prof. Q. J. Zhang at Carleton University, Ottawa, ON, Canada, for valuable discussions and insights throughout this work.
Funding
This work is supported by the Fundamental Research Funds for Universities in Tianjin (No. 2016CJ13), partly supported by the Key project of Tianjin Natural Science Foundation (No. 16JCZDJC38600), National Natural Science Foundation of China (No. 61601494, 61602346), and the Research Forums Cooperation Project of ZTE Corporation (2016ZTE0409).
Availability of data and materials
The training and test data of the microwave transistor is obtained from measurement and can be shared if it is necessary.
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The authors have contributed jointly to all parts on the preparation of this manuscript. LZ (first author) and JZ contributed to the structure and sensitivity analysis of the general NeuroSM model. LZ (third author), WL and LP contributed to the training algorithm development. HW and DL contributed to the analysis of simulation results. All authors read and approved the final manuscript.
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Correspondence to Lin Zhu.
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Keywords
 Artificial neural network
 Space mapping
 NeuroSM
 Microwave device modeling