CONSTRUCTION OF BEHAVIOURAL MODELS FOR MICROWAVE DEVICES FROM TIME-DOMAIN LARGE-SIGNAL MEASUREMENTS TO SPEED-UP HIGH-LEVEL DESIGN SIMULATIONS

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1 CONSTRUCTION OF BEHAVIOURAL MODELS FOR MICROWAVE DEVICES FROM TIME-DOMAIN LARGE-SIGNAL MEASUREMENTS TO SPEED-UP HIGH-LEVEL DESIGN SIMULATIONS D. Schreurs, J. Wood, N. Tufillaro, L. Barford, and D.E. Root K.U.Leuven, Div. ESAT-TELEMIC, Kasteelpark Arenberg 1, B-31 Leuven-Heverlee, Belgium Fax: , Phone: Agilent Technologies Microwave Technology Center, Santa Rosa, CA 9543, USA Agilent Laboratories, Palo Alto, CA 9434, USA SHORT TITLE: NONLINEAR BEHAVIORAL MODELS FROM TIME DATA ABSTRACT We developed behavioural models for microwave devices from time-domain large-signal measurements. For the presented examples, the dynamical model is defined by representing the terminal currents as function of the embedded voltages. When using this type of models as building blocks of higher level designs, the simulation speed is significantly improved. KEYWORDS microwave devices, large-signal measurements, device modelling, time series analysis, optimisation I. INTRODUCTION The recent availability of vectorial large-signal measurement setups [1-4] makes it possible to develop new measurement based non-linear modelling approaches that are not limited to the use of only static (DC) and S-parameter data. Some examples of such modelling techniques include parametric equivalent-circuit model extraction [5-7], and black-box model identifications in the frequency domain [7-8]. In this work, we develop a time domain black-box modelling procedure, that is based on non-linear system identification, using techniques developed in non-linear time-series analysis (NLTSA) [9-11]. One advantage of this technique is that the resulting model should be transportable: in other words, usable in a range of environments, and not restricted to a small domain of applicability, for example, a single bias condition. A further advantage 1

2 of this time-domain technique is that it is not restricted to the modelling of only weakly non-linear phenomena, unlike some frequency-domain methods such as Volterra Series analysis. The model is described directly by timedifferential equations that are reconstructed from measured data. By this means, all the observable dynamics of the device are determined. Finally, this black-box modelling principle is applicable to any device type, regardless its complexity, because no physical preknowledge is required. Since only the observable dynamics are captured, the model size of especially circuits will in general by significantly smaller compared to the case where these circuits are represented by separate models for each of the constituting building blocks. This enables the construction of a compact, accurate, and transportable dynamical model [12,13]. In the next section, we describe in detail the developed modelling procedure. The different steps are clarified by applying them to a microwave transistor, a HEMT, and an amplifier MMIC. The obtained models are validated by examining the DC, small-signal and large-signal behaviour (Section III). Finally, we will give an indication about the gain in simulation time when using this type of model representation. II. METHODOLOGY To introduce the principle, we take a HEMT as an example. Its simplified large-signal equivalent scheme is shown in Figure 1. We neglect the non-quasi-static effects and the extrinsic parasitic network to simplify the following equations. The terminal currents and can be expressed by! " #$ % & (1) (2) ' $( with and the terminal voltages, corresponding to and respectively. By taking the partial derivatives of the charges ) and ) towards the voltages and, we obtain: #* ( ',+- #$ &. /+ 1 $ % '! #,+ 1 /+ 1 (3) (4) or in other words, the terminal currents are a function of the terminal voltages and the first derivatives of the terminal voltages: 2 %! * ' -6 * % -6 ' 7 3(84 * ' -6 * % -6 ' 7 (5) (6) 2

3 E The generalised form of this equation for a two-port device is: 9 :(;<=?> 9D ; <=CB&D.;<=CFE B*: ; <=CFE B'D; <=CHG B*:%;<=C I!I!I CJE 9 : ; <=C1E 9D; <=C!I I!I ; <=CB&D.;<=C B*: ; <=C E B'D; <=C E B*:%;<=C I!I!I C G 9 : ; <=C E 9D; <=C!I I!I E K (7) (8) The objective of the modelling procedure is to find the functional :(;I = Since the black-box modelling approach supposes that no physical background information is available, we first have to determine the independent variables of the : ;LIM= D ;LIM=, which are the state variables. In our method, these state variables are estimated from time domain large-signal measurements. The model is hence built from time domain data, obtained by performing vectorial large-signal measurements using the Non-linear Network Measurement System (NNMS) [4]. At the start of the modelling process, operating bounds for the model are established by defining the minimum and maximum values of the state variables. These bounds define the operation region within the state space for which the model is to be developed and used. To enable practical identification of the device dynamics, the measured time domain data need to sample this operation region efficiently. However, since the actual state variables are unknown at the start, we begin by defining the B*: B&D minimum and maximum and voltages. If it turns out after the embedding process (see further) that there is B*: E B'D not yet sufficient variation in the other state variables (,,... ), it might be necessary to iterate between this measurement and the embedding part. The advantage of applying a large-signal excitation to the device instead of the conventional small-signal excitation (bias-dependent S-parameters) is that the device is characterised under closer-to-use conditions. The instantaneous voltage trajectory can sample an extensive region of the device s ;#B : CB D = plane, which is otherwise unreachable by conventional measurements. This is illustrated by Figure 2, where the time domain waveform of B D ;<= is plotted as function of the B*:(;<= time domain waveform. In this example, the HEMT device was excited by a single tone signal at the gate and by a second periodic signal at a different fundamental frequency at the drain. The ;#B$:%CB&D = area is divided in a grid with 5 mv x 1 mv sections. The purpose of the data gathering process is to have a minimum number of time domain data in each of the sections. For the given example, we notice that one largesignal measurement crosses 28 of these sections. Hence, all the practical ;#B : CB D = area can be sufficiently covered with a minimum number of vectorial large-signal measurements. This can be achieved by suitable variation of the parameters of the measurement system: DC bias, input powers, input frequencies, etc. [5]. It is important to point out that the proposed procedure in this work does not require multiple trajectories through exactly the same ;NB*:%CB&D2= grid points, which is a requirement for the direct extraction method [5,7]. 3

4 P W V T V P U T P U Q U Q U U T T We performed similar measurements on an amplifier MMIC. In this case, we measured at the fundamental frequencies covering the amplifier s bandwidth and varied the input powers, because the amplifier s optimal operation fixes the DC bias condition. The next step consists in determining the level of the device s dynamics, which corresponds to the orders of voltage and current time derivatives, that have to be taken into account in Equations (7) and (8). This can be accomplished by the so-called embedding technique, based on the false nearest neighbour principle [14]. To illustrate the idea, we plotted for a HEMT the time domain waveform of O!P Q RS as a function of the time-domain waveform of T*U QRS (Figure 3). We note that O!P.QRS is not a single-valued function of T*U QRS, which indicates that T*U is not the only independent variable. If we would plot, in case of the idealised HEMT (Figure 1), this O!P Q RS characteristic in a five-dimensional space, by which the different dimensions are defined by O P T U T,, P T, V U, and V P, we would obtain a single-valued function. Hence, the basic principle of the embedding technique is to unfold the characteristics of the dependent variables QO variables X T$U W T T*U W U%W O P S in an increasing dimensional space by increasing the number of independent W*Y T*U WHY W!Z!Z Z [ until a single-valued function for each current is obtained. This principle is based on a theorem of Takens, with an extension to the driven case by Stark [15]. It implies that these embedded models can be faithful to the dynamics of the original system. In particular, deterministic prediction is possible from an embedded model that will mimic the actual dynamics. The results of applying this technique to actual measured data of the studied HEMT and amplifier MMIC are displayed in Tables 1 and 2. The first column represents the dimensionality of the state space. The numbers in the second and third columns are, respectively for O and OP, the normalised number of data points that have false neighbours, which is an indication to check whether a single-valued function can be obtained for that dimensionality of the state space. The last column lists the state variables that have been taken into account for this calculation. When performing measurements at fundamental frequencies up to 5 GHz, we found that it is necessary to include T Y U state variables up to for O and O P T in case of the HEMT and up to V U for O and V MMIC. P for O P in case of the amplifier Finally, the functional relationships \ Z S and \(P Q Z S of Equations (7) and (8) are determined by fitting the measured time domain terminal currents, using the independent variables determined in the preceding step. In this work, we use multivariate polynomials to describe \ Z S and \(P Q Z S, but other types of fitting functions can also be used. A 4

5 least-squares fitting procedure is used to obtain the multivariate polynomial coefficients. III. RESULTS We implemented the obtained behavioural models in the Agilent ADS microwave circuit simulator by means of a symbolically defined device (SDD). The SDD can determine the time-derivatives of the terminal voltages at each time-step in the simulation, hence enabling the functional forms for the currents to be evaluated. Model validation starts by testing limiting cases, such as DC or small-signal operation. We note that both DC and small-signal (= S-parameters) data were not explicitly used to construct the model. Further validation is achieved by comparing the simulated model performance with measured large-signal data. The DC I-V characteristics predicted by the HEMT model are shown in Figure 4. The DC behaviour of the model arises from effectively setting all the derivative terms to zero in the functional equations for ] ^%_L`Ma and ](b _` a. The resulting I-V curves are then determined by the static non-linearities in the functions of c$^ and c&b. The Figure shows that the dynamical HEMT model predicts well the static behaviour of the HEMT in the region of operation. Next, the HEMT behavioural model was used to generate the small-signal S-parameters, as function of the applied DC bias at several frequencies. The HEMT small-signal equivalent circuit parameters were then extracted using a typical equivalent circuit [16]. In Figures 5 and 6, we show two examples of the small-signal equivalent circuit parameters: the transconductance d e and the total gate capacitance fhg. The parameters were extracted at 2 GHz, which is well above the frequencies at which the behavioural model time domain data were measured. The extracted parameters display the general trends of the expected variations with the applied bias as described by the physics of the HEMT. Some capacitive elements, for example f-ij, are less well determined. One of the reasons could be the rather low frequencies that were used in the measurements: the fundamental frequency is less than 5 GHz due to the current bandwidth limitation of the NNMS. This low frequency means that the capacitive component of k b is significantly smaller than the in-phase current contributions, and hence that the extraction of this capacitive component is more difficult. Finally, the model was validated using large-signal measurements. We used the behavioural model to predict the time-dependent output currents k ^ _ la and kb._ la as functions of the drive voltages c*^%_la and c&b._la and compared the predictions with measured values. The simulations were carried out at the same excitation conditions as the measurements.the input signal was a two-tone excitation, with frequencies of 4. GHz and 4.5 GHz. Figure 7 presents the excellent agreement between the large-signal simulation and vector-corrected time-domain measurements that 5

6 can be obtained with this new modelling technique. We performed similar validations on the behavioural model we obtained for the amplifier MMIC. Figure 8 shows the very good agreement between the large-signal simulation and corresponding measurements under one-tone excitation. The proposed modelling method is based on large-signal time domain data of the device-under-test. These data can be obtained by either performing vectorial large-signal measurements or by simulating a conventional model of the device using harmonic-balance or a time-domain analysis. Here the terminal voltages and currents, and their higher order time derivatives, are determined directly in the simulator. The excitation design and model generation process follow the same principles as for the models produced from measured data. In this case too, the purpose is to construct a lower-dimensional behavioural model of the device of which the validity range is determined by the observable dynamics. The advantage of such lower-dimensional behavioural models in especially the case of circuits is that the simulation time is significantly reduced compared with the simulation of the full transistor-level representation of the circuit. The gain in simulation speed depends on the circuit s complexity and the type of simulations, but our preliminary results show over tenfold reduction in simulation time. This is illustrated by Figure 9, where we compare the simulation results of the behavioural model of an amplifier MMIC with the results of the corresponding full transistor-level model representation. It has to be noted that there were no iterations during the behavioural model creation process, which implies that there is still room for accuracy improvement. This one-cycle model generation took about two hours, of which the data gathering step, consisting of simulating the transistor-level model of this MMIC, consumed 75 % of the time, the embedding and function fitting took 2 % and the model implementation 5 %. The particular power sweep shown on Figure 9 took 14 sec. in case of the transistor-level model, whereas the behavioural model based simulation only took 14 sec. Such an improvement in simulation speed is of great importance when simulating high-level designs, such as (sub)systems, which consist of ICs as building blocks. CONCLUSIONS We have presented a methodology for developing time-domain black-box models for non-linear microwave devices, directly from vectorial large-signal measurements, or simulated data. The advantages of this method are 6

7 that it is not restricted to weakly non-linear systems, and that the dynamics of the device are determined directly from the time-series data, resulting in a compact, accurate and transportable model. The resulting model of the HEMT device shows excellent prediction of large-signal performance, and displays physically realistic behaviour under limiting vases of DC and small-signal (linear) conditions. Moreover, we showed that this method is not only applicable to microwave transistors, but also to ICs, because no physical preknowledge is required. Since the observable dynamics are determined directly from the large-signal time-series data of the IC, the model does not need to include explicitly the internal, unobservable dynamics of the individual transistors in the IC. This results in a compact model that simulates the IC behaviour accurately and quickly. ACKNOWLEDGEMENTS D. Schreurs is supported by the Fund for Scientific Research-Flanders as a post-doctoral fellow. This work has been performed in the framework of a visiting scientist position held by D. Schreurs at the Agilent Technologies Microwave Technology Center. We thank the management of Agilent Technologies, Inc. for their support of this work. 7

8 REFERENCES [1] F. van Raay and G. Kompa, A new on-wafer large-signal waveform measurement system with 4 GHz harmonic bandwidth, IEEE MTT-S Int. Microwave Symp. Digest, 1992, pp [2] M. Demmler et al., A vector corrected high power on-wafer measurement system with a frequency range for the higher harmonics up to 4 GHz, Proc. 24th European Microwave Conference, 1994, pp [3] C. Wei et al., Waveform characterization of microwave power heterojunction bipolar transistors, IEEE MTT-S Int. Microwave Symp. Digest, 1995, pp [4] J. Verspecht et al., Accurate on wafer measurement of phase and amplitude of the spectral components of incident and scattered voltage waves at the signal ports of a nonlinear microwave device, IEEE MTT-S Int. Microwave Symp. Digest, 1995, pp [5] D. Schreurs, Overview of non-linear device modelling methods based on vectorial large-signal measurements, Proc. European Gallium Arsenide and related III-V compounds Application Symp., 1999, pp [6] M. Curras-Francos et al., Direct extraction of nonlinear FET I-V functions from time domain large signal measurements, Electronics Letters 34 (1998) pp [7] D. Schreurs and J. Verspecht, Large-signal modelling and measuring go hand-in-hand: accurate alternatives to indirect S-parameter methods, International Journal of RF and Microwave Computer-Aided Engineering 1 (2), pp [8] J. Verspecht et al., System level simulation benefits from frequency domain behavioral models of mixers and amplifiers, Proc. European Microwave Conference, 1999, pp [9] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, [1] J. Hellerstein, An introduction to modeling dynamic behavior with time series analysis, Performance Evaluation of Computers and Communication Systems, Springer-Verlag, 1993, pp [11] M. Casdagli, A dynamical systems approach to modeling input-output systems, Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity, Proc. Vol. XII, eds. M.Casdagli and S. Eubank, Addison Wesley, [12] D. Walker and N. Tufillaro, Phase-space reconstruction using input-output time series data, Phys. Rev. E 6 (1999), pp [13] D. Walker et al., Constructing transportable behavioral models for nonlinear electronic devices, Phys. Lett. A 255 (1999), pp [14] M. Kennel et al., Determining embedding dimension for phase-space reconstruction using a geometrical construction, Phys. Rev. A 45 (1992), pp [15] J. Stark, Delay embeddings and forced systems, J. Nonlinear Science 9 (1999), pp [16] G. Dambrine et al., A new method for determing the small-signal equivalent circuit, IEEE Trans. Microwave Theory Tech. 36 (1988), pp

9 Figure Captions Fig. 1: Simplified large-signal equivalent scheme of a HEMT. Fig. 2: Example coverage of the mn&o pnrqs plane by a single vectorial large-signal measurement under two-tone excitation. Fig. 3: Measured tq mvuls time domain waveform of a HEMT as function of the corresponding n&o mvuls time domain waveform. Fig. 4: DC simulation of the dynamically modelled t q of a HEMT as function of n q with n o ranging between -1.2 V and V. Fig. 5: Small-signal w2x of the HEMT derived from model-generated S-parameters at 2 GHz (n q (=nyz ) ranges from.5 V to 4.5 V). Fig. 6: Small-signal { of the HEMT derived from model-generated S-parameters at 2 GHz (nrq (=nyz ) ranges from.5 V to 4.5 V). Fig. 7: Comparison of the measured (x) and modelled ( Fig. 8: Comparison of the measured (x) and modelled ( ) t2omvu}s (left) and tq mvuls (right) of a HEMT under two-tone excitation. ) t2omvuls (left) and tq mvu}s (right) of an amplifier MMIC at 3 GHz. Fig. 9: Gain versus input power of an amplifier MMIC. The x are the simulation results using the transistor level model for the MMIC; the solid line represents the simulation results of the corresponding behavioural model, obtained without any iteration during the modelling process. Table Captions Table 1: Normalised number of false nearest neighbours (fnn) for t2o and tq of a HEMT as function of the state variables taken into consideration. Table 2: Normalised number of false nearest neighbours (fnn) for t o and t q of an amplifier MMIC as function of the state variables taken into consideration. 9

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13 dimension fnn  à fnn ÂÄ state variables Å# ÃÇÆÈÆ2ÂÄ)É + Ê$Ã, Ê&Ä Å# ÃÇÆÈÆ2ÂÄ)É + Ê$Ã, Ê&Ä, Ê$Ã Ë Å# à ÆÈÆ2Â Ä É + Ê Ã, Ê Ä, Ê Ë Ã, Ê Ë Ä Å# à ÆÈÆ2Â Ä É + Ê Ã, Ê Ä, Ê Ë Ã, Ê Ë Ä, Ê Ì Ã Å# ÃÇÆÈÆ2ÂÄ)É + Ê$Ã, Ê&Ä, Ê$à Ë, Ê&Ä Ë, Ê$à Ì, Ê&Ä Ì Table 1: 13

14 dimension fnn Í2Î fnn ÍÏ state variables ÐNÍ Î8ÑÈÑ2ÍÏ)Ò + Ó*Î, Ó&Ï ÐNÍ Î8ÑÈÑ2ÍÏ)Ò + Ó*Î, Ó&Ï, Ó*Î Ô ÐNÍ Î ÑÈÑ2Í Ï Ò + Ó Î, Ó Ï, Ó Ô Î, Ó Ô Ï ÐNÍ Î ÑÈÑ2Í Ï Ò + Ó Î, Ó Ï, Ó Ô Î, Ó Ô Ï, Ó Õ Î ÐNÍ Î8ÑÈÑ2ÍÏ)Ò + Ó*Î, Ó&Ï, Ó*Î Ô, Ó&Ï Ô, Ó$Î Õ, Ó&Ï Õ Table 2: 14

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