Improved VHDL-based Modeling Techniques

VHDL-AMS EXTENSIONS FOR FREQUENCY DOMAIN HARMONIC BALANCE SIMULATION By Gennady Serdyuk, e-mail: [email protected] Doug Goodman, e-mail: [email protected], 85704, phone: (520) 742-3300 Ridgetop Group, Inc. PO Box 35441 Tucson, Arizona 85740, USA Abstract - VHDL has been shown to be an effective means of modeling complex electronic devices. Initially targeted as handling digital devices, it was later extended to Analog and Mixed Signal devices. In this paper, the authors show how VHDL can be extended further to accommodate microwave and RF modeling for use with Harmonic Balance Simulators. The authors refer to this extended modeling capability as VHDL-FD, and provide an example of applying it toward simulation of a Gilbert Cell mixer and the new RinconTM simulator from Ridgetop.

I. Introduction Microwave and RF Designers have long sought to achieve higher and higher levels of performance and accuracy in their simulation tools. Due to the special considerations in the Microwave and RF region, there has been a complicated evolution ranging from the application of linearized approaches of TOUCHSTONE and SPICE on nonlinear circuits to more advanced, up-to-date approaches, applying harmonic balance and complex timedomain techniques for fast simulation of nonlinear analog circuits. To improve overall performance capabilities, Electronic Design Automation (EDA) tool development has been concentrated in different, but related areas - initially simulation techniques were of primary interest with a limited set of primitive models. As Designers became more comfortable with their use, the emphasis shifted later to modeling of more complex microwave devices, and appropriate description languages were required to provide the level of model and simulation accuracy deemed necessary by designers. In the low-frequency area, the SPICE circuit description format is considered a de-facto standard but in high frequency area (above, say 100MHz) this has not been the case. The wide variety of RF and microwave elements has demanded more advanced description languages to accurately describe their behavior over varying conditions. To that end, most EDA suppliers created their own proprietary description languages to describe these components. On the surface, however, it appeared to be a quite general situation lending itself to a standardization effort. In the field of digital circuits both proprietary and standards-based languages were created, but later most useful and popular among them - VHDL and Verilog emerged into the public domain and became widely-used standards. Demands of simulating complex analog, digital and mixed digital-analog circuits led to development of extensions of these languages - VHDL-AMS and VerilogAMS. However, the Microwave and RF simulation area remains without standard description language. This paper proposes extensions of a standard language to accommodate the needs of microwave and RF design applications.

II. Merging VHDL-AMS and Harmonic Balance Apart from Harmonic Balance (HB), a technique that simulates the circuit in the frequency-domain (to be further described later), there are certain time-domain techniques that are also useful, but carry certain limitations. These

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time-domain methods include Microwave SPICE, Shooting Technique and SpectreRF. Using SPICE (Microwave SPICE) – perhaps the most traditional and one of the oldest methods - uses numerical integration of component equations and Kirchoff's laws in the time domain and creates a solution for each time step. The time steps can then be graphed so designers can examine behavior of the overall circuit under various drive conditions. Spice suffers from all usual drawbacks of time-domain techniques; long computation times due to large time constant differences found in microwave circuits, and an inability to handle transmission lines well, including long computation times and having to use lumped element approximations. Among the advantages of Spice are the ability to solve simple circuits and its relative simplicity - almost everybody knows SPICE and can apply it quickly. Table 1. Comparison of different simulation and modeling techniques Simulator Type

Relative Computation Time

Harmonic Balance

Accu -racy

Model Library

New Modeling Capability

Average

Excellent

Proprietary

Limited

Rincon Simulator: HB & VHDL-FD

Average

Excellent

Open

Extensive

Spice

Long

Poor

Short

Medi -um

TM

SpectreRF

Spice Primitives Proprietary

Comments

Model limitations Newly introduced

None

Widely understood

None

Limited library

A new approach utilizing the Shooting approach (1) has also been introduced to deal with the differing time constants issue. The shooting approach eliminates excessive computations while the steady-state solution is being found. There is also an appropriate technique to find the transient response with different time constants present (such as with SpectreRF). These tools are adequate for their tasks, but the library of components with complex frequency dependency is lacking. Both aforementioned techniques are suitable for ordinary differential equations (ODE) – but not for equations with complex frequencydependent components, which are not described by ODE. Harmonic Balance overcomes both limitations found in the Shooting or SpectreRF approaches - neither different time constants nor complex frequency-dependent passive components impact Harmonic Balance’s ability to accurately solve the circuit equations to provide meaningful results.

2001ѝ11thѝInternationalѝConferenceѝ“Microwaveѝ&ѝTelecommunicationѝTechnology”ѝ(CriMiCo’2001).ѝ10-14ѝSeptember,ѝSevastopol,ѝCrimea,ѝUkraineѝ ©ѝ2001:ѝCriMiCo’2001ѝOrganizingѝCommittee;ѝWeberѝCo.ѝISBN:ѝ966-7968-00-6.ѝIEEEѝCatalogѝNumber:ѝ01EX487

Harmonic Balance Technique The Harmonic Balance simulation approach has been in service for many years from sources such as Agilent and others. Harmonic Balance equations usually are formulated in the form:

F ( X ) ≡ L (ω , τ i ) ⋅ X + + N ( X , jω X , e −

j ωτ

i

X ) + I = 0 (1 )

where: ω - angular frequency

τi

- explicit delay component in circuit;

L( ω , τ i )

- complex linear operator, represents linear

part of the circuit; X - complex spectrum of variables; jωX - image of variables’ derivatives;

e − jωτ i X - image of delayed variable component; N ( X , jωX , e − jωτ i X ) - complex response of nonlinear subcircuit; I - vector of free sources in frequency domain (complex amplitudes). X usually contains all necessary frequency components- fundamentals and products with orders ranging from a value of 5 up to 20. Most complex calculations are contained in the N term of the equation. There is no closed form formula to calculate response of the nonlinear elements in frequency domain, excluding only some special cases (e.g. polynomials). To deal with this, frequency transformation techniques are applied. With frequency transformation techniques, the response FFT is typically used: first, compute the timedomain waveform of stimulus via a inverse FFT, then compute its time-domain response using nonlinear functions of element(s)and then calculate its frequency domain response with the forward FFT. Multi-dimensional FFT’s are used to allow transformation of quasi-periodic waveforms. The Newton method is used to solve the Harmonic Balance equation. Modern implementations use modifications, which provide the capability to handle systems of quite large size. The Krylov subspace approach [6] is an example of this. To simplify a depiction of its function, assume that the usual Newton method is applied. Then, according to (1), the sequence of steps

X

i+1

= X

Si = − J J =

i

−1 i

+ ki ⋅ Si;

(2)

Fi ;

(3)

dF ; dX

should converge to the solution of equation (1). This is guaranteed only if initial point X0 is close enough to solution point [2]. If not, different globalization strategies are helpful [3]. Usually it consists of choosing proper value of ki to allow the sequence (2) to converge to a solution, where F=0. As usual, the described approach carries both advantages and drawbacks that must be assessed.

Harmonic Balance’s ability to obtain a steady-state solution is often mentioned among the main advantages of HB technique. It is follows directly from formulation of equation (1). Another key advantage is its ability to handle circuits of any linear circuit, including black boxes with measured frequency responses. Referring to the term L( ω , τ i ) * X in (1), L( ω , τ i ) can represent any possible linear transformation of X in the complex plane, so any imaginable linear circuit (as long as Kirchoff's laws are satisfied at its ports) may be represented by L( ω , τ i ) . This includes elements with complex frequency-dependent behavior, like microstrip lines, lines under layered substrates, under anisotropic substrates, etc. Usage of the multidimensional FFT in HB allows for the solution of circuits under quasiperiodic excitation, so different time constants are not an issue for it either. The main disadvantage is a large computational time necessary to provide the solution. The HB task has approximately a cubic algorithm complexity, which is defined as a "bottle neck" of the algorithm - solving (3) to define the step. Usage of sparse matrices and inexact Newton techniques can decrease the solution time. Another timeconsuming procedure is the calculation the response of the nonlinear subcircuit term N in (1). It requires many nonlinear function calculations along with forward and reverse FFT. Another one disadvantage is a flipside of the appropriate advantage - this method is not too suitable if transient is in interest. HB -based envelope techniques [7], mixed shooting/time integration scheme like in Spectre RF and/or direct time-domain integration are more suitable in this case. VHDL-AMS and Frequency Domain The needs of Microwave and RF engineers for a standard description language point to a frequency domain version of VHDL that is compatible with a Harmonic Balance Simulator. With the advent of VHDL-AMS, which provides for description of analog and mixed signal models, the foundation has been laid for a further extensions of the effort into the Frequency domain. This is referred to as VHDL-FD. Referring to the VHDL-AMS Language Reference Manual (LRM) [5] the VHDL language may be divided onto three large components: digital , analog and circuitdescriptive component, which handles circuit decomposition into elements or vice versa - combining elements and sub blocks. This third component supports the construction ability of the language. The digital component is implemented using signals, processes and means of their interconnection - concurrent statements, sequential statements. The analog component is implemented via quantities, terminals (like special case of structure of quantities with conservation semantics) and simultaneous statements. The descriptive component is represented by component instantiation statement and serves as the infrastructure of language for describing the topology of the circuit. To establish the suitability of VHDL-like language with Harmonic Balance, there are three questions should to be answered: • What is the method of circuit representation in VHDL-AMS? • What class of circuits can VHDL-AMS currently represent?


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• What extensions are required for the microwave world? For analog circuit representation, VHDL-AMS uses two techniques – basic element representation and lowlevel (with respect to design hierarchy) equation-level representation. The equation-level representation builds a modeling basis for component library design. Operating with equations (simultaneous statements in the terminology of the language) the designer is able to build the particular component desired. Values of the circuit - voltages and currents - along with their derivatives with respect to time, integrals and delays are in the designer's arsenal. The designer can build equations of any complexity, using functions that are needed. The VHDL-AMS language allows the use of standard, physics-based equations along with branching "if" statements and procedural statements. Being supported by equations, designer can build higher-level blocks and connect them to each other in order to create more complex units. Two ways are available to provide this: connecting using "quantities" and "terminals". Quantities are standard variables in the scope of VHDL-AMS. Several blocks can share the same variables and connection via quantities facilitates this. It is convenient to describe signal-flow diagrams, simple closed systems, etc. using quantities. Therefore, conservative semantic structures are not necessary. Terminals carry out additional work - they assume preservation of conservation laws - like Kirchoff's laws in electrical engineering. Terminals contain two quantities: an “across” variable and a “through” variable. The “across” quantity acts like voltage of node (or branch) and “through” quantity - like incident current of the node. Terminals are useful then it is necessary to represent circuit with component interconnection. Figure 1 describes the idea of terminal and “across” and “through” quantities.

• Derivatives of variables (they are mapped onto 'DOT attribute) • Integrals of variables ('INTEG attribute) • Delayed values of variables ('DELAYED attribute) These values may be combined by means of algebraic functions to form the equations. This mechanism creates the ability to express any system of DAE - Differential Algebraic Equation sets during modeling. In terms of circuit simulations it is equivalent to incorporating nonlinear and common linear reactive elements like inductors and capacitors. This set is usually suitable for low-frequency applications. However, the means for modeling of distributed systems are not yet available with VHDL-FD, except as lumped circuit approximations. It will be necessary to create the proper extensions to accommodate frequencydomain modeling capabilities having distributed reactive elements. However, while not precluded, it is not an initial VHDL objective [5]. To allow frequency domain modeling, it is necessary to operate with quantities in the frequency-domain and time-domain together in one module. If it is assumed that VHDL-AMS defines quantities in time domain by default, the new attributes should be introduced to represent the same quantity in frequency domain. It is proposed, that the 'FD (frequency domain) extension be provided. Under this scenario, an extended description capability is enabled: ui'FD == Zk(FREQUENCY)*iin'FD; -- complex arithmetic where ui'FD and iin'FD - images of ui and iin in frequency domain (complex values) and Zk(FREQUENCY) some complex value of impedance (dependent on FREQUENCY). This notion allows usage of frequency-domain modeling. If the model in the time-domain is described as: i == C*v'dot;

-- (a)

it may be simply and automatically transformed into frequency-domain: i'FD== *math_j*math_2_pi*FREQUENCY*v'FD; -- (b)

Fig. 1. VHDL: Depicting concept of through and across variables As implemented, both quantities and terminals may be used together, providing an effective mechanism for mixed, system/component level description Another useful feature of behavioral approach - it is multidisciplinary. It is possible to describe and simulate mixed systems – such as electro-mechanical, electrohydraulic and others. These extremely useful characteristics provide a significant inducement to utilize VHDL-AMS as the basis language for adaptation in microwave design realm. We refer to this form as VHDL-FD. Let us consider the classes of circuits, which may be represented by means of analogous subset of VHDLAMS. Language means allow to use in equations the following items: • Variables itself (quantities in terminology of VHDL-AMS)

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The model (a) can be simply represented in timedomain too, but many complex frequency-domain models (e.g. with dispersion) have a very complex time-domain description. Using frequency-domain extensions, as shown in (b), it is possible to extend the modeling capabilities of VHDL-AMS up to high-frequency bands. The Authors understand that most of the advantages of such VHDL-AMS extension can be available for frequency-domain simulation techniques, such as for Harmonic Balance. But we believe, that this approach is viable in time-domain simulation as well and can be utilized in time-domain simulators via convolutions or equivalent transformations.

III. Experimental Results Ridgetop Group has created a working Simulator TM called Rincon employing the VHDL-FD extensions with the Harmonic Balance Simulator. Very encouraging results have been found and the basis for further work has been identified. Perhaps the best indication is to illustrate its performance on a typical microwave circuit, such as a Gilbert Cell Mixer.


Fig. 2: Gilbert Mixer Cell.

Fig. 4 (a,b). Mixer at high LO level of 100 mV (a) and at low LO level of 10 mV and at low bias voltage (b)

V. References [1] Shooting Technique Reference www.Cadence.com [2] Ortega J.M., Rheinboldt W.C., Iterative Solution of Nonlinear Equations of Several Variables. Academic Press, New Tork, 1970. [3] J.E. Dennis, Jr, R.B. Schnabel, ”Numerical Methods for Unconstrained Optimization and Nonlinear Equations”, Prentice Hall, Englewood Cliffs, 1983. [4] Envelope technique reference www.Agilent.com [5] VHDL-AMS Language Reference Manual from the IEEE [6] V. Rizzoli et al., “Fast and Robust Inexact Newton Approach to the Harmonic-Balance Analysis of Nonlinear Microwave Circuits”, IEEE Microwave and Guided Wave Lett., Vol. 7, No. 10, Oct. 1997, pp.359-361. [7] Harmonic Balance -based envelope techniques reference http://www.Agilent.com

Fig 3 (a,b). Mixer spectrum and waveform at low LO level of 10 mV. The Mixer was simulated under two different excitations. Two different LO drive voltages (10 mV and 100 mV) under different biases were tested. Drive waveforms and spectra are presented in the figures 3 and 4.

IV. Conclusion It has been shown that a new approach to Harmonic Balance simulation is required to take full advantage of standardized modeling capabilities in microwave design CAD. Frequency-domain extensions of VHDL-AMS broadens the class of circuits that may be represented with this modeling language and allows using of "black box" components in simulations - such as measured frequency-domain data or frequency-domain design specifications. This greatly extends the capabilities available to Microwave and RF Designers. VHDL-FD, integrated with TM the new Rincon Harmonic Balance Simulator, has been demonstrated to be a powerful tool for rigorous analysis, as shown with the Gilbert Cell mixer. 2001ѝ11thѝInternationalѝConferenceѝ“Microwaveѝ&ѝTelecommunicationѝTechnology”ѝ(CriMiCo’2001).ѝ10-14ѝSeptember,ѝSevastopol,ѝCrimea,ѝUkraineѝ ©ѝ2001:ѝCriMiCo’2001ѝOrganizingѝCommittee;ѝWeberѝCo.ѝISBN:ѝ966-7968-00-6.ѝIEEEѝCatalogѝNumber:ѝ01EX487

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