Behavioral Modeling and Transient Analysis with Analog Insydes

This paper presents some new extensions of our Mathematica toolbox Analog Insydes for modeling and analysis of nonlinear circuits. Circuits are described by means of hierarchical netlists containing top-level netlists, subcircuits, and symbolic device model descriptions from which circuit equations can be set up in sparse tableau or modified nodal formulation. Some examples of Analog Insydes’ nonlinear behavioral modeling capabilities are demonstrated. Transient circuit analysis is performed by application of a dedicated numerical DAE (Differential and Algebraic Equation) solver. The implementation of the solver is highlighted because most computer algebra systems provide solvers for systems of differential equations only. 1. THE NETLIST FORMAT This section gives a brief introduction to the netlist format of Analog Insydes [1]. Analog Insydes provides mechanisms for describing circuits hierarchically in terms of netlists and subcircuits. The hierachically structured circuit description comprises a top-level netlist with subcircuit or model references, and a set of subcircuit or model definitions. A netlist consists of a sequence of netlist entries which either can be a reference to a primitive circuit element or to a subcircuit. Netlist entries must be lists of three components as there are where • the reference designator identifies a particular circuit element, • the connectivity list specifies the nodes of the circuit to which the terminals of the element are connected, • and the value field specifies the numerical quantity or symbolic expression of a circuit element. 2. BEHAVIORAL MODELS Models which are based on sets of mathematical equations describing the behavior of a circuit element or an analog building block are frequently called (analog) behavioral models, or ABMs. A behavioral model is similar to a subcircuit object in that it constitutes a box which is connected to a circuit through electrical ports. The difference is that the interior of a behavioral model box is implemented in terms of algebraic or differential equations rather than a netlist. 2.1. ABM IMPLEMENTATION Analog Insydes’ behavioral modeling capabilities allow for modeling arbitrary nonlinear and dynamic element characteristics. The full syntax for an ABM definition is shown below. The Model function takes eight arguments which must all be written in Mathematica's option syntax keyword -> value. The Name argument identifies an entire group, or class, of different subcircuit implementations of a non-primitive object. The Selector argument then selects one particular member from this class. The Scope {reference designator, {connectivity}, value field} Model[ Name -> subcircuit class name, Selector -> selector, Scope -> scope, Ports -> {portnodes}, Parameters -> {parameters}, Translation -> {parameter translation rules}, Variables -> {variables}, Definition -> {equations} ] argument specifies whether the model definition is limited to the circuit in which the definition is made or whether it is stored in the global subcircuit database and made accessible to other circuits. The Ports argument defines the port nodes of a subcircuit structure which serve as external connection points. The Parameters argument specifies which symbols in a subcircuit definition denote formal model parameters that can be accessed from a netlist on a higher hierarchy level. The Translation argument defines functional relations between formal subcircuit parameters and internal element values. For behavioral model definitions, the right-hand side of the Definition argument must be a list of symbolic device equations. The Variables argument serves to specify the symbols which are unknowns of the model equations. Behavioral models defined with the Model command can be referenced from a netlist on a higher hierarchy level by a netlist entry whose value field contains the options Model and Selector To reference a subcircuit object the subcircuit classname and the selector must be the same identifiers as those given as arguments to the Name and Selector parameters in the corresponding Model statement. 2.2. ABM CAPABILITIES In this section we demonstrate the large variety of device modeling capabilities of Analog Insydes using Chua’s circuit (see Figure 1), a well-known example for a simple nonlinear autonomous oscillator circuit which exhibits chaotic behavior. The circuit consists of a resistor R, two capacitors C1 and C2, an inductor L, and an active nonlinear conductance NL. The values of the circuit elements are assumed to be R = 1.43 Ω, C1 = 111 mF, C2 = 1 F, and L = 143 mH [2]. The active nonlinear conductance NL has an ArcTan-shaped voltage-current relation as illustrated in Figure 2. To observe chaotic oscillations it is not necessary to use an exact ArcTan shape. Many other functions which resemble such a voltage-current relation qualitatively can be used as well. Within Mathematica the voltage-current relation can be described in various ways: • ArcTan function (see Figure 2) • piecewise linear (PWL) function using Mathematica’s data type InterpolatingFunction (see Figure 3) • piecewise linear function, implemented procedurally using Mathematica’s control structure Which (see Figure 3) • polynomial approximation (see Figure 4) The next step is now the implementation of the different voltage-current relations of the nonlinear conductance as a library of analog behavioral models within Analog Insydes. Therefore, we define the model class ChuaNL containing four different models which are then stored in the global subcircuit/model database by applying the function ExpandSubcircuits. {instance name, {connectivity}, Model -> classname, Selector -> selector}