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2.6 Modeling and Analysis of Nonlinear Circuits 97
equations cannot be set up in matrix form. Therefore, whenever a netlist contains behavioral models,
you must call CircuitEquations with the option MatrixEquation −> False, or as in our case,
you need to switch the AnalysisMode to e.g. DC for setting up nonlinear static equations which
automatically implies the setting MatrixEquation −> False.
In[5]:= dnwmna = CircuitEquations[diodeNetwork,
AnalysisMode −> DC];
DisplayForm[dnwmna]
Out[6]//DisplayForm=
V$1 V$2
V$1 V$2
I$V0
⩵⩵ 0, I$AC$D1
⩵⩵ 0,
R1
R1
V$1 ⩵⩵ V0, I$AC$D1 ⩵⩵ 1 V$2
Vt Is$D1,
V$1, V$2, I$V0, I$AC$D1, DesignPoint
Here, we have set up a system of nonlinear modified nodal equations in the unknowns V$1, V$2,
I$V0, and I$AC$D1. The latter symbol has been created automatically from the port current identifier
Current[A, C] in the definition of the diode model. All port branch voltages have been replaced
by corresponding differences of node voltages.
Generally, a port current Current[x, y] in a model instance MX will be denoted by symbols of the
form I$xy$MX. A port voltage Voltage[x, y] will be denoted by V$xy$MX, provided that branch
voltages appear as unknowns in the selected analysis method. To examine both effects we set up
the sparse tableau equations.
In[7]:= CircuitEquations[diodeNetwork,
Formulation −> SparseTableau, AnalysisMode −> DC
] // GetVariables
Out[7]= V$V0, V$R1, V$AC$D1, I$V0, I$R1, I$AC$D1
We use the command GetVariables (Section 3.6.7) to extract the list of variables from the equation
system. As you can see, the corresponding variables are called V$AC$D1 and I$AC$D1
Solving Nonlinear Equations
Solving nonlinear circuit equations analytically is, unfortunately, mathematically impossible in the
general case. However, in many applications it is possible to reduce the original set of equations by
eliminating a number of variables. This may already yield some qualitative insight into the behavior
of a nonlinear circuit. On the other hand, we can always assign values to symbolic parameters and
solve the equations numerically using NDAESolve (Section 3.7.5).
Let’s derive an expression which relates the diode current
I$AC$D1 to the input voltage V0 by eliminating all other variables. For this task, Analog Insydes
provides the function CompressNonlinearEquations (Section 3.12.2) which removes equations and
variables from a nonlinear DAEObject that are irrelevant for solving for a set of given variables. The
option setting EliminateVariables −> All additionally allows for eliminating variables that occur
linear somewhere in the equations.