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18 1. A Short Introduction
Setting Up Circuit Equations for Approximation
To prepare the simplification process we set up the system of circuit equations again. As against
the previous call to the function CircuitEquations, where the circuit equations were given in the
modified nodal analysis format, they are set up in sparse tableau formulation this time. Sparse
tableau is the preferred equation formulation when performing linear approximation tasks. Details
about the Formulation option are given in Section 3.5.1.
In[13]:= staAC741 = CircuitEquations[op741,
ElementValues −> Symbolic, Formulation −> SparseTableau,
ModelLibrary −> "BasicModels‘"]
Out[13]= DAEAC, 402 402
The returned equation system is a DAEObject with equations and variables.
In[14]:= Statistics[staAC741]
Number of equations : 402
Number of variables : 402
Number of entries : 161604
Number of non−zero entries : 1392
Complexity estimate : 1.6e21
Although this sparse tableau system is more than 10 times bigger than the modified nodal system,
it usually produces better approximation results.
Performing Matrix Approximation
Now we call the matrix approximation routine on the symbolic circuit equations of the ΜA741 circuit.
This routine simplifies the equations based on numerical constraints with respect to a given output
variable. The required information is provided in form of several sampling points, specified in
combination with a maximum error constraint each. The approximation routine then processes the
matrix such that the sought output function is located within the defined error range around the
given sampling points. For further information please refer to Section 3.11.3.
To capture the first corner frequency, we place sampling points at Hz and at Hz with a maximum
error of each.
In[15]:= samplingpoints = {{s −> 2. Pi I 1., MaxError −> 0.3},
{s −> 2. Pi I 10., MaxError −> 0.3}}
Out[15]=
s 6.28319 , MaxError 0.3, s 62.8319 , MaxError 0.3
With the given setup, the approximation routine can be called. The computation is carried out
with respect to the output voltage across the load resistor R13, which in this case is equivalent
to the sought transfer function. Again, based on the automatic naming scheme (Section 2.4.1), the
corresponding variable is called V$R13 (note that sparse tableau equations consist of branch voltages
instead of node voltages).