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2.8 Linear Symbolic Approximation 139
In[22]:= DisplayForm[dvdmna]
Out[22]//DisplayForm=
1
R1
1
R1
0 1
V$1 0
1 1
R1
R1
1
R2
1
R3
1
R3
0 V$2 0
. ⩵⩵
0
1
1
R3
R3
1
R4
0 V$3 0
1 0 0 0
I$V0 V0
Since we are interested in a simplified solution for the voltage at the output node 3, we specify V$3
as the variable for which the equations are to be approximated.
In[23]:= approxdvd = ApproximateMatrixEquation[dvdmna,
V$3, dpdvd];
DisplayForm[approxdvd]
Out[24]//DisplayForm=
0 0 0 1
1 1
R1
R1 V$1 0
1
R2
0 0 V$2 0
0
1 1
R3
R3
. ⩵⩵
1
R4
0 V$3 0
1 0 0 0 I$V0 V0
The result is a matrix equation from which all numerical irrelevant symbolic terms have been
removed. We can now compute a simplified transfer function directly by solving the approximated
equations.
In[25]:= Solve[approxdvd, V$3]
R2 R4 V0
Out[25]= V$3
R1 R2 R3 R4
In this case, we obtain identical results from both equation-based and solution-based approximation.
However, as opposed to the latter, equation-based approximation does not require an exact symbolic
solution to be computed first. In such a small example this advantage makes nearly no difference,
but it will turn out to be a key factor in larger applications.
Approximating the Equations of the Amplifier
As an example for a slightly larger application let’s continue with our well-known common-emitter
amplifier once more. Again, we start by setting up a design point for
ApproximateMatrixEquation, reusing the numerical values we defined for simplifying the exact
solution with ApproximateTransferFunction. For matrix approximation, however, we still need
some more numerical information. While the SAG algorithm approximates the coefficients of a
transfer function independently, the SBG method uses the total value of a transfer function in
a design point as reference quantity. Therefore, we must specify one or more particular frequency
points at which ApproximateMatrixEquation should monitor the change in the value of the transfer
function caused by the removal of terms.
Here, we might be interested in computing a simplified expression describing the passband voltage
gain of the amplifier. The Bode plot above shows that the passband extends from about Hz to
several MHz, so we choose a design-point value for the operating frequency in the middle of the
passband, e.g. at f ⩵ kHz. The corresponding value for the Laplace frequency s is then given by