Download - Wolfram Research

134 2. Tutorial
than R3 and R4: R1 R2 ≪ R3 R4. Under these conditions we can discard the product term R1 R2 in
the denominator of the transfer function because it is the product of two small quantities and thus
contributes little to the total value of the expression as compared to the other terms:
In[10]:= simptfdvd = tfdvd /. R1*R2 −> 0
Out[10]=
R2 R4
R1 R3 R2 R3 R1 R4 R2 R4
The resulting function can now be factored and simplifies to a more meaningful form.
In[11]:= Factor[simptfdvd]
Out[11]=
R2 R4
R1 R2 R3 R4
This approximate formula holds for all sets of resistor values for which the condition R1 R2 ≪ R3 R4
is satisfied, but no longer in the general case. Symbolic approximation thus always implies a trade-off
between low expression complexity on the one hand and generality and precision on the other.
Numerical Reference Values: Design Points
Deciding on which terms to discard from a symbolic expression on the basis of vague information
such as "X is much larger than Y" may be feasible for humans but is virtually impossible to do for a
computer. This is particularly true when an expression contains a large number of symbols in nontrivial
combinations. To enable a computer to reduce a symbolic expression to its dominant content
we must provide input which allows for clear yes-or-no decisions on whether a term is important
or negligible. This input can be given as a set of accompanying numerical reference values for the
symbols, known as a design point, based on which the contributions of compound symbolic terms
can be compared numerically. Design-point values need not always be the exact element values for
which a circuit works within specifications. However, the reference values should reflect the relative
magnitude relations among the symbols in a realistic way.
In the case of the double voltage divider we could express the conditions on the relative magnitudes
of the resistors by an (arbitrary) numerical assignment of design-point values such as R1 ⩵ R2 ⩵
and R3 ⩵ R4 ⩵ . We rewrite the netlist of the double voltage divider keeping both a symbolic
element value and an associated design-point value together in each netlist entry.
In[12]:= doubleVoltageDivider =
Netlist[
{V0, {1, 0}, Symbolic −> V0, Value −> 1.},
{R1, {1, 2}, Symbolic −> R1, Value −> 10.},
{R2, {2, 0}, Symbolic −> R2, Value −> 10.},
{R3, {2, 3}, Symbolic −> R3, Value −> 1000.},
{R4, {3, 0}, Symbolic −> R4, Value −> 1000.}
]
Out[12]= NetlistRaw, 5
Now, we set up the corresponding equations using the option setting ElementValues −> Symbolic.
The design-point values are then automatically stored in the DAEObject and can be extracted via
GetDesignPoint (Section 3.6.12).