LaPlace Transform (qualitative)
The method of LaPlace Transforms is a sophisticated mathematical 'trick' by which the integro–differential
equations of a circuit analysis (in particular) are transformed into algebraic equations much easier to solve,
and for which the solution can be transformed 'back' to provide the solution for the actual circuit problem.
While there is the added work of making the transformation and ultimately the reverse transformation, the
net gain in simplicity makes this added effort well worthwhile. There is a certain formal difficulty in the
forward and reverse transformation processes; they involve the integration of complex variables.
Fortunately, because the transformation mathematics can be quite involved in some cases, it is not necessary
actually to do the mathematics in order to make effective use of the LaPlace Transform; we simply use
tables which record previously determined relationships, in much the same way as a Table of Integrals is
used. (And in much the same way as for a table of integrals the results for some common transformations
and operations can be memorized for convenience.)
If we call the circuit variable y(t) in the time domain it can be transformed into a different function Y(s) of a
complex variable s, and vice versa, by a complicated integration which provides a unique relationship
between y(t) and Y(s). These two functions are different views of the same mathematical object; given one
the other can be determined. A particular reason for going to the trouble of making the transformation (and
the inverse transformation) is the fact that operations of integration and differentiation on y(t) are
transformed into algebraic operations on Y(s). The advantages of this relationship alone are so great that
they considerably outweigh the added effort in making the transformation of y(t) to Y(s), and then later the
reverse transformation from the s-plane back to the time domain.
For the present purpose only a few function transform and operation pairs extracted from the many in a
more complete table will suffice:
Circuits LaPlace Transform
1
M H Miller
Additional pairs can be obtained from a more
extensive table, or in many instances derived
rather easily. For example the transform for a
sinusoidal function may be derived as shown to
the right. The 'displacement' relation provides the
first expression, and Euler's Theorem provides
the rest. In practice transforms such as these are
available in tables.
Inductive reasoning provides the transforms for tn as shown below:
One place where a formal change takes place is in the volt-ampere relations for capacitors
and inductors. Thus in the time domain we have i = C (dv/dt). Note: Here we use lower case characters i,v
for a time domain variable, upper case I,V for the corresponding frequency domain variable. From the
'Transform Operations Table' we note that the corresponding transformed expression is
i = C (dv/dt) à I = C[sV - v(0+)].
Similarly for an inductor (although there is not one in this particular illustration)
v = L(di/dt) àV = L[ sI - i(0-)]
The transform volt-ampere relation is an algebraic expression (in s) and in the transform plane solving for
the transform circuit variables is an algebraic process. Note the implicit inclusion of a constant of
integration; this is what the initial condition of the capacitor voltage provides. The initial condition is
included conveniently in the capacitor icon as a battery in
series with the (transform) capacitor. Note that the
voltage across the capacitor in the time domain
corresponds to the voltage across both the capacitor and
battery icons in the frequency domain.
Circuits LaPlace Transform
2
M H Miller