Civil Engineering Reference
In-Depth Information
Once the Padé approximant to the transfer function has been computed‚
the final step is to compute a response to an applied stimulus. For a unit-step
stimulus and an approximation of order
this involves finding the
inverse Laplace transform of
A first step is to find the roots of the denominator‚ which reduces to finding
the roots of the polynomial For a polynomial of order
closed-form solutions for the roots exist; for higher order polynomials‚
numerical techniques are essential. Specifically‚ the closed-form formula for
the roots of a quadratic equation is widely known and is not reproduced in this
topic. The formulæ for the roots of cubic and quartic equations are provided in
Appendix A. For this reason‚ and because higher-order AWE approximations
are often unstable‚ it is common to select Next‚ given these poles‚ the
approximant is represented as a sum of partial fractions‚ and the corresponding
residues are calculated. The final step of inverse Laplace transforming this sum
yields the response as a sum of exponentials.
A typical approach to finding an reduced order system is to start with a
desired order and to compute the corresponding approximant. The stability
of this approximant is easily checked using‚ for example‚ the Routh-Hurwitz
criterion [Kuo93]. If the solution is unstable‚ the order is successively reduced
until a stable solution is found. The first order approximant is guaranteed to
be stable for an RLC system‚ and therefore this procedure will definitely result
in a solution; whether the level of accuracy of the solution is satisfactory or not
is another matter altogether.
Example:
For the RC line shown in Figure 3.3‚ the moments of the voltage at
node 5 were calculated earlier as
Matching these
to a [1/2] Padé approximant‚ we obtain
which leads to the equations
Solving these‚ we obtain the [1/2] Padé approximant to the transfer function
as
A comparison of this with the exact response to a step excitation was shown
earlier‚ in Figure 3.4(b).
