Civil Engineering Reference
In-Depth Information
Combining this with Equation (2.42) yields
Since the values at time
are unknown‚ we drop all subscripts
to obtain the
approximated characteristic equation at
as
This corresponds to a constant voltage source‚ and therefore‚ the circuit may
be analyzed by replacing the capacitor by a constant voltage source.
Alternatively‚ one could use the Backward Euler formula‚
Substituting the characteristic equation for the linear capacitor and dropping
the
subscripts‚ we obtain the following approximation:
This corresponds to a conductance of value
in parallel with a current source
of value
The stamps for these elements may be used to solve the
circuit at time
If the Trapezoidal rule is used instead‚ it is easy to verify that for the linear
capacitor‚ this results in the equation
which again is a conductance in parallel with a current source.
All three methods use the information from one previous time step‚ but from
Equations (2.47) and (2.48)‚ we can see that the Backward Euler only requires
the voltage across the capacitor at the previous time step‚ while the Trapezoidal
Rule requires both the voltage and current from the previous time point.
Stability
While the preceding discussion makes these formulæ all look superficially sim-
ilar, they have vastly different characteristics in terms of
numerical stability.
We explore the concept of numerical stability with the aid of a test equation,
where
is an arbitrary complex number. As
the solution
tends
to zero when
and it tends to
when
At the very
