eldo_user

Speed and Accuracy
Speed and Accuracy in Eldo
Speed and Accuracy in Eldo
The system of equations that represent the behavior of a circuit cannot be solved analytically,
apart from trivial cases. Thus a simulator has to use numerical algorithms to find an
approximate solution.
In the case of transient analysis, the problem to solve is to find the solution of a DAE
(Differential Algebraic Equations) system. To simplify, and deliberately using loose notations,
the ‘solution’ that the simulator tries to find is a set of N voltage waveforms vn(t), where n
ranges from 1 to N—N being the number of nodes in the circuit—and t represents the time
variable. These voltages are the solution of f(v(t), q’(v(t)), t)=0, where f() and q() are non-linear
functions. Equation f()=0 is nothing but the expression of the Kirchoff law, that is, the sum of
all currents entering a node is null, at any time. Function q() models the dynamic part of the
circuit, that is, the generally bias-dependent charges in the circuit.
This system is differential and non-linear. To solve it numerically, time is discretized, and the
equations are solved at discrete time points. The time-derivatives are approximated using a socalled
integration scheme or method, using current and previous values of the solution v(ti),
v(ti-1),…, v(ti-m). The number m of previous time points used to approximate the timederivatives
depends on the ‘order’ of the integration method. In all cases, an approximation
error is introduced in this process.
Once the time-derivatives have been eliminated, the system to solve is ‘simply’ non-linear.
Many numerical methods exist to solve this numerically; they typically requiring a certain
number of iterations, starting from an initial guess v0, and each iteration providing, hopefully, a
better estimate vj of the solution vexact. This iterative process will normally converge, although
some criterion are needed to decide when to stop the iterative process and accept the last
estimate as the ‘solution’ at the current time point. Again, an approximation error is introduced
here. One of the commonly used methods to solve non-linear systems of equations is the socalled
Newton-Raphson method. It is commonly used mainly because of its generality and also
for its relative robustness.
In summary, primarily two types of error are involved in the resolution of the system:
• Errors due to the numerical integration process.
• Errors due to the numerical resolution of non-linear equations.
Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246
Time Step Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248
Newton Iterations Accuracy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252
Global Tuning of the Accuracy—EPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253
Global Tuning of the Accuracy—TUNING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255
Eldo® User's Manual, 15.3 1245