next up previous contents
Next: 4.3 The Full-Newton Algorithm Up: 4. Simulator Coupling Previous: 4.1.2 Modified Two-Level Newton

4.2 The Quasi Full-Newton Algorithm

In [39] a method was proposed which was termed full-Newton algorithm. However, this approach is very similar to the two-level method proposed in the same paper hence it is termed ``quasi'' full-Newton in this thesis. The basic idea will be demonstrated for the circuit shown in Fig. 4.1. The combined device and circuit equations read
f(x, V) = 0 (4.13)
I(x, V) + G . (V - VS) = 0   . (4.14)

Applying Newton's method to (4.13) and (4.14) one gets
$\displaystyle \mathbb {J}$x . $\displaystyle \Delta$x + $\displaystyle \mathbb {J}$V . $\displaystyle \Delta$V = - f(x, V) (4.15)
$\displaystyle {\frac{\partial I}{\partial \mathbf{x}}}$ . $\displaystyle \Delta$x + $\displaystyle {\frac{\partial I}{\partial V}}$ . $\displaystyle \Delta$V + G . $\displaystyle \Delta$V = - I(x, V) - G . (V - VS)   . (4.16)

Rearranging (4.15) yields

$ \Delta$x = $ \mathbb {J}$x-1 . $ \left(\vphantom{-\mathbf{f}(\mathbf{x},V) -\mathbb{J}_{V} \cdot \Delta V}\right.$ - f(x, V) - $ \mathbb {J}$V . $ \Delta$V$ \left.\vphantom{-\mathbf{f}(\mathbf{x},V) -\mathbb{J}_{V} \cdot \Delta V}\right)$ (4.17)

which can be rewritten as
$\displaystyle \Delta$x = $\displaystyle \Delta$$\displaystyle \hat{x}$ - $\displaystyle \mathbb {J}$x-1 . $\displaystyle \mathbb {J}$V . $\displaystyle \Delta$V (4.18)
$\displaystyle \Delta$$\displaystyle \hat{x}$ = - $\displaystyle \mathbb {J}$x-1 . f(x, V) (4.19)

Substituting (4.18) in (4.16) yields

$ \left(\vphantom{ -\frac{\partial I}{\partial \mathbf{x}}\cdot\mathbb{J}_{\mathbf{x}}^{-1}\cdot\mathbb{J}_{V}+\frac{\partial I}{\partial V} +G \cdot }\right.$ - $ {\frac{\partial I}{\partial \mathbf{x}}}$ . $ \mathbb {J}$x-1 . $ \mathbb {J}$V + $ {\frac{\partial I}{\partial V}}$ + G . $ \left.\vphantom{ -\frac{\partial I}{\partial \mathbf{x}}\cdot\mathbb{J}_{\mathbf{x}}^{-1}\cdot\mathbb{J}_{V}+\frac{\partial I}{\partial V} +G \cdot }\right)$$ \Delta$V = - $ {\frac{\partial I}{\partial \mathbf{x}}}$ . $ \Delta$$ \hat{x}$ - I(x, V) - G . (V - VS)   . (4.20)

This equation can be rewritten as

$ \left(\vphantom{G_{\mathit{eq}} + G}\right.$Geq + G$ \left.\vphantom{G_{\mathit{eq}} + G}\right)$ . $ \Delta$V = - IT - G . (V - VS) (4.21)

with
Geq = - $\displaystyle {\frac{\partial I}{\partial \mathbf{x}}}$ . $\displaystyle \mathbb {J}$x-1 . $\displaystyle \mathbb {J}$V + $\displaystyle {\frac{\partial I}{\partial V}}$ (4.22)
IT =    $\displaystyle {\frac{\partial I}{\partial \mathbf{x}}}$ . $\displaystyle \Delta$$\displaystyle \hat{x}$ + I(x, V)   . (4.23)

Equation (4.21) is similar in form to that obtained by the two-level Newton algorithm. Hence, similar methods can be used to embed distributed devices into a circuit simulator and to provide a decoupling between both simulators even for this quasi full-Newton algorithm.


next up previous contents
Next: 4.3 The Full-Newton Algorithm Up: 4. Simulator Coupling Previous: 4.1.2 Modified Two-Level Newton
Tibor Grasser
1999-05-31