4.1 The Two-Level Newton Algorithm

Figure 4.1: a) Original circuit and b) linearized companion model for the two-level Newton algorithm

The problem of a two-level device and circuit simulation is demonstrated by an example circuit as shown in Fig. 4.1 where V_S is a constant voltage source and G is a linear conductance. The non-linear device is a diode whose characteristic is determined by the doping profile and the physical models used in the device simulation. The DC characteristic is shown in Fig. 4.2. In Fig. 4.1a the original circuit is shown. When using Newton's method to solve the non-linear circuit equations, a linear companion circuit [9] is solved at each iteration until some convergence criterion is met. The companion circuit for arbitrary non-linear two-terminal devices is shown in Fig. 4.1b. It consists of a linear conductance G_eq and a current source I_eq which depend on the operating point. They are derived from the non-linear characteristic as shown in Fig. 4.2 and read

$$\left.\vphantom{ \frac{{\mathrm{d}} I}{{\mathrm{d}} V} }\right. {\frac{{\mathrm{d}} I}{{\mathrm{d}} V}} \left.\vphantom{ \frac{{\mathrm{d}} I}{{\mathrm{d}} V} }\right\vert _{k}^{}$$ G^{k + 1}_eq = (4.1)

I^{k + 1}_eq = I^k - G^{k + 1}_eq ^. V^k (4.2)

with k being the iteration number of the Newton algorithm. For compact models, G_eq and I_eq are evaluated analytically. As this is not possible for distributed devices, the device simulator has to be called for each Newton iteration. The node voltages are initialized to values supplied by the user and default to zero. After this initialization, the device simulator has to be called to calculate the companion model.

For the calculation of the conductances an approach as proposed in [39] is well suited and will be outlined in the following section.

Figure 4.2: Non-linear characteristic of the diode and the quantities for the linearized companion model