6.2.4 Local Limiting

In conventional circuit simulation it is common practice to limit the contact voltages before evaluating the compact model. This is illustrated for a simple diode model represented by the equation

$$\left(\vphantom{ \exp \left( \frac{V}{V_{T}} \right) - 1 }\right. \left(\vphantom{ \frac{V}{V_{T}} }\right. {\frac{V}{V_{T}}} \left.\vphantom{ \frac{V}{V_{T}} }\right) \left.\vphantom{ \exp \left( \frac{V}{V_{T}} \right) - 1 }\right)$$ (6.3)

Figure 6.1: Horizontal and vertical projection of the current solution for a diode.

with I_S being the saturation current and V_T = k_B ^. T/q the thermal voltage. At the Newton iteration step k equation (6.3) is linearized around the current solution point V^k. The solution of the linearized system is found as V^{k + 1}. However, directly using V^{k + 1} (vertical projection) can easily lead to overflow of the exponential function in (6.3) as shown in Fig. 6.1. To overcome this problem, horizontal projection has been introduced. The appropriate expressions are easily derived using Fig. 6.1. The current change evaluates to

$$\left(\vphantom{ V^{k+1} - V^{k} }\right. \left.\vphantom{ V^{k+1} - V^{k} }\right) \left.\vphantom{ \frac{{\mathrm{d}} I}{{\mathrm{d}} V} }\right. {\frac{{\mathrm{d}} I}{{\mathrm{d}} V}} \left.\vphantom{ \frac{{\mathrm{d}} I}{{\mathrm{d}} V} }\right\vert _{V=V^{k}}^{}$$ (6.4)

Inserting (6.3) gives

$${\frac{V^{*} - V^{k}}{V_{T}}} {\frac{\Delta V }{V_{T}}}$$ = (6.5)

$$\Delta$$ = V^{k + 1} - V^k (6.6)

and finally

$$\left(\vphantom{ 1 + \frac{ \Delta V}{V_{T}} }\right. {\frac{\Delta V}{V_{T}}} \left.\vphantom{ 1 + \frac{ \Delta V}{V_{T}} }\right)$$ (6.7)

However, horizontal projection is only useful for V^{k + 1} > V_crit. Below V_crit vertical projection obviously delivers better results. Equation (6.7) can be generalized to arbitrary non-linearities [32]

$${\frac{\mathrm{sign}\left(\Delta V\right)}{k}} \left(\vphantom{ 1 + k \cdot \vert\Delta V \vert}\right. \Delta \left.\vphantom{ 1 + k \cdot \vert\Delta V \vert}\right)$$ (6.8)

with k being a new scaling factor which can be optimized for each single device in the course of iteration progress. A larger k gives stronger damping and hence increases reliability of the algorithm for the price of longer simulation times. It must be pointed out that this mixed method of updating is not amenable to an easy description in terms of matrices, Jacobians, etc., and hence convergence theorems for this method are impossible to formulate. However, actual experience with this procedure in handling all types of circuits during the last decades has proven its effectiveness. As the voltages are damped in an individual manner for each device this damping algorithm will be termed local in the following. When investigating (6.3) one notices that $\partial$I/$\partial$V can become very small for V $\ll$ 0 which could result in a singular matrix. To prevent this occurrence a small leakage conductance G_min of typically 10^-9- 10^-12 S is placed in shunt with each junction. Alternatively G_min can be connected between each device node and ground which is similar from an electrical point of view but improves diagonal dominance compared to a shunt conductance. In addition, the exponential characteristic is replaced by a linear characteristic for bias voltages smaller than a few V_T. Although the above mentioned precautions are guided by numerical reasoning it should be noticed that they do not have any negative impact on the value of the solution as they might as well be justified on physical grounds.