C. The Gummel-Poon BJT Model

The Gummel-Poon model is a compact model for bipolar junction transistors (BJT) which also takes into account effects of low currents and at high-level injection. Fig. C.1 shows the equivalent circuit of the Gummel-Poon model.

Figure C.1: Equivalent circuit of the Gummel-Poon model.

The model distinguishes four operating regions. The base and collector current as they are implemented in SPICE [45] are as follows.

• normal active region: $\ensuremath{V_{\mathrm{be}}}> -\frac{5\cdot\ensuremath{n_{\mathrm{f}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$ and $\ensuremath{V_{\mathrm{bc}}}\leq -\frac{5\cdot\ensuremath{n_{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$

  $$\frac{\ensuremath{I_{\mathrm{s}}}}{\ensuremath{q_{\mathrm{b}}}}\c... ...ight)\cdot\ensuremath{V_{\mathrm{bc}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ I_c =   (C1)

  $$\ensuremath{I_{\mathrm{b}}}=\, \ensuremath{I_{\mathrm{s}}}\cdot\left(\frac{1}{\ensuremath{\beta_... ...math{n_{\mathrm{el}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}\right) - 1\right)$$

  $$- C_{4}\cdot\ensuremath{I_{\mathrm{s}}}+ \left(\frac{\ensuremath{... ...c}}}}{\ensuremath{\beta_{\mathrm{r}}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ (C2)

  
  

• inverse region: $\ensuremath{V_{\mathrm{be}}}\leq -\frac{5\cdot\ensuremath{n_{\mathrm{f}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$ and $\ensuremath{V_{\mathrm{bc}}}> -\frac{5\cdot\ensuremath{n_{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$

  $$-\frac{\ensuremath{I_{\mathrm{s}}}}{\ensuremath{q_{\mathrm{b}}}}\... ..._{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}\right) - 1\right)\right)$$ I_c =  

  $$- C_{4}\cdot\ensuremath{I_{\mathrm{s}}}\cdot\left(\exp\left(\frac... ...ight)\cdot\ensuremath{V_{\mathrm{bc}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ (C3)

  $$\ensuremath{I_{\mathrm{b}}}=\, -\ensuremath{I_{\mathrm{s}}}\cdot\left(\frac{1}{\ensuremath{\beta... ..._{B}}}\cdot T}\right) - 1\right)\right) - C_{2}\cdot\ensuremath{I_{\mathrm{s}}}$$

  $$+ C_{4}\cdot\ensuremath{I_{\mathrm{s}}}\cdot\left(\exp\left(\frac... ...c}}}}{\ensuremath{\beta_{\mathrm{r}}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ (C4)

  
  

• saturated region: $\ensuremath{V_{\mathrm{be}}}> -\frac{5\cdot\ensuremath{n_{\mathrm{f}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$ and $\ensuremath{V_{\mathrm{bc}}}> -\frac{5\cdot\ensuremath{n_{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$

  $$\frac{\ensuremath{I_{\mathrm{s}}}}{\ensuremath{q_{\mathrm{b}}}}\c... ..._{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}\right) - 1\right)\right)$$ I_c =  

  $$- C_{4}\cdot\ensuremath{I_{\mathrm{s}}}\cdot\left(\exp\left(\frac... ...ight)\cdot\ensuremath{V_{\mathrm{bc}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ (C5)

  $$\ensuremath{I_{\mathrm{b}}}=\, \ensuremath{I_{\mathrm{s}}}\cdot\left(\frac{1}{\ensuremath{\beta_... ..._{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}\right) - 1\right)\right)$$

  $$+ C_{2}\cdot\ensuremath{I_{\mathrm{s}}}\cdot\left(\exp\left(\frac... ...math{n_{\mathrm{cl}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}\right) - 1\right)$$

  $$+ \left(\frac{\ensuremath{V_{\mathrm{be}}}}{\ensuremath{\beta_{\m... ...c}}}}{\ensuremath{\beta_{\mathrm{r}}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ (C6)

  
  

• off region: $\ensuremath{V_{\mathrm{be}}}\leq -\frac{5\cdot\ensuremath{n_{\mathrm{f}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$ and $\ensuremath{V_{\mathrm{bc}}}\leq -\frac{5\cdot\ensuremath{n_{\mathrm{r}}}\cdot\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}$

  $$\frac{\ensuremath{I_{\mathrm{s}}}}{\ensuremath{\beta_{\mathrm{r}}... ...ight)\cdot\ensuremath{V_{\mathrm{bc}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ I_c =   (C7)

  $$\ensuremath{I_{\mathrm{b}}}=\, -\ensuremath{I_{\mathrm{s}}}\cdot\frac{\ensuremath{\beta_{\mathrm... ...c}}}}{\ensuremath{\beta_{\mathrm{r}}}}\right)\cdot\ensuremath{G_{\mathrm{min}}}$$ (C8)

  
  

In these equations G_min is the minimum conductance which is automatically switched in parallel to each pn-junction.