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98 2. Tutorial
In[8]:= dnwsol = CompressNonlinearEquations[dnwmna, I$AC$D1,
EliminateVariables −> All];
DisplayForm[dnwsol]
Out[9]//DisplayForm=
I$AC$D1 ⩵⩵ 1
I$AC$D1 R1V0
Vt
Is$D1, I$AC$D1, DesignPoint
The implicit equation is what we have been looking for. Without resorting to approximation methods
such as Taylor series we cannot simplify the result any further and solve for the diode current
analytically.
2.6.5 Multi-Dimensional Models
The Ebers-Moll Transistor Model
The procedure for modeling one-dimensional nonlinear element characteristics can be easily extended
to the multi-dimensional case. Let’s demonstrate multi-dimensional device modeling on a practical
example by defining nonlinear DC models for a bipolar junction transistor (see Figure 6.4).
IC
VBC
C
IB
B
VCE
VBE
E
IE
Figure 6.4: NPN transistor
Our considerations will be based on the BJT model introduced by Ebers and Moll which expresses
the relations between the transistor currents and voltages I C , I E , V BE , and V BC as
I C
I E
⩵ I S e V BEV t
I S
Α r
e V BCV t
⩵ I S
Α f
e V BEV t
I S e V BCV t
In these equations, the parameter I S represents the transport saturation current, Α r and Α f denote the
large-signal reverse and forward current gains of the common base configuration, and V t designates
the thermal voltage. In integrated circuit design, the saturation current is usually expressed as the
product of the transport saturation current density J S , which is a process parameter, and the emitter
area A, which is a design parameter: