A simple model that describes dc (large-signal) behavior [14] of a bipolar junction transistor (BJT)
is the Ebers-Moll model [11]. The model has been used in a number of analytical studies. Field-
effect transistors (FETs) do not possess such a simple, mathematically tractable, large-signal model.
Nonetheless, many of the theoretical results related to BJT circuits have been extended to include
circuits with FETs [67].
Two important albeit simple attributes of BJT and FET transistors are their “passivity” [17] and
“no-gain” [66] PROPERTIES. These properties have proved instrumental in establishing theoretical results
dealing with dc OPERATING points as well as in DESIGNING algorithms for solving equations describing
transistor circuits [53]. When considering their dc behavior, transistors are passive devices, which
implies that at any dc operating POINT the net power delivered to the device is nonnegative. They are
also no-gain and, hence, are incapable of producing voltage or current gains. Subsequently, passivity
is a consequence of the no-gain property.
By using the Ebers-Moll transistor model, the large-signal dc behavior of an arbitrary circuit
containing n/2 bipolar transistors may be described with an equation of the form
QT F(v) + P v + c = 0. (1)
The real n × n matrices P and Q and the real n-vector c, where
P v + Qi + c = 0, (2)
describe the linear multiport that connects the nonlinear transistors. The real matrix T, a block
diagonal matrix with 2 × 2 diagonal blocks of the form
Ti =
-
1 −αi+1
−αi 1

, (3)
and
F(v) ≡ (f1(v1),...,fn(vn))T (4)
capture the presence of the nonlinear elements. The controlled-source current-gains αk, k = 1, 2, lie
within the open interval (0, 1). The FUNCTIONS fk: R1 → R1 are continuous and strictly monotone
increasing. Typically,
i = fk(v) ≡ mk(enkv − 1), (5)
where the real numbers mk, nk are positive when modeling a pnp transistor and negative for an npn
transistor. They satisfy the reciprocity condition:
miαi = mi+1αi+1, for i odd. (6)
The nonlinear elements are described via the equation
i = T F(v). (7)
Hence,
AF(v) + Bv + c = 0, (8)
where A = QT and B = P. This equation represents a general description of an arbitrary nonlinear
transistor circuit. Its solutions are the circuit’s dc operating points.
The determinant det(AD + B) is the Jacobian of the mapping AF(v) + Bv + c evaluated at the
point v, where
D = diag(d1, d2,...,dn), (9)
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