1.2 Four Basic Two-Terminal Circuit Elements
11
As already stated in Example 1.2, a nonlinear resistor is said to be currentcontrolled if it is possible to explicitly write v = ˆ
v(i), i.e., the voltage is a
(single-valued) function of the current (Fig. 1.3a). Similarly, a nonlinear resistor is
voltage-controlled if it is possible to explicitly write i = ˆ
i(v), i.e., the current is a
(single-valued) function of the voltage (Fig. 1.3b).
The power delivered to the resistor at time t by the remainder of the circuit to
which it is connected is given by p(t) = v(t)i(t). Obviously, p(t) ≥ 0 if and only
if v(t) and i(t) have the same sign for all t. We call a two-terminal resistor passive
if and only if its characteristic lies in the closed first and third quadrants of the v − i
plane or the i − v plane. A resistor is said to be active if it is not passive.
Consider a current-controlled nonlinear resistor v = ˆ
v(i). The resistor is passive
if and only if i ˆ
v(i) ≥ 0 for any i. It is said to be eventually passive if it absorbs
power for all large values of the current, i.e., there exists ˜
i > 0 such that i ˆ
v(i) ≥ 0
for any |i| ≥ ˜
i.
Let us consider an operating point P = (i P , ˆ
v(i P )), i.e., a point on the
characteristic of a current-controlled nonlinear resistor. The slope R(i P ) of the
characteristic at P is named small-signal or differential resistance of the resistor
at P , i.e.,
R(i P ) = ˆ
v
(i P ) =
d ˆ
v(i)
di
i P
.
The resistor is said to be locally passive at P if we have R(i P ) ≥ 0. Otherwise,
if R(i P ) < 0, it is said to be locally active at P .
In the particular case where the CR is a horizontal line, i.e., v(t) = E = const, or
a vertical line, i.e., i(t) = I = const, the resistor coincides with a (constant) voltage
source or a current source, respectively. A voltage source such that E = 0 is called
a short circuit, whereas a current source such that I = 0 is called an open circuit.
It is remarkable that several commercially available two-terminal devices are
described by a nonlinear function of voltage and current, e.g., pn-junction diodes,
zener diodes, varistors, tunnel diodes, etc. Such devices can be realistically modeled
in a broad range of voltages, currents, and frequencies, as two-terminal nonlinear
resistors.
Example 1.5 (pn-Junction Diode) The pn-junction diode is modeled by a voltagecontrolled nonlinear resistor with CR given by the “diode junction law”
i(t) = I s (exp (v(t)/V T ) − 1)
where I s is the reverse saturation current (a constant on the order of microamperes)
and V T is the thermal voltage (V T = 26 mV at room temperature). The characteristic is reported in Fig. 1.4. Note that the pn-junction diode has a monotone strictly
increasing characteristic i = ˆ
i(v), hence the differential resistance is always positive
and the pn-junction diode is locally passive at any operating point P . Moreover, the
characteristic is also current controlled in the interval (−I s , +∞).
11
As already stated in Example 1.2, a nonlinear resistor is said to be currentcontrolled if it is possible to explicitly write v = ˆ
v(i), i.e., the voltage is a
(single-valued) function of the current (Fig. 1.3a). Similarly, a nonlinear resistor is
voltage-controlled if it is possible to explicitly write i = ˆ
i(v), i.e., the current is a
(single-valued) function of the voltage (Fig. 1.3b).
The power delivered to the resistor at time t by the remainder of the circuit to
which it is connected is given by p(t) = v(t)i(t). Obviously, p(t) ≥ 0 if and only
if v(t) and i(t) have the same sign for all t. We call a two-terminal resistor passive
if and only if its characteristic lies in the closed first and third quadrants of the v − i
plane or the i − v plane. A resistor is said to be active if it is not passive.
Consider a current-controlled nonlinear resistor v = ˆ
v(i). The resistor is passive
if and only if i ˆ
v(i) ≥ 0 for any i. It is said to be eventually passive if it absorbs
power for all large values of the current, i.e., there exists ˜
i > 0 such that i ˆ
v(i) ≥ 0
for any |i| ≥ ˜
i.
Let us consider an operating point P = (i P , ˆ
v(i P )), i.e., a point on the
characteristic of a current-controlled nonlinear resistor. The slope R(i P ) of the
characteristic at P is named small-signal or differential resistance of the resistor
at P , i.e.,
R(i P ) = ˆ
v
(i P ) =
d ˆ
v(i)
di
i P
.
The resistor is said to be locally passive at P if we have R(i P ) ≥ 0. Otherwise,
if R(i P ) < 0, it is said to be locally active at P .
In the particular case where the CR is a horizontal line, i.e., v(t) = E = const, or
a vertical line, i.e., i(t) = I = const, the resistor coincides with a (constant) voltage
source or a current source, respectively. A voltage source such that E = 0 is called
a short circuit, whereas a current source such that I = 0 is called an open circuit.
It is remarkable that several commercially available two-terminal devices are
described by a nonlinear function of voltage and current, e.g., pn-junction diodes,
zener diodes, varistors, tunnel diodes, etc. Such devices can be realistically modeled
in a broad range of voltages, currents, and frequencies, as two-terminal nonlinear
resistors.
Example 1.5 (pn-Junction Diode) The pn-junction diode is modeled by a voltagecontrolled nonlinear resistor with CR given by the “diode junction law”
i(t) = I s (exp (v(t)/V T ) − 1)
where I s is the reverse saturation current (a constant on the order of microamperes)
and V T is the thermal voltage (V T = 26 mV at room temperature). The characteristic is reported in Fig. 1.4. Note that the pn-junction diode has a monotone strictly
increasing characteristic i = ˆ
i(v), hence the differential resistance is always positive
and the pn-junction diode is locally passive at any operating point P . Moreover, the
characteristic is also current controlled in the interval (−I s , +∞).