4.1 First-Order Circuits
137
Fig. 4.7 Dynamic route of
circuit with inductor and
tunnel diode
not vanish at Q B , hence, Q B is a forward impasse points and solutions of the circuit
cannot be continued forward in time after they reach Q B . Similarly, it can be seen
that solutions staring with voltages less than v B , reach in finite backward time point
Q A and they cannot be further prolonged backward in time. Since Q A is not an EP,
such a point represents a backward impasse point.
Example 4.4 (Multiple Operating Point Paradox) Consider the resistive circuit with
a dc voltage source E and a tunnel diode in Fig. 4.8a (cf. Example 2.8 in Ch. 2).
The dc solutions of the circuit, which are named dc operating points, are obtained
by intersecting the load line E = ri + v with the diode characteristic i = g(v).
Suppose E and R are chosen so that there are three distinct operating points ¯
v 1 , ¯
v 2 ,
and ¯
v 3 (Fig. 4.8b).
The existence of multiple solutions leads to a paradoxical situation since in any
actual single laboratory experiment on the corresponding physical circuit we would
observe only one operating point! The considered circuit is clearly not a realistic
model of the physical circuit. In fact, in physical circuits, parasitic inductances and
capacitances always exist. When the circuit has a unique solution, parasitics can
often be neglected without significant errors. Instead, in circuits exhibiting multiple
operating points, one or more of these parasitic elements cannot be neglected, no
matter how small their values are.
To solve the previous paradox, suppose to insert in parallel to the tunnel diode
a small parasitic inductance C p modeling the electric field between the diode
terminals as shown in Fig. 4.8c. We then obtain a dynamic first-order nonlinear
circuit obeying the SE
dv C
dt
= −g(v C ) +
E − v C
R
where v C is the capacitor voltage and we let C = 1. It is easily seen that there
are three EPs ¯
v C 1 = ¯
v 1 , ¯
v C 2 = ¯
v 2 , and ¯
v C 3 = ¯
v 3 that coincide with the operating
137
Fig. 4.7 Dynamic route of
circuit with inductor and
tunnel diode
not vanish at Q B , hence, Q B is a forward impasse points and solutions of the circuit
cannot be continued forward in time after they reach Q B . Similarly, it can be seen
that solutions staring with voltages less than v B , reach in finite backward time point
Q A and they cannot be further prolonged backward in time. Since Q A is not an EP,
such a point represents a backward impasse point.
Example 4.4 (Multiple Operating Point Paradox) Consider the resistive circuit with
a dc voltage source E and a tunnel diode in Fig. 4.8a (cf. Example 2.8 in Ch. 2).
The dc solutions of the circuit, which are named dc operating points, are obtained
by intersecting the load line E = ri + v with the diode characteristic i = g(v).
Suppose E and R are chosen so that there are three distinct operating points ¯
v 1 , ¯
v 2 ,
and ¯
v 3 (Fig. 4.8b).
The existence of multiple solutions leads to a paradoxical situation since in any
actual single laboratory experiment on the corresponding physical circuit we would
observe only one operating point! The considered circuit is clearly not a realistic
model of the physical circuit. In fact, in physical circuits, parasitic inductances and
capacitances always exist. When the circuit has a unique solution, parasitics can
often be neglected without significant errors. Instead, in circuits exhibiting multiple
operating points, one or more of these parasitic elements cannot be neglected, no
matter how small their values are.
To solve the previous paradox, suppose to insert in parallel to the tunnel diode
a small parasitic inductance C p modeling the electric field between the diode
terminals as shown in Fig. 4.8c. We then obtain a dynamic first-order nonlinear
circuit obeying the SE
dv C
dt
= −g(v C ) +
E − v C
R
where v C is the capacitor voltage and we let C = 1. It is easily seen that there
are three EPs ¯
v C 1 = ¯
v 1 , ¯
v C 2 = ¯
v 2 , and ¯
v C 3 = ¯
v 3 that coincide with the operating