Direct Current and Transient Analysis 127
reach a steady state instantaneously. In contrast, if storing energy elements are
present in a circuit, such as inductors or capacitors, and switching occurs, then
transient takes place due to the trapped network voltages and currents, and it takes
time to settle into the new stable values.
In brief, transients exist in a circuit if and only if, at any time, a sudden change is
made (switched) and energy-storing devices are present.
R.2.89 Transient solutions involve differential equations in which initial conditions must
be considered. These solutions require the inclusion of the energy-storing devices
(capacitors and inductors) as well as the trapped conditions.
R.2.90 Transient solutions involve either growth or decay time exponentials, and are
dependent on the exponential coeffi cient referred as the network time constant (τ).
R.2.91 The time constant can be either RC or L/R or a combination or superposition of
time constants for the case of more complex circuits.
R.2.92 The time constant (τ) is the time the circuit requires to complete 63.2% of the change
(or discharge) that ultimately takes place. Practical considerations are associated
with the time constant, for example, four or fi ve time constants is the time that a
circuit needs to reach the steady-state conditions, and after that time, all transients
die out or are no longer present.
R.2.93 The transient solution of a given circuit depends on the number of independent
energy-storing elements (capacitors and inductors), and not on the number of the
network loops or nodes. The reader should observe that capacitors or inductors in
series or parallel are not independent, since they can easily be combined. Similarly,
a delta (∆) or Y connection consisting of three inductors or three capacitors, one in
each branch, leads to just two independent energy-storing elements, since the third
element depends on the value of the other two.
R.2.94 Electric circuits with one capacitor or inductor lead to fi rst-order ordinary differ-
ential equations, regardless of the structure or complexity of the circuit. Complex
circuits with only one type of energy-storing device can be analyzed by using
Thevenin’s theorem, assuming that the load is the energy-storing element.
R.2.95 Let us start the transient analysis by considering the RL series circuit shown in
Figure 2.27. The switch shown in the circuit closes at t = 0 (it is open for t < 0), and
let us assume that the coil is discharged, that is, i(0) = 0. Then by applying KVL, a
fi rst-order linear differential equation is obtained, and then solved in terms of the
unknown current i(t), illustrated as follows:Ri t Ldi t
dt() V
()
FIGURE 2.26
Load RL is increased to 15 Ω from 5 Ω (potentiometer) for maximum power transfer of the network of R.2.83.VTH = 60 V R
L = 15 ΩRTH = 15 Ω