PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Direct Current and Transient Analysis 135

Observe that the term (similar to the parallel case) α^2 − w^2 o can be positive, zero,

or negative. Then the respective solutions are

Overdamped

Critical damped

Underdamped

R.2.115 The natural overdamped RLC series circuit response occurs for the condition α^2 > w (^) o^2.
Then the roots s 1 and s 2 are real, unequal, and negative numbers, resulting in a
solution of the form i(t) = A 1 e−s^1 t + A 2 e−s^2 t, where the constants A 1 and A 2 depend
on the initial network conditions ( usually given by i(t = 0) and
di(t)

dt
(^) 
t = 0
(^) ).
R.2.116 The natural critical-damped RLC series circuit response occurs for the condition
α^2 = w (^) o^2. Then the roots s 1 and s 2 are real (−α) and repeated, resulting in a solution
of the form i(t) = A 1 e−αt + A 2 te−αt, where the constants A 1 and A 2 can be evaluated
from the system’s initial conditions.
R.2.117 The natural underdamped RLC series circuit response occurs for the condition
α^2 < w (^) o^2. Then the roots s 1 and s 2 are a complex conjugate pair, resulting in a solu-
tion of the form
it e A wt A wt
()
t[ cos( 12 dd) sin( )]
where wd = (^) √

___

w (^) o^2 − α 2 and A 1 and A 2 are constants that depend on the initial
network conditions.
R.2.118 The transient analysis of a circuit that contains more than one loop can be described
by a set of differential equations. The set of differential equations may consist of
either loop or node equations where the unknowns may be either the loop cur-
rents or the node voltages. For example, the set of two loop differential equations
(using KVL) for the circuit shown in Figure 2.33 is illustrated, where the two loop
currents defi ne the system’s transient response.

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FIGURE 2.33

Transient electrical network of R.2.118.

R 1

R 3

R 2

L 3

L 1

L 2

i 1 (t) i 2 (t)