PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

134 Practical MATLAB® Applications for Engineers

where

wwdo
^22 

where A 1 and A 2 are constants that can be evaluated from the initial network
conditions.

R.2.114 Let us now turn our attention to the source-free RLC series circuit shown in
Figure 2.32, assuming for simplicity that all the initial conditions are zero.
The loop differential equation of the circuit of Figure 2.32 is given by

1

0

C

i t dt L

di t

dt

() Ri t

()

∫   ()
.

Differentiating every term of the preceding equation with respect to t yields

L

dit

dt

R

di t

dt C

it

2

2

1

0

() ()


()

or

dit

dt

R

L

di t

dt

it

2

2

1

0

() ()


().

CL

The preceding equation is a second-order, linear, homogeneous differential

equation, and the auxiliary equation is given by

s

R

L

(^2) s

1


0

LC

and the roots of this equation are

s

R

L

R

(^12) L
2
22

1

,




 





 LC

where α = _ 2 RL is referred as the neper frequency, wo = ___ √___ 1 LC is the resonant

frequency, then s1,2 = −α ± (^) √

___

α^2 − w (^) o^2 are referred to as the complex network
frequencies.
FIGURE 2.32
Source free series RLC of R.2.114.
R
L
C