Page 48 - Circuit Analysis II with MATLAB Applications

Chapter 1 Second Order Circuits

1.9 Summary

x Circuits that contain energy storing devices can be described by integrodifferential equations and
upon differentiation can be simplified to differential equations with constant coefficients.

x A second order circuit contains two energy storing devices. Thus, an RLC circuit is a second order
circuit.

x The total response is the summation of the natural and forced responses.

x If the differential equation describing a series RLC circuit that is excited by a constant (DC) volt-
age source is written in terms of the current, the forced response is zero and thus the total
response is just the natural response.

x If the differential equation describing a parallel RLC circuit that is excited by a constant (DC) cur-
rent source is written in terms of the voltage, the forced response is zero and thus the total
response is just the natural response.

x If a circuit is excited by a sinusoidal (AC) source, the forced response is never zero.
x The natural response of a second order circuit may be overdamped, critically damped, or under-
damped depending on the values of the circuit constants.

s
s
x For a series RLC circuit, the roots and are found from
1
2
2
2
2
s s = – D S r D – Z = – D S r E S if D ! Z 2 0

1
S
2
S
0
or
2
2
2
s s = – D S r Z – D = – D S r Z nS if Z ! D 2 S

2
0
0
1
S
where
R
1
2
2
D = ------ Z = ----------- E = D – Z 2 0 Z nS = Z – D S 2
S
0
S
0
S
2L
LC
2 2
s
s
If D ! Z 0 , the roots and are real, negative, and unequal. This results in the overdamped nat-
2
1
S
ural response and has the form
s t s t
i t = k e 1 + k e 2
2
1
n
2
If D = Z 2 0 , the roots and are real, negative, and equal. This results in the critically dampeds 1 s 2
S
natural response and has the form
– D t
S
i t = e k + k t
2
1
n
1-36 Circuit Analysis II with MATLAB Applications
Orchard Publications