Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

12.4 
StAndArd FormAtS For Second-order cIrcuIt Zero-Input reSponSe 
Second-order circuits embody all the three kinds of responses possible in any linear system. It is a 
widely studied and applied response. Hence certain standard formats have emerged in dealing with 
second-order system response in the areas of circuit theory, linear control systems, linear electronic 
circuits, linear system analysis, etc. We have already introduced many terms associated with this 
standardised format and we complete the rest in this section.
A second-order circuit will have a homogeneous differential equation of the following type where 
x(t) is some circuit variable chosen as the describing variable for the circuit.
d x
dt
a
dx
dt
a x
t
x
x
dx
dt
x
o
2
2
1
2
0
0
0
0
+
+
=
≥
=
= ′
+
−
−
for
with 
and
o
( )
(
)
(12.4-1)
The coefficients of this differential equation, a
1
and a
2
, will be decided by circuit parameters like R, 
L and C. The same differential equation will be obtained quite independent of which particular circuit 
variable happens to be chosen as the describing variable x(t).
The first standard format for this equation is 
d x
dt
dx
dt
x
t
x
x
dx
dt
n
n
o
2
2
2
0
2
0
0
0
+
+
=
≥
=
=
+
−
−
xw
w
for
with 
and
( )
(
)
′′
=
=
x
a
a
a
o
n
where
and 
w
x
2
1
2
2
(12.4-2)
The second standard format is 
d x
dt
Q
dx
dt
x
t
x
x
dx
dt
n
n
o
2
2
2
0
0
0
0
+
+
=
≥
=
= ′
+
−
−
w
w
for
with 
and
( )
(
)
xx
a
Q
a
a
o
n
where 
and
w
x
=
=
=
2
2
1
1
2
(12.4-3)

---

12.18


SeriesandParallel
RLC
Circuits
The 
x
factor was seen to be responsible for damping of oscillations in series RLC circuit and was 
called damping factor. 
w
n
was seen to be the angular frequency of voltage and current sinusoidal 
oscillations in an undamped LC circuit and was called undamped natural frequency. These 
interpretations for 
x
and 
w
n
are applicable to any second-order circuit or linear system.
The new factor that has appeared in Eqn. 12.4-3 – the Q factor – is called the quality factor of the 
second-order circuit or system and is inversely proportional to its damping factor. Damping factor and 
Quality factor are used interchangeably in linear system related studies with electrical filter designers 
preferring quality factor to damping factor.
Reduced damping leads higher values for Q factor. An undamped system has a quality factor that 
tends to infinity.
Four cases are identified depending on the value of