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12.11 H I G H E R - O R D E R C I R C U I T S *
To close this chapter, we briefly consider the analysis of circuits having an order
higher than two. The important message here is that the methods of analysis
developed earlier in this chapter for second-order circuits are perfectly applica-
ble to the analysis of higher-order circuits. We will illustrate this through the
analysis of the circuit shown in Figure 12.63. Since the circuit has two inde-
pendent capacitors and two independent inductors, it is a fourth-order circuit.
Despite this, it readily submits to both a node analysis with the node voltages
as the primary unknowns, and a state-variable analysis with the states as the
primary unknowns.
Consider first the node analysis of the circuit shown in Figure 12.63, carried
out using v 1 and v 2 and as the two unknown node voltages. To begin, we write
KCL at Nodes #1 and #2 in terms of these voltages. This yields
C 1
dv 1 ( t )
dt
R
( v 1 ( t ) − v 2 ( t )) +
L 1
t
−∞
( v 1 (
t ) − v IN (
t )) d
t = 0 (12.288)
for Node #1, and
C 2
dv 2 ( t )
dt
dv IN ( t )
dt
R
( v 2 ( t )− v 1 ( t ))+
L 2
t
−∞
v 2 (
t ) d
t − i IN = 0 (12.289)
for Node #2. To treat these equations simultaneously, we use Equa-
tion 12.288 to determine v 2 in terms of v 1 , and then substitute the result
R
L 1
C 1 L 2
C 2
i IN
v IN v 1 v 2
i L 1
i L 2
vC 1
v C 2
v IN
Node 1 Node 2
F I G U R E 12.63 A fourth-order
circuit.
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into Equation 12.289. This yields
v 2 ( t ) = RC 1
dv 1 ( t )
dt
R
L 1
t
−∞
( v 1 (
t ) − v IN ( t )(
t )) d
t (12.290)
d 4 v 1 ( t )
dt 4
RC 1
RC 2
d 3 v 1 ( t )
dt 3
L 1 C 1
L 2 C 2
d 2 v 1 ( t )
dt 2
RC 1 L 2 C 2
RC 2 L 1 C 1
dv 1 ( t )
dt
L 1 C 1 L 2 C 2
v 1 ( t ) =
RC 1
d
3 v IN ( t )
dt 3
L 1 C 1
d
2 v IN ( t )
dt 2
RC 2 L 1 C 1
dv IN ( t )
dt
L 1 C 1 L 2 C 2
v IN ( t ) +
RC 1 C 2
d 2 i IN ( t )
dt
Note that in deriving Equation 12.291 we differentiated Equation 12.289 twice
and divided it by RC 1 C 2 prior to the substitution of Equation 12.290. Finally,
to complete the node analysis, we solve Equation 12.291 for v 1 , substitute the
result into Equation 12.290 to determine v 2 , and then use the two node voltages
to determine any other branch variables of interest. For brevity, however, we
will not carry out these remaining steps. Instead, we note that to do so requires
initial conditions for v 1 , and its first, second, and third derivatives. Most likely,
this information will be determined from the state variables specified at the
initial time, which takes additional work.
Consider next a state-variable analysis of the circuit shown in Figure 12.63.
To carry out this analysis we determine the state equation for each capacitor
and inductor. This yields,
C 1
dv C1 ( t )
dt
= i C1 ( t ) =
R
( v IN ( t ) − v C1 ( t ) − v C2 ( t )) + i L1 ( t ) (12.292)
C 2
dv C2 ( t )
dt
= i C2 ( t ) =
R
( v IN ( t ) − v C1 ( t ) − v C2 ( t )) + i L2 ( t ) − i IN ( t ) (12.293)
L 1
di L1 ( t )
dt
= v L1 ( t ) = v IN ( t ) − v C1 ( t ) (12.294)
L 2
di L2 ( t )
dt
= v L2 ( t ) = v IN ( t ) − v C2 ( t ). (12.295)
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