118
3 RLC Networks Equations and Analysis Methods
C 1
C 2
+
−
v 1
i1
N
vC 1
iC 1
+
−
+
−
v2
i2
vC 2
iC 2
+
−
(a)
v 1
v 2
+
−
v 1
i1
N
+
−
v2
i2
(b)
Fig. 3.9 (a) Decomposition of a linear network with two capacitors and (b) resistive circuit
obtained by replacing each capacitor with a voltage source
Remark 3.5 If there is a loop formed exclusively by capacitors and independent
voltage sources, then the capacitor voltages are dependent variables and the SE
representation thus described does not exist. Yet, it may be in general still possible
to write an SE using a reduced number of state variables. Systematic methods
to write the SEs when there are such loops (and, dually, there are cut-sets made
by inductors and current sources only) are discussed in [2]. Such methods, which
can be applied also to nonlinear RLC networks, usually involve, however, a huge
amount of algebra. An alternative and more effective technique is to use suitable
circuit transformations that enable to eliminate such undesired capacitor loops (and
inductor cut-sets) [4, Th. 2].
Consider now the configuration with a capacitor and an inductor as in Fig. 3.10a,
where the two-port network N contains the linear resistors and the independent
voltage and current sources. We have
i C 1 = C 1
dv C 1
dt
= −i 1 , v C 1 = v 1
3 RLC Networks Equations and Analysis Methods
C 1
C 2
+
−
v 1
i1
N
vC 1
iC 1
+
−
+
−
v2
i2
vC 2
iC 2
+
−
(a)
v 1
v 2
+
−
v 1
i1
N
+
−
v2
i2
(b)
Fig. 3.9 (a) Decomposition of a linear network with two capacitors and (b) resistive circuit
obtained by replacing each capacitor with a voltage source
Remark 3.5 If there is a loop formed exclusively by capacitors and independent
voltage sources, then the capacitor voltages are dependent variables and the SE
representation thus described does not exist. Yet, it may be in general still possible
to write an SE using a reduced number of state variables. Systematic methods
to write the SEs when there are such loops (and, dually, there are cut-sets made
by inductors and current sources only) are discussed in [2]. Such methods, which
can be applied also to nonlinear RLC networks, usually involve, however, a huge
amount of algebra. An alternative and more effective technique is to use suitable
circuit transformations that enable to eliminate such undesired capacitor loops (and
inductor cut-sets) [4, Th. 2].
Consider now the configuration with a capacitor and an inductor as in Fig. 3.10a,
where the two-port network N contains the linear resistors and the independent
voltage and current sources. We have
i C 1 = C 1
dv C 1
dt
= −i 1 , v C 1 = v 1