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3 RLC Networks Equations and Analysis Methods
Property 3.1 A set of sufficient conditions for the existence of the hybrid representation (3.54) of the considered RLC circuit is as follows.
1. There is no loop formed exclusively by capacitors, inductors, and/or independent
voltage sources. Furthermore, there is no cut-set formed exclusively by capacitors, inductors, and/or independent current sources.
2. Each voltage-controlled (but not current-controlled) resistor is in parallel with a
capacitor. Each current-controlled (but not voltage-controlled) resistor is in series
with an inductor.
3. Each remaining two-terminal resistor is strongly passive or else it is either in
parallel with a capacitor or in series with an inductor.
A nonlinear resistor v = ˆ
v R (i) is said to be strongly passive if there exist
constants 0 < γ < γ such that γ ≤ ( ˆ
v R (i 1 ) − ˆ
v R (i 2 ))/(i 1 − i 2 ) ≤ γ for
any i 1 = i 2 .
Note that the conditions in Property 3.1 are couched in topological terms and as
such can be usually checked by inspection on a given circuit. When these conditions
fail, the SE representation of a nonlinear RLC circuit is not guaranteed to exist
globally due for example to the presence of impasse points (see Chap. 4).
Remark 3.7 In the treatment we assumed that capacitors, inductors, and resistors
are uncoupled. It would be not difficult to extend the treatment to multiterminal
or multiport capacitors, inductors, or resistors. The interested readers can find a
detailed discussion in [4].
Example 3.7 Consider the network in Fig. 3.16a with a linear capacitor C 1 , a
nonlinear flux-controlled inductor L 1 defined by i L 1 = ϕ 3
L 1
, and a linear inductor
L 2 , that are connected to a resistive networks with a nonlinear resistor i R 2 = v 2
R 2
.
It can be immediately checked that the conditions for the existence of the SE
representation are satisfied. By replacing C 1 with a voltage source and L 1 , L 2
with current sources, we obtain the network in Fig. 3.16b. By analyzing the linear
memoryless network in Fig. 3.16b we easily obtain the following.
KCL at cut-set C yields
i a 1 = v
2
a 1
+
v a 1
R 3
− i b 1 − i b 2 −
sin(2t)
R 3
.
We can also write the two KVLs
v b 1 = v a 1 + R 1 i b 1
and
v b 2 = v a 1 − sin(2t).
We have v C 1 = v a 1 , i L 1 = i b 1 = ϕ 3
L 1
, and i L 2 = i b 2 , while i C 1 = −i a 1 , v L 1 = −v b 1 ,
and v L 2 = −v b 2 . By substitution, we obtain the third-order non-autonomous SE in
3 RLC Networks Equations and Analysis Methods
Property 3.1 A set of sufficient conditions for the existence of the hybrid representation (3.54) of the considered RLC circuit is as follows.
1. There is no loop formed exclusively by capacitors, inductors, and/or independent
voltage sources. Furthermore, there is no cut-set formed exclusively by capacitors, inductors, and/or independent current sources.
2. Each voltage-controlled (but not current-controlled) resistor is in parallel with a
capacitor. Each current-controlled (but not voltage-controlled) resistor is in series
with an inductor.
3. Each remaining two-terminal resistor is strongly passive or else it is either in
parallel with a capacitor or in series with an inductor.
A nonlinear resistor v = ˆ
v R (i) is said to be strongly passive if there exist
constants 0 < γ < γ such that γ ≤ ( ˆ
v R (i 1 ) − ˆ
v R (i 2 ))/(i 1 − i 2 ) ≤ γ for
any i 1 = i 2 .
Note that the conditions in Property 3.1 are couched in topological terms and as
such can be usually checked by inspection on a given circuit. When these conditions
fail, the SE representation of a nonlinear RLC circuit is not guaranteed to exist
globally due for example to the presence of impasse points (see Chap. 4).
Remark 3.7 In the treatment we assumed that capacitors, inductors, and resistors
are uncoupled. It would be not difficult to extend the treatment to multiterminal
or multiport capacitors, inductors, or resistors. The interested readers can find a
detailed discussion in [4].
Example 3.7 Consider the network in Fig. 3.16a with a linear capacitor C 1 , a
nonlinear flux-controlled inductor L 1 defined by i L 1 = ϕ 3
L 1
, and a linear inductor
L 2 , that are connected to a resistive networks with a nonlinear resistor i R 2 = v 2
R 2
.
It can be immediately checked that the conditions for the existence of the SE
representation are satisfied. By replacing C 1 with a voltage source and L 1 , L 2
with current sources, we obtain the network in Fig. 3.16b. By analyzing the linear
memoryless network in Fig. 3.16b we easily obtain the following.
KCL at cut-set C yields
i a 1 = v
2
a 1
+
v a 1
R 3
− i b 1 − i b 2 −
sin(2t)
R 3
.
We can also write the two KVLs
v b 1 = v a 1 + R 1 i b 1
and
v b 2 = v a 1 − sin(2t).
We have v C 1 = v a 1 , i L 1 = i b 1 = ϕ 3
L 1
, and i L 2 = i b 2 , while i C 1 = −i a 1 , v L 1 = −v b 1 ,
and v L 2 = −v b 2 . By substitution, we obtain the third-order non-autonomous SE in