3.3 State Equations
119
C 1
L 2
+
−
v 1
i1
N
vC 1
iC 1
+
−
+
−
v2
i2
vL 2
iL 2
−
+
(a)
v 1
i 2
+
−
v 1
i1
N
+
−
v2
i2
(b)
Fig. 3.10 (a) Decomposition of a linear network with a capacitor and an inductor and (b) resistive
circuit obtained by replacing the capacitor with a voltage source and the inductor with a current
source
and
v L 2 = L 2
di L 2
dt
= −v 2 , i L 2 = i 2 .
Suppose to replace the capacitor by a voltage sources v 1 and the inductor by
a current source i 2 , as shown in Fig. 3.10b. If the resistive circuit thus obtained is
uniquely solvable for any v 1 and i 2 , and for any value of the currents and voltages
impressed by the independent sources, i.e., v 1 and i 2 are independent variables, a
standard two-port representation theorem for N yields the hybrid representation [1]
i 1 (t) = h 11 v 1 (t) + h 12 i 2 (t) + i s 1 (t)
(3.47)
v 2 (t) = h 21 v 1 (t) + h 22 i 2 (t) + v s 2 (t)
(3.48)
where h 11 , h 12 , h 21 , h 22 are constants and i s 1 (t), v s 2 (t) are time functions that
depend on the independent sources within N.