5.2 Forms of Kirchhoff Laws in the Flux-Charge Domain
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2. Second Form of Kirchhoff Laws in the (ϕ, q)-domain.
Suppose that −∞ < t 0 < +∞ is a finite initial instant and we are interested
in applying Kirchhoff laws to analyze a circuit for t ≥ t 0 . If we integrate KCL
and KVL in the interval [t 0 , t], where t ≥ t 0 , we obtain
k
q k (t) =
k
q k (t 0 )
(5.5)
for any cut-set and any t ≥ t 0 and
k
ϕ k (t) =
k
ϕ k (t 0 )
(5.6)
around any loop and for any t ≥ t 0 , where
q k (t 0 ) =
t 0
−∞
i k (τ )dτ
and
ϕ k (t 0 ) =
t 0
−∞
v k (τ )dτ
are the initial charge and initial flux at t 0 , respectively.
3. Third Form of Kirchhoff Laws in the (ϕ, q)-domain.
Suppose again that t 0 is a finite initial instant and define for any t ≥ t 0 the
incremental charge and incremental flux of a two-terminal element as follows:
q k (t; t 0 ) = q k (t) − q k (t 0 ) =
t
t 0
i k (τ )dτ
and
ϕ k (t; t 0 ) = ϕ k (t) − ϕ k (t 0 ) =
t
t 0
v k (τ )dτ
respectively. Then, we can write
k
q k (t; t 0 ) = 0
(5.7)
for any cut-set and any t ≥ t 0 and
k
ϕ k (t; t 0 ) = 0
(5.8)
around any loop and for any t ≥ t 0 , respectively.