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3 RLC Networks Equations and Analysis Methods
which reduces to Tellegen’s theorem [1]
v
T i = 0
(3.27)
by using (3.8) and (3.23).
Remark 3.1 Tellegen’s theorem can also be proved by means of the reduced
incidence matrix A. In fact, deriving v T = v T
e A from (3.12), and multiplying by
i, it follows that
v
T i = v
T
e Ai = 0
(3.28)
due to KCL equations Ai = 0 in (3.10).
3.2 Tableau Analysis
The tableau analysis is based on the network equations obtained by combining KCL
and KVL equations with the branch characteristics expressing the CRs of circuits
elements composing the network. The following nonlinear integro-differentialalgebraic equation describes the k-th (α k , β k )-element
f k (v
(α k )
k , i
(β k )
k ) = 0
(3.29)
where v
(α k )
k
and i
(β k )
k
(k = 1, 2, . . . , b) are the higher-order integral or derivative
of the branch voltage and current, respectively, as defined in Chap. 1. Combining
CRs and Kirchhoff equations we have the following three forms of the tableau
equations:
• cut-set tableau equations: basic variables are the twig voltages v t
⎧
⎨
⎩
Qi = 0
v = Q T v t
f(v (α) , i (β) ) = 0
(3.30)
• node tableau equations: basic variables are the node-to-datum voltages v e
⎧
⎨
⎩
Ai = 0
v = A T v e
f(v (α) , i (β) ) = 0
(3.31)
• loop tableau equations: basic variables are the link currents i l
3 RLC Networks Equations and Analysis Methods
which reduces to Tellegen’s theorem [1]
v
T i = 0
(3.27)
by using (3.8) and (3.23).
Remark 3.1 Tellegen’s theorem can also be proved by means of the reduced
incidence matrix A. In fact, deriving v T = v T
e A from (3.12), and multiplying by
i, it follows that
v
T i = v
T
e Ai = 0
(3.28)
due to KCL equations Ai = 0 in (3.10).
3.2 Tableau Analysis
The tableau analysis is based on the network equations obtained by combining KCL
and KVL equations with the branch characteristics expressing the CRs of circuits
elements composing the network. The following nonlinear integro-differentialalgebraic equation describes the k-th (α k , β k )-element
f k (v
(α k )
k , i
(β k )
k ) = 0
(3.29)
where v
(α k )
k
and i
(β k )
k
(k = 1, 2, . . . , b) are the higher-order integral or derivative
of the branch voltage and current, respectively, as defined in Chap. 1. Combining
CRs and Kirchhoff equations we have the following three forms of the tableau
equations:
• cut-set tableau equations: basic variables are the twig voltages v t
⎧
⎨
⎩
Qi = 0
v = Q T v t
f(v (α) , i (β) ) = 0
(3.30)
• node tableau equations: basic variables are the node-to-datum voltages v e
⎧
⎨
⎩
Ai = 0
v = A T v e
f(v (α) , i (β) ) = 0
(3.31)
• loop tableau equations: basic variables are the link currents i l