Theory of impedance networks: The two-point impedance and LC resonances

Details Theory of impedance networks: The two-point impedance and LC resonances Theory of impedance networks: The two-point impedance and LC resonances

2006-01-23

arxiv.org/abs/math-ph/0601048v6

J. Phys. A: Math. Gen. 39 (2006) 8579-8591

Physics

Mathematical Physics

21 pages, 3 figures; v4: typesetting corrected; v5: Eq. (63) corrected

Scientific paper

10.1088/0305-4470/39/27/002

We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation L u_a = lambda_a u_a^* with orthonormal vectors u_a, the effective impedance between nodes p and q of the network is Z = Sum_a [u_{a,p} - u_{a,q}]^2/lambda_a where the summation is over all lambda_a not identically equal to zero and u_{a,p} is the p-th component of u_a. For networks consisting of inductances (L) and capacitances (C), the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of lambda_a. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.

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