The two-point impedance and LC resonances

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 Luα = λα u ∗ α with orthonormal vectors ua, the effective impedance between nodes p and q of the network is Zpq = ∑ α(uαp − uα q)/λα where the summation is over all λα not identically equal to zero and uαp is the p-th component of uα. For networks consisting of inductances L and capacitances C, the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of λα. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.