Theory of impedance networks: 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 = λαuα with orthonormal vectors ua , the effective impedance between nodes p and q of the network is Zpq = ∑ α(uαp−uαq)/λα , where the summation is over all λα not identically equal to zero and uαp is the pth 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. PACS numbers: 01.55.+b, 02.10.Yn, 84.30.Bv