Closed-Form Expressions for the Input Impedance of Some 2-D Fractal Circuit Networks

We derive closed form expressions for the input impedance of two-dimensional (2-D) infinite ladder-tree and tree-ladder networks using the combination of results for the input impedance of an infinite 1-D ladder network and an infinite 1-D tree network. We show that the effect of the number of branches in the tree network can always be absorbed via impedance scaling resulting in a universal formula derived in this brief. The meaningful number of branches of the tree network is shown to be either two branches; i.e., a binary tree, or four branches; i.e., a quaternary tree. Special cases of component choices are investigated, and both circuit simulations and experimental results are provided to validate the theory.