Semiconductor-device limitations to system miniaturization have receded, but exposed by their improvement numerous "ancillary" barriers which continue to preoccupy nearly every electronics industry. Prominent among these obstacles are package parasitics and heat, which have come to the fore as conventional circuits are applied in modern regimes of frequency and integration density. To an ever increasing extent, integration limits are symptoms of the fundamental frequency- and size-scaling limits of passive components. Power inductors and transformers, in particular, are challenging to miniaturize because of their poor performance when scaled down in size, and the difficulty of fabricating them with available planar processes. A family of approximating networks for transmission lines, the focus of this work, enables miniaturization by internally circulating energy and exchanging delay fidelity for bulk energy storage. These multi-resonant components are substantially smaller than their lumped counterparts, in particular requiring less inductance, and enforce useful waveform symmetries that can be traded for higher power or higher efficiency.