  Compact inductor-capacitor (LC) resonators, in contrast to coplanar waveguide
(CPW) resonators, have a simple lumped-element circuit representation but
usually call for sophisticated finite-element method (FEM) simulations for an
accurate modelling. Here we present an simple analytical model for a family of
coplanar LC resonators where the electrical properties are directly obtained
from the circuit geometry with a satisfying accuracy. Our experimental results
on $10$ high-internal-quality-factor resonators ($Q_{\rm i}\gtrsim 2\times
10^{5}$), with frequency ranging roughly from $300\,{\rm MHz}$ to $1\,{\rm
GHz}$, show an excellent consistency with both the derived analytical model and
detailed FEM simulations. These results showcase the ability to design
sub-gigahertz resonators with less than $2\%$ deviation in the resonance
frequency, which has immediate applications, for example, in the implementation
of ultrasensitive cryogenic detectors. The achieved compact resonator size of
the order of a square millimeter indicates a feasible way to integrate hundreds
of microwave resonators on a single chip for realizing photonic lattices.
