page_title: Loop-gap resonator
section_title: Toroidal LGR
hierarchical_section_title: Loop-gap resonator / Resonant Frequency and Quality Factor / Toroidal LGR
caption_attribution_description: English: The two halves of a copper loop-gap resonator.
context_page_description: A loop-gap resonator is an electromagnetic resonator that operates in the radio and microwave frequency ranges. The simplest LGRs are made from a conducting tube with a narrow slit cut along its length. The LGR dimensions are typically much smaller than the free-space wavelength of the electromagnetic fields at the resonant frequency. Therefore, relatively compact LGRs can be designed to operate at frequencies that are too low to be accessed using, for example, cavity resonators. These structures can have very sharp resonances making them useful for electron spin resonance experiments, and precision measurements of electromagnetic material properties.
context_section_description: In some applications requiring high quality factors, the electromagnetic shielding provided by a concentric circular waveguide surrounding a cylindrical LGR can be bulky and awkward to work.  A toroidal LGR can be used for high- measurements without requiring additional electromagnetic shielding.  In the toroidal geometry the two ends of a cylindrical LGR are joined to form a completely closed structure.  In this case, the magnetic field is completely confined within the bore of the resonator and there is no radiative power loss.  The toroidal LGR consists of two halves that are bolted together along the outer diameter of the structure.
  Like the cylindrical LGR, the toroidal LGR can be modelled as a series  circuit.  In general, the effective capacitance, inductance, and resistance of the toroidal LGR will differ from the expressions given above for the cylindrical LGR.  However, in limit that the radius of the torus is large compared to the bore radius , the capacitance, inductance, and resistance of the toroidal LGR are approximated by the expressions above if one takes  to be equal to the circumference of the torus.
  The toroidal LGR is particularly convenient when characterizing the electromagnetic properties of liquid samples or particles suspended in a liquid.  In these cases, the bore of the toroidal LGR can be partially filled with the liquid sample without requiring a special sample holder.  This setup allows one to characterize the magnetic properties of, for example, a ferrofluid.  Alternatively, if the liquid sample is nonmagnetic, the entire toroidal LGR can be submerged in the liquid (or gas).  In this case, the dielectric properties of the sample only modify the effective capacitance of the resonator and the changes in  and  can be used to determine the complex permittivity of the sample.
caption: Photograph of the two halves of a copper toroidal loop-gap resonator.  Also visible are an inductive coupling loop suspended within the resonator bore and a so-called extended split-ring resonator placed in the LGR bore.[10]
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