Use of a tapered transmission line as an Ideal transformer

A tapered transmission line [1, 2] has an impedance that slowly varies spatially from source end to load end. Use of a tapered line yields a nearly ideal transformer for frequencies above the transmission-line cutoff frequency, which is related to the taper rate and phase velocity in the transmission medium. Use of a material with high permeability and high permittivity reduces the wavelength in a transmission line, and so extends the use to lower frequencies. Calculations for an exponential line utilizing a high-permeability high-permittivity material include the range of impedance transformation, transformer ratio, phase velocity, input impedance, power reflection coefficient, power deposition, and thermal loading.