As a contribution to increasing the areal density of digital data stored on a magnetic recording medium, this thesis provides mathematical analyses of various magnetic recording heads. Each of the heads considered here is for use in a perpendicular recording system, writing to or reading from a multi-layer medium which includes a high magnetic permeability layer between the data storage layer and the substrate. The exact two-dimensional analysis is performed in each case by one of two methods: either Fourier analysis or conformal mapping. The types of heads analysed include conventional styles but particular emphasis is placed on the effects of the novel idea of potential grading across the pole pieces. Exact head fields are derived for thin film heads with both constant and linearly varying pole potentials, single pole heads with linearly and arbitrarily varying pole potentials and shielded magnetoresistive heads, all in the presence of a magnetic underlayer. These and other published solutions are used to derive output characteristics for perpendicular replay heads, which are compared with published theoretical and experimental results where possible. The Fourier solutions obtained are in the form of infinite series dependent on at least one set of coefficients which are determined by infinite systems of linear equations. Approximations to the potentials in the head face planes, independent of these coefficients, are derived from the exact Fourier solutions. The accuracy of these approximations is demonstrated when they are used to estimate the vertical field components and the spectral response functions. Heads with graded pole potentials are found to have more localised vertical field components than the corresponding constant potential heads. They are also better suited for use with thin media for 'in contact' recording.

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