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Impedance matching using a Hermite line is studied. The problem formulation leads to a non-linear differential equation which, after solving with appropriate boundary conditions, yields a formula suitable for impedance matching on microwave integrated circuits (MIC). Numerical examples are considered. The results show tremendous reduction in matching component size which could be achieved if Hermite lines are used instead of uniform transmission lines. This is a considerable advantage in the miniaturisation of solid-state circuits.