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In the present paper a novel mathematical model of physical processes of transient electromagnetic waves excitation and propagation in a biconical transmission line with radially inhomogeneous magneto-dielectric filling is proposed. The model is based on time domain mode expansions over spherical waves. The basis functions of the mode expansions are calculated analytically. The mode expansion coefficients are governed by Klein-Gordon-Fock equation with coefficients depending on a radial spatial coordinate. The explicit finite difference time domain computational scheme is derived to calculate the mode expansion coefficients. Dependences of cutoff frequencies of higher modes of TE and TM waves on the line geometry and dielectric filling are studied. In order to calculate electromagnetic field in the line with higher accuracy, just finite number of terms in the mode expansions is required. Electromagnetic field excited by the transient electric ring current is calculated in both homogeneous and radially inhomogeneous biconical transmission line. It is shown that there is a possibility to increase the bandwidth of the line via introduction of partial dielectric filling without changing the line geometrical size.