Long electromagnetic transients occur in electrical systems because of switching and impulse actions As a result, the simulation time of such processes can be long, which is undesirable. Simulation time is significantly increased if the circuit in the study is complex, and also if this circuit is described by a rigid system of state equations. Modern requests of design engineers require an increase in the speed of calculations for realizing a real-time simulation. This work is devoted to the development of a unified spectral method for calculating electromagnetic transients in electrical circuits based on the representation of solution functions by series in algebraic and orthogonal polynomials. The purpose of the work is to offer electrical engineers a method that can significantly reduce the time for modeling transients in electrical circuits. Research methods. Approximation of functions by orthogonal polynomials, numerical methods for integrating differential equations, matrix methods, programming and theory of electrical circuits. Obtained results. Methods for calculating transients in electrical circuits based on the approximation of solution functions by series in algebraic polynomials as well as in the Chebyshev, Hermite and Legendre polynomials, have been developed and investigated. The proposed method made it possible to convert integro-differential equations of state into linear algebraic equations for images of time-dependent functions. The developed circuit model simplifies the calculation method. The images of true current functions are interpreted as direct currents in the proposed equivalent circuit. A computer program for simulating the transient process in an electrical circuit was developed on the basis of the described methods. The performed comparison of methods made it possible to choose the best method and a way to use it. The advantages of the presented method over other known methods are to reduce the simulation time of electromagnetic transients (for the considered examples by more than 6 times) while ensuring the required accuracy. The calculation of the process in the circuit over a long time interval showed a decrease and stabilization of errors, which indicates the prospects for using research methods for calculating complex electrical circuits.