In this paper, a procedure for the analysis and design of nonlinear circuits based on the Carleman linearization is presented. Since this procedure starts with the general network equations which are derived by Kirchhoff's law, it can be used for a variety of circuits. The proposed procedure is applied for oscillator and RF mixer circuits which are a crucial part of every wireless communication system. Based on the general network equations, a transformation into equivalent polynomial differential equations is performed using the not widely known method of Kerner. This polynomial differential equation is further transformed into an equivalent infinite-dimensional system of linear differential equations. Since the infinite-dimensional linear system cannot be solved in general, an approximation by a finite one is performed using a self-consistent technique. The obtained finite linear system is used for the approximation of the infinite-dimensional linear system on a predefined interval. The approximation is utilized for the design and analysis process of the circuit, which is a challenging problem until now.