Time-variant inductors exist in many industrial applications, including sensors and actuators. In some applications, this characteristic can be deleterious, for example, resulting in inductive loss through eddy currents in motors designed for high efficiency operation. Therefore, it is important to investigate the electrical dynamics of systems with time-variant inductors. However, circuit analysis with time-variant inductors is nonlinear, resulting in difficulties in obtaining a closed form solution. Typical numerical algorithms used to solve the nonlinear differential equations are time consuming and require powerful processors. This investigation proposes a nonlinear method to analyze a system model consisting of the time-variant inductor with a constraint that the circuit is powered by DC sources and the derivative of the inductor is known. In this method, the Norton equivalent circuit with the time-variant inductor is realized first. Then, an iterative solution using a small signal theorem is employed to obtain an approximate closed form solution. As a case study, a variable inductor, with a time-variant part stimulated by a sinusoidal mechanical excitation, is analyzed using this approach. Compared to conventional nonlinear differential equation solvers, this proposed solution shows both improved computation efficiency and numerical robustness. The results demonstrate that the proposed analysis method can achieve high accuracy.