The growing penetration of inverter-based resources and associated controls necessitates system-wide electromagnetic transient (EMT) analyses. EMT tools and methods today were not designed for the scale of these analyses. In light of the emerging need, there is a great deal of interest in developing new techniques for fast and accurate EMT simulations for large power grids; the foundations of which will be built on current tools and methods. However, we find that educational texts covering the fundamentals and inner workings of current EMT tools are limited. As such, there is a lack of introductory material for students and professionals interested in researching the field. To that end, in this tutorial, we introduce the principles of EMT analyses from the circuit-theoretic viewpoint, mimicking how timedomain analyses are performed in circuit simulation tools like SPICE and Cadence. We perform EMT simulations for two examples, one linear and one nonlinear, including induction motor (IM) from the first principles. We anticipate that by the end of this monograph, the readers will have a basic understanding of how power grid EMT tools work.