10.1.1 Parallel RC Circuit, Step Input

• Capacitor charging transient

• To find vc(t), we must solve a non-homogeneous, linear first-order ordinary differential equation with constant coefficients: 

• To solve this equation, we will use the method of homogeneous and particular solutions because this method can be readily extended to higher order equations

• The method states that the solution to the non-homogeneous differential equation can be obtained by summing together the homogeneous solution and the particular solution

• Total solution (vCH(t) is called the homogeneous solution and vCP(t) is called the particular solution)

• When dealing with circuit responses, the homogeneous solution is also called the natural response of the circuit because it depends only on the internal energy storage properties of the circuit and not on external inputs. The particular solution is also called the forced response or the forced solution because it depends on the external inputs to the circuit.

• The root of the characteristic equation, s, is also called the natural frequency of the system.

• Complete solution: