10.1.2 RC Discharge Transient

• RC discharge transient:

• Properties of exponentials: aid to sketching waveform for decaying exponential in simple RC & RL transients

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:

Chapter 10 First-Order Transients in Linear Electrical Networks

• Systems with one storage element (capacitor/inductor) are described by first-order differential equations

9.5 Energy, Charge and Flux Conservation

• Circuit analysis with initial charge/flux linkage (switch closed at t = 0s):

• Energy is lost when v1 is not equal to v2:

Inductors Series and Parallel Connection

• Series Connection:

• Parallel Connection:

9.2.1 Capacitors Series and Parallel Connections

• Series connection:

• Parallel connection:

9.1.2 Inductors

• Idealized toroidal inductor ( u = magnetic permeability)

• Magnetic flux density, B

• Magnetic flux density across the core

• Total magnetic flux linkage (Weber/Wb)

• Inductance (Webers/Ampere, Henry)

• For linear time-invariant inductor:

• Inductor memory property:

• Inductor energy property: