# Transients Analysis Figure 5.2, 5.3 Transients Analysis Transients RC CIRCUIT

## Transients Analysis Figure 5.2, 5.3 Transients Analysis Transients RC CIRCUIT

Feldman, Mark, Executive Vice President has reference to this Academic Journal, PHwiki organized this Journal Transients Analysis Solution to First Order Differential Equation Consider the general Equation Let the initial condition be x(t = 0) = x( 0 ), then we solve the differential equation: The complete solution consists of two parts: the homogeneous solution (natural solution) the particular solution ( as long as ced solution) The Natural Response Consider the general Equation Setting the excitation f (t) equal to zero, It is called the natural response.

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The Forced Response Consider the general Equation Setting the excitation f (t) equal to F, a constant as long as t 0 It is called the as long as ced response. The Complete Response Consider the general Equation The complete response is: the natural response + the as long as ced response Solve as long as , The Complete solution: called transient response called steady state response WHAT IS TRANSIENT RESPONSE Figure 5.1

Figure 5.2, 5.3 Circuit with switched DC excitation A general model of the transient analysis problem In general, any circuit containing energy storage element Figure 5.5, 5.6 Figure 5.9, 5.10 (a) Circuit at t = 0 (b) Same circuit a long time after the switch is closed The capacitor acts as open circuit as long as the steady state condition (a long time after the switch is closed).

(a) Circuit as long as t = 0 (b) Same circuit a long time be as long as e the switch is opened The inductor acts as short circuit as long as the steady state condition (a long time after the switch is closed). Why there is a transient response The voltage across a capacitor cannot be changed instantaneously. The current across an inductor cannot be changed instantaneously. Figure 5.12, 5.13 5-6 Example

Transients Analysis 1. Solve first-order RC or RL circuits. 2. Underst in addition to the concepts of transient response in addition to steady-state response. 3. Relate the transient response of first-order circuits to the time constant. Transients The solution of the differential equation represents are response of the circuit. It is called natural response. The response must eventually die out, in addition to there as long as e referred to as transient response. (source free response) Discharge of a Capacitance through a Resistance Solving the above equation with the initial condition Vc(0) = Vi

Discharge of a Capacitance through a Resistance Exponential decay wave as long as m RC is called the time constant. At time constant, the voltage is 36.8% of the initial voltage. Exponential rising wave as long as m RC is called the time constant. At time constant, the voltage is 63.2% of the initial voltage. RC CIRCUIT as long as t = 0-, i(t) = 0 u(t) is voltage-step function

RC CIRCUIT Solving the differential equation Complete Response Complete response = natural response + as long as ced response Natural response (source free response) is due to the initial condition Forced response is the due to the external excitation. Figure 5.17, 5.18 5-8 a). Complete, transient in addition to steady state response b). Complete, natural, in addition to as long as ced responses of the circuit

Circuit Analysis as long as RC Circuit Apply KCL vs is the source applied. Solution to First Order Differential Equation Consider the general Equation Let the initial condition be x(t = 0) = x( 0 ), then we solve the differential equation: The complete solution consits of two parts: the homogeneous solution (natural solution) the particular solution ( as long as ced solution) The Natural Response Consider the general Equation Setting the excitation f (t) equal to zero, It is called the natural response.

The Forced Response Consider the general Equation Setting the excitation f (t) equal to F, a constant as long as t 0 It is called the as long as ced response. The Complete Response Consider the general Equation The complete response is: the natural response + the as long as ced response Solve as long as , The Complete solution: called transient response called steady state response Example Initial condition Vc(0) = 0V

Example Initial condition Vc(0) = 0V in addition to Energy stored in capacitor If the zero-energy reference is selected at to, implying that the capacitor voltage is also zero at that instant, then Power dissipation in the resistor is: pR = V2/R = (Vo2 /R) e -2 t /RC RC CIRCUIT Total energy turned into heat in the resistor

RL CIRCUIT where L/R is the time constant DC STEADY STATE The steps in determining the as long as ced response as long as RL or RC circuits with dc sources are: 1. Replace capacitances with open circuits. 2. Replace inductances with short circuits. 3. Solve the remaining circuit.

## Feldman, Mark Executive Vice President

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