rl transient

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RL Transient

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  • Transients Analysis

  • Solution to First Order Differential Equation Consider the general EquationLet 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 (forced solution)

  • The Natural Response Consider the general EquationSetting the excitation f (t) equal to zero, It is called the natural response.

  • The Forced Response Consider the general EquationSetting the excitation f (t) equal to F, a constant for t 0 It is called the forced response.

  • The Complete Response Consider the general EquationThe complete response is: the natural response + the forced response Solve for , The Complete solution:called transient responsecalled steady state response

  • WHAT IS TRANSIENT RESPONSEFigure 5.1

  • Figure 5.2, 5.3Circuit with switched DC excitationA general model of the transient analysis problem

  • In general, any circuit containing energy storage elementFigure 5.5, 5.6

  • Figure 5.9, 5.10(a) Circuit at t = 0(b) Same circuit a long time after the switch is closedThe capacitor acts as open circuit for the steady state condition(a long time after the switch is closed).

  • (a) Circuit for t = 0(b) Same circuit a long time before the switch is openedThe inductor acts as short circuit for 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, 5.135-6Example

  • Transients Analysis1. Solve first-order RC or RL circuits.

    2. Understand the concepts of transient response and steady-state response.

    3. Relate the transient response of first-order circuits to the time constant.

  • TransientsThe solution of the differential equation represents are response of the circuit. It is called natural response.

    The response must eventually die out, and therefore referred to as transient response. (source free response)

  • Discharge of a Capacitance through a ResistanceSolving the above equation with the initial conditionVc(0) = Vi

  • Discharge of a Capacitance through a Resistance

  • Exponential decay waveformRC is called the time constant.At time constant, the voltage is 36.8%of the initial voltage.Exponential rising waveformRC is called the time constant.At time constant, the voltage is 63.2% of the initial voltage.

  • RC CIRCUIT for t = 0-, i(t) = 0u(t) is voltage-step function

    R

    C

    +

    V

    C

    -

    i(t)

    t = 0

    +

    _

    V

    R

    C

    +

    V

    C

    -

    i(t)

    t = 0

    +

    _

    Vu(t)

  • RC CIRCUIT Solving the differential equation

  • Complete ResponseComplete response = natural response + forced responseNatural response (source free response) is due to the initial conditionForced response is the due to the external excitation.

  • Figure 5.17, 5.185-8a). Complete, transient and steady state responseb). Complete, natural, and forced responses of the circuit

  • Circuit Analysis for RC CircuitApply KCL vs is the source applied.

    DC

    Vs

    +Vc-

    + VR -

    R

    C

    iR

    iC

  • Solution to First Order Differential Equation Consider the general EquationLet 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 (forced solution)

  • The Natural Response Consider the general EquationSetting the excitation f (t) equal to zero, It is called the natural response.

  • The Forced Response Consider the general EquationSetting the excitation f (t) equal to F, a constant for t 0 It is called the forced response.

  • The Complete Response Consider the general EquationThe complete response is: the natural response + the forced response Solve for , The Complete solution:called transient responsecalled steady state response

  • Example

    Initial condition Vc(0) = 0V

    DC

    100V

    +Vc-

    + VR -

    100 k ohms

    0.01 microF

    iR

    iC

  • Example

    Initial condition Vc(0) = 0Vand

    DC

    100V

    +Vc-

    + VR -

    100 k ohms

    0.01 microF

    iR

    iC

  • Energy stored in capacitorIf the zero-energy reference is selected at to, implying that thecapacitor voltage is also zero at that instant, then

  • Power dissipation in the resistor is:pR = V2/R = (Vo2 /R) e -2 t /RCRC CIRCUITTotal energy turned into heat in the resistor

    R

    C

  • RL CIRCUITSInitial condition i(t = 0) = Io

    L

    R

    -VR+

    +VL-

    i(t)

  • RL CIRCUITSInitial condition i(t = 0) = Io

  • RL CIRCUITPower dissipation in the resistor is:pR = i2R = Io2e-2Rt/LRTotal energy turned into heat in the resistorIt is expected as the energy stored in the inductor is

  • RL CIRCUIT where L/R is the time constant

  • DC STEADY STATEThe steps in determining the forced response for 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.

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