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Transients Analysis

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)

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Transients Analysis

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  1. Transients Analysis

  2. 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 (forced solution)

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

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

  5. The Complete Response Consider the general Equation Solve for , • The complete response is: • the natural response + • the forced response The Complete solution: called transient response called steady state response

  6. WHAT IS TRANSIENT RESPONSE Figure 5.1

  7. A general model of the transient analysis problem Circuit with switched DC excitation Figure 5.2, 5.3

  8. In general, any circuit containing energy storage element Figure 5.5, 5.6

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

  10. (a) Circuit for t = 0(b) Same circuit a long time before the switch is opened The inductor acts as short circuit for the steady state condition (a long time after the switch is closed).

  11. Why there is a transient response? • The voltage across a capacitor cannot be changed instantaneously. • The current across an inductor cannot be changed instantaneously.

  12. Example Figure 5.12, 5.13 5-6

  13. Transients Analysis 1. 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.

  14. Transients The 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)

  15. ic iR Discharge of a Capacitance through a Resistance Solving the above equation with the initial condition Vc(0) = Vi

  16. Discharge of a Capacitance through a Resistance

  17. Exponential decay waveform RC is called the time constant. At time constant, the voltage is 36.8% of the initial voltage. Exponential rising waveform RC is called the time constant. At time constant, the voltage is 63.2% of the initial voltage.

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

  19. RC CIRCUIT Solving the differential equation

  20. Complete response = natural response + forced response Natural response (source free response) is due to the initial condition Forced response is the due to the external excitation. Complete Response

  21. Figure 5.17, 5.18 a). Complete, transient and steady state response b). Complete, natural, and forced responses of the circuit 5-8

  22. Circuit Analysis for RC Circuit Apply KCL vsis the source applied.

  23. 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 (forced solution)

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

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

  26. The Complete Response Consider the general Equation Solve for , • The complete response is: • the natural response + • the forced response The Complete solution: called transient response called steady state response

  27. Example Initial condition Vc(0) = 0V

  28. Example Initial condition Vc(0) = 0V and

  29. 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

  30. RC CIRCUIT Power dissipation in the resistor is: pR = V2/R = (Vo2 /R)e -2 t /RC Total energy turned into heat in the resistor

  31. RL CIRCUITS Initial condition i(t = 0) = Io

  32. RL CIRCUITS Initial condition i(t = 0) = Io

  33. RL CIRCUIT Power dissipation in the resistor is: pR = i2R = Io2e-2Rt/LR Total energy turned into heat in the resistor It is expected as the energy stored in the inductor is

  34. RL CIRCUIT where L/R is the time constant

  35. DC STEADY STATE The 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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