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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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solution to first order differential equation
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)
the natural response
The Natural Response

Consider the general Equation

Setting the excitation f (t) equal to zero,

It is called the natural response.

the forced response
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.

the complete response
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

slide7

A general model of the transient analysis problem

Circuit with switched DC excitation

Figure 5.2, 5.3

slide9

(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).

slide10

(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).

why there is a transient response
Why there is a transient response?
  • The voltage across a capacitor cannot be changed instantaneously.
  • The current across an inductor cannot be changed instantaneously.
slide12

Example

Figure 5.12, 5.13

5-6

transients analysis1
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.

transients
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)

discharge of a capacitance through a resistance

ic

iR

Discharge of a Capacitance through a Resistance

Solving the above equation

with the initial condition

Vc(0) = Vi

slide17

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.

rc circuit
RC CIRCUIT

for t = 0-, i(t) = 0

u(t) is voltage-step function

rc circuit1
RC CIRCUIT

Solving the differential equation

complete response
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
slide21

Figure 5.17, 5.18

a). Complete, transient and steady state response

b). Complete, natural, and forced responses of the circuit

5-8

circuit analysis for rc circuit
Circuit Analysis for RC Circuit

Apply KCL

vsis the source applied.

solution to first order differential equation1
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)
the natural response1
The Natural Response

Consider the general Equation

Setting the excitation f (t) equal to zero,

It is called the natural response.

the forced response1
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.

the complete response1
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

example
Example

Initial condition Vc(0) = 0V

example1
Example

Initial condition Vc(0) = 0V

and

energy stored in capacitor
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

slide30

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

rl circuits
RL CIRCUITS

Initial condition

i(t = 0) = Io

rl circuits1
RL CIRCUITS

Initial condition

i(t = 0) = Io

rl circuit
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

rl circuit1
RL CIRCUIT

where L/R is the time constant

dc steady state
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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