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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc. Chapter 4 Transien ts
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Chapter 4 Transients

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Chapter 4 Transients. Chapter 4 Transients. 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. - PowerPoint PPT Presentation
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Page 1: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Chapter 4Transients

Page 2: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Chapter 4Transients

1. Solve first-order RC or RL circuits.

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

Page 3: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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

4. Solve RLC circuits in dc steady-state conditions.

5. Solve second-order circuits.

6. Relate the step response of a second-order system to its natural frequency and damping ratio.

Page 4: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Transients

The time-varying currents and voltages resulting from the sudden application of sources, usually due to switching, are called transients. By writing circuit equations, we obtain integrodifferential equations.

Page 5: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 6: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Discharge of a Capacitance through a Resistance

0

R

tv

dt

tdvC CC

0 tvdt

tdvRC C

C

stC Ketv

0 stst KeRCKse

Page 7: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

RCs

1

RCtC Ketv RCt

iC eVtv

iC Vv 0

Page 8: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 9: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

The time interval τ = RC iscalled the time constant of

the circuit.

tssC eVVtv

Page 10: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 11: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 12: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

DC STEADY STATE

The steps in determining the forced response for RLC circuits with dc sources are:

1. Replace capacitances with open circuits.

2. Replace inductances with short circuits.

3. Solve the remaining circuit.

Page 13: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 14: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 15: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

RL CIRCUITS

The steps involved in solving simple circuits containing dc sources, resistances, and one energy-storage element (inductance or capacitance) are:

Page 16: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

1. Apply Kirchhoff’s current and voltage laws to write the circuit equation.

2. If the equation contains integrals, differentiate each term in the equation to produce a pure differential equation.

3. Assume a solution of the form K1 + K2est.

Page 17: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

4. Substitute the solution into the differential equation to determine the values of K1 and s . (Alternatively, we can determine K1 by solving the circuit in steady state as discussed in Section 4.2.)

5. Use the initial conditions to determine the value of K2.

6. Write the final solution.

Page 18: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 19: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 20: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

RL Transient Analysis

LtReKti 22

Time constant is

R

L

Page 21: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 22: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 23: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 24: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 25: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 26: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

RC AND RL CIRCUITS WITH GENERAL

SOURCES

The general solution consists of two parts.

Page 27: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

The particular solution (also called the forced response) is any expression that satisfies the equation.

In order to have a solution that satisfies the initial conditions, we must add the complementary solution to the particular solution.

Page 28: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

The homogeneous equation is obtained by setting the forcing function to zero.

The complementary solution (also called the natural response) is obtained by solving the homogeneous equation.

Page 29: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Step-by-Step Solution

Circuits containing a resistance, a source,

and an inductance (or a capacitance)

1. Write the circuit equation and reduce it to a first-order differential equation.

Page 30: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

2. Find a particular solution. The details of this step depend on the form of the forcing function. We illustrate several types of forcing functions in examples, exercises, and problems.

3. Obtain the complete solution by adding the particular solution to the complementary solution given by Equation 4.44, which contains the arbitrary constant K.

4. Use initial conditions to find the value of K.

Page 31: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 32: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 33: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 34: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 35: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 36: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 37: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Page 38: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

SECOND-ORDER CIRCUITS

tvvdttiC

tRidt

tdiL sC

t

01

0

L

R

2

LC

10

Page 39: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

tftidt

tdi

dt

tid 2

02

2

2

Page 40: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

0

20

21 s

20

22 s

Page 41: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

1. Overdamped case (ζ > 1). If ζ > 1 (or equivalently, if α > ω0), the roots of the characteristic equation are real and distinct. Then the complementary solution is tsts

c eKeKtx 2121

In this case, we say that the circuit is overdamped.

Page 42: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

2. Critically damped case (ζ = 1). If ζ = 1 (or equivalently, if α = ω0 ), the roots are real and equal. Then the complementary solution is

tstsc teKeKtx 11

21

In this case, we say that the circuit is critically damped.

Page 43: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

3. Underdamped case (ζ < 1). Finally, if ζ < 1 (or equivalently, if α < ω0), the roots are complex. (By the term complex, we mean that the roots involve the square root of –1.) In other words, the roots are of the form

nn jsjs 21 and in which j is the square root of -1 and the natural frequency is given by

220 n

Page 44: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

In electrical engineering, we use j rather than i to stand for square root of -1, because we use i for current.

For complex roots, the complementary solution is of the form

teKteKtx nt

nt

c sincos 21

In this case, we say that the circuit is underdamped.

Page 45: Chapter 4 Transients

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.