Chapter 4 Transients
description
Transcript of Chapter 4 Transients
![Page 1: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/1.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
Chapter 4Transients
![Page 2: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/2.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/3.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/4.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/5.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 6: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/6.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/7.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/8.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 9: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/9.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/10.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 11: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/11.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 12: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/12.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/13.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 14: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/14.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 15: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/15.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/16.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/17.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/18.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 19: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/19.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 20: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/20.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/21.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 22: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/22.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 23: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/23.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 24: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/24.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 25: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/25.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 26: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/26.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/27.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/28.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/29.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/30.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/31.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 32: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/32.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 33: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/33.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 34: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/34.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 35: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/35.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 36: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/36.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 37: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/37.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 38: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/38.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/39.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/40.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/41.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/42.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/43.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/44.jpg)
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](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/45.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 46: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/46.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 47: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/47.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 48: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/48.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 49: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/49.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 50: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/50.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 51: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/51.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 52: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/52.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 53: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/53.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 54: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/54.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.
![Page 55: Chapter 4 Transients](https://reader036.fdocuments.in/reader036/viewer/2022082214/56815b13550346895dc8bfc0/html5/thumbnails/55.jpg)
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.