Rizzoni 6e SM CH06 - 전기전자공학개론 6장입니다 PDF

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6.PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers andChapter 6: Frequency Response and System Concepts –Instructor NotesChapter 6 can be covered immediately following Chapter 4, or after completing Chapter 5. There is no direct dependence of Ch...

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G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Chapter 6:

Frequency Response and System Concepts – Instructor Notes

Chapter 6 can be covered immediately following Chapter 4, or after completing Chapter 5. There is no direct dependence of Chapter 6 on Chapter 5. Modularity is a recurrent feature of this book, and we shall draw attention to it throughout these Instructor Notes. Section 6.1 introduces the notion of sinusoidal frequency response and motivates the use of sinusoidal signals; the Fourier Series method of representing signals is described in detail in Section 6.2. Further, the text and examples also illustrate the effect of a multi-components signal propagating through a linear system. Four examples accompany this presentation. Section 6.3 introduces filters, and outlines the basic characteristics of low-, high- and band-pass filters. The concept of resonance is treated in greater depth than in the previous edition, and a connection is made with the natural response of second order circuits, which may be useful to those instructors who have already covered transient response of second-order circuits. Four detailed examples are included in this section, Further, the boxes Focus on Measurements: Wheatstone bridge filter (pp. 315-317), Focus on Measurements: AC line interference filter (pp. 317319), and Focus on Measurements: Seismic displacement transducer (pp. 319-322) touch on additional application examples. The first and last of these boxes can be linked to related material in Chapters 2, 3, and 4. The instructor who has already introduced the operational amplifier as a circuit element will find that section 8.3, on active filters, is an excellent vehicle to reinforce both the op-amp concept and the frequency response ideas. Another alternative (employed by this author) consists of introducing the op-amp at this stage, covering sections 8.1 through 8.3. Finally, Section 6.4 covers Bode plots, and illustrates how to create approximate Bode plots using the straight-line asymptotic approximation. The box Focus on Methodology: Bode Plots (p. 327) clearly outlines the method, which is further explained in two examples. The homework problems present several frequency response, Fourier Series, filter and Bode plot exercises of varying difficulty. The instructor who wishes to use one of the many available software aids (e.g., MATLAB® or Electronics Workbench® ) to analyze the frequency response of more complex circuits and to exploit more advanced graphics capabilities, will find that several advanced problems lend themselves nicely to such usage. A number of new application oriented problems have been introduced in the 5th Edition, including problems related to loudspeaker crossover networks (6.64, 6.66 and 6.69), and 60-Hz line noise filtering (6.68). The 5th Edition of this book includes 7 new problems; some of the 4th Edition problems were removed, increasing the end-of-chapter problem count from 76 to 81.

Learning Objectives for Chapter 6 1. 2.

3. 4.

Understand the physical significance of frequency domain analysis, and compute the frequency response of circuits using AC circuit analysis tools. Compute the Fourier spectrum of periodic signals using the Fourier series representation, and use this representation in connection with frequency response ideas to compute the response of circuits to periodic inputs. Analyze simple first- and second-order electrical filters, and determine their frequency response and filtering properties. Compute the frequency response of a circuit and its graphical representation in the form of a Bode plot.

6.1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Problem 6.1

Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.. Assume: ฀ ฀ = 0.5 ฀฀ and ฀ ฀ = 200 ฀฀Ω.

Known quantities:

฀ ฀ = 0.5 ฀฀, ฀ ฀ = 200 ฀฀Ω

Find:

(a) Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.1.

(b) Plot the magnitude and phase of the circuit for frequencies between 10 and 107 rad/s on graph paper, with a linear scale for frequency. (c) Repeat part (b), using semilog paper. Place the frequency on the logarithmic axis. (d) Plot the magnitude response on semilog paper with magnitude in decibels.

Analysis:

(a) Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.1. ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) can be determine by the voltage division:

Convert to the frequency domain:

Substitute known values:

฀฀ ฀฀฀฀฀฀฀฀ = ฀฀฀฀฀฀ ฀ ฀ + ฀฀

฀฀฀฀฀฀ = ฀฀฀฀฀฀ (฀฀฀฀) ฀฀฀฀฀฀ + ฀฀

฀฀฀฀฀฀฀฀ (฀฀฀฀)

฀฀฀฀฀฀. ฀฀

฀฀฀฀฀฀ ฀ ฀ + ฀฀฀฀฀฀. ฀฀ 6.2 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6 (b) Plot the magnitude and phase of the circuit for frequencies between ฀฀฀฀ and ฀฀฀฀฀฀ rad/s on graph paper, with a linear scale for frequency. Convert the result from (a) into a manageable format using the complex conjugate of the denominator: ฀฀฀฀฀฀฀฀ (฀฀฀฀) ฀฀฀฀฀฀ (฀฀฀฀)

Magnitude: ฀฀฀฀฀฀฀฀ (฀฀฀฀)

�

฀฀฀฀฀฀ (฀฀฀฀)

� = ��

=

(0.5฀฀)2 + ฀฀฀฀100 ฀฀ 4฀฀10 + (0.5฀฀)2

0.5฀฀2

4฀฀10 + (0.5฀฀)

Phase: ∠

100 ฀฀ ฀฀

�+� 2 2

� 2

2

4฀฀10 + (0.5฀฀)

฀฀฀฀฀฀฀฀ (฀฀฀฀) 100 ฀฀ ฀฀ = atan � � (฀฀฀฀) ฀฀฀฀฀฀ (0.5฀฀)2 100฀฀ � = atan � 0.25฀฀

Plots: Magnitude 1.4

1.2

Magnitude

1

0.8

0.6

0.4

0.2

0

0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 6

x 10

6.3 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Phase (degrees) 90 80 70

Phase

60 50 40 30 20 10 0 0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 6

x 10

(c) Repeat part (b), using semilog paper. Place the frequency on the logarithmic axis. Magnitude 1.4

1.2

Magnitude

1

0.8

0.6

0.4

0.2

0 1 10

2

10

3

10

4

10 Frequency (rad/s)

5

10

6

10

7

10

6.4 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Phase (degrees) 90 80 70

Phase

60 50 40 30 20 10 0 1 10

2

10

3

10

4

10 Frequency (rad/s)

5

10

6

10

7

10

(d) Plot the magnitude response on semilog paper with magnitude in decibels. Magnitude 50 45 40

Magnitude

35 30 25 20 15 10 5 0 1 10

2

10

3

10

4

10 Frequency (rad/s)

5

10

6

10

7

10

6.5 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Problem 6.2 Repeat the instructions of Problem 6.1 for the circuit of Figure P6.2.

Known quantities: Given in figure.

Find:

(a) Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.2.

(b) Plot the magnitude and phase of the circuit for frequencies between 10 and 107 rad/s on graph paper, with a linear scale for frequency. (c) Repeat part (b), using semilog paper. Place the frequency on the logarithmic axis. (d) Plot the magnitude response on semilog paper with magnitude in decibels.

Analysis:

(a) Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.2. Determine ฀฀฀฀฀฀฀฀ (฀฀฀฀) and ฀฀฀฀฀฀ (฀฀฀฀) by using voltage division: ฀฀฀฀฀฀฀฀

Convert to frequency domain:

฀฀฀฀ ฀ ฀ + ฀฀ = ฀฀฀฀฀฀ ฀฀฀฀ ฀ ฀ +฀ ฀ + ฀ ฀

1 ฀ ฀ ฀฀฀฀฀฀ 1 ฀฀+ ฀฀฀฀฀฀ ฀฀฀฀฀฀฀฀ (฀฀฀฀) = ฀฀฀฀฀฀ (฀฀฀฀) 1 ฀ ฀฀฀฀฀฀ ฀ ฀฀+ 1 ฀฀+ ฀฀฀฀฀฀

Simplify and solve for ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀):

6.6 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6 1 ฀฀฀฀฀฀฀฀ (฀฀฀฀) = ฀฀฀฀฀฀ (฀฀฀฀) 2 + ฀฀฀฀฀฀฀฀

Substitute known values:

฀฀฀฀฀฀฀฀ (฀฀฀฀) ฀฀ = ฀฀฀฀฀฀(฀฀฀฀) ฀ ฀ + ฀฀฀฀฀฀. ฀฀

(b) Plot the magnitude and phase of the circuit for frequencies between ฀฀฀฀ and ฀฀฀฀ ฀฀ rad/s on graph paper, with a linear scale for frequency.

Determine the equations for magnitude in phase. First, simplify ฀฀฀฀฀฀฀฀ (฀฀฀฀ )/฀฀฀฀฀฀ (฀฀฀฀) by multiplying by the complex conjugate of the denominator: ฀฀฀฀฀฀฀฀ (฀฀฀฀) 2 − 0.1฀฀฀฀ = ฀฀฀฀฀฀ (฀฀฀฀) 4 + 0.01฀฀ 2

Magnitude:

2 2 2 ฀฀฀฀฀฀฀฀ (฀฀฀฀) 0.1฀฀ � � = �� � � + � ฀฀฀฀฀฀ (฀฀฀฀) 4 + 0.01฀฀ 2 4 + 0.01฀฀ 2

Phase: ∠ Plots:

฀฀฀฀฀฀฀฀ (฀฀฀฀) −0.1฀฀ ) = arctan( 2 ฀฀฀฀฀฀ (฀฀฀฀)

Magnitude 0.45 0.4 0.35

Magnitude

0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 x 10

6

6.7 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Phase (degrees) -20 -30 -40

Phase

-50 -60 -70 -80 -90 -100

0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 x 10

6

(c) Repeat part (b), using semilog paper. Place the frequency on the logarithmic axis. Magnitude 0.45 0.4 0.35

Magnitude

0.3 0.25 0.2 0.15 0.1 0.05 0 1 10

10

2

3

10

4

10 10 Frequency (rad/s)

5

6

10

10

7

6.8 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Phase (degrees) -20 -30 -40

Phase

-50 -60 -70 -80 -90 -100 1 10

10

2

3

10

4

5

10 10 Frequency (rad/s)

10

6

10

7

(d) Plot the magnitude response on semilog paper with magnitude in decibels. Magnitude 0

-20

Magnitude

-40

-60

-80

-100

-120 1 10

2

10

3

10

4

10 Frequency (rad/s)

5

10

6

10

7

10

6.9 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Problem 6.3 Repeat the instructions of Problem 6.1 for the circuit of Figure P6.3.

Known quantities: Values in Figure P6.3.

Find:

(a) Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.3.

(b) Plot the magnitude and phase of the circuit for frequencies between 10 and 107 rad/s on graph paper, with a linear scale for frequency. (c) Repeat part (b), using semilog paper. Place the frequency on the logarithmic axis. (d) Plot the magnitude response on semilog paper with magnitude in decibels.

Analysis:

(a) Determine the frequency response ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀) for the circuit of Figure P6.3.

To determine the frequency response, place an imaginary voltage source at ฀฀฀฀฀฀ and determine ฀฀฀฀฀฀฀฀ . For this problem, ฀฀฀฀฀฀฀฀ may be determined from the voltage division of the top node: where ฀฀1 is: and ฀฀฀฀฀฀ is:

C ฀฀฀฀฀฀฀฀ = ฀฀1 1000 Ω + ฀฀

฀฀฀฀฀฀ ฀฀1 = ฀฀฀฀฀฀ 2000 Ω + ฀฀฀฀฀฀ 2000 Ω ∗ (1000 Ω ∗ ฀฀) ฀฀฀฀฀฀ = 2000 Ω + (1000 Ω + ฀฀) 6.10

PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Put ฀฀฀฀฀฀ into the frequency domain:

2฀฀6 Ω ∗ ฀฀ = 3000 Ω + ฀฀ 1 2฀฀6 Ω ∗ ฀฀฀฀฀฀ ฀฀฀฀฀฀ (฀฀฀฀) = 1 3000 Ω + ฀฀฀฀฀฀

Simplify: 1 2฀฀6 ∗ ฀฀฀฀฀฀ ฀฀฀฀฀฀ ฀฀฀฀฀฀ (฀฀฀฀) = ∗ 1 3000 + ฀฀฀฀฀฀ ฀฀฀฀฀฀ =

= Substitute ฀฀฀฀฀฀ (฀฀฀฀) into ฀฀1 (฀฀฀฀):

฀฀฀฀฀฀ 2฀฀6 ∗฀฀฀฀฀฀ ฀฀฀฀฀฀ 3000 Ω ∗ jωC + ฀฀฀฀฀฀ 1 + 3000 ฀฀฀฀฀฀ 2฀฀6

2฀฀6 1 + 3000 ฀฀฀฀฀฀ ฀฀1 (฀฀฀฀) = ฀฀฀฀฀฀ (฀฀฀฀) 2฀฀6 2000 + 1 + 3000 ฀฀฀฀฀฀

Simplify: 2฀฀6 1 + 3000 ฀฀฀฀฀฀ 1 + 3000 ฀฀฀฀฀฀ ฀฀1 (฀฀฀฀) = ฀฀฀฀฀฀ (฀฀฀฀ ) ∗ 2฀฀6 1 + 3000 ฀฀฀฀฀฀ 2000 + 1 + 3000 ฀฀฀฀฀฀ 1 + 3000 ฀฀฀฀฀฀ 2฀฀6 ∗1 + 3000 ฀฀฀฀฀฀

= ฀฀฀฀฀฀ (฀฀฀฀ ) 1 + 3000 ฀฀฀฀฀฀ 2000 ∗ (1 + 3000 ฀฀฀฀฀฀) + 2฀฀6 ∗ 1 + 3000 ฀฀฀฀฀฀

1 + 3000 ฀฀฀฀฀฀ 2฀฀6 ∗ 1 + 3000 ฀฀฀฀฀฀ = ฀฀฀฀฀฀ (฀฀฀฀ ) 1 + 3000 ฀฀฀฀฀฀ 2000 ∗ (1 + 3000 ฀฀฀฀฀฀) + 2฀฀6 ∗ 1 + 3000 ฀฀฀฀฀฀ 2฀฀6 = ฀฀฀฀฀฀ (฀฀฀฀) 2000 + ฀฀฀฀6฀฀6 ∗ ฀฀ + 2฀฀6

2฀฀6 ≈ ฀฀฀฀฀฀ (฀฀฀฀) 2฀฀6 + ฀฀฀฀60 6.11

PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Substitute ฀฀1 (฀฀฀฀) into ฀฀฀฀฀฀฀฀ (฀฀฀฀):

1 ฀฀฀฀฀฀

฀฀฀฀฀฀฀฀ (฀฀฀฀) = ฀฀1 (฀฀฀฀) 1 1000 + ฀฀฀฀฀฀ Simplify: 1 ฀฀฀฀฀฀

฀฀฀฀฀฀฀฀ (฀฀฀฀) = ฀฀1 (฀฀฀฀) 1 1000 + ฀฀฀฀฀฀

฀฀฀฀฀฀ ฀฀฀฀฀฀

฀฀฀฀฀฀ ∗ ฀฀฀฀฀฀

= ฀฀1 (฀฀฀฀) ฀฀฀฀฀฀ 1000 ∗ ฀฀฀฀฀฀ + ฀฀฀฀฀฀

Plug ฀฀1 (฀฀฀฀) into this equation:

1 = ฀฀1 (฀฀฀฀) 1 + 0.01฀฀฀฀

2฀฀6 1 ฀฀฀฀฀฀฀฀ (฀฀฀฀) = ฀฀฀฀฀฀ (฀฀฀฀) 2฀฀6 + ฀฀฀฀601 + 0.01฀฀฀฀

Simplify and solve for ฀฀฀฀฀฀฀฀ (฀฀฀฀)/฀฀฀฀฀฀ (฀฀฀฀):

฀฀฀฀฀฀฀฀ (฀฀฀฀) 2฀฀6 ≈ ฀฀฀฀฀฀ (฀฀฀฀) 2฀฀6 + 2฀฀4฀฀฀฀

Simplify: ฀฀฀฀฀฀฀฀ (฀฀฀฀) ฀฀ ≈ ฀฀฀฀฀฀(฀฀฀฀) ฀ ฀ + ฀฀. ฀฀฀฀฀฀฀฀

(b) Plot the magnitude and phase of the circuit for frequencies between ฀฀฀฀ and ฀฀฀฀ ฀฀ rad/s on graph paper, with a linear scale for frequency. First, get ฀฀฀฀฀฀฀฀ (฀฀฀฀ )/฀฀฀฀฀฀ (฀฀฀฀) into a manageable form:

1 ฀฀฀฀฀฀฀฀ (฀฀฀฀) 1 − 0.01฀฀฀฀ = ∗ ฀฀฀฀฀฀ (฀฀฀฀) 1 + 0.01฀฀฀฀ 1 − 0.01฀฀฀฀ ≈

1 − 0.01 ฀฀฀฀

1 + 0.0001฀฀ 2

Calculate magnitude of complex expression: 6.12 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6 2 1 ฀฀฀฀฀฀฀฀ (฀฀฀฀) 2 0.01฀฀ � � = �� � � + � ฀฀฀฀฀฀ (฀฀฀฀) 2 1 + 0.0001฀฀ 1 + 0.0001฀฀ 2

Phase response:

∠

฀฀฀฀฀฀฀฀ (฀฀฀฀) ฀฀฀฀฀฀ (฀฀฀฀)

−0.01฀฀ 1 − 0.0001฀฀ 2 � = arctan � 1 1 − 0.0001฀฀ 2

= arctan(−0.01฀฀)

Plots: Magnitude 1 0.9 0.8

Magnitude

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 x 10

6

6.13 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Phase (degrees) 0 -10 -20

Phase

-30 -40 -50 -60 -70 -80 -90

0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 6

x 10

(c) Repeat part (b), using semilog paper. Place the frequency on the logarithmic axis. Magnitude 1 0.9 0.8

Magnitude

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 10

10

2

10

3

4

10 Frequency (rad/s)

10

5

10

6

10

7

6.14 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.

G. Rizzoni, Principles and Applications of Electrical Engineering, 6th Edition Problem solutions, Chapter 6

Phase (degrees) 0 -10 -20

Phase

-30 -40 -50 -60 -70 -80 -90 1 10

2

10

3

10

4

5