Electronic Communications by Dennis Roddy, John Coolen .pdf

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Electronic

Communications

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This page is intentionally left blank.

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Electronic

Communications

Fourth Edition

Dennis Roddy John Coolen

Lakehead University, Ontario

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Copyright © 2014 Dorling Kindersley (India) Pvt. Ltd.

Licensees of Pearson Education in South Asia

No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent.

This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time.

ISBN 9788177585582 eISBN 9789332538030

Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India Registered Office: 11 Local Shopping Centre, Panchsheel Park, New Delhi 110 017, India

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The major additions to the fourth edition are new chapters on digital signals and digital communications.

Many of the other chapters have been considerably expanded, and new problem sets have been added.

Mathcad¹has been introduced as a tool for problem solving, but the problems are formulated in such a way that these can be solved using other computer packages, or a good calculator. Although there are many powerful software packages available for circuit and system analysis, most are too highly specialized to be included here. A more general program such as Mathcad requires the user to be able to formulate the physical concepts in mathematical terms, which requires a good understanding of the underlying theory. Also, algebraic manipulation, with its attendant errors, can usually be avoided, as can simplifying assumptions which limit accuracy in certain instances.

The text is intended for use in the final year of technology programs in telecommunications. A one-term course covering the basics of communications systems might include material from chapters 1, 2, 3, 4, 8, 9, 10, 11, and 12. Material of a more difficult nature, such as that on the fast Fourier transform, could be omit- ted without breaking the continuity, and instructors who wished to include more on receiver principles might choose to make their own selection from these chapters and from chapters 5, 6, and 7. A second one-term course on transmission and propagation could be based on chapters 13, 14, 15, 16, with material dealing with systems being selected from the remaining chapters 17, 18, 19, and 20. There is more material in this book than can be reasonably covered in the final year of a technology program: The aim has been to make it useful as a reference text for technology graduates in the work force, as well as for students currently in college programs. It is also hoped that the book will provide a useful “bridging text” for those graduate technologists who continue with engineering degree studies.

The authors would like to thank the following reviewers for their valuable suggestions and input:

Donald Stenz, Milwaukee School of Engineering; Alvis Evans, Tarrant County Jr. College; Allan Smith, Louisiana Tech University; Dr. Lester Johnson, Savannah State College; Warren Foxwell, DeVry Institute — Lombard; Donald Hill, RETs Electronics Institute; Shakti Chatterjee, DeVry Institute — Columbus; and Hassan Moghbelli, Purdue University — Calumet.

Dennis Roddy John Coolen

The publishers would like to thank K. C. Raveendranathan, Professor and Head, Department of Electronics and Communication, Government Engineering College, Barton Hill, Trivandrum, for his valuable suggestions and inputs in enhancing the content of this book to suit the requirements of Indian universities.

Preface

1Mathcad is a trademark of Mathsoft Inc.

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vii 1 PASSIVE CIRCUITS, 1

1.1 Introduction, 1 1.2 Attenuator Pads, 1 1.3 Series Tuned Circuit, 9 1.4 Parallel Tuned Circuit, 15 1.5 Self-capacitance of a Coil, 17 1.6 Skin Effect, 19

1.7 Mutual Inductance, 20

1.8 High-frequency Transformers, 22 1.9 Tapped Inductor, 27

1.10 Capacitive Tap, 29

1.11 Maximum Power Transfer and Impedance Matching, 32 1.12 Low-frequency Transformers, 34 1.13 Passive Filters, 36

Problems 46 2 WAVEFORM SPECTRA, 51

2.1 Introduction, 51

2.2 Sinusoidal Waveforms, 51 2.3 General Periodic Waveforms, 53 2.4 Trigonometric Fourier Series for

a Periodic Waveform, 53 2.5 Fourier Coefficients, 55 2.6 Spectrum for the Trigonometric

Fourier Series, 55 2.7 Rectangular Waves, 56 2.8 Sawtooth Waveform, 59 2.9 Pulse Train, 59

2.10 Some General Properties of Periodic Waveforms, 62

2.11 Exponential Fourier Series, 62 2.12 Approximate Formulas for the

Fourier Coefficients, 64 2.13 Energy Signals and Fourier

Transforms, 66

2.14 Fast Fourier Transform, 68 2.15 Inverse Fast Fourier

Transform, 71

2.16 Filtering of Signals, 73 2.17 Power Signals, 74

2.18 Bandwidth Requirements for Analog Information Signals, 76 Problems, 77

3 DIGITAL LINE WAVEFORMS, 82 3.1 Introduction, 82

3.2 Symbols, Binits, Bits, and Bauds, 82

3.3 Functional Notation for Pulses, 84

3.4 Line Codes and Waveforms, 85 3.5 M-ary Encoding, 95

3.6 Intersymbol Interference, 96 3.7 Pulse Shaping, 96

Problems, 101 4 NOISE, 105

4.1 Introduction, 105 4.2 Thermal Noise, 105 4.3 Shot Noise, 116 4.4 Partition Noise, 116 4.5 Low Frequency or Flicker

Noise, 116 4.6 Burst Noise, 117 4.7 Avalanche Noise, 117 4.8 Bipolar Transistor Noise, 118 4.9 Field-effect Transistor Noise, 118 4.10 Equivalent Input Noise

Generators and Comparison of BJTs and FETs, 118

4.11 Signal-to-noise Ratio, 120

13 TRANSMISSION LINES AND CABLES, 407

13.2 Primary Line Constants, 408 13.3 Phase Velocity and Line

Wavelength, 409

13.4 Characteristic Impedance, 410 13.5 Propagation Coefficient, 412 13.6 Phase and Group Velocities, 415 13.7 Standing Waves, 417

13.8 Lossless Lines at Radio Frequencies, 419

13.9 Voltage Standing-wave Ratio, 420 13.10 Slotted-line Measurements at

Radio Frequencies, 421

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13.11 Transmission Lines as Circuit Elements, 424

13.12 Smith Chart, 428

13.13 Time-domain Reflectometry, 438 13.14 Telephone Lines and Cables, 440 13.15 Radio-frequency Lines, 443 13.16 Microstrip Transmission Lines, 443 13.17 Use of Mathcad in Transmission

Line Calculations, 446 Problems, 450 14 WAVEGUIDES, 453

14.2 Rectangular Waveguides, 453 14.3 Other Modes, 464

Problems, 467

15 RADIO-WAVE PROPAGATION, 468 15.1 Introduction, 468

15.2 Propagation in Free Space, 468 15.3 Tropospheric Propagation, 473 15.4 Ionospheric Propagation, 482 15.5 Surface Wave, 493

15.6 Low Frequency Propagation and Very Low Frequency Propagation, 495

15.7 Extremely Low Frequency Propagation, 498

15.8 Summary of Radio-wave Propagation, 503

Problems, 503 16 ANTENNAS, 505

16.2 Antenna Equivalent Circuits, 505 16.3 Coordinate System, 509

16.4 Radiation Fields, 510 16.5 Polarization, 510 16.6 Isotropic Radiator, 512

16.7 Power Gain of an Antenna, 513 16.8 Effective Area of an Antenna, 515 16.9 Effective Length of an

Antenna, 516

16.10 Hertzian Dipole, 518 16.11 Half-wave Dipole, 520 16.12 Vertical Antennas, 523 16.13 Folded Elements, 526 16.14 Loop and Ferrite-rod

Receiving Antennas, 527 16.15 Nonresonant Antennas, 529 16.16 Driven Arrays, 530

16.17 Parasitic Arrays, 534 16.18 VHF–UHF Antennas, 536 16.19 Microwave Antennas, 538

Problems, 545

17 TELEPHONE SYSTEMS, 548 17.1 Wire Telephony, 548

17.2 Public Telephone Network, 561 Problems, 573

18 FACSIMILE AND TELEVISION, 576 18.1 Introduction, 576

18.2 Facsimile Transmission, 576 18.3 Television, 593

18.4 Television Signal, 606 18.5 Television Receivers, 608 18.6 Television Transmitters, 612 18.7 High-definition Television, 614

Problems, 618

19 SATELLITE COMMUNICATIONS, 620 19.1 Introduction, 620

19.2 Kepler’s First Law, 620 19.3 Kepler’s Second Law, 621 19.4 Kepler’s Third Law, 622 19.5 Orbits, 622

19.6 Geostationary Orbit, 623 19.7 Power Systems, 624 19.8 Attitude Control, 624

19.9 Satellite Station Keeping, 626 19.10 Antenna Look Angles, 627 19.11 Limits of Visibility, 635 19.12 Frequency Plans and

Polarization, 637 19.13 Transponders, 638

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19.14 Uplink Power Budget Calculations, 642 19.15 Downlink Power Budget

Calculations, 646 19.16 Overall Link Budget

Calculations, 647

19.17 Digital Carrier Transmission, 648 19.18 Multiple-access Methods, 649

Problems, 650

20 FIBER-OPTIC COMMUNICATIONS, 654

20.2 Principles of Light Transmission in a Fiber, 654

20.3 Losses in Fibers, 668 20.4 Dispersion, 673

20.5 Light Sources for Fiber Optics, 682 20.6 Photodetectors, 691

20.7 Connectors and Splices, 694 20.8 Fiber-optic Communication

Link, 698 Problems, 701 APPENDIX

A Logarithmic Units, 704

B The Transverse Electromagnetic Wave, 709

INDEX, 713

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Passive Circuits

1.1 Introduction

A passive electric networkis defined in the IEEE Standard Dictionary of Electrical and Electronic Terms as an electric network containing no source of energy. Passive networks contain resistors, inductors, and capacitors connected in various ways. The properties of the network are independent of the energy sources energizing the network.

In this chapter, circuits that are of particular relevance to electronic communications are examined. It is assumed that the student has a thorough understanding of ac and dc circuit theory.

An objective of the chapter is to explain how circuits function, and for this reason specific computer packages for circuit analysis have been avoided. Even so, it should be recognized that the computer approach to circuit analysis generally eliminates much algebraic manipulation, which is a common source of errors.

Examples and problems are set up so that they also can be solved without the aid of specific computer packages or programs, but the student will find that a programmable calculator or a personal computer, along with a versatile program such as Mathcad/MATLAB, is an extremely useful tool for problem solving.

1.2 Attenuator Pads

An attenuator padis a resistive network that is used to introduce a fixed amount of attenuation between a source and a load. Referring to Fig. 1.2.1, let I_LOrepresent the load current without the network inserted, and I_Lthe load current with the network inserted; then the insertion lossof the network is defined as the ratioI_L/I_LO.

In addition to providing attenuation of the signal, the pad usually has to provide input and output matching. Again referring to Fig. 1.2.1, this means that the input resistance R_INmust be equal to the source resistance R_S, and the output resistance R_OUTmust be equal to the load resistance R_L. In evaluating R_INthe load must be connected, and in evaluating R_OUTthe source must be connected.

▲ ▲

1

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From Fig. 1.2.1(a) the reference load current I_LOis seen to be given by

(1.2.1) The quantities I_L,R_IN, and R_OUTshown in Fig. 1.2.1(b) have to be evaluated for specific attenuator circuits. Commonly employed circuits are the T-attenuator and the pi-attenuator, analyzed in the following sections.

The T-Attenuator

The network resistors that make up a T-attenuator are shown as R₁, R₂, and R₃ in Fig. 1.2.2. The name T-attenuator arises because the circuit is configured like the letter T. Applying Kirchhoff’s voltage law to the loop consisting of the source and R₁and R₂yields

EI₁ (R_SR₁R₃) I_L R₃ (1.2.2)

Applying Kirchhoff’s voltage law to the loop consisting of R₂,R_L, and R₃yields

0 I₁ R₃I_L (R₂R₃R_L) (1.2.3)

Equations (1.2.2) and (1.2.3) may be solved for I_Lto give

(1.2.4) Combining Eqs. (1.2.1) and (1.2.4) gives for the insertion loss

(1.2.5) The insertion loss (IL) is usually quoted in decibels. This will be denoted by IL dB, which by definition is

IL dB 20 log₁₀IL (1.2.6)

The negative sign is to show that attenuation occurs; that is, the insertion loss will come out as a positive number of decibels.

R₃(R_SR_L)

(R_SR₁R₃) (R₂R₃R_L) R²₃ I_L I_L

I_LO

I_L ER₃

(R_SR₁R₃) (R₂R₃R_L) R²₃ I_LO E

R_SR_L

Figure 1.2.1 Source and load (a) connected directly and (b) connected through an attenuator pad.

R_s Rs

I_L I_LO

R_L

R_L RIN ROUT

E E

(a) (b)

+ + –

–

Attenuator Pad

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The equations for input resistance and output resistance are found from inspection of Fig. 1.2.2. For the input resistance, the source is removed and the load left connected. The input resistance is that seen looking to the right into the network:

(1.2.7) For the output resistance, the load is removed and the source left connected (with the source emf at zero). The output resistance is that seen looking to the left into the network:

(1.2.8) Given the problem of designing an attenuator to meet specified values of input resistance, output resistance, and insertion loss, Eqs. (1.2.5), (1.2.7), and (1.2.8) can be solved for R₁,R₂, and R₃. Because the insertion loss is usually quoted in decibels, it must first be converted to a current ratio using Eq. (1.2.6) before being substituted into the equations.

Unfortunately, explicit expressions for R₁,R₂, and R₃cannot be obtained, and numerical methods must be employed. Furthermore, not all combinations of input resistance, output resistance, and insertion loss can be met in a given attenuator design, and where the design fails, some of the resistor values will come out negative.

R_OUTR₂ R₃(R₁R_S) (R₁R₃R_S) R_INR₁ R₃ (R₂R_L)

(R₂R₃R_L) Figure 1.2.2 T-attenuator.

R_s

I_L I1

R_L

R1 R2

R3

E ⁺–

EXAMPLE 1.2.1

Determine the resistor values for a T-attenuator that must provide 14-dB attenuation between a 75- source and a 50-load. The attenuator must also provide input and output matching.

SOLUTION From Eq. (1.2.6), the insertion loss is IL10^0.70.2 The three equations to be solved are then

50 R₂ R₃ (R₁R_S) (R₁R₃R_S) 75 R₁ R₃ (R₂R_L)

(R₂R₃R_L)

0.2 R₃(R_SR_L)

(R_SR₁R₃)(R₂R₃R_L)R²₃

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Exercise 1.2.1 Repeat the preceding calculations for an insertion loss of 1 dB.

(Ans. R₁106 ,R₂ 79 , and R₃450 .)

In the preceding exercise, the negative value for R₂shows that the attenuator is physically unrealizable.

Although the attenuator design in general is best achieved by computational methods, as shown by the preceding results, there is one particular case of practical importance that allows for an analytical solution. This is when the attenuator has to provide a specified attenuation between a matched source and load. This results in a symmetrical attenuator in which R₁R₂and R_INR_OUT. Let R₁R₂Rand R_INR_OUTR_o. Then either Eq. (1.2.7) or (1.2.8) gives

(1.2.9)

From this, we have

(1.2.10) From Eq. (1.2.1),I_LOE/2R_o, and because R_INR_o, the input current to the attenuator is also equal to I_LO. Applying the current divider rule to the center node gives I_L I_LOR₃/(R₃ R R_o), and therefore the insertion loss IL is

(1.2.11)

The right-hand side of this is seen to be the same as that for Eq. (1.2.10), and therefore

(1.2.12) (1.2.13)

This allows the resistor Rto be determined for a given insertion loss. Once Ris known,R₃can be determined from Eq. (1.2.11), and it is left as an exercise for the student to show that

(1.2.14) R₃ 2R_o(IL)

1 (IL)²

∴RR_o1IL 1IL R_oR

RR_o IL

IL R₃

R₃RR_o R_oR

RR_o R₃ R₃RR_o R_oR R₃(RR_o)

R₃RR_o

Numerical methods must be employed, for example, using a computer or programmable calculator. The results obtained using Mathcad are R₁56, R₂29, and R₃25, respectively, the values being rounded off to integer values.

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The Pi-Attenuator

The network resistors making up the pi-attenuator are shown as R_A,R_B, and R_Cin Figure 1.2.3. The name pi-attenuator arises because the circuit is configured like the Greek letter . A direct analysis of the circuit may be carried out to find the input resistance, output resistance, and insertion loss in terms of the network resistors. Alternatively, Eqs. (1.2.5), (1.2.7), and (1.2.8) may be transformed through dualityto give, for the pi-network,

(1.2.15)

(1.2.16)

(1.2.17) In these equations, conductance G is the reciprocal of resistance R; thus G_C 1/R_C, and so on.

Although the equations are most easily solved in terms of conductance, resistor values will usually be specified. The computations therefore generally involve the extra steps of converting source resistance and load resistance into conductance values and, once the network conductances are found, converting these back into resistor values.

Alternatively, a T-attenuator may be designed to meet the specified values of insertion loss, input resistance, and output resistance, and the resulting R₁,R₂, and R₃values converted to R_A,R_B, and R_Cvalues using the Y–transformation.

G_OUTG_B G_C(G_AG_S) G_AG_CG_S G_ING_A G_C(G_BG_L)

G_BG_CG_L

ILG_C G_SG_L

(G_SG_AG_C) (G_BG_CG_L)G²_C EXAMPLE 1.2.2

A T-type attenuator is required to provide a 6-dB insertion loss and to match 50-input and output. Find the resistor values.

SOLUTION From Eq. (1.2.6), 6 dB gives an insertion loss ratio of 0.5:1. Therefore, from Eq. (1.2.13),

From Eq. (1.2.14),

R₃ 100 0.5

1 0.5² 66.67 R50 10.5

10.516.67

R_L I_L

R_B R_A

R_C R_S

E + –

Figure 1.2.3 Pi-attenuator.

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The equations obtained using the Y–transformation are

(1.2.18) (1.2.19) (1.2.20) There is no particular advantage favoring the -configuration or the T-configuration, except in specific situations where one type may yield more practical values of resistors than the other. It also follows that if one type is physically unrealizable so also will be the other.

As with the T-attenuator, an analytical solution for the specific case of equal input and output resistances may be obtained. Denoting the insertion loss (as a current ratio, less than unity) as IL, and R_S R_LR_o, the resulting equations are

(1.2.21) (1.2.22) R_CR_o1 (IL)²

2(IL) R_AR_BR_o 1IL

1IL R_CR_AR₂

R₃ R_BR_AR₂ R₁

R_A R₁R₂R₁R₃R₂R₃ R₂

EXAMPLE 1.2.3

A pi-attenuator is required to provide a 6-dB insertion loss and to match 50-input and output. Find the resistor values.

SOLUTION From Eq. (1.2.6), an insertion loss of 6 dB gives IL 0.5. From Eq. (1.2.21),

From Eq. (1.2.22),

R_C50 10.5²

2 0.5 37.5 R_AR_B50 10.5

10.5150

The L-Attenuator

The T- and pi-attenuators described so far are made up of three resistors. The value of each resistor can be chosen independently of the others, thus enabling the three design criteria of input resistance, output resistance, and insertion loss to be met. In many situations the only function of the pad is to provide matching between source and load, and although attenuation will be introduced, this may not be a critical design parameter. This allows a simpler type of pad to be designed, requiring only two resistors; it is known as an L-pad because the network configuration resembles an inverted letter L.

Figure 1.2.4 shows the L-attenuator, and it will be seen that this can be derived from either the T- or the pi-attenuator simply by the removal of one of the resistors. For convenience, it is assumed that the T-network

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forms the basis, so the resistor labels are the same as those used in the T-network, and the T-network equations can be used to evaluate these. Exactly the same resistor values would result by using the pi-network as the basis.

As shown in Fig. 1.2.4, different configurations are required depending on whether R_SR_Lor R_SR_L. Figure 1.2.4(a) shows the circuit required for the condition R_SR_L. Inspection of the circuit shows that

So

and therefore

(1.2.23) Also,

which gives

(1.2.24) The 1/R₃can be eliminated from Eqs. (1.2.23) and (1.2.24), giving

∴ 1

R_SR₁ 1

R_SR₁ 2 R_L 1

R_SR₁ 1 R_L 1

R_L 1 R₃R₁ 1

R_L 1

R₃ 1 R₁R_S R_LR_OUT R₃(R₁R_S)

R₃R₁R_S 1

R_SR₁ 1 R₃ 1

R_L R_SR₁ R₃R_L

R₃R_L R_SR_inR₁ R₃R_L

R₃R_L

R_S

I_L

E

R_L

(a) +–

R1

R3

RS

IL

E

R_L

(b) +–

R1

R3

R2

Figure 1.2.4 L-attenuator for (a) R_SR_Land (b) R_SR_L.

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or

(1.2.25) Adding Eqs. (1.2.23) and (1.2.24) and simplifying for R₃gives

(1.2.26) With R₂0, Eq. (1.2.5) gives

(1.2.27) These equations are for the situation shown in Fig. 1.2.4(a), where the source resistance is greater than the load resistance.

IL R₃(R_SR_L)

(R_SR₁R₃)(R₃R_L)R²₃ R₃ R²_SR²₁

R₁ R₁R_S(R_SR_L)

∴R²₁R²_SR_SR_L

∴ 2R_S R²_SR²₁ 2

R_L

EXAMPLE 1.2.4

Design an L-attenuator to match a 75-source to a 50-load, and determine the insertion loss.

SOLUTION From Eq. (1.2.25),

From Eq. (1.2.26),

From Eq. (1.2.27),

In decibels, this is 20 log 0.528 5.54 dB.

IL 86.6 (7550)

(7543.386.6) (86.650)86.6²0.528 R₃ 75²43.3²

43.3 86.6 R₁75 (7550)43.3

For the condition R_SR_L, Fig. 1.2.4(b) applies. Inspection of the circuit yields

(1.2.28) (1.2.29) R_OUTR₂R₃R_S

R₃R_S R_IN R₃(R₂R_L)

R₂R₃R_L

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These equations may be solved to yield

(1.2.30) (1.2.31) Equation (1.2.30) is used to find R₂, and then Eq. (1.2.31) to find R₃. The insertion loss is obtained from Eq. (1.2.5), with R₁0, as

(1.2.32)

IL R₃(R_SR_L)

(R_SR₃) (R₂R₃R_L)R²₃ R₃ R²_LR²₂

R₂

R₂R_L(R_LR_S)

EXAMPLE 1.2.5

Design an L-attenuator to match a 10-source to a 50-load, and determine the insertion loss.

SOLUTION From Eq. (1.2.30),

and from Eq. (1.2.31)

From Eq. (1.2.32),

In decibels, this is 20 log 0.318 9.95 dB.

IL 11.18 (1050)

(1011.18) (44.7211.1850)11.18²0.318 R₃ 50²44.72²

44.72 11.18 R₂50 (5010)44.72

1.3 Series Tuned Circuit

Impedance of a Series Tuned Circuit

The series tuned circuit consists of a coil connected in series with a capacitor, as shown in Figure 1.3.1.

Resistance rmust be included since in a practical circuit there will always be resistance, mostly that of the coil.

Denoting by Xthe total reactance of the circuit, equal to L1/C,the impedance is given by

(1.3.1) The magnitude of the impedance is

(1.3.2) Z_s

^r²X²

rj

^L¹^C

Z_srjX

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The phase angle of the impedance is

(1.3.3) An examination of the impedance equation shows that at high frequencies such that L1/Cthe inductive term dominates and Xis positive. At low frequencies such that L 1/C the capacitive term dominates and Xis negative. Figure 1.3.2 shows the impedance plot for a series circuit for which C57 pF, L263 µH, and r21.5 .

Since Xcan vary from positive to negative, there must exist a frequency at which it is zero. This frequency is known as the series resonant frequency.

Series Resonant Frequency

Series resonance occurs when the reactive part of the impedance is zero or, equivalently, the phase angle is zero, as shown by Eq. (1.3.3). The magnitude of the impedance is a minimum at resonance, equal to r,from Eq. (1.3.2).

Denoting the series resonant frequency as so2f_so, then for resonance

from which

(1.3.4) Equation (1.3.4) shows that by adjustment of either Lor C(or both) the circuit can be brought into resonance with the applied frequency, a process known as tuning,and the circuit is also referred to as a series tuned circuit. The usefulness of the series tuned circuit is that it permits signals at one frequency to be selected in preference to those at other frequencies, a property referred to as frequency selectivity.

Series Q-Factor

The Q-factor (which stands for quality factor) can be defined as the ratio of inductive reactance at resonance to resistance in a tuned circuit. (The concept was originally applied to coils to indicate that a high reactance relative to resistance was desirable.) Normally, any series resistance associated with the capacitor in the tuned circuit is negligible, but, if significant, it is included in the total series resistance. The Q-factor can therefore be expressed as

(1.3.5) Q_s soL

r f_so 1

2LC

soL 1 soC0 sarctan X

r Figure 1.3.1 Series tuned circuit.

+

– V

I C

L r

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Since soL1/soC,the Q-factor can also be expressed as

(1.3.6) The subscript ssignifies series Q-factor. The Q-factor is an important parameter used in specifying the behavior of tuned circuits, so much so that instruments known as Q-meters are routinely used to measure Q.

The Q-meter allows the Q-factor of a coil to be measured at a specific frequency and tuning capacitance.

Assuming that Q-meter measurements yield so,Cand Q,then Land rare easily found from Eqs. (1.3.5) and (1.3.6). Also, by combining Eqs. (1.3.4). (1.3.5), and (1.3.6), it is easily shown that

(1.3.7) The significance of Eq. (1.3.7) is that it shows Q_sis constant to the extent that L, C,and rare constant.

This holds reasonably well for frequencies about resonance (but see Sections 1.5 and 1.6 for a discussion of ways in which Qmay vary with frequency).

The Q-factor is also referred to as the voltage magnification factorbecause it gives the ratio of reactive voltage magnitude to applied voltage at resonance. This follows because the current at resonance is V/r,where

Q_s1 r

^C^L

Q_s 1 soC_r

Figure 1.3.2 Impedance magnitude and phase angle as functions of frequency for L 263 µH,C 57 pF, and r21.5 .

70 60 50

, 40 30 20 10

120 90 60 30 0 –30 –60 –90

–120–20 –10 0 10 20

0

–20 –10 0

Frequency shift from resonance, kHz

Phase angle, °Ž

(a)

(b)

10 20

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Vis the applied voltage, and the magnitude of the voltage across Lis (V/r) _soLVQ,and across Cit is (V/r) 1/soCVQ.

The magnitude of either reactive voltage is seen to be Qtimes the applied voltage, and this can reach comparatively high levels. Although the total reactive voltage at resonance is zero, it is possible to make use of the voltage magnification by coupling into the inductive or capacitive voltage separately, and use is made of this in filters and coupled circuits, as described later.

Note that the voltage rating of the reactive elements must take into account the expected high voltage at resonance, and that the inductive voltage is not the same as the voltage across the inductor, which includes the voltage across r.

Impedance in Terms of Q

Equation (1.3.1) for impedance can be rewritten as

Land Ccan be eliminated through the use of Eqs. (1.3.5) and (1.3.6) to give

(1.3.8) Denoting the frequency variable by y,

(1.3.9) allows the impedance to be expressed as

Z_sr(1 jyQ_s) (1.3.10)

(1.3.11)

stan¹yQ_s (1.3.12)

These impedance relationships enable the performance of the circuit to be readily gauged in terms of the Q-factor. The higher the Q-factor is, the greater the impedance magnitude at a given frequency off resonance and the sharper the phase change. The frequency selectivity of the circuit is also highly Qdependent.

Relative Response

The relative response of the circuit is the ratio of the current at any given frequency to the current at resonance. For a constant applied voltage V,the current in general is V/ Z_sand at resonance it is V/r.Hence the relative response is

(1.3.13) The relative response determines the frequency selectivity of the circuit, which is how well it discriminates between wanted and unwanted signals. A measure of this is the 3dB bandwidth described in the following section.

1 1jyQ A_r r

Z_s

Z_sr1jyQ_s

y

so so

Z_sr

¹^j

^so ^so

^Q^s

Z_sr

¹^j

^r^L ¹^C^r

(26)

Relative Response in Decibels

The relative response in decibels is the magnitude of A_rexpressed as a decibel voltage ratio:

(1.3.14)

The curve of Fig. 1.3.2 is plotted as a relative response curve in Fig. 1.3.3.

The 3dB Bandwidth

The main function of a tuned circuit is frequency selection, that is, the ability to select frequencies at or near resonance while rejecting other frequencies. A useful measure of the selectivity is the 3-dB bandwidth. This is the frequency band spanned by the 3-dB points on the resonance curve, as shown in Fig. 1.3.3.

At the 3-dB points, the magnitude of the relative response is , and comparing this with Eq. (1.3.13), it is seen that 2 1 (y₃Q_s)²or y₃ 1/Q_s, where y₃is yevaluated at the 3-dB frequencies. These are denoted as f3and f3in Fig. 1.3.3. This last expression written in full is

(1.3.15)

or f²₃f²_so f_sof₃

Q_s f₃

f_sof_so f₃

1 Q_s

1² 10log(1 (yQ)²)

A_r dB 20 log 1

¹ (yQ)²

Figure 1.3.3 Relative response in decibels for a series tuned circuit.

0

–2

–4

–3-dB Bandwidth

Relative response, dB

Frequency shift from resonance, kHz –6

–8

–10

–12

–20 –10

f3 f3

0 10 20

(27)

Referring to Fig. 1.3.3, it is seen that f3is less than f_so, and therefore

Also,fsis seen to be greater than f_so, and therefore

Hence

or (1.3.16)

However, as can be seen from Fig. 1.3.3,f3f3is the 3-dB bandwidth, and therefore

(1.3.17) Thus, for the series tuned circuit of Example 1.3.2 for which Q100 and f_so1.3 MHz, the 3-dB bandwidth is 13 kHz.

Series Tuned Wavetrap

One function of a tuned circuit is to select a signal at a wanted frequency while rejecting signals at other frequencies. One way in which this may be achieved is to use the tuned circuit as a wavetrap,meaning simply that it traps signals at the resonant wavelength. Figure 1.3.4 shows a simple wavetrap circuit. By tuning the series tuned circuit to resonate at the unwanted frequency, the unwanted signal will be shunted away from the load resistor R_L.

B_{3 dB} f_so Q_s f3f3 f_so

Q_s

∴(f3f3) (f3f3) f_so

Q_s (f3f3) f²3f²3 f_so

Q_s (f3 f3) f²3f²_so f_sof3

Q_s f²3f²_so f_sof3

Q_s

Figure 1.3.4 Simple wavetrap circuit.

f_so f2

I2

r C

L

R_L I1

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1.4 Parallel Tuned Circuit

The parallel tuned circuit is shown in Fig. 1.4.1. The inductor has inductance Land resistance r.The capacitor has capacitance Cand is assumed to have negligible resistance. This represents quite accurately most parallel tuned circuits. As will be shown, the resonant frequency and the Q-factor of the parallel tuned circuit are, for all practical purposes, equal to those of the series tuned circuit, but the impedance is the inverse, being very high at resonance and decreasing as the frequency departs from the resonant value.

Impedance of a Parallel Tuned Circuit

Denoting the capacitive branch impedance by Z_C, and the inductive branch by Z_L, then from Fig. 1.4.1 the parallel impedance is given by

(1.4.1) Now,Z_C1/jCand Z_LrjLjL.The approximation introduced here is that the inductive reactance will be very much greater than the resistance at high frequencies, which is normally the case. It will also be seen that the denominator is equal to the impedance of the same components connected in series, or Z_sZ_LZ_C, which from Eq. (1.3.10) is Z_sr(1 jyQ_s). Combining these expressions gives, to a very close approximation:

(1.4.2)

where R_Dis known as the dynamic impedance:

(1.4.3) The parallel impedance is seen to equal the dynamic impedance when the complex term in the denominator is equal to unity. This corresponds to the resonant condition for the parallel tuned circuit. The subscript D,for

“dynamic,” is used to emphasize that the expression applies only for alternating currents at resonance, and the symbol Ris to show that at resonance the impedance is purely resistive.

R_D L Cr R_D

(1jyQ_s) Z_P LC

r (1jyQ_s) Z_p Z_LZ_C

Z_LZ_C Figure 1.4.1 Parallel tuned circuit.

I

V C

L r –

+

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Before examining the resonant conditions in more detail, it is left as an exercise for the student to show that, in terms of Q-factor, alternative forms of the equation for dynamic impedance are

(1.4.4) In these expressions the single subscript ois used for resonant frequency, and the subscript sis dropped from Q, for reasons which will be apparent shortly.

The form Q/oCis particularily useful because each of the quantities involved can be obtained directly from Q-meter measurements. The form Q²ris interesting in that it shows clearly the relationship between the series and parallel impedances at resonance. For example, if Q100 and r20 , then when connected as a series circuit the impedance at resonance would be 20 purely resistive, while connected as a parallel circuit the impedance at resonance would be 200 kpurely resistive. Note again the distinction, however, that the 20 is the physical resistance of the coil, which opposes direct as well as alternating currents (but see Section 1.6), while the 200 kis a “dynamic resistance” applicable only to alternating current at resonance.

In summary, it is seen that the parallel circuit offers a high impedance, and the series circuit a low impedance at resonance, and the parallel impedance varies with frequency as the inverse of the series impedance.

Furthermore, it may be shown that if I_ois the input current at resonance to a parallel circuit, the magnitude of the current in the capacitive branch is I_oQand in the inductive branch I_oQ.Thus, whereas the series resonant circuit exhibits voltage magnification, the parallel resonant circuit exhibits current magnification.

Parallel Resonant Frequency and Q-Factor

Parallel resonance occurs when the reactive part of the impedance is zero. This requires the imaginary term jyQsin the impedance equation to be equal to zero. Since this is the same term as occurs in the series impedance equation, the resonant frequency must be the same for both circuits, and the subscript scan be dropped.

Furthermore, by definingthe Q-factor as Q _oL/r1/oCr,the Q-factor can be used for these ratios wher- ever they appear in equations, whether for series or parallel circuits, and no subscript is required.

It will be recalled that the expression for parallel impedance involved the approximation , which is true for most cases of practical interest, and this is a required condition for the simplified relationships between parallel and series circuits to hold true.

Relative Response of the Parallel Tuned Circuit

When the parallel tuned circuit is fed from a constant current source I,the voltage in general is given by VIZ_pIR_D/(1 jyQ) and at resonance by V_oIR_D. The relative response A_ris the ratio V/V_o and is seen therefore to be given by

(1.4.5) This shows that the relative response for the parallel tuned circuit is identical to that for the series tuned circuit, given by Eq. (1.3.13).

It also follows that the 3-dB bandwidth will be given by the same expression as for the series tuned circuit, or B3 dBf_o/Q.Note carefully, however, that the relative response for the parallel circuit is defined in terms of voltages, while that for the series circuit is defined in terms of currents.

A_r 1 1jyQ

L Q²r

Q oC R_D _oLQ

(30)

A Simple Parallel Tuned Wavetrap

The tuned circuit in Fig. 1.3.4 may be connected as a parallel tuned circuit, and tuned to resonate at the wantedfrequency. In this way it becomes a wavetrapfor the unwanted frequency.

The parallel tuned circuit may also be used as a wavetrap, as shown in Fig. 1.4.2. In this situation the wanted and unwanted signals appear as emfs in series. The parallel tuned wavetrap is connected in series with the load and is tuned to resonate at the unwanted frequency, so it presents a high impedance to this current component (recall that the series circuit was connected in parallel with the load).

1.5 Self-capacitance of a Coil

In addition to resistance, an inductor has capacitance distributed between the turns of the winding. A reasonably accurate circuit representation using lumped (as distinct from distributed) components is shown in Fig. 1.5.1(a).

This figure shows that the coil in fact appears as a parallel tuned circuit, which can be represented as an impedance Z_Leffr_effjL_effand which can be evaluated by the methods described in Section 1.4. The coil has a self-resonant frequency given by and for the coil to behave as an inductor with series resistance and inductance, the frequency of operation must be below the self-resonant frequency of the coil.

A useful approximate expression may be derived for L_efffor frequencies such that |yQ|²1 (note that y /sr srL/in this context). For typical circuits this would include frequencies up to about 90%

off_sr. Considering the coil as a parallel tuned circuit, then Eq. (1.4.2) can be used to represent this:

(1.5.1) The subscript sr is used to denote the coil self-resonant conditions. From this it is seen that r_effR_Dsr/(yQ_sr)². Recalling that , this may be substituted to give

(1.5.2) r_eff r

y² R_Dsr²srr

R_Dsr

(yQ_sr)² (1jyQ_sr) R_Dsr

1(yQ_sr)² (1jyQ_sr) Z_p R_Dsr

(1jyQ_sr) sr1LC_o,

r L

C

R_L E1

E2

Figure 1.4.2 Parallel tuned wavetrap.

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The effective inductance is obtained by equating L_eff R_Dsr/yQ_sr. Again recalling that R_DsrQ_srsrLgives

(1.5.3) After simplifying, this gives

(1.5.4) The effective Q-factor of the coil at angular frequency may be defined as Q_eff L_eff/r_eff. Using Eqs. (1.5.2) and (1.5.4) gives, after simplifying,

(1.5.5) Care must be taken in how the effective circuit values are used. If the coil forms part of a parallel tuned circuit, as shown in Fig. 1.5.1(b), then C_ois simply absorbed in the total tuning capacitance and the circuit behaves as a normal parallel tuned circuit, with the total tuning capacitance, which includes C_o, resonating with the actual inductance L.The operating frequency must be below the self-resonant frequency of the coil.

When the coil is used as part of a series tuned circuit, as shown in Fig. 1.5.1(c), then L_effis the inductance to which the external capacitor must be tuned for resonance, and Q_effwill be the effective Q-factor, assuming that the losses in the capacitor are negligible. The bandwidth of the series circuit, assuming Q_eff remains sensibly constant over the bandwidth range, is given by

(1.5.6) It is worth noting that most Q-meters measure Q_eff.

B_{3 dB eff} f_o Q_eff Q_effQ 1

^sr

²

L_eff L 1(sr)² L_eff srL

y

Figure 1.5.1 (a) Coil with self-capacitance C_o. (b) In a parallel tuned circuit C_ois absorbed in the total tuning capacitance.

(c) In a series tuned circuit,C_oappears separate from C_T. Co

L

r Co

C_T

C_T= tuning capacitance

(a) (b) (c)

L

r Ctotal

Ctotal = C_o + C_T

L

r

(32)

1.6 Skin Effect

The self-induced emf in a conductor resulting from the rate of change of flux linkages opposes the current flow that gives rise to the flux (Lenz’s law). Normally it is assumed that all the flux links with all the conductor.

However, the actual flux linkages increase toward the core of the conductor, since the magnetic flux within the conductor only links with the inner section; in Fig 1.6.1(a), for example, flux line 1links with the complete conductor, while flux line 2links only with the section of radius a.The self-induced emf is greatest at the center of the conductor, which experiences the greatest flux linkages, and becomes less toward the outer circumference.

This results in the current density being least at the center and increasing toward the outer circumference, since the induced emf opposes the current flow (Lenz’s law). The lower current density at the center results in lower magnetic flux there, which tends to offset the effect producing the nonuniform distribution, and, in this way, equi- librium conditions are established. The overall effect, however, is that the current tends to flow near the surface of the conductor, this being referred to as the skin effect.Because the current is confined to a smaller cross section of the conductor, the apparent resistance of the conductor increases. The increase is more noticeable for thick conductors and at high frequencies (where the rate of change of flux linkages is high). Equally important is the fact that the resistance becomes dependent on frequency.

With coils, a special type of wire called Litzendraht wire(Litz wire,for short) is often used to reduce skin effect. Litz wire is made up of strands insulated from each other and wound in such a way that each strand changes position between the center and outer edge over the length of the wire [Fig. 1.6.1(b)]. In this manner, each strand, on average, has equal induced emfs, so that over the complete cross section (made up of the many cross sections of the individual strands), the current density tends to be uniform.

EXAMPLE 1.5.1

A coil has a series resistance of 5 , a self-capacitance of 7 pF, and an inductance of 1 µH. Determine the effective inductance and effective Q-factor when the coil forms part of a series tuned circuit resonant at 25 MHz.

SOLUTION The self-resonant frequency of the coil is

The Q-factor of the coil, excluding self-capacitive effects, is

Hence

and

Q_effQ

¹

²⁵⁶⁰

²

²⁶

L_eff 10⁶

1 (2560)² 1.21 H Q 2 25 10⁶ 10⁶

5 31.4