Добавил:
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
Bardwell J.Math and physics for the 802.11 wireless LAN engineer.pdf
Скачиваний:
41
Добавлен:
23.08.2013
Размер:
3.45 Mб
Скачать

Math and Physics for the 802.11 Wireless LAN Engineer

A Discussion of What Every LAN Engineer Should Know About 802.11

by Joseph Bardwell

Math and Physics for the 802.11 Wireless LAN Engineer” is a discussion of physics and electromagnetic wave theory as applied to 802.11 wireless networking. Mr. Joseph Bardwell has written a readable paper that provides an explanation of how Maxwellʼs wave equations, Fresnel Zone calculations, and many other complicated engineering topics can be readily understood, and how they come into play in the realm of wireless network design and implementation. While there is an allusion to Calculus, the paper is written so that anyone with a high school algebra background can easily follow the

math. Youʼll learn about how electromagnetic waves are affected by the environment and youʼll be introduced to some of the more esoteric quantum mechanical characteristics of particle/wave duality. Ever wonder how a radio signal can go around the corner of a building in the absence of any reflective surfaces to bounce from? Youʼll find out.

October 2003

Table of Contents

Math and Physics for the 802.11 Wireless LAN Engineer

1

About the Author

1

Section 1: Introduction

2

Are You the Professor, or the Chauffeur?

2

Purpose and Perspective

2

Apprehensive Attitudes Resulting from Lack of Knowledge

4

What You’ll Learn in this Paper

5

A Note to the Reader Familiar with the Subject

5

Section 2: Electricity and Electromagnetic Fields

7

Electrical Force

7

Ohm’s Law

8

Resistance and Reactance

9

Power Measurement

10

Watts, Milliwatts, Decibels, and dBm Units of Measurement

10

Magnetic Fields

12

Figure 2.1 The Magnetic Field Surrounding a Current Carrying Conductor

12

Zeno’s Paradoxes

13

Bardwell’s ERP Paradox

14

Section 3: The Electromagnetic Spectrum

15

Figure 3.1 The Electromagnetic Spectrum

15

The Shape of the Electromagnetic Field

16

Figure 3.2 The Spherical Radiation Pattern of a Theoretical Isotropic Radiator

16

Figure 3.3 The Doughnut-Shape of the Electromagnetic Radiation Pattern

16

Particles and Waves

17

Figure 3.4 A Beam of Light Reflecting From the Surface of a Mirror

18

Figure 3.5 A Beam of Light Manifesting Fresnel Diffraction

19

Figure 3.6 A 15-mile Span Using 6 Antennae and 2 Repeaters

19

Figure 3.7 Monthly Sunspot Activity Since 1950

20

The Electromotive Force

20

Scalar and Vector Measurement Metrics

21

Figure 3.8 Hiking in the Las Trampas Wildlife Refuge

22

Measuring the Characteristics of the Electromagnetic Field

23

Differentiation of Functions with One Independent Variable

26

Figure 3.9 Position Versus Time and the Rate of Change

26

Figure 3.10 The Notation for Differentiation

27

 

 

Differentiation of Functions With More Than One Independent Variable

28

Magnetic Flux Density (B) and the Vector Potential (A)

28

Figure 3.11 Partial Differentiation to Compute the Components of B

28

Figure 3.12 Basic Maxwell Wave Equations in Vector Form

29

Section 4: Electromagnetic Field Propagation

30

Time Symmetry and the Reciprocity Theorem

30

Practical Considerations Related to Antenna Reciprocity

31

Figure 4.1 Correct and Incorrect 802.11 Access Point Antenna Orientation

32

Transmitters and Receivers with Different Power Levels

32

Propagation of Electromagnetic Waves in Space

33

Figure 4.2 The Radiating Elements of a Dipole Antenna

33

Figure 4.3 Wavefront Formation with a Dipole Radiator

34

Figure 4.4 The Electromagnetic Field Surrounding a Dipole Antenna

34

Coupling and Re-radiation

35

Representing the Direction of Field Propagation

35

The Transverse Wavefront

36

Figure 4.5 Surface Area Defined On the Spherical Wavefront

36

Figure 4.6 An 802.11 NIC Encounters a Flat, Planar Wavefront

36

The Electromagnetic Field Pattern

37

Polar Coordinate Graphs of Antennae Field Strength

37

Figure 4.7 The Elevation Cut View of Antennae in a Warehouse

38

Figure 4.8 The Azimuth Cut View of a Directional Antenna

38

Figure 4.9 Polar Coordinate Graphs for an Omni-Directional Antenna

39

Figure 4.10 Vertical and Horizontal Cuts of an Apple

39

Figure 4.11 Close-up View of the Elevation Cut Polar Coordinate Graph

40

Figure 4.12 The Omni-Directional Elevation Cut Seen in the Warehouse

40

Figure 4.13 Polar Coordinate Graphs for a Directional Antenna

41

Figure 4.14 The Elevation Cut Rotated to the Left

41

Figure 4.15 The Directional Antennaʼs Elevation Cut Seen in the Warehouse

42

The “E” Graph and the “H” Graph

42

Half-Power Beam Width

42

Figure 4.16 Antenna Field Pattern and Half Power Beam Width Measurement

43

Half-Power Beamwidth on a Polar Coordinate Graph

43

Figure 4.17 Identifying Half-Power Beamwidth (HPBW) Points

43

Figure 4.18 Horizontal and Vertical Beamwidth for a Directional Antenna

44

Fulland Half-Wavelength Antennae Beamwidth

44

Figure 4.19 The Field Pattern for a Full Wavelength Dipole Antenna

44

Figure 4.20 The Field Pattern for a Half-Wavelength Dipole Antenna

45

Use of the Unit Vector

45

802.11 Site Considerations Related to Beamwidth

45

A Challenging Beamwidth Question

46

Figure 4.21 The Client and the Access Point Are Within Each Otherʼs HPBW Zone

46

 

 

Signal Strength and Reduced Data Rate

46

Figure 4.22

User #1 Is Outside the Beamwidth Angle of the Access Point

47

Physical Measurements Associated With the Polar Coordinate Graph

48

Figure 4.23

The Polar Elevation Cut as it Relates to a Real-World Situation

48

RF Modeling and Simulation

49

Figure 4.24

Results of an RF Simulation

49

Section 5: Electromagnetic Field Energy

51

The Particle Nature of the Electromagnetic Field

51

Field Power and the Inverse Square Law

51

Figure 5.1 Determining the Surface Area of a Sphere

52

Electric Field Strength Produced By An Individual Charge

53

Figure 5.2 The Strength of the Electric Field for an Individual Charge

53

Time Delay and the Retarded Wave

54

Figure 5.2 (repeated) The Strength of the Electric Field for an Individual Charge

54

The Derivative of the Energy With Respect To Time

55

Effective Radiated Power

55

The Near Field and the Far Field

56

Figure 5.3 The Far Field Transformation of the Field Strength

57

Signal Acquisition from the Spherical Wavefront

57

Figure 5.4 The Spherical Presentation of the Wavefront

58

Figure 5.5 An Impossible Antenna of Unreasonable Length

58

The Boundary Between the Near Field and the Far Field

59

Figure 5.6 Out of Phase Signals Meeting a Vertical Antenna

60

Figure 5.7 A Close View of the Out of Phase Waves

60

Characteristics of the Far Field

61

Considerations Concerning Near Field Interaction

61

The Reactive Near Field and the Radiating Near Field

62

Antenna Gain and Directivity

62

Figure 5.8 A Spherical Versus a Toroidal Radiation Pattern

64

Phased Array Design Concepts

65

Figure 5.9 Top-View of Canceling Fields Parallel to the Two Radiators

65

Figure 5.10 Top-View of Augmenting Fields Perpendicular to the Two Radiators

66

Figure 5.11 A Multiple Element Phased Array Field Pattern

66

Parasitic Element Design Concepts

67

Figure 5.12 The Yagi-Uda Antenna

67

Antenna Beamwidth and the Law of Reciprocity

67

Figure 5.13 The Depiction of an Antennaʼs Beamwidth

67

Section 6: The Huygens-Fresnel Principle

69

Figure 6.1

A Spherical Wavefront from an Isotropic Radiator

69

Figure 6.2

Each New Point Source Generates a Wavelet

69

Applying the Huygens-Fresnel Principle in the 802.11 Environment

70

Figure 6.3 An Obstruction Causes the Wavefront to Bend

71

 

 

Diffraction of the Expanding Wavefront

71

How Interference Relates To Diffraction

72

Figure 6.4 Wavelets Combining Out of Phase at the Receiver

72

Figure 6.5 The Critical Angle at Which the Wave is 180O Out of Phase

73

Figure 6.6 The Effect of an Obstruction on the Received Wavelets

74

Figure 6.7 The Receiverʼs Location Determines the Obstructions Affect

74

Fresnel Zones

75

Figure 6.8 The Oval Volume of a Fresnel Zone

75

Figure 6.9 Multiple Fresnel Zones Built Up Around the Central Axis

75

Fresnel Zones are not Related to Antenna Gain or Directivity

75

Calculating the Radius of the Fresnel Zones

76

Obstructions in the First Fresnel Zone

76

Figure 6.10 Interior Obstructions in the First Fresnel Zone

76

Practical Examples of the Fresnel Zone Calculation

77

The Fresnel Construction

78

Figure 6.11 The Pythagorean Construction of the First Fresnel Zone

78

Figure 6.12 Two Triangles Are Constructed Between Transmitter and Receiver

79

Dealing with an Unfriendly Equation

85

One More Equation

87

The Erroneous Constant of Proportionality

88

Figure 6.13 The Typical Presentations of the Fresnel Zone Equations

88

Concluding Thoughts

89

Appendix A

90

The Solution To Zeno’s and Bardwell’s Paradoxes

90

Appendix B

91

Trigonometric Relationships: Tangent, Sine, and Cosine

91

Figure B.1: Trigonometric Relationships In Right Triangles

91

Figure B.2: The Basic Trigonometric Relationships in a Right Triangle

91

Appendix C

92

Representational Systems for Vector Description

92

Figure C.1 Vectors Represented Using Cylindrical Coordinates

92

Figure C.2 The Spherical Coordinate System

93

Appendix D

94

Electromagnetic Forces at the Quantum Level

94

Appendix E

95

Enhanced Bibliography

95

 

 

Math and Physics for the 802.11 Wireless LAN Engineer

About the Author

Joseph Bardwell has been active in the computer industry since the 1970ʼs and is a widely recognized lecturer, technical consultant, and co-author of the recent book “Troubleshooting Campus Networks” (Wiley, 2002). He has been a contributor to the development of the WildPackets AiroPeek NXtm wireless LAN analyzer and the original Network General Sniffertm protocol analyzer. He was

named as one of the “Service 25” a list of key, influential people in the network services industry for his pioneering work in developing technical certification for network analysis. Over the past several years hundreds of people across the country have heard him speak in the “ZEN and the Art of Network Maintenance” seminar series that he developed. As Mr. Bardwell says in his seminars “When you know what the magician knows, itʼs not magic anymore.”

Mr. Bardwell is the Chief Scientist and President of Connect802 Corporation (www.Connect802.com), providing turnkey Wi-Fi networking solutions and predictive RF modeling and simulation for 802.11 network design. He is a certified RF designer. Mr. Bardwell may be contacted via email at: joe@Connect802.com.

Math and Physics for the 802.11 Wireless LAN Engineer

1

Copyright 2003 - Joseph Bardwell

Соседние файлы в предмете Электротехника