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 |
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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 |
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Signal Strength and Reduced Data Rate |
46 |
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Figure 4.22 |
User #1 Is Outside the Beamwidth Angle of the Access Point |
47 |
Physical Measurements Associated With the Polar Coordinate Graph |
48 |
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Figure 4.23 |
The Polar Elevation Cut as it Relates to a Real-World Situation |
48 |
RF Modeling and Simulation |
49 |
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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 |
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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 |
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Figure 6.3 An Obstruction Causes the Wavefront to Bend |
71 |
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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 |
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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 |
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Copyright 2003 - Joseph Bardwell