- •Math and Physics for the 802.11 Wireless LAN Engineer
- •About the Author
- •Section 1: Introduction
- •Are You the Professor, or the Chauffeur?
- •Purpose and Perspective
- •Apprehensive Attitudes Resulting from Lack of Knowledge
- •What You’ll Learn in this Paper
- •A Note to the Reader Familiar with the Subject
- •Section 2: Electricity and Electromagnetic Fields
- •Electrical Force
- •Resistance and Reactance
- •Power Measurement
- •Watts, Milliwatts, Decibels, and dBm Units of Measurement
- •Magnetic Fields
- •Figure 2.1 The Magnetic Field Surrounding a Current Carrying Conductor
- •Zeno’s Paradoxes
- •Bardwell’s ERP Paradox
- •Section 3: The Electromagnetic Spectrum
- •Figure 3.1 The Electromagnetic Spectrum
- •The Shape of the Electromagnetic Field
- •Figure 3.2 The Spherical Radiation Pattern of a Theoretical Isotropic Radiator
- •Figure 3.3 The Doughnut-Shape of the Electromagnetic Radiation Pattern
- •Particles and Waves
- •Figure 3.4 A Beam of Light Reflecting From the Surface of a Mirror
- •Figure 3.5 A Beam of Light Manifesting Fresnel Diffraction
- •Figure 3.6 A 15-mile Span Using 6 Antennae and 2 Repeaters
- •Figure 3.7 Monthly Sunspot Activity Since 1950
- •The Electromotive Force
- •Scalar and Vector Measurement Metrics
- •Figure 3.8 Hiking in the Las Trampas Wildlife Refuge
- •Measuring the Characteristics of the Electromagnetic Field
- •Differentiation of Functions with One Independent Variable
- •Figure 3.9 Position Versus Time and the Rate of Change
- •Figure 3.10 The Notation for Differentiation
- •Differentiation of Functions With More Than One Independent Variable
- •Magnetic Flux Density (B) and the Vector Potential (A)
- •Figure 3.11 Partial Differentiation to Compute the Components of B
- •Figure 3.12 Basic Maxwell Wave Equations in Vector Form
- •Section 4: Electromagnetic Field Propagation
- •Time Symmetry and the Reciprocity Theorem
- •Practical Considerations Related to Antenna Reciprocity
- •Figure 4.1 Correct and Incorrect 802.11 Access Point Antenna Orientation
- •Transmitters and Receivers with Different Power Levels
- •Propagation of Electromagnetic Waves in Space
- •Figure 4.2 The Radiating Elements of a Dipole Antenna
- •Figure 4.3 Wavefront Formation with a Dipole Radiator
- •Figure 4.4 The Electromagnetic Field Surrounding a Dipole Antenna
- •Coupling and Re-radiation
- •Representing the Direction of Field Propagation
- •The Transverse Wavefront
- •Figure 4.5 Surface Area Defined On the Spherical Wavefront
- •Figure 4.6 An 802.11 NIC Encounters a Flat, Planar Wavefront
- •The Electromagnetic Field Pattern
- •Polar Coordinate Graphs of Antennae Field Strength
- •Figure 4.7 The Elevation Cut View of Antennae in a Warehouse
- •Figure 4.8 The Azimuth Cut View of a Directional Antenna
- •Figure 4.9 Polar Coordinate Graphs for an Omni-Directional Antenna
- •Figure 4.10 Vertical and Horizontal Cuts of an Apple
- •Figure 4.11 Close-up View of the Elevation Cut Polar Coordinate Graph
- •Figure 4.12 The Omni-Directional Elevation Cut Seen in the Warehouse
- •Figure 4.13 Polar Coordinate Graphs for a Directional Antenna
- •Figure 4.14 The Elevation Cut Rotated to the Left
- •Figure 4.15 The Directional Antenna’s Elevation Cut Seen in the Warehouse
- •The “E” Graph and the “H” Graph
- •Half-Power Beam Width
- •Figure 4.16 Antenna Field Pattern and Half Power Beam Width Measurement
- •Half-Power Beamwidth on a Polar Coordinate Graph
- •Figure 4.17 Identifying Half-Power Beamwidth (HPBW) Points
- •Figure 4.18 Horizontal and Vertical Beamwidth for a Directional Antenna
- •Figure 4.19 The Field Pattern for a Full Wavelength Dipole Antenna
- •Figure 4.20 The Field Pattern for a Half-Wavelength Dipole Antenna
- •Use of the Unit Vector
- •802.11 Site Considerations Related to Beamwidth
- •A Challenging Beamwidth Question
- •Figure 4.21 The Client and the Access Point Are Within Each Other’s HPBW Zone
- •Signal Strength and Reduced Data Rate
- •Figure 4.22 User #1 Is Outside the Beamwidth Angle of the Access Point
- •Physical Measurements Associated With the Polar Coordinate Graph
- •Figure 4.23 The Polar Elevation Cut as it Relates to a Real-World Situation
- •RF Modeling and Simulation
- •Figure 4.24 Results of an RF Simulation
- •Section 5: Electromagnetic Field Energy
- •The Particle Nature of the Electromagnetic Field
- •Field Power and the Inverse Square Law
- •Figure 5.1 Determining the Surface Area of a Sphere
- •Electric Field Strength Produced By An Individual Charge
- •Figure 5.2 The Strength of the Electric Field for an Individual Charge
- •Time Delay and the Retarded Wave
- •Figure 5.2 (repeated) The Strength of the Electric Field for an Individual Charge
- •The Derivative of the Energy With Respect To Time
- •Effective Radiated Power
- •The Near Field and the Far Field
- •Figure 5.3 The Far Field Transformation of the Field Strength
- •Signal Acquisition from the Spherical Wavefront
- •Figure 5.4 The Spherical Presentation of the Wavefront
- •Figure 5.5 An Impossible Antenna of Unreasonable Length
- •The Boundary Between the Near Field and the Far Field
- •Figure 5.6 Out of Phase Signals Meeting a Vertical Antenna
- •Figure 5.7 A Close View of the Out of Phase Waves
- •Characteristics of the Far Field
- •Considerations Concerning Near Field Interaction
- •The Reactive Near Field and the Radiating Near Field
- •Antenna Gain and Directivity
- •Figure 5.8 A Spherical Versus a Toroidal Radiation Pattern
- •Phased Array Design Concepts
- •Figure 5.9 Top-View of Canceling Fields Parallel to the Two Radiators
- •Figure 5.10 Top-View of Augmenting Fields Perpendicular to the Two Radiators
- •Figure 5.11 A Multiple Element Phased Array Field Pattern
- •Parasitic Element Design Concepts
- •Figure 5.12 The Yagi-Uda Antenna
- •Antenna Beamwidth and the Law of Reciprocity
- •Figure 5.13 The Depiction of an Antenna’s Beamwidth
- •Section 6: The Huygens-Fresnel Principle
- •Figure 6.1 A Spherical Wavefront from an Isotropic Radiator
- •Figure 6.2 Each New Point Source Generates a Wavelet
- •Applying the Huygens-Fresnel Principle in the 802.11 Environment
- •Figure 6.3 An Obstruction Causes the Wavefront to Bend
- •Diffraction of the Expanding Wavefront
- •How Interference Relates To Diffraction
- •Figure 6.4 Wavelets Combining Out of Phase at the Receiver
- •Figure 6.5 The Critical Angle at Which the Wave is 180O Out of Phase
- •Figure 6.6 The Effect of an Obstruction on the Received Wavelets
- •Figure 6.7 The Receiver’s Location Determines the Obstructions Affect
- •Fresnel Zones
- •Figure 6.8 The Oval Volume of a Fresnel Zone
- •Figure 6.9 Multiple Fresnel Zones Built Up Around the Central Axis
- •Fresnel Zones are not Related to Antenna Gain or Directivity
- •Calculating the Radius of the Fresnel Zones
- •Obstructions in the First Fresnel Zone
- •Figure 6.10 Interior Obstructions in the First Fresnel Zone
- •Practical Examples of the Fresnel Zone Calculation
- •The Fresnel Construction
- •Figure 6.11 The Pythagorean Construction of the First Fresnel Zone
- •Figure 6.12 Two Triangles Are Constructed Between Transmitter and Receiver
- •Dealing with an Unfriendly Equation
- •One More Equation
- •The Erroneous Constant of Proportionality
- •Figure 6.13 The Typical Presentations of the Fresnel Zone Equations
- •Concluding Thoughts
- •Appendix A
- •The Solution To Zeno’s and Bardwell’s Paradoxes
- •Appendix B
- •Trigonometric Relationships: Tangent, Sine, and Cosine
- •Figure B.1: Trigonometric Relationships In Right Triangles
- •Figure B.2: The Basic Trigonometric Relationships in a Right Triangle
- •Appendix C
- •Representational Systems for Vector Description
- •Figure C.1 Vectors Represented Using Cylindrical Coordinates
- •Figure C.2 The Spherical Coordinate System
- •Appendix D
- •Electromagnetic Forces at the Quantum Level
- •Appendix E
- •Enhanced Bibliography
of accurate transmission relative to noise and errors and adjusts its data transmission rate accordingly. This change in data rate is transparent to the user of the radio (except, of course, they may see
their files being transferred more slowly). Most 802.11 radio designers use more than simple signal strength measurements to determine when to shift to lower data rates.
A Challenging Beamwidth Question
The Professor and the Chauffeur are discussing 802.11 networking. The Chauffeur has been learning, little by little, as he has had to sit through the Professorʼs lectures many times. The Chauffeur makes the following statement:
The ideal situation would be to have a user who is inside the HPBW of the access point antenna and the access point inside the HPBW of the userʼs antenna.
The Chauffeur illustrates his thought with the following diagram:
Figure 4.21 The Client and the Access Point Are Within Each Other’s HPBW Zone
The Professor says, “Interesting idea, but Iʼm sorry to say that youʼre completely wrong.” The reason the Chauffeur is wrong is because the reciprocity theorem says that “If I can hear you, you can hear me (for the same input power).” Hence, if the client can hear the AP then the AP can hear the client, or vice versa. Considering the location of 802.11 client machines relative to the HPBW of the access point antenna is appropriate. Confirming that the access point is within the HPBW of the client antenna is unnecessary. This assumes, of course, that the client and the access point have the same input power to their respective antennae.
Signal Strength and Reduced Data Rate
As a general rule of thumb, when an 802.11 radio is receiving at less than -70 dBm the range is being approached where noise and environmental considerations are probably forcing the data rate to drop. Remember that the power is going to drop in alignment with the Inverse Square Law.
Assume that youʼre 10 feet away from the transmitting antenna of a typical access point rated at 100 mW which is 20 dBm. You measure the signal strength along the perpendicular center-line of the antenna (at the point of strongest signal). You find that the signal strength at this location is -10 dBm
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(which is, by the way, not unrealistic). If you now move up or down in space, to the point where you meet the extension of the HPBW angle, you can expect the measure -13 dBm (half of what was measured at the center-line). By the time you get 100 feet away from the antenna you can expect your signal strength reading (at the center-line) to fall to less than -60 dBm. Now itʼs time to be concerned about being outside the HPBW angle since youʼre only 10 dB away from dropping below -70 dBm. Fortunately, the area inside the HPBW angle has grown as you got further away from the apex of the angle and you may find that you simply canʼt position a receiver outside the HPBW when youʼre 100 feet away from the antenna.
Itʼs very important that you recognize that the preceding description assumed the measurement of - 60 dBm at a distance of 100 feet. While this is a plausible value for some interior environments it is not to be construed to be a generally applicable rule-of-thumb or suggested guideline. As they say in automobile advertisements “Your mileage may vary.”
We can use the basic trigonometric relationships to make some general comments about access point placement in 802.11 wireless networks. Consider a typical access point with a 1-wavelength antenna (roughly 12.5 cm). This dipole will have a HPBW of roughly 47º. Figure 4.22 (below) shows an access point mounted on a 9-foot high ceiling with a user sitting at their desk near the access point. The annotations on the right-hand side of the figure are explained following the figure.
Figure 4.22 User #1 Is Outside the Beamwidth Angle of the Access Point
The ceiling in the figure (above) is 9 feet high. The top of the userʼs desk, where their notebook computerʼs antenna is located, is 3 feet high. The antenna, then, is 6 feet below the ceiling. Because the access point is mounted at the ceiling line (point A) the ceiling forms the centerline of the beamwidth angle. The bottom half of the beamwidth angle is 23.5 degrees (below the horizontal). The tangent of 23.5 degrees is 0.4348 (obtained using a scientific calculator). The tangent is the ratio of the side opposite the angle to the side adjacent the angle (Law of Tangents). In this case, thatʼs the line CB (the 6 foot distance from ceiling to desktop) divided by the unknown distance from the wall, line AC. Rearranging the equation for the Law of Tangents to solve for the length of line AC yields: AC = CB / tangent or AC = 6 / 0.4348 and the answer (13.799) is rounded to 13.8 feet. User #1, therefore, is underneath the beamwidth angle and outside the 47O HPBW of the antenna. User #2 is beyond 13.8 feet from the wall and is inside the HPBW angle.
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