Wave Propagation

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NANYANG TECHNOLOGICAL UNIVERSITY

School of Electrical and Electronic Engineering

Laboratory Experiment

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Laboratory Report
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For
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Experiment NO. 315
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WAVE PROPAGATION
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EE3071 Laboratory 3

Location: S1-B4a-03

AY 2011/2012

Name: EMIR NUROV
Matriculation number: U0920108K
Group: LA03

1. Introduction

1.1 Propagation in Free-Space

1.1.1 Friis Transmission Equation

To begin the derivation of the Friis Equation, consider two antennas in free space (no obstructions nearby) separated by a distance R: Figure 1. Transmit (Tx) and Receive (Rx) Antennas separated by R. Assume that  Watts of total power are delivered to the transmit antenna. For the moment, assume that the transmit antenna is omnidirectional, lossless, and that the receive antenna is in the far field of the transmit antenna. Then the power density p (in Watts per square meter) of the plane wave incident on the receive antenna a distance R from the transmit antenna is given by:
If the transmit antenna has an antenna gain in the direction of the receive antenna given by, then the power density equation above becomes: The gain term factors in the directionality and losses of a real antenna. Assume now that the receive antenna has an effective aperture given by Aer. Then the power received by this antenna (Pr) is given by:
Since the effective aperture for any antenna can also be expressed as: The resulting received power can be written as:
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This is known as the Friis Transmission Formula. It relates the free space path loss, antenna gains and wavelength to the received and transmits powers. This is one of the fundamental equations in antenna theory, and should be remembered (as well as the derivation above).

1.1.2 Radiation Pattern and Far Field Distance

The distance to the far-field involves the wavelength and the size of the transducer or array. For example, the distance between a projector being evaluated and a measuring Hydrophone must be considerably greater than the projector size to prevent the different distances from the center and ends of the projector from introducing Significant phase and amplitude differences at the measuring hydrophone. This is particularly important when beam pattern measurements are made and the transducer is rotated. The far-field is established for a projector or hydrophone array of length L at the so-called Rayleigh distance, r> L2/2γ. This distance may be understood by considering an acoustic wave arriving at a hydrophone array of length L from a small source at a distance r from the ends of the array. If the source is far from the array, the arc of the curved wavefront will fall along L and all the hydrophone elements of the array will receive an approximately in-phase wave. The difference between a spherical wave from a nearby small source and a plane wave from a distant small source may be measured by the “sagitta” distance δ = r (1 – cosθ). As the source distance r grows, the angle θ becomes small so that δ = r θ2/2 and θ = L/2r...
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