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## EE 542 Antennas & Propagation for Wireless Communications

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**EE 542Antennas & Propagation for Wireless Communications**Topic 3 - Basic EM Theory and Plane Waves**Outline**• EM Theory Concepts • Maxwell’s Equations • Notation • Differential Form • Integral Form • Phasor Form • Wave Equation and Solution (lossless, unbounded, homogeneous medium) • Derivation of Wave Equation • Solution to the Wave Equation – Separation of Variables • Plane waves O. Kilic EE542**E, H**J EM Theory Concept The fundamental concept of em theory is that a current at a point in space is capable of inducing potential and hence currents at another point far away. O. Kilic EE542**Introduction to EM Theory**• The existence of propagating em waves can be predicted as a direct consequence of Maxwell’s equations. • These equations satisfy the relationship between the vector electric field, E and vector magnetic field, H in time and space in a given medium. • Both E and H are vector functions of space and time; i.e. E (x,y,z;t), H (x,y,z;t.) O. Kilic EE542**What is an Electromagnetic Field?**• The electric and magnetic fields were originally introduced by means of the force equation. • In Coulomb’s experiments forces acting between localized charges were observed. • There, it is found useful to introduce E as the force per unit charge. • Similarly, in Ampere’s experiments the mutual forces of current carrying loops were studied. • B is defined as force per unit current. O. Kilic EE542**Why not use just force?**• Although E and B appear as convenient replacements for forces produced by distributions of charge and current, they have other important aspects. • First, their introduction decouples conceptually the sources from the test bodies experiencing em forces. • If the fields E and B from two source distributions are the same at a given point in space, the force acting on a test charge will be the same regardless of how different the sources are. • This gives E and B meaning in their own right. • Also, em fields can exist in regions of space where there are no sources. O. Kilic EE542**Maxwell’s Equations**• Maxwell's equations give expressions for electric and magnetic fields everywhere in space provided that all charge and current sources are defined. • They represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. • These set of equations describe the relationship between the electric and magnetic fields and sources in the medium. • Because of their concise statement, they embody a high level of mathematical sophistication. O. Kilic EE542**Notation: (Time and Position Dependent Field Vectors)**O. Kilic EE542**Notation: Sources and Medium**O. Kilic EE542**Maxwell’s Equations – Physical Laws**• Faraday’s Law Changes in magnetic field induce voltage. • Ampere’s Law Allows us to write all the possible ways that electric currents can make magnetic field. Magnetic field in space around an electric current is proportional to the current source. • Gauss’ Law for Electricity The electric flux out of any closed surface is proportional to the total charge enclosed within the surface. • Gauss’ Law for Magnetism The net magnetic flux out of any closed surface is zero. O. Kilic EE542**Differential Form of Maxwell’s Equations**Faraday’s Law: (1) Ampere’s Law: (2) Gauss’ Law: (3) (4) O. Kilic EE542**Constitutive Relations**Constitutive relations provide information about the environment in which electromagnetic fields occur; e.g. free space, water, etc. permittivity (5) permeability (6) Free space values. O. Kilic EE542**Time Harmonic Representation - Phasor Form**• In a source free ( ) and lossless ( ) medium characterized by permeability m and permittivity e, Maxwell’s equations can be written as: O. Kilic EE542**Examples of del Operations**• The following examples will show how to take divergence and curl of vector functions O. Kilic EE542**Example 1**O. Kilic EE542**Solution 1**O. Kilic EE542**Example 2**Calculate the magnetic field for the electric field given below. Is this electric field realizable? O. Kilic EE542**Solution**O. Kilic EE542**Solution continued**O. Kilic EE542**Solution continued**To be realizable, the fields must satisfy Maxwell’s equations! O. Kilic EE542**Solution Continued**These fields are NOT realizable. They do not form em fields. O. Kilic EE542**Time Harmonic Fields**• We will now assume time harmonic fields; i.e. fields at a single frequency. • We will assume that all field vectors vary sinusoidally with time, at an angular frequency w; i.e. O. Kilic EE542**Time Harmonics and Phasor Notation**Using Euler’s identity The time harmonic fields can be written as Phasor notation O. Kilic EE542**Phasor Form**Information on amplitude, direction and phase Note that the E and H vectors are now complex and are known as phasors O. Kilic EE542**Time Harmonic Fields in Maxwell’s Equations**With the phasor notation, the time derivative in Maxwell’s equations becomes a factor of jw: O. Kilic EE542**Maxwell’s Equations in Phasor Form (1)**O. Kilic EE542**Maxwell’s Equations in Phasor Form (2)**O. Kilic EE542**Phasor Form of Maxwell’s Equations (3)**Maxwell’s equations can thus be written in phasor form as: Phasor form is dependent on position only. Time dependence is removed. O. Kilic EE542**Examples on Phasor Form**Determine the phasor form of the following sinusoidal functions: • f(x,t)=(5x+3) cos(wt + 30) • g(x,z,t) = (3x+z) sin(wt) • h(y,z,t) = (2y+5)4z sin(wt + 45) • V(t) = 0.5 cos(kz-wt) O. Kilic EE542**Solutions**a) O. Kilic EE542**Solutions**b) O. Kilic EE542**Solution**c) O. Kilic EE542**Solution**d) O. Kilic EE542**Example**• Find the phasor notation of the following vector: O. Kilic EE542**Solution**O. Kilic EE542**Example**• Show that the following electric field satisfies Maxwell’s equations. O. Kilic EE542**Solution**O. Kilic EE542**The Wave Equation (1)**If we take the curl of Maxwell’s first equation: Using the vector identity: And assuming a source free, i.e. and lossless; i.e. medium: O. Kilic EE542**The Wave Equation (2)**Define k, which will be known as wave number: O. Kilic EE542**Wave Equation in Cartesian Coordinates**where O. Kilic EE542**Laplacian**O. Kilic EE542**Scalar Form of Maxwell’s Equations**Let the electric field vary with x only. and consider only one component of the field; i.e. f(x). O. Kilic EE542**Possible Solutions to the Scalar Wave Equation**Energy is transported from one point to the other Standing wave solutions are appropriate for bounded propagation such as wave guides. When waves travel in unbounded medium, traveling wave solution is more appropriate. O. Kilic EE542**The Traveling Wave**• The phasor form of the fields is a mathematical representation. • The measurable fields are represented in the time domain. Let the solution to the a-component of the electric field be: Traveling in +x direction Then O. Kilic EE542**Traveling Wave**As time increases, the wave moves along +x direction O. Kilic EE542**Standing Wave**Then, in time domain: O. Kilic EE542**Standing Wave**Stationary nulls and peaks in space as time passes. O. Kilic EE542**To summarize**• We have shown that Maxwell’s equations describe how electromagnetic energy travels in a medium • The E and H fields satisfy the “wave equation”. • The solution to the wave equation can be in various forms, depending on the medium characteristics O. Kilic EE542**The Plane Wave Concept**• Plane waves constitute a special set of E and H field components such that E and H are always perpendicular to each other and to the direction of propagation. • A special case of plane waves is uniform plane waves where E and H have a constant magnitude in the plane that contains them. O. Kilic EE542**Plane Wave Characteristics**amplitude Frequency (rad/sec) phase polarization Wave number, depends on the medium characteristics Direction of propagation amplitude phase O. Kilic EE542