JAPAN - MALAYSIA TECHNICAL INSTITUTE

MANPOWER DEPARTMENT

MINISTRY OF HUMAN RESOURCE

INFORMATION SHEET

				Date: 22/10/2007
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DEPARTMENT	ELECTRONICS ENGINEERING TECHNOLOGY			Ref. No	EDL-04
SECTION	ELECTRONICS INDUSTRY			Code	TKE 5504
COURSE	ADV. DIP. IN ELECTRONICS ENG. TECHNOLOGY			NOSS Ref	NOT AVAILABLE
BLOCK	OPTICAL ELECTRONICS
UNIT	INTRODUCTION TO ELECTROMAGNETIC WAVES			SEM	3
SEGMENT	WAVES PROPAGATION UNBOUNDED
DUTY	-	TASK	-
TITLE: WAVES PROPAGATION UNBOUNDED INSTRUCTION AIMS: Upon completion of this course, the students will be able to understand: 1. Time Harmonic Fields 2. Plane wave Propagation in Lossless Media 3. Wave Polarization

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Introduction

A time-varying electric field E(t) produces a time-varying magnetic field H(t) and, conversely, a time-varying magnetic field produces an electric field. This cyclic pattern generates electromagnetic (EM) waves capable of propagating through free space and in material media. When its propagation is guided by a material structure, such as a transmission line, the EM wave is said to be traveling in a guided medium. Earth's surface and ionosphere constitute parallel boundaries of a natural guiding structure for the propagation of short-wave radio transmissions in the HF band* (3 to 30 MHz); the ionosphere is a good reflector at these frequencies, thereby allowing the waves to zigzag between the two boundaries. EM waves also can travel in unbounded media; light waves emitted by the sun and radio transmissions by antennas are typical examples.

When we discussed wave propagation on a transmission line in, we dealt with voltages and currents. We will focus our attention on wave propagation in unbounded media. We will consider both lossless and lossy media. Wave propagation in a lossless medium (perfect dielectric such as air) is similar to that on a lossless transmission line. In a lossy medium characterized by a nonzero conductivity, such as water, part of the power carried by the EM wave gets converted into heat, just like what happens to a wave propagating on a lossy transmission line. When energy is emitted by a source, such as an antenna, it expands outwardly from the source in the form of spherical waves. Even though the antenna may radiate more energy along some directions than along others, the spherical wave travels at the same speed in all directions and therefore expands at the same rate. To an observer very far away from the source, the wave front of the spherical wave appears approximately planar, as if it were part of a uniform plane wave with uniform properties at all points in the plane tangent to the wave front. Plane-wave propagation can be accommodated by Cartesian coordinates, which are easier to work with mathematically.

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Time-Harmonic Fields

In the time-varying case, the electric and magnetic fields, E, D, B, and H, and their sources, the charge density ρ_v and current density J, are each, in general, a function of the spatial coordinates (x,y, z) and the time variable t. If their time variation is a sinusoidal function with an angular frequency ω₀, each of these quantities can be represented by a time-independent phasor that depends on (x,y,z) only.

Definition 14

The vector phasor corresponding to the instantaneous field E(x, y, z;t) is defined according to E(x, y, t) = Re

Mathematical Form 4

Also known as Maxwell’s Equation

(1)

(2)

(3)

(4)

Derived Formula 6

In a medium with conductivity σ, the current density J is related to E by J = σE. Equation (4) of mathematical form 4 can be written as,

= jω(ε – j)

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Definition 15

The complex permittivity ε_c is defined as

ε_c

Derived Formula 7

Applying definition 15 we can say that

Mathematical Form 4 – 1^st Modified version

If a medium is free of charge then the term J and ρ_vcan be ignored from the equation.

(1)

(2)

Mathematics Definition 3

Mathematics Formula 3

Derived Formula 8

Go to Mathematical Form 4 of equation (2), you get by taking curl

Replace then

Applying mathematics formula and definition 3 we get,

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Derived Formula 9

Derived Formula 8 is rewritten by introducing propagation constant γ such that

(1)

Similarly (2)

Plane Wave Propagation in Lossless Media

Definition 16

The propagation properties of an electromagnetic wave, such as its phase velocity up and wavelength it, are governed by the angular frequency ω and the three constitutive parameters of the medium: ε,μ and σ. If the medium is nonconducting (σ = 0), the wave does not suffer any attenuation as it travels through the medium, and hence the medium is said to be lossless. In a lossless medium ε_c = ε in which case

γ² = -ω²με

Definition 17

γ² = -ω²με

γ² + ω²με = 0

By introducing k = ω

γ² = -k²

k is defined the wave number.

Derived Formula 10

Applying Derived Formula 10 and Definition 17 we get,

Facts 7

A plane wave has no electric or magnetic field components along its direction and propagation.

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Definition 18

The intrinsic impedance of a lossless medium is defined as

Facts 8

The electric and magnetic fields of a transverse electromagnetic wave (TEM wave) are perpendicular to each other and both are perpendicular to the direction of wave travel

Figure 14

Figure of a TEM wave

Mathematical Form 5

The expression for TEM wave in electric and magnetic field respectively is described by,

In a general case may be a complex quantity composed of a magnitude | | and a phase angle ⁺ that is

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Mathematical Form 6

The instantaneous electric and magnetic fields in a EM wave are given by

Definition 19

Because E(z, t) and H(z, t) exhibit the same functional dependence on z and t, they are said to be in phase; when the amplitude of one of them is a maximum, the amplitude of the other is a maximum also. This in phase property is characteristic of waves propagating in lossless media. Their time variation is defined by the oscillation frequency f = ω/2π, and their spatial variation is characterized by the wavelength λ.

Formula 10

Formula 11

Dimension 9

If the medium is vacuum, ε = ε₀ and μ = μ₀ in which case the phase velocity and intrinsic impedance become

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Example 1

The electric field of a 1-MHz plane wave traveling in the +z-direction in air points along the x-direction. If the peak value of E is 1.2π (mV/m) and E is maximum at t = 0 and z = 50 m, obtain expressions for E(z,t) and H(z, t), and then plot these variations as a function of z at t = 0.

Solution

At f = 1 MHz the wavelength in air is given by

λ = = = 300 meter

wave number k = 2π/300.

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General Relation between E and H

Phasor for magnetic field presented byis interrelated to the electric field phasor shown by by

Mathematical Form 7

= (1)

(2)

Rule 1

When we rotate the four fingers of the right hand from the direction of E toward the direction of H, the thumb will point in the direction of wave travel, .

Relations given by equation (1) and (2) from mathematical form 5 are valid for both lossless and lossy media.

Let us apply equation 1 mathematical form 7 to mathematical form 5. The direction of propagation = and . Hence

= =

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Mathematical Form 8

For a wave traveling in the –z direction with an electric field given by

apply equation 1 of mathematical form 7 gives

In general a uniform plane wave traveling in the +z direction may have both x and y components in which case is given by

(1)

and the associated magnetic field is given by

(2)

Derived Formula 11

Apply equation (1) of mathematical form 7,

(1)

Equate equation (2)(mathematical form 8) with equation (1) yields

(2)

You may considered electromagnetic wave as comprising of two waves one with

() components and another with () components.

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EXERCISE 1

A 10-MHz uniform plane wave is traveling in a nonmagnetic medium with m= m₀ and e_r = 9. Find (a) the phase velocity, (b) the wavenumber, (c) the wavelength in the medium, and (d) the intrinsic impedance of the medium.

Ans. (a) m_p = 1 x 10⁸ m/s, (b) k = 0.2p rad/m, (c) l = 10 m, (d) h = 125.67 W.

EXERCISE 2

The electric field phasor of a uniform plane wave traveling in a lossless medium with an intrinsic impedance of 188.5 W is given by = 10e^-j4^p^y (mV/m). Determine (a) the associated magnetic field phasor and (b) the instantaneous expression for E (y, t) if the medium is nonmagnetic ( m= m₀).

Ans. (a) = 53e^-j4^p^y (mA/m), (b)E(y, t) = 10cos(6p x 10⁸t - 4py) (mV/m).

EXERCISE 3

If the magnetic field phasor of a plane wave traveling in a medium with intrinsic impedance h = 100 W is given by = ( 10 + 20) e^-j4x (mA/m), find the associated electric field phasor.

Ans. =(- + 2)e-^j4x(V/m).

EXERCISE 4

Repeat Exercise 3 for a magnetic field given by H = (10e^-j3x – 20e^j3x) (mA/m).

Ans. = - (e^-j3x + 2e^j3x) (V/m).

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Definition 20

The polarization of a uniform plane wave describes the shape and locus of the tip of the E vector (in the plane orthogonal to the direction of propagation) at a given point in space as a function of time. In the most general case, the locus of E is an ellipse, and the wave is called elliptically polarized. Under certain conditions, the ellipse may degenerate into a circle or a segment of a straight line, in which case the polarization state is then called circular or linear, respectively.

Derived Formula 12

The z-components of the electric and magnetic fields of a z-propagating plane wave are both zero. Hence, the electric-field phasor (z) may consist of an x-component, _x(z), and a y-component, _y(z):

(z) = _x(z) + _y(z) (1)

with

_x(z) = _x0e^-jkz (2a)

_y(z) = y₀e^-jkz (2b)

where E_x0 and E_y0 are the complex amplitudes of _x(z) and _y(z), respectively. For the sake of simplicity, the plus sign superscript has been suppressed throughout; the negative sign in e^-jkz is sufficient to remind us that the wave is traveling in the positive z-direction.

The two amplitudes E_x0 and E_y0 are, in general, complex quantities, with each characterized by a magnitude and a phase angle. The phase of a wave is defined relative to a reference condition, such as z = 0 and t = 0 or any other combination of z and t. Wave polarization depends on the phase of E_y0 relative to that of E_x0, but not on the absolute phases of E_x0 and E_y0. Hence, for convenience, we will choose the phase of E_x0 as our reference (thereby assigning E_x0 a phase angle of zero), and we will denote the phase of E_y0, relative to that of E_x0, as d. Thus, d is the phase difference between the y-component of and its x-component. Accordingly, we define x0 andy0 as

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E_x0 = a_x (3a)

E_y0 = a_ye^j^d (3b)

Where a_x = |E_x0| and a_y = |E_y0| are the magnitudes of E_x0 and E_y0 respectively. Thus by definition, a_x and a_y may not assume negative values. Inserting E_x0 and E_y0 from equation 3 into equation 2 yields total electric field phasor given by

(z) = (a_x + a_ye^j^d)e^-jkz (4)

and the corresponding instantaneous field is

E(z,t) = Re [(z) e^j^wt]

= a_x cos (wt – kz) + a_y cos (wt – kz + d) (5)

Definition 21

Linear Polarization

The polarization state of a wave traveling in the z-direction is determined by tracing the tip of E(z,t) as a function of time in a plane orthogonal to the direction of wave travel. For convenience and without loss of generality, we usually choose the z = 0 plane. A wave is said to be linearly polarized if E_x (z, t) and E_y (z, t) are in phase (i.e., d = 0) or out of phase (d = p). This is because, at a specified value of z, say z = 0, the tip of E(0,t) traces a straight line in the x-y plane. At z = 0 and for d = 0 or n, Eq. (5) from Derived Formula 12 simplifies to

E(0,t) = (a_x + a_y) coswt (in-phase) (1)

E(0,t) = (a_x – a_y) coswt (out-of-phase) (2)

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Derived Formula 13

Left Hand Circular Polarization

Consider special case when the magnitudes of the x and y components of (z) are equal and the phase difference d = p/2. For a_x = a_y = a and d = p/2, equation (4) and (5) from Derived Formula 12 become

(z) = (a + ae^j^p^/2)e^-jkz

= a( + j)e^-jkz

E (z,t) = Re [(z) e^j^wt]

= acos(wt – kz) + a cos (wt – kz + p/2)

= acos(wt – kz) – a cos (wt – kz ) (1)

The corresponding modulus (intensity) and inclination angle are given by

|E(z,t)| = [a²cos²(wt – kz) + a²sin²(wt – kz)]^1/2

= a (2)

y(z,t) = tan^-1

= tan^-1

= - (wt – kz) (3)

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Figure 15

Definition 22

Right-Hand Circular (RHC) Polarization

The trace of E as a function of t is shown in Fig. 15(b) for z = 0. For RHC polarization, the fingers of the right hand point in the direction of rotation of E when the thumb is along the propagation direction. Figure 16 depicts a right-hand circularly polarized wave radiated by a helical antenna. Note that polarization handedness is defined in terms of the rotation of E as a function of time in a fixed plane orthogonal to the direction of propagation, which is opposite of the direction of rotation of E as a function of distance at a fixed point in time.

For a_x = a_y = a and d = -p/2, we have

|E(z,t)| = a, y = (wt – kz)

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Figure 16

In the most general case, where a_x ¹ 0, a_y ¹ 0, and d ¹ 0, the tip of E traces an ellipse in the x-y plane, and the wave is said to be elliptically polarized. The shape of the ellipse and its handedness (left-hand or right-hand rotation) are determined by the values of the ratio (a_y/a_x) and the polarization phase difference d.

The polarization ellipse shown in Fig. 16 has a major axis a_x along the ~-direction and a minor axis a_halong the h-direction. The rotation angle g is defined as the angle between the major axis of the ellipse and a reference direction, chosen here to be the x-axis, with g being bounded within the range -p/2 < g < p/2. The shape of the ellipse and its handedness are characterized by the ellipticity angle c, defined as follows:

tanc = ± = ±

with the plus sign corresponding to left-handed rotation and the minus sign corresponding to right-handed rotation. The limits for c are -p/4 £ c £ p/4. The quantity R = a_x/a_h is called the axial ratio of the polarization ellipse, and it varies between 1 for circular polarization and ¥ for linear polarization. The polarization angles g and c are related to the wave parameters a_x, a_y, and d by

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tan 2g = (tan 2y₀) cosd ( - p/2 £ g £ p/2 ) (1)

sin 2c = (tan 2y₀) sind ( - p/2 £ c £ p/2 ) (2)

where y₀ is an auxiliary angle defined by

tan y₀ = ( 0 £ y₀ £ )

Sketches of the polarization ellipse are shown in Fig. 17 for various combinations of the angles ( g, c). The ellipse reduces to a circle for c = ±45° and to a line for c = 0. Positive values of c, corresponding to sin d > 0, are associated with left-handed rotation, and negative values of c, corresponding to sin d <>

Since the magnitudes a_x and a_y are, by definition, nonnegative numbers, the ratio ay/ax, may vary between zero for an x-polarized linear polarization and ¥ for a y-polarized linear polarization. Consequently, the angle y₀ is limited to the range 0 £ y₀ £ 90°. Application of Eq. (1) leads to two possible solutions for the value of y, both of which fall within the defined range from -p/2 to p/2. The correct choice is governed by the following rule:

g > 0 if cos d > 0

g <> d <>

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Figure 17