David K. Chengpds22.egloos.com/pds/201110/26/66/Fundamentals_of... · 2011. 10. 26. · 7-2 Plane Waves in Lossless Media A uniform plane wave is a particular solution of Maxwell’s

Fundamentals of Engineering ElectromagneticsDavid K. Cheng

© 2005 Pearson Education

Chapter 7: Plane Electromagnetic Waves


7-2 Plane Waves in Lossless Media

A uniform plane wave is a particular solution of Maxwell’s equations with E assuming the same direction, same magnitude, and same phase in infinite planes perpendicular to the direction of propagation (similarly for H)If we are far from a source, the wavefront (surface of constant phase) becomes almost sphericalA very small portion of the surface of a giant sphere is very nearly a plane


7-2 Plane Waves in Lossless Media (con’t)

The source-free wave equations in nonconducting simple media becomes a homogenous vector Helmholtz’sequation:

Where k is the wavenumber.



In a medium characterized by and µ,we have, from Eq. (6-82a),



In Cartesian coordinates, Eq. (7-3) is equivalent to three scalar Helmholtz’s equations, one each in the components, Ex, Ey, and Ez. Writing it for the component, Ex, we have



Consider a uniform plane wave characterized by a uniform Ex (uniform magnitude and constant phase) over plane surfaces perpendicular to z; that is,

which is an ordinary differential equation because Ex, a phasor, depends only on z.



The solution of Eq. (7-6) is readily seen to be

Where and are arbitrary constant that must bedetermined by boundary conditions.

(7-7)



Let us now examine the first phasor term on the right side of Eq. (7-7) and write

For a cosine reference, the instantaneous expression of E in Eq. (7-8) is





If we fix our attention on a particular point (a point of a particular phase) on the wave, we set cos ( t - kz)= a constant, or

t – kz = A constant phase,

Wavenumber k bears a definite relation to the wavelength.



Eq.(7-10) assures that the velocity of propagation of an equiphase front (the phase velocity) equals to that of lightWavenumber k bears a definite relation to the wavelength

as has been noted in Eqs (6-82a and b).



The associated magnetic field H can be found from the x E equation (6-80a):



which leads to



Thus is the only nonzero component of Hcorresponding to the E in Eq. (7-8), and since

Eq. (7-12b) yields



We have introduced a new quantity, , in Eq. (7-13):= µ/k, or

which is called the intrinsic impedance of the medium.



(z) is in phase with (z), and we can write the instantaneous expression for H as


7-2.1 Doppler Effect

The electromagnetic wave emitted by T in air at a reference time t = 0 will reach R at

At a later time t = t, T has moved to the new position T’,and the wave emitted by T’ at that time will reach R at


7-2.1 Doppler Effect (con’t)

The time elapsed at R, t’, corresponding to t at T is

which is not equal to t.


7-2.1 Doppler Effect (con’t)

If t represents a period of the time-harmonic source –that is, if t = 1/f – then the frequency of the received wave at R is

for the usual case of (u/c)2 << 1.


7-2.2 Transverse Electromagnetic Waves

Examine the propagating of a uniform plane wave along an arbitrary direction that does not necessarily coincide with a coordinate axis.

Instead of the E(z) in Eq. (7-8), let us consider

Which represents the y-directed electric intensity of a uniform plane wave propagating in both +x and +zdirections.


7-2.2 Transverse Electromagnetic Waves (con’t)

If we define a wavenumber vector, k, as

And a radius vector R from the origin to an arbitrary point



Eq. (7-20) can be written succinctly as

The situation is illustrated in Fig. 7-4.





The magnetic field H associated with the electric field in Eq. (7-23) is, from Eq. (6-80a),



Eq. (7-24) can be put in a more general form:

Thus, if E of a uniform plane wave propagating in a given direction is known, the associated H can be easily found from Eq. (7-25).


7-2.3 Polarization of Plane Waves

The polarization of a uniform plane wave describes the time-varying behavior of the electric field intensity vector at a given point in spaceIf the E vector of a plane wave is fixed in the x direction, the wave is said to be linearly polarizedIn some cases the direction of E of a plane wave at a given point may change with time


7-2.3 Polarization of Plane Waves (con’t)

Consider the superposition of two linearly polarized wavesIn phasor notation we have

Where E10 and E20 are real numbers denoting the amplitudes of the two linearly polarized waves.



In examining the direction change of E at a given point as t changes, it is convenient to set z = 0. We have



Equation for an ellipse:





When E20 = E10, the instantaneous angel that E makes with the x-axis at z = 0 is

Which indicates that E rotates at a uniform rate with an angular velocity in a counterclockwise direction.



If we start with an E2(z), which leads E1(z) by 90° ( π /2 rad) in time phase, Eqs. (7-26) and (7-27) will be, respectively,

Comparing eq.(7-31) and (7-27), E will rotate with an angular velocity ω in a clockwise direction; this is a left-hand or negative circularly polarized wave.



If E2(z) and E1(z) are in space quadrature but in time phase, the instantaneous expression for E at z = 0 is

(7-32)


7-3 Plane Waves in Lossy Media

The time-harmonic x H equation (6-80b) should then be changed to

with


7-3 Plane Waves in Lossy Media (con’t)

In treating such media it is customary to include the effects of both the damping and the ohmic losses in the imaginary part of a complex permittivity :

Where both and may be functions of frequency.



Alternatively, we may define an equivalent conductivity representing all losses and write



The ratio / is called a loss tangent because it is a measure of the power loss in the medium:

The quantity in eq.(7-39) may be called the loss angle



A medium is said to be a good conductor if A medium is said to be a good insulator if



To study the time-harmonic behavior in lossy media, the real k in eq.(7-3) is replaced by a complex wavenumber kc



Examine the solution of the homogeneous Helmholtz’s equation

With the following definition of propagation constant γ, such that



Since γ is complex,

Or

Where α and β are real and imaginary parts of γ



For a lossless medium, σ = 0, α = 0, and β = k = ω



For a uniform plane wave propagating in the +z direction and characterized by E = axEx and H = ayHy

The solution of eq.(7-45b) is

α is called an attenuation constant, and β is called a phase constant


7-3.1 Low-Loss Dielectrics

A low-loss dielectric is a good but imperfect insulatorThe attenuation constant

and the phase constant


7-3.1 Low-Loss Dielectrics (con’t)

The intrinsic impedance of a low-loss dielectric is a complex quantity.


7-3.1 Low-Loss Dielectrics (con’t)

The phase velocity is obtained from the ratio / . Using Eq. (7-48), we have

which is slightly lower than its value when the medium is lossless.


7-3.2 Good Conductors

A good conductor is a medium for which / >> 1.Under this condition it is convenient to use Eq. (7-43) and neglect 1 in comparison with / . We write


7-3.2 Good Conductors (con’t)

For a good conductor,

The intrinsic impedance of a good conductor is

which has a phase angle if 45°.



The phase velocity in a good conductor is



The wavelength of a plane wave in a good conductor is



The distance through which the amplitude of a traveling plane wave decreased by a factor of e-1 or 0.368 is called the skin depth of the depth of penetration of a conductor:



Since = β for a good conductor, can also be written as


7-4 Group Velocity

The relation between and the phase constant, β, is

The phenomenon of signal distortion caused by a dependence of the phase velocity on frequency is called dispersionLossy dielectric is a dispersive medium


7-4 Group Velocity (con’t)

A group velocity is the velocity of propagation of the wave packet envelopeConsider the simplest case of a wave packet that consists of two traveling waves having equal amplitude and slightly different angular frequencies


7-4 Group Velocity (con’t)

In the limit that ∆ → 0, we have the formula for computing the group velocity in a dispersive medium:


7-5 Flow of Electromagnetic Power and the Poynting Vector

Electromagnetic waves carry with them electromagnetic powerEnergy is transported through space to distant receiving points by electromagnetic wavesThe relationship between the rate of such energy transfer and the electric and magnetic field intensities associated with a traveling electromagnetic wave can be derived by considering the curl equations


7-5 Flow of Electromagnetic Power and the Poynting Vector (con’t)

The following identity of vector operations can be verified in a straightforward manner by using Cartesian coordinates:

Substitution of Eqs. (7-61) and (7-62) in Eq. (7-63) yields



Equation (7-64) can then be written as

Which is a point-function relationship.



An integral form of Eq. (7-65) is obtained by integrating both sides over the volume of concern:



The quantity ( ) is a vector representing the power flow per unit area. Define

Quantity is known as the Poynting vector, which is the power density vector associated with an electromagnetic field.



Equation (7-66) may be written in another form:

where


7-5.1 Instantaneous and Average Power Densities

For the phasor,

The instantaneous expression is


7-5.1 Instantaneous and Average Power Densities (con’t)

For a uniform plane wave propagating in a lossy medium in the +z-direction, the associated magnetic field intensity phasor is



The corresponding instantaneous expression for H(z) is



The instantaneous expression for the Poynting vector or power density vector, from Eq. (7-72) and (7-74), is



From Eq. (7-76), we obtain the time-average Poyntingvector,



For wave propagation in lossless media, → is real, = 0, and = 0, Eq.(7-77) reduces to



In general case we may not be dealing with a wave propagating in the z-direction. We write

Which is a general formula for computing the average power density in a propagating wave.


7-6 Normal Incidence of Plane Waves at Plane Boundaries

Consider the incident wave (Ei, Hi) in medium 1 travels In the +z-direction toward medium 2.


7-6 Normal Incidence of Plane Waves at Plane Boundaries (con’t)

The incident electric and magnetic field intensity phasorsare (aki = az):



Because of the medium discontinuity at z = 0, the incident wave is partly reflected back into medium 1 and partly transmitted into medium 2.

a)


7-6 Normal Incidence of Plane Waves at Plane Boundaries (con’t)b) For transmitted wave



At the dielectric interface z = 0 the tangential components (the x-components) of the electric and magnetic field intensities must be continuous. We have



Solving Eqs. (7-90) and (7-91), we obtain

The ratio Er0/Ei0 and Et0/Ei0 are called reflection coefficient and transmission coefficient, respectively.



The ratios Er0/Ei0 and Er0/Ei0 are called reflection coefficients and transmission coefficients, respectivelyIn terms of the intrinsic impedances they are



Reflection and transmission coefficients are related by the following equation:



The total field in medium 1 (E1, H1) is the sum of the incident and reflected field. From Eqs. (7-84) and (7-86), we have

(7-97)



The ratio of maximum value to the minimum value of the electric field intensity of a standing wave is called the standing-wave ratio (SWR), S.



An inverse relation of Eq. (7-98) is



The magnetic field intensity in medium 1 is obtained by combining Hi(z) and Hr(z) in eq.(7-85) and (7-87)



In medium 2, (Et, Ht) constitute the transmitted wave propagating in +z-direction. Eq.(7-88) and (7-95) implies

From eq.(7-89), we obtain


7-6.1 Normal Incidence on a Good Conductor

Consider the incident field vector phasor given in eq.(7-84) and (7-85)


7-6.1 Normal Incidence on a Good Conductor (con’t)

The incident wave is totally reflected with a phase reversal, and no power is transmitted across a perfectly conducting boundary. We have



Eq.(7-112) and (7-113) show that E1(z) and H1(z) are in time quadrature.



The instantaneous expressions corresponding to the electric and magnetic field intensity phasors obtained in eq.(7-112) and (7-113)

Both E1(z,t) and H1(z,t) possess zeros and maxima at fixed distances from the conducting boundary for all t


7-7 Oblique Incidence of Plane Waves at Plane Boundaries


7-7 Oblique Incidence of Plane Waves at Plane Boundaries (con’t)

The plane containing the normal to the boundary surface and the wavenumber vector ak is called the plane of incidenceThree angles are in evidence

Angle of incidence θi

Angle of reflection θr

Angle of refraction θt



Since both the incident and the reflected waves propagate in medium 1 with the same phase velocity up1, the distances OA’ and AO’ must be equal. Thus,

it assures that the angle of reflection is equal to the angle of incidence, which is Snell’s law of reflection.



In medium 2, the transmitted wave to travel from O to Bequals the time for the incident wave to travel from A to O’

Where n1 and n2 are the indices of refraction for media 1 and 2, respectively. Eq.(7-117) is known as Snell’s law of refraction



For media with equal permeability, , Eq. (7-117) becomes

Where and are the intrinsic impedances of the media.


7-7.1 Total Reflection

The angle of incidence is called the critical angle.


7-7.1 Total Reflection (con’t)

When θi is larger than the critical angle, from Eq. (7-118), we have

which does not yield a real solution for .



Although in Eq. (7-121) is still real, becomes imaginary when >1:



In medium 2 the unit vector akt in the direction of propagation of a typical transmitted (refracted) wave, as shown in Fig. 7-10, is

(7-123)



Both Et and Ht vary spatially in accordance with the following factor:

(7-124)



When Eqs. (7-118) and (7-119) for > are used, the expression in Eq. (7-124) becomes

where

(7-125)



For θi > θc an evanescent wave exists along the interface, which is attenuated exponentially (rapidly) in medium 2 in the normal direction (z-direction)This wave is tightly bound to the interface and is called a surface wave




7-7.2 The Ionosphere

In he earth’s upper atmosphere, there exist layers of ionized gases called the ionosphere.Ionized gases with equal electron and ion densities are called plasmasAnalysis has shown that the effect of the ionosphere or plasma on wave propagation can be studied on the basis of an effective permittivity :


7-7.2 The Ionosphere (con’t)

where is called the plasma angular frequency and



From Eqs.(7-42) and (7-1300 we obtain the propagation constant as

(7-132)



If the value of e, m, and are substituted into Eq. (7-131), we find a very simple formula for the plasma (cutoff) frequency:

(7-133)


7-7.3 Perpendicular Polarization


7-7.3 Perpendicular Polarization (con’t)

For oblique incidence with perpendicular polarization, Ei is perpendicular to the plane of incidence, as illustrated in Fig. 7-14. Noting that

We have, from Eqs. (7-23) and (7-25),



For the reflected wave,

The reflected electric and magnetic fields are

(7-137)



For the transmitted wave,

we have

(7-140)



From Eiy (x, 0) + Ery (x, 0) = Ety (x, 0) we have

Similarly, from Hix (x, 0) + Hrx (x, 0) = Htx (x, 0) we require



Eq. (7-143) and (7-144) can now be written simply as



From which Er0 and Et0 can be found in terms of Ei0. The reflection and transmission coefficients are



and are related in the following way:

If medium 2 is a perfect conductor, = 1. The tangential E field on the surface of the conductor vanishes, and no energy is transmitted across a perfectly conducting boundary

(7-149)


7-7.4 Parallel Polarization


7-7.4 Parallel Polarization (con’t)

The incident and reflected electric and magnetic field intensity phasors in medium 1 are:



The transmitted electric and magnetic field intensity phasors in medium 2 are



Continuity requirements for the tangential components of E and H at z = 0 lead again to Snell’s laws of reflection and refraction, as well as to the following two equations:



Solving for Er0 and Et0 in terms of Ei0, we obtain



It is easy to verify that


7-7.5 Brewster Angle of No Reflection

The numerator of expression for reflection coefficient in eq.(7-158) is the difference of two termsConsider the that makes Γ║=0 for no reflection

Denoting this particular by , we require


7-7.5 Brewster Angle of No Reflection (con’t)

Squaring both sides of Eq. (7-161) and using Eq. (7-117), we obtain

The angle is known as the Brewster angle of no reflection for the case of parallel polarization.



And alternative form for Eq. (7-163) is



Mathematically we could find a formula for , the angle of incidence that would make Γ┴ vanish. Setting eq.(7-147) to zero, the condition

In conjunction with Snell’s law of reflection, and eq.(7-117) yield



Notice the difference in formulas for Brewster angles for perpendicular and parallel polarizations for unpolarizedwave.When an unpolarized wave incident upon an boundary at the Brewster angle in eq.(7-164), only the component with perpendicular polarization will be reflected. Brewster angle is also referred to as a polarization angle


David K. Chengpds22.egloos.com/pds/201110/26/66/Fundamentals_of... · 2011. 10. 26. · 7-2 Plane Waves in Lossless Media A uniform plane wave is a particular solution of Maxwell’s

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