Questions and Answers of Principles Communications Systems

For a dipole antenna of overall length, 2ℓ = 1.5λ,(a) Evaluate the locations in θ at which the zeros and maxima in the E-plane pattern occur;(b) Determine the sidelobe level, as per the
For a dipole antenna of overall length 2„“ = 1.3Î», determine the locations in Î¸ and the peak intensity of the sidelobes, expressed as a fraction of the main lobe
For a dipole antenna of overall length 2ℓ = λ, evaluate the maximum directivity in decibels, and the half-power beamwidth.
The radiation field of a certain short vertical current element is Eθs = (20/r) sin θ e−j10πr V/m if it is located at the origin in free space.(a) Find Eθs at P(r = 100, θ = 90◦, ϕ =
Find the zeros in θ for the E-plane pattern of a dipole antenna for which(a) ℓ = λ;(b) 2ℓ = 1.3λ. Use Figure 14.8 as a guide.
A monopole antenna extends vertically over a perfectly conducting plane, and has a linear current distribution. If the length of the antenna is 0.01λ, what value of I0 is required to(a) Provide a
A dipole antenna in free space has a linear current distribution with zero current at each end, and with peak current I0 at the enter. If the length d is 0.02λ, what value of I0 is required to(a)
Show that the chord length in the E-plane plot of Figure 14.4 is equal to b sin Î¸, where b is the circle diameter. dA = r-dQ = r² sinOd@do
A short current element has d = 0.03λ. Calculate the radiation resistance that is obtained for each of the following current distributions:(a) Uniform, I0;(b) Linear, I (z) = I0(0.5d −
Consider the term in Eq. (14) (or in Eq. (10)) that gives the 1/r 2 dependence in the Hertzian dipole magnetic field. Assuming this term dominates and that kr << 1, show that the resulting
Write the Hertzian dipole electric field whose components are given in Eqs. (15) and (16) in the near zone in free space where kr << 1. In this case, only a single term in each of the two
Two short antennas at the origin in free space carry identical currents of 5 cos ωt A, one in the az direction, and one in the ay direction. Let λ = 2π m and d = 0.1 m. Find Es at the distant
Prepare a curve, r vs. θ in polar coordinates, showing the locus in the ϕ = 0 plane where(a) The radiation field |Eθs| is one-half of its value at r = 104 m, θ = π/2;(b) Average radiated power
A short dipole-carrying current I0 cos ωt in the az direction is located at the origin in free space.(a) If k = 1 rad/m, r = 2 m, θ = 45◦, ϕ = 0, and t = 0, give a unit vector in rectangular
The mode field radius of a step index fiber is measured as 4.5 μm at free-space wavelength λ = 1.30 μm. If the cutoff wavelength is specified as λc = 1.20 μm, find the expected mode field radius
Is the mode field radius greater than or less than the fiber core radius in single-mode step index fiber?
A step index optical fiber is known to be single mode at wavelengths λ > 1.2μm. Another fiber is to be fabricated from the same materials, but it is to be single mode at wavelengths λ >
An asymmetric slab waveguide is shown in Figure 13.26. In this case, the regions above and below the slab have unequal refractive indices, where n1> n3> n2.(a) Write, in terms of the
In a symmetric slab waveguide, n1 = 1.50, n2 = 1.45, and d = 10μm.(a) What is the phase velocity of the m = 1 TE or TM mode at cutoff?(b) How will your part (a) result change for higher-order modes
A symmetric slab waveguide is known to support only a single pair of TE and TM modes at wavelength λ = 1.55μm. If the slab thickness is 5 μm, what is the maximum value of n1 if n2 = 3.30?
Consider a transform-limited pulse of center frequency f = 10 GHz, and of full-width 2T = 1.0 ns. The pulse propagates in a lossless single-mode rectangular guide which is air-filled and in which the
Show that the group dispersion parameter, d2Î²/dÏ‰2, for a given mode in a parallel-plate or rectangular waveguide is given bywhere Ï‰c is the radian cutoff frequency
Integrate the result of Problem 13.22 over the guide cross section, 0 < x < a, 0 < y < b, to show that the average power in watts transmitted down the guide is given aswhere
Using the relation (S) = 1/2 Re{EsÃ— Hˆ—s} and Eqs. (106) through (108), show that the average power density in the TE10 mode in a rectangular waveguide is given by
An air-filled rectangular waveguide is to be constructed for single-mode operation at 15 GHz. Specify the guide dimensions, a and b, such that the design frequency is 10 percent higher than the
Two rectangular waveguides are joined end-to-end. The guides have identical dimensions, where a = 2b. One guide is air-filled; the other is filled with a lossless dielectric characterized by
A rectangular waveguide has dimensions a = 6 cm and b = 4 cm.(a) Over what range of frequencies will the guide operate single mode?(b) Over what frequency range will the guide support both TE10 and
In the guide of Figure 13.25, it is found that m = 1 modes propagating from left to right totally reflect at the interface, so that no power is transmitted into the region of dielectric constant
A parallel-plate guide is partially filled with two lossless dielectrics (Figure 13.25) where ˆˆ'r1= 4.0, ˆˆ'r2= 2.1, and d = 1 cm. At a certain frequency, it is found that the
The cutoff frequency of the m = 1 TE and TM modes in an air-filled parallel-plate guide is known to be fc1 = 7.5 GHz. The guide is used at wavelength λ = 1.5 cm. Find the group velocity of the m = 2
For the guide of Problem 13.14, and at the 32 GHz frequency, determine the difference between the group delays of the highest-order mode (TE or TM) and the TEM mode. Assume a propagation distance of
A d = 1 cmparallel-plate guide is made with glass (n = 1.45) between plates. If the operating frequency is 32 GHz, which modes will propagate?
A lossless parallel-plate waveguide is known to propagate the m = 2 TE and TM modes at frequencies as low as 10 GHz. If the plate separation is 1 cm, determine the dielectric constant of the medium
A parallel-plate guide is to be constructed for operation in the TEM mode only over the frequency range 0 < f < 3 GHz. The dielectric between plates is to be teflon (∈'r = 2.1). Determine the
A parallel-plate waveguide is known to have a cutoff wavelength for the m = 1 TE and TM modes of λc1 = 4.1 mm. The guide is operated at wavelength λ = 1.0 mm. How many modes propagate?
Two micro-strip lines are fabricated end-to-end on a 2-mm-thick wafer of lithium niobate (∈'r = 4.8). Line 1 is of 4 mm width; line 2 (unfortunately) has been fabricated with a 5 mm width.
A microstrip line is to be constructed using a lossless dielectric for which ∈'r = 7.0. If the line is to have a 50 Ω characteristic impedance, determine(a) ∈r,eff;(b) w/d.
A transmission line constructed from perfect conductors and an air dielectric is to have a maximum dimension of 8 mm for its cross section. The line is to be used at high frequencies. Specify the
Pertinent dimensions for the transmission line shown in Figure 13.2 are b = 3 mm and d = 0.2 mm. The conductors and the dielectric are nonmagnetic.(a) If the characteristic impedance of the line is
The transmission line in Fig. 6.8 is filled with polyethylene. If it were filled with air, the capacitance would be 57.6 pF/m. Assuming that the line is lossless, find C, L, and Z0.
Each conductor of a two-wire transmission line has a radius of 0.5 mm; their center-to-center separation is 0.8 cm. Let f = 150 MHz, and assume σ and σc are zero. Find the dielectric constant of
Find R, L, C, and G for a two-wire transmission line in polyethylene at f = 800 MHz. Assume copper conductors of radius 0.50 mm and separation 0.80 cm. Use ∈'r = 2.26 and σ/(ω∈') = 4.0 ×
Two aluminum-clad steel conductors are used to construct a two-wire transmission line. Let σAl = 3.8 × 107 S/m, σSt = 5 × 106 S/m, and μSt = 100μH/m. The radius of the steel wire is 0.5 in.,
Find R, L, C, and G for a coaxial cable with a = 0.25 mm, b = 2.50 mm, c = 3.30 mm, ∈'r = 2.0, μr = 1, σc = 1.0 × 107 S/m, σ = 1.0 × 10−5 S/m, and f = 300 MHz.
The conductors of a coaxial transmission line are copper (σc = 5.8 × 107 S/m), and the dielectric is polyethylene (∈'r = 2.26, σ/ω∈' = 0.0002). If the inner radius of the outer conductor is 4
A T = 20 ps transform-limited pulse propagates through 10 km of a dispersive medium for which β2 = 12 ps2/km. The pulse then propagates through a second 10 km medium for which β2 = −12 ps2/km.
A T = 5 ps transform-limited pulse propagates in a dispersive medium for which β2 = 10 ps2/km. Over what distance will the pulse spread to twice its initial width?
Over a small wavelength range, the refractive index of a certain material varies approximately linearly with wavelength as n(λ) = na + nb(λ − λa), where na, nb and λa are constants, and where
Using Eq. (79) in Chapter 11 as a starting point, determine the ratio of the group and phase velocities of an electromagnetic wave in a good conductor. Assume conductivity does not vary with
Show how a single block of glass can be used to turn a p-polarized beam of light through 180◦, with the light suffering (in principle) zero reflective loss. The light is incident from air, and the
In the Brewster prism of Figure 12.18, determine for s-polarized light the fraction of the incident power that is transmitted through the prism, and from this specify the dB insertion loss, defined
A Brewster prism is designed to pass p-polarized light without any reflective loss. The prism of Figure 12.18 is made of glass (n = 1.45) and is in air. Considering the light path shown, determine
A dielectric waveguide is shown in Figure 12.17 with refractive indices as labeled. Incident light enters the guide at angle Ï• from the front surface normal as shown. Once inside, the
The 50-MHz plane wave of Problem 12.12 is incident onto the ocean surface at an angle to the normal of 60◦. Determine the fractions of the incident power that are reflected and transmitted for(a)
You are given four slabs of lossless dielectric, all with the same intrinsic impedance, η, known to be different from that of free space. The thickness of each slab is λ/4, where λ is the
A uniform plane wave in free space is normally incident onto a dense dielectric plate of thickness λ/4, having refractive index n. Find the required value of n such that exactly half the incident
A left-circularly polarized plane wave is normally incident onto the surface of a perfect conductor.(a) Construct the superposition of the incident and reflected waves in phasor form.(b) Determine
A right-circularly polarized plane wave is normally incident from air onto a semi-infinite slab of plexiglas (∈'r = 3.45, ∈"r = 0). Calculate the fractions of the incident power that are
A 50-MHz uniform plane wave is normally incident from air onto the surface of a calm ocean. For seawater, σ = 4 S/m, and ∈'r = 78.(a) Determine the fractions of the incident power that are
A 150-MHz uniform plane wave is normally incident from air onto a material whose intrinsic impedance is unknown. Measurements yield a standing wave ratio of 3 and the appearance of an electric field
In Figure 12.1, let region 2 be free space, while Î¼r1= 1, ˆˆ"r1= 0, and ˆˆ'r1is unknown. Find ˆˆ'r1if(a) The amplitude of Eˆ’1 is one-half that
Region 1, z < 0, and region 2, z > 0, are both perfect dielectrics (μ = μ0, = 0). A uniform plane wave traveling in the az direction has a radian frequency of 3 × 1010 rad/s. Its
A wave starts at point a, propagates 1 m through a lossy dielectric rated at 0.1 dB/cm, reflects at normal incidence at a boundary at which Γ = 0.3 + j0.4, and then returns to point a. Calculate the
The semi-infinite regions z < 0 and z > 1 m are free space. For 0 < z < 1 m, ∈'r = 4, μr = 1, and r = 0. A uniform plane wave with ω = 4 × 108 rad/s is traveling in the az direction
In the beam-steering prism of Example 12.8, suppose the antireflective coatings are removed, leaving bare glass-to-air interfaces. Calcluate the ratio of the prism output power to the input power,
The region z < 0 is characterized by ∈'r = μr = 1 and r = 0. The total E field here is given as the sum of two uniform plane waves, Es = 150 e−j10zax + (50 ∠ 20◦) ej10zax V/m.(a) What is
A 10 MHz uniform plane wave having an initial average power density of 5 W/m2 is normally incident from free space onto the surface of a lossy material in which ∈"2/∈'2 = 0.05, ∈'r2 = 5, and
A uniform plane wave in region 1 is normally incident on the planar boundary separating regions 1 and 2. If ∈"1 = ∈"2 = 0, while ∈'r1 = μ3r1 and ∈'r2 = μ3r2, find the ratio ∈'r2/∈'r1
The plane z = 0 defines the boundary between two dielectrics. For z < 0, ∈r1 = 9, ∈"r1 = 0, and μ1 = μ0. For z > 0, ∈'r2 = 3, ∈"r2 = 0, and μ2 = μ0. Let E+x1 = 10 cos(ωt − 15z)
Given a general elliptically polarized wave as per Eq. (93):Es = [Ex0ax + Ey0ejϕay ]e−jβz(a) Show, using methods similar to those of Example 11.7, that a linearly polarized wave results when
Given a wave for which Es = 15e−jβzax + 18e−jβzejϕay V/m in a medium characterized by complex intrinsic impedance, η(a) Find Hs;(b) Determine the average power density in W/m2.
A linearly polarized uniform plane wave, propagating in the forward z direction, is input to a lossless anisotropic material, in which the dielectric constant encountered by waves polarized along
In an anisotropic medium, permittivity varies with electric field direction, and is a property seen in most crystals. Consider a uniform plane wave propagating in the z direction in such a medium,
Consider a left circularly polarized wave in free space that propagates in the forward z direction. The electric field is given by the appropriate form of Eq. (100). Determine(a) The magnetic field
A uniform plane wave in free space has electric field vector given by Es = 10e−jβxaz + 15e−jβxay V/m.(a) Describe the wave polarization.(b) Find Hs.(c) Determine the average power density in
The planar surface z = 0 is a brass-Teflon interface. Use data available in Appendix C to evaluate the following ratios for a uniform plane wave having ω = 4 × 1010 rad/s:(a) αTef/αbrass;(b)
The dimensions of a certain coaxial transmission line are a = 0.8 mm and b = 4 mm. The outer conductor thickness is 0.6 mm, and all conductors have σ = 1.6 × 107 S/m.(a) Find R, the resistance per
A good conductor is planar in form, and it carries a uniform plane wave that has a wavelength of 0.3 mm and a velocity of 3 × 105 m/s. Assuming the conductor is nonmagnetic, determine the frequency
A hollow tubular conductor is constructed from a type of brass having a conductivity of 1.2 × 107 S/m. The inner and outer radii are 9 and 10 mm, respectively. Calculate the resistance per meter
The inner and outer dimensions of a coaxial copper transmission line are 2 and 7 mm, respectively. Both conductors have thicknesses much greater than δ. The dielectric is lossless and the operating
Consider the power dissipation term, ʃ E · Jdv, in Poynting’s theorem (Eq. (70)). This gives the power lost to heat within a volume into which electromagnetic waves enter. The term pd = E · J is
Let η = 250 + j30Ω and jk = 0.2 + j2m−1 for a uniform plane wave propagating in the az direction in a dielectric having some finite conductivity. If |Es| = 400 V/m at z = 0, find(a) (S) at z = 0
Given a 100-MHz uniform plane wave in a medium known to be a good dielectric, the phasor electric field is Es = 4e−0.5ze−j20zax V/m. Determine(a) ∈';(b) ∈";(c) η;(d) Hs;(e) S;(f) The
Voltage breakdown in air at standard temperature and pressure occurs at an electric field strength of approximately 3 × 106 V/m. This becomes an issue in some high-power optical experiments, in
The cylindrical shell, 1 cm < ρ < 1.2 cm, is composed of a conducting material for which σ = 106 S/m. The external and internal regions are nonconducting. Let Hϕ = 2000 A/m at ρ = 1.2 cm.
In a medium characterized by intrinsic impedance η = |η|ejϕ, a linearly polarized plane wave propagates, with magnetic field given as Hs = (H0yay + H0zaz)e−αx e−jβx. Find(a) Es;(b) E(x,
Let jk = 0.2 + j1.5 m−1 and η = 450 + j60Ω for a uniform plane propagating in the az direction. If ω = 300 Mrad/s, find μ, ∈', and ∈" for the medium.
A certain nonmagnetic material has the material constants ∈'r = 2 and ∈"/∈' = 4 × 10−4 at ω = 1.5 Grad/s. Find the distance a uniform plane wave can propagate through the material before(a)
A 10 GHz radar signal may be represented as a uniform plane wave in a sufficiently small region. Calculate the wavelength in centimeters and the attenuation in nepers per meter if the wave is
Uniform current sheets are located in free space as follows: 8az A/m at y = 0, −4az A/m at y = 1, and −4az A/m at y = −1. Find the vector force per meter length exerted on a current filament
Two conducting strips, having infinite length in the z direction, lie in the xz plane. One occupies the region d/2 < x < b + d/2 and carries surface current density K = K0az; the other is
A current of −100az A/m flows on the conducting cylinder ρ = 5 mm, and +500az A/m is present on the conducting cylinder ρ = 1 mm. Find the magnitude of the total force per meter length that is
(a) Use Eq. (14), Section 8.3, to show that the force of attraction per unit length between two filamentary conductors in free space with currents I1az at x = 0, y = d/2, and I2az at x = 0, y =
Two circular wire rings are parallel to each other, share the same axis, are of radius a, and are separated by distance d, where d << a. Each ring carries current I. Find the approximate force
A current of 6 A flows from M(2, 0, 5) to N(5, 0, 5) in a straight, solid conductor in free space. An infinite current filament lies along the z axis and carries 50 A in the az direction. Compute the
A solenoid is 25 cm long, 3 cm in diameter, and carries 4 A dc in its 400 turns. Its axis is perpendicular to a uniform magnetic field of 0.8 Wb/m2 in air. Using an origin at the center of the
A solid conducting filament extends from x = −b to x = b along the line y = 2, z = 0. This filament carries a current of 3 A in the ax direction. An infinite filament on the z axis carries 5 A in
Assume that an electron is describing a circular orbit of radius a about a positively charged nucleus.(a) By selecting an appropriate current and area, show that the equivalent orbital dipole moment
The hydrogen atom described in Problem 8.16 is now subjected to a magnetic field having the same direction as that of the atom. Show that the forces caused by B result in a decrease of the angular
Calculate the vector torque on the square loop shown in Figure 8.15 about an origin at A in the field B, given(a) A(0, 0, 0) and B = 100ay mT;(b) A(0, 0, 0) and B = 200ax + 100ay mT;(c) A(1, 2, 3)

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