Solutions fundamentals of applied electromagnetics, 5e ulaby

Assuming the velocity of wave propagation on the line is c, for which of the following situations is it reasonable to ignore the presence of the transmission line in the solution of the

Chapter 1: Introduction: Waves and Phasors

Lesson #1

Chapter — Section: Chapter 1

Topics: EM history and how it relates to other fields

Highlights:

• EM in Classical era: 1000 BC to 1900

• Examples of Modern Era Technology timelines

• Concept of “fields” (gravitational, electric, magnetic)

• Static vs dynamic fields

• The EM Spectrum

Special Illustrations:

• Timelines from CD-ROM

Timeline for Electromagnetics in the Classical Era

ca 900 Legend has it that while walking

BC across a field in northern Greece, a

shepherd named Magnus experiences

a pull on the iron nails in his sandals

by the black rock he was standing on

The region was later named Magnesia

and the rock became known as

magnetite [a form of iron with

permanent magnetism]

ca 600 Greek philosopher Thales

BC describes how amber,

after being rubbed

with cat fur, can pick

1785 Charles-Augustin de

Coulomb (French) demonstrates that

the electrical force between charges is proportional to the inverse of the square of the distance between them

1800 Alessandro Volta

(Italian) develops the first electric battery

1820 Hans Christian Oersted

(Danish) demonstrates the interconnection between electricity and magnetism through his discovery that an electric current in a wire causes a compass needle to orient itself perpendicular to the wire

CHAPTER 1 3

Chapter 1

Section 1-3: Traveling Waves

have a differential pressure p x
t 10 N/m2at x 0 and t 50 µs If the reference phase of p x
t is 36 , find a complete expression for p x
t The velocity of sound

 A cos 126 rad 031A

it follows that A 10 031 3236 N/m2 So, with t in (s) and x in (m),

Figure P1.2: (a) Pressure wave as a function of distance at t  0 and (b) pressure

wave as a function of time at x 0

that completes 180 vibrations per minute If it is observed that a given crest, ormaximum, travels 300 cm in 10 s, what is the wavelength?

Solution:

f 

18060

033

 01 m 10 cm

the same frequency, but y2 t leads y1 t by a phase angle of 60 If

Figure P1.4: Plots of y1 t and y2 t.

y x
t 15 sin 05t

06x (m)

Determine the phase velocity and the wavelength and then sketch y x
t at t  2 s

over the range from x 0 to x 2λ

Solution: The given wave may be rewritten as a cosine function:

2 λ x

1.5

1 0.5

0 -0.5

-1 -1.5

where x  0 is the end of the string, which is tied rigidly to a wall, as shown in

Fig 1-21 (P1.6) When wave y1 x
t arrives at the wall, a reflected wave y2 x
t is

generated Hence, at any location on the string, the vertical displacement yswill bethe sum of the incident and reflected waves:

ys x
t y1 x
t

y2 x
t

and the fact that the end of the string cannot move

2λ x 0 atωt π 4 and atωt π 2

Solution:

same angular frequencyω, and since y2 x t is traveling on the same string as y1 x t ,

Figure P1.6: Wave on a string tied to a wall at x 0 (Problem 1.6).

the two waves must have the same phase constantβ Hence, with its direction being

in the negative x-direction, y2 x
t is given by the general form

Since the string cannot move at x 0, the point at which it is attached to the wall,

ys 0
t 0 for all t Thus,

Figure P1.6: (c) Plots of y1, y2, and ysversus x atωt π 2.

and what is the corresponding value of ys?

and what is the corresponding value of ys?

Solution:

x-direction.

2 sin x sin y cos x

string in the negative x-direction, given that ymax 40 cm,λ 30 cm, f  10 Hz,and

Solution: For a wave traveling in the negative x-direction, we use Eq (1.17) with

ω  2πf  20π (rad/s), β 2π λ  2π 03  20π 3 (rad/s), A  40 cm, and x

assigned a positive sign:

CHAPTER 1 11

20 vibrations in 50 s The wave peak is observed to travel a distance of 2.8 m alongthe string in 50 s What is the wavelength?

does y2 t lead or lag y1 t, and by what phase angle?

20πz (V), where z is the distance in

meters from the generator

(a) Find the frequency, wavelength, and phase velocity of the wave.

Solution:

β  20πrad/m From Eq (1.29a), f  ω 2π  2 109Hz  2 GHz; from

Eq (1.29b),λ 2π β 01 m From Eq (1.30),

 081 Np/m

to have an amplitude of 98.02 (V/m) at a depth of 10 m and an amplitude of 81.87(V/m) at a depth of 100 m What is the attenuation constant of sea water?

α

018

90  2 10

3 (Np/m)

CHAPTER 1 13

Section 1-5: Complex Numbers

result in rectangular form:

Solution: (Note: In the following solutions, numbers are expressed to only two

decimal places, but the final answers are found using a calculator with 10 decimalplaces.)

(a) Express z1and z2in polar form.

Solution: (Note: In the following solutions, numbers are expressed to only two

decimal places, but the final answers are found using a calculator with 10 decimalplaces.)

(a) Determine the product z1z2in polar form.

2in polar form

connected to a series RC load as shown in Fig 1-19 If R 1 MΩand C 200 pF,

obtain an expression for vc t , the voltage across the capacitor

Solution: In the phasor domain, the circuit is a voltage divider, and

Vc  Vs

1 jωC R

1 jωC 

Vs1

where t is expressed in seconds.

the following phasors:

(b) Obtain the corresponding phasor-domain equation.

(c) Solve the equation to obtain an expression for the phasor current I.

Vs(t)

C i

where x is the distance along the string in meters and y is the vertical displacement.

Determine: (a) the direction of wave travel, (b) the reference phase φ0, (c) the frequency, (d) the wavelength, and (e) the phase velocity.

of 1 (µW/m2) at a distance of 2 m from the laser gun and an intensity of 0.2

(µW/m2) at a distance of 3 m Given that the intensity of an electromagneticwave is proportional to the square of its electric-field amplitude, find the attenuationconstantαof fog

Solution: If the electric field is of the form

Using (1) for Vs and replacing R1, R2, L andωwith their numerical values, we have

• CD-ROM Modules 2.1-2.4, Configurations A-C

• CD-ROM Demos 2.1-2.4, Configurations A-C

• CD-ROM Modules 2.1-2.4, Configurations D and E

• CD-ROM Demos 2.1-2.4, Configurations D and E

Lessons #10 and 11

Chapter — Section: 2-9

Topics: Smith chart

Highlights:

• Structure of Smith chart

• Calculating impedances, admittances, transformations

• Locations of maxima and minima

Special Illustrations:

• Example 2-10

• Example 2-11

1946, and by the 1970s microwave ovens had become standard household items

CHAPTER 2 33

Chapter 2

Sections 2-1 to 2-4: Transmission-Line Model

source with an oscillation frequency f Assuming the velocity of wave propagation

on the line is c, for which of the following situations is it reasonable to ignore the

presence of the transmission line in the solution of the circuit:

an inner conductor diameter of 05 cm and an outer conductor diameter of 1 cm,

filled with an insulating material where µ  µ0, εr  45, andσ 10

copper strips separated by a 0.15-cm-thick layer of polystyrene Appendix B gives

yields the same telegrapher’s equations given by Eqs (2.14) and (2.16)

Solution: The voltage at the central upper node is the same whether it is calculated

from the left port or the right port:

Figure P2.4: Transmission line model.

Recognizing that the current through the G C branch is i z
t

i z

∆z
t (fromKirchhoff’s current law), we can conclude that

From both of these equations, the proof is completed by following the steps outlined

in the text, ie rearranging terms, dividing by∆z, and taking the limit as∆z
0

Solution: From Eq (2.22),

Section 2-5: The Lossless Line

features: (1) it is dispertionless (µp is independent of frequency) and (2) its

characteristic impedance Z0 is purely real Sometimes, it is not possible to design

a transmission line such that R
ωL and G
ωC , but it is possible to choose the

dimensions of the line and its material properties so as to satisfy the condition

R C  L G (distortionless line)

Such a line is called a distortionless line because despite the fact that it is not lossless,

it does nonetheless possess the previously mentioned features of the loss line Showthat for a distortionless line,

Solution: Using the distortionless condition in Eq (2.22) gives

jωC 

L C

up 25 108(m/s), find the line parameters andλat 100 MHz

(Np/m), andβ 075 rad/m Find the line parameters R , L , G , and C

α  002 Np/m, and β  075 rad/m Since Z0 is real and α  0, the line isdistortionless From Problem 2.6,β ω
L C and Z0 

found to have a maximum magnitude of 1.5 V and a minimum magnitude of 0.6 V.Find the magnitude of the load’s reflection coefficient.

Solution: From the definition of the Standing Wave Ratio given by Eq (2.59),

lossless coaxial line with characteristic impedance of 50Ω The radius of the innerconductor is 1.2 mm

(a) What is the radius of the outer conductor?

(b) What is the phase velocity of the line?

b ae Z0 εr 60 12 mm e50 225 60 42 mm

j50 Ω The wavelength is 8 cm Find:

(a) the reflection coefficient at the load,

(b) the standing-wave ratio on the line,

(c) the position of the voltage maximum nearest the load,

(d) the position of the current maximum nearest the load.

were noted: distance of first voltage minimum from the load 3 cm; distance of firstvoltage maximum from the load 9 cm; S 3 Find ZL.

9 cm 3 cm 6 cm λ 4

or λ  24 cm Accordingly, the first voltage minimum is at min  3 cm 

1 

2

4  05Hence,Γ 05 e

first minimum from the load 4 cm; distance of second minimum from the load 

14 cm, voltage standing-wave ratio 15 If the line is lossless and Z0  50Ω, findthe load impedance

λ 2 lmin 1  lmin 0 

 20 cmand

lossless transmission line with characteristic impedance Z0, with Z0chosen such that

the standing-wave ratio is the smallest possible What should Z0be?

the value of Z0which gives the minimum possible S also gives the minimum possible

Γ, and, for that matter, the minimum possible Γ2 A necessary condition for aminimum is that its derivative be equal to zero:

of the range may be local minima or maxima without the derivative being zero there,

the endpoints (namely Z0 0Ωand Z0 ∞ Ω) should be checked also

voltage standing wave ratio of 3 Find all possible values of ZL.

Section 2-6: Input Impedance

transmission line 2.5 m in length is terminated with an impedance ZL 40

j20 Ω.Find the input impedance

terminated in a load impedance as shown in Fig 2-38 (P2.18) FindΓ, S, and Zin.

l = 0.35λ

Figure P2.18: Loaded transmission line

Solution: From Eq (2.49a),

line terminated in a short circuit appears as an open circuit.

line is a maximum the input impedance is purely real

which is real, provided Z0is real

impedance Zg  50 Ω is connected to a 50-Ω lossless air-spaced transmissionline The line length is 5 cm and it is terminated in a load with impedance

ZL 100

j100 Ω Find

(a) Γat the load

up c, and from the expression for vg t we concludeω 2π 109rad/s Therefore

(a) λon the line,

(b) the reflection coefficient at the load,

(c) the input impedance,

CHAPTER 2 45

Since this exceeds 2π(rad), we can subtract 2π, which leaves a remainderβl 04π

connected in parallel through a pair of transmission lines, and the combination isconnected to a feed transmission line, as shown in Fig 2.39 (P2.23(a)) All lines are

CHAPTER 2 47

0.2λ

75 Ω(Antenna)

75 Ω(Antenna)

Figure P2.23: (a) Circuit for Problem 2.23

Section 2-7: Special Cases

of a lossless 50-Ω transmission line terminated in a short circuit to construct an

equivalent load with reactance X  40Ω If the phase velocity of the line is 075c,

what is the shortest possible line length that would exhibit the desired reactance at itsinput?

On a lossless short-circuited transmission line, the input impedance is always purely

imaginary; i.e., Zinsc jXinsc Solving Eq (2.68) for the line length,

for which the smallest positive solution is 805 cm (with n 0)

long (in wavelengths) should the line be in order for it to appear as an open circuit atits input terminals?

nπ , then Zinsc j∞ Ω Hence,

unknown characteristic impedance was measured at 1 MHz With the line terminated

in a short circuit, the measurement yielded an input impedance equivalent to an

inductor with inductance of 0.064 µH, and when the line was open circuited, the

measurement yielded an input impedance equivalent to a capacitor with capacitance

of 40 pF Find Z0of the line, the phase velocity, and the relative permittivity of theinsulating material

Zinsc jωL j2π 106 0064 10 6 j04Ω

 1 0652  24 For other values

of n, upis very slow andεris unreasonably high

line, which itself is preceded by anotherλ 4 section of a 100-Ωline What is the inputimpedance?

between the transmitter and a tower-mounted half-wave dipole antenna The antennaimpedance is 73Ω You are asked to design a quarter-wave transformer to match theantenna to the line

(a) Determine the electrical length and characteristic impedance of the

quarter-wave section

between the wires is made of polystyrene withεr  26, determine the physicallength of the quarter-wave section and the radius of the two wire conductors

(a) For a match condition, the input impedance of a load must match that of the

transmission line attached to the generator A line of electrical length λ 4 can beused From Eq (2.77), the impedance of such a line should be

d 2a

d 2a

d 2a

2

1 731

and whose solution is a d 744 25 cm 744 336 mm

ZL  50

j25 Ω The time-average power transferred from the generator into the

load is maximum when Zg Z

L
where Z

Lis the complex conjugate of ZL To achieve

this condition without changing Zg, the effective load impedance can be modified by adding an open-circuited line in series with ZL, as shown in Fig 2-40 (P2.29) If the line’s Z0 100 Ω, determine the shortest length of line (in wavelengths) necessaryfor satisfying the maximum-power-transfer condition

input impedance of the line is Zin 

j25Ω, thereby cancelling the imaginary part

of ZL (once ZL and the input impedance the line are added in series) Hence, using

Eq (2.73),

j100 cotβl j25

CHAPTER 2 51

+ -

generator with Vg 300 V and Zg 50Ωto a load ZL Determine the time-domaincurrent through the load for:

Figure P2.30: Circuit for Problem 2.30(a).

Using Eq (2.66) gives

Section 2-8: Power Flow on Lossless Line

ZL 75Ωthrough a 50-Ωlossless line of length l 015λ

1 2

L
How does Pincompare to PL? Explain

average power dissipated in Zg Is conservation of power satisfied?

Solution:

CHAPTER 2 55

(c)

Pin

12



ViI

i

12



VLI

L

12

connected to a generator with Vg 250 V and Zg 50Ω, how much average power

is delivered to each antenna?

ZL 1  75 Ω The same is true for line 3 At junction C–D, we now have two 75-Ω

impedances in parallel, whose combination is 75 2 375 Ω Line 1 is λ 2 long.Hence at A–C, input impedance of line 1 is 37.5Ω, and

+-

Generator

λ/2λ/2

incident power, the average reflected power, and the average power transmitted intothe infinite 100-Ω line The λ 2 line is lossless and the infinitely long line isslightly lossy (Hint: The input impedance of an infinitely long line is equal to itscharacteristic impedance so long asα 0.)

Solution: Considering the semi-infinite transmission line as equivalent to a load

(since all power sent down the line is lost to the rest of the circuit), ZL Z1  100Ω.Since the feed line is λ 2 in length, Eq (2.76) gives Zin  ZL  100 Ω and

to have an amplitude of 98.02 (V/m) at a depth of 10 m and an amplitude of 81.87(V/m) at a depth of 100 m What is the attenuation constant of sea water?

α... distance of m from the laser gun and an intensity of 0.2

(µW/m2) at a distance of m Given that the intensity of an electromagneticwave is proportional to the square of its... velocity of wave propagation

on the line is c, for which of the following situations is it reasonable to ignore the

presence of the transmission line in the solution of the