ECE320 Fall 2017

9 The Edward S. Rogers Sr. Department

'

of Electrical & Computer Engineering

UNIVERSITY OF TORONTO ,

ECE320: Fields and Waves Final Exam - December 16, 2017

Instructors: Sean Hum and Costas Sarris

Last name: ...................................................................................... .

First name: ...................................................................................... .

Student number: ................................................................................ .

Instructions

• Duration: 150 minutes (9:30 to 12:00 noon).

• Exam Paper Type: D. Closed book. Students ca~ use a ·one-page, double-sided aid sheet

• Calculator Type: 2. All non-programmable electronic calculators are allowed.

• Give units in your answers.

• A compass and a ruler are allowed .

. • Do NOT tear out the Smith Charts.

• Only answers that are fully justified will be given full credit.

Marks:

Ql: /20 II Q2: /20 11

Q3: /20 11

Q4:

TOTAL: /100 I

/20 11

Q5: /20

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Fall 2017

USEFUL FORMULAS

TRANSMISSION LINES

Reflection coefficient for a load impedance ZL:

Reflection coefficient at a distanced from a load impedance ZL:

ZL-Zo . rd= Z Z exp(-J2/3d) = fexp(-j2/3d)

L + 0

Impedance transformation at a distanced from a load impedance ZL:

Z. _ Z I+ r _ Z ZL cos(/3d) + jZ0 sin(/3d) in - 0 - 0

1 - r Zo cos(/3d) + jZL sin(/3d)

Input impedance of a short-circuited stub:

Zsc = j Zo tan((3d)

Input admittance of an open-circuited stub:

Yoe = jYo tan(/3d)

Voltage standing wave ratio (SWR):

ELECTROMAGNETIC WAVES

Propagation constant k = w..JEii,

Speed of light in vacuum c = 3 x 108 mis

Dielectric permittivity of vacuum: Eo = 8.8542 x 10-12 F/m

Magnetic permeability of vacuum: µ0 = 41r x 10-7 Him

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ECE320

ECE320

I . . . d ~ ntnns1c wave 1mpe ance: 'TJ = V ~ MAXWELL'S EQUATIONS - INTEGRAL FORM

Gauss law (electric field): fp D · dS = 1 Pvdv ·s V

Gauss law (magnetic field): fp B • dS = 0 s

Faraday law: i E · dl = _i f B. dS c dt ls

Ampere-Maxwell law: i H . dl = / J . dS + i f D . dS c ls dt ls

MAXWELL'S EQUATIONS - DIFFERENTIAL FORM WITH PHASORS

Gauss law (electric field): v' · D = Pv

Gauss law (magnetic field): v' • B = 0

Faraday law: v' x E = -jwB

Ampere-Maxwell law: v' x H = j + jwD

BOUNDARY CONDITIONS

n · (D2 - D1) = Ps

n · (B2 - B1) = 0

n x (E2 - E1) = o n x (H2 - H1) = Js

PLANE WAVE INCIDENCE AT MATERIAL INTERFACES

Snell's law: n 1 sin ei = n2 sin Bt

Reflection/transmission coefficients:

Brewster angle:

r..1

Tj_

'TJ2 cos et - 'TJ1 cos ei 'T/2 COS 0t + 'TJI COS 0i

2TJ2 cos Bi 'TJ2 COS 0t + 'TJI COS 0i

ry2/ cos Bt - ryi/ cos Bi

'TJ2 / cos Bt + TJi/ cos 0i

2'T)2/ COS0t

sin0B,..l = 1 - (µ1 E2/ µ2E1)

1 - (µi/ µ2) 2

Fall 2O17

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Fall 2017

RECTANGULAR WAVEGUIDES

sinBs,11 = 1 - (µ2ci/ µ1 E2)

1 - (ci/c2)2

Propagation constant of ( m, n) TE/TM mode in a rectangular waveguide of width a and height b:

(3 _ ✓ 2 (m,1r)2 (n1r)2 - WEµ- - - -a b

Cut-off frequency of ( m, n) TE/TM mode in a rectangular waveguide of width a and height b:

ECE320

TE/TM modes in a rectangular waveguide of width a and height b (0 ::; x ::; a, 0 ::; y ::; b), filled with a

dielectric material with constitutive parameters E, µ. In the following: k~ = ( -:1r) 2 + ( nt) 2.

TE modes TM modes

E _ jwµ (n1r) H, m1rx . n1ry -jf3z x - k't b o cos a sm b e

E -j/3 (m7r) E m1rx . n1ry -j(3z x = -- -- acos--sm--e k't a a b

E -jwµ (m'Ti) H . m1rx n'Tiy -j(3z E -j/3 (n'Ti) E . m'li"X n1ry -j(3z = -- - as1n--cos--e = -- -- osm--cos--e Y k2 a a b y k2 b a b

C C

Ez =0 - m1rx n1ry .13 Ez = Ea sin -- sin --e-J z

a b

Hx = -Ey/ZrE Hx = -Ey/ZTM

Hy= Ex/ZTE Hx = Ex/ZrM

- ffi'li"X n'Tiy "(3 Hz= 0 Hz = Ho cos -- cos --e-J z

a b

ZTE = rJ/ JI - Uc/ !)2 ZTM = rJJl - Uc! !)2

1 pF = 10-12 F 1 nH = 10-9 H 1 MHz= 106 Hz 1 GHz= 109 Hz

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ECE320 Fall 2017

Question 1 ;

Consider the dielectric-dielectric interface shown in Figure 1. The medium in the region z· ~ 0 is free space, while the medium in the region.z > 0 is silica (Er = _3.8). Both media are lossless. A 1 GHz plane wave is

· incident fr<?m the bottom left as shown, such that the electric and magnetic· field vectors are contained within a plane shown by the. dotted_ line, which also extends out of the page. Similar planes are shown for th_e reflected , and transmitted fields. The angle of incidence is 0i = 30°.

.,.

EQ plane contai_ning Er, Hr

X

.. z = o·- ··-,

Figure 1 ·· Dielectric media interface and plane waves

z

A plane wave is incident upon the interface as shown in Figure 1. The incident field is polarized such that its· electric and magnetic field vectors lie in a plane as shown. The electric-field is described by a phasor

E i [ '/n3A. 2A' A] [• 'k''(i v13 )] . = v 0x + y - z exp - J o 2x + ~ z (1)

(a) Determine the value ofk0 in equation (1).

(2 pts)

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Fall 2017 ECE320

(b) Write the phaser expression for the incident magnetic field Hi, evaluating all known constants in the ex­pression.

(4 pts)

(c) Calculate the vector wavenumber of the transmitted electric field, kt.

(3 pts)

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ECE320

(d) Determine the reflection and transmission coefficients associated with:

(i) the x-component of the electric field;

(ii) the y-component of the electric field;

(iii) the z-component of the electric field.

Fall 2017

(5 pts)

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Fall 2017 ECE320

( e) Write the phasor expression for the reflected electric field, Er.

(4 pts)

(f) For what value of 0i does only a y-component of the reflected electric field exist?

(2 pts)

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ECE320 Fall 2017

Question 2

A complex load ZL = 30-±- j50 n terminates a 50 n transmission line as shown in Figure 2. The line is excited by a voltage source V9 = 5L'.0°. The frequency of the source is 100 MHz. The speed of light along the transmission line is 2 x 108 mis, and the line is 4 m long.

Zo

Zo, /3

z = -f z=O

Figure 2 Terminated transmission line

(a) Knowing that the input voltage to the transmission line is

V(-f) = v0+ exp(j/3£) + rv/ exp(-j/3£)

determine the value of the constant V/. (3 pts)

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Fall 2017 ECE320

(b) Determine the standing wave ratio along the line, and the location(s) of all maxima, in metres, in the standing wave pattern over the region -f, ::; z ::; 0.

(4 pts)

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ECE320 Fall 2017

(c) The load is to be matched with an open-circuit shunt stub circuit, as shown in Figure 3. Determine the distance between the stub and the load d such that d is the smallest possible value, and the length fl,5 of the stub. Express all answers in metres.

(6 pts)

d

ZoJ3

Figure 3 Open circuited shunt stub circuit

( d) Determine the fraction of the incident power delivered to the load Z L in part ( c ).

(1 pt)

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Fall 2017 ECE320

(e) A different circuit, shown in Figure 4, is used to produce Zin =:= Zo. Determine the distance between the resistor and the load dR in metres such that dR is minimized, and the resistor value R in Ohms.

(4 pts)

Zo, /3 R

Figure 4 Resistive matching circuit

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ECE320 Fall 2017

(f) Determine the fraction of the incident power delivered to the load ZL in part (e).

(2 pts)

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Fall 2017

Question 3

A transmission line has the following per unit length parameters: R' 0.001 Sim, and C' = 88.9 pF/m.

ECE320

5 0/m, L' 0.5 µHim, G'

(a) The line is excited with a 50 MHz signal. Is the line low-loss or lossy? Determine the phase velocity and wavelength of the line at 50 MHz.

(4 pts)

(b) Determine the attenuation constant of the line. By what percentage is the voltage signal reduced to over a length of 3 m?

(2 pts)

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ECE320 Fall 2017

(c) A lossless version of this line is connected to a generator and load as shown in Figure 5. The line length is f = 3 m, the transient time through the line is T = 20 ns, and its characteristic impedance is 75 n.

500

Z0 ,u 675 n

Figure 5 Transmission line circuit

Graph the voltage measured at a position z = 1.5 m for 0 :S t :S 100 ns if vg(t) = 20u(t), where u(t) is a unit step function. Indicate the exact times of all discontinuities and the values for all voltages in the waveform.

(8 pts)

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Fall 2017 ECE320

(d) Repeat part (c) if v9 (t) = 20[u(t) - u(t - T)], and T = 5 ns.

(4 pts)

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ECE320 Fall 2017

(e). If the line was lossy, what would happen to the pulse in part (d) as it travels down the line besides being reduced in amplitude?

(2 pts)

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Fall 2017 ECE320

Question 4

A rectangular tunnel with reinforced concrete walls can be modeled as an air-filled (Er = 1) rectangular waveg­uide. The waveguide shown below has width a= 7 m and height b = 4.5 m.

(y) i-----7 m-----y=b ..,._ __________ ..

4.5m

y=O x=O x=a (x)

(a) What is the mode with the lowest cut-off frequency ("dominant") mode of this waveguide? Calculate its cut-off frequency (in MHz).

(4 pts)

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ECE320 Fall 2017

(b) Analyze the electric field of the dominant mode of the waveguide (which you found in part (a)) in terms of plane (TEM) waves. In a diagram, show how this mode propagates along the waveguide.

(4 pts)

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Fall 2017 ECE320

(c) Draw the electric field vector of the dominant mode of the waveguide (which you found in part (a)), at a cross-section of the waveguide. You can use the figure below to draw the field vector.

20

(y)

y=b

y=O x=O x=a (x)

(4 pts)

ECE320 Fall 2017

(d) An AM radio station transmitting at f = l MHz generates a vertical electric field of magnitude IEyl 0.025 V/m, measured at the entrance of the tunnel, at x = a/2, y = b/2. The signal of the radio station is quickly reducing in strength, as one travels down the tunnel. Can you explain why?

(2 pts)

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Fall 2017 ECE320

(e) A car radio stops detecting the signal of the AM radio station, when the amplitude of the electric field transmitted by the station drops to 10-4 V /m. Determine the distance from the entrance of the tunnel, beyond which the car radio cannot detect the signal of the radio station. You can assume that this distance can be estimated by studying the propagation/attenuation characteristics of the dominant mode in the tunnel.

(6 pts)

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ECE320 Fall 2017

Question 5

The following questions are independent from each other and equally weighted (2.5 pts each). Provide clear and concise explanations for all your answers.

1. The reflection coefficient at the input of a loss-less transmission line of length l = l .5>. is:

r Zin - Zo ( . o) · = z z = 0.5exp -J60 .

· in+ 0

Hence, the load is ZL = RL + jXL, with XL > 0 .

..,....,...,..,............, ......... -[ = 1.5,),,.....,... ___ _

True False

Justification:

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Fall 2017 ECE320

. . . 2. The following diagrams show the wavevectors for incident and transmitted plane waves ki and kt at the

interface'between,air and a dielectric with Er = 9, µr = l. According to (a), 0t < ei. According to (b), ei < 0t, Which one of the two is correct ?' .•

)l '.J

24

Air Er~ 1

; ,· , ,· ;

Answer:

,· ;

(x)

. (a)

',

Air Er~ 1

, .,

' .

;

•;

,' ,·

(b)

•·

•,

ECE320 Fall 2017

3. In total internal reflection, the transmitted electric and magnetic field components are zero.

True False

Justification:

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Fall 2017 ECE320

26

4. The following diagram shows the incidence of a plane wave with an electric field: Ei = Ef1 + Ei at the interface between air and a dielectric with Er = 9, µr = l. The reflected electric field has only a perpendicular component: .Er = E:;_. Is this possible?' If yes, determine the arigle of incidence for this to happen. If !lot, why not? '

Answer:

..

Air Er= 1

(x)

e. \/, ) ,·· v~: . ,

Ei" ~ Ef1 + Ei

ECE320 Fall 2017

5. A linearly polarized plane wave propagating in the z-direction has the phasor form:

E = x.Eo exp (-jkz).

Can you write it as a superposition of two circularly polarized waves? If yes, derive the two circularly polarized waves, in phasor form. If not, why not ?

Answer:

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·Fall 2017 ECE320

28

6. A plane wave is propagating along the z-axis, in a conducting medium with skin depth 85 • Then, the ratio:

is equal to:

a) exp (-6)

b) exp (-3)

c) exp ( +3)

IE(z = 58s)I IE(z = 28s)I

Choose the correct answer and briefly explain.

ECE320 Fall 2017

7. The time-averaged value of the Poynting vector, < S > for an evanescent mode in a rectangular waveg­uide is zero.

True False

Justification:

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Fall 2017 ECE320

8. It is not possible for the phase velocity of propagating waveguide modes to exceed the speed of light.

True False

Justification:

30

NAME ................................ .

STUDENT# .......................... .

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0040 30 20 15 10

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0.9 0.8 0.7 0.6 0.5

0.0 0.1 0.2 03 0.4 0.5

ORIGIN

The Complete Smith Chart Black Magic Design

RADIALLY SCALED PARAMITERS

TOW ARD LOAD-> 2.5 2 1.6 1.6 1.4 1.2 ).) I JS JO 7 5

6 5 I 11 ).) 1.2 13 1.4 1.6 1.8 2

7 8 9 JO 12 14 0.1 0.2 0.4 0.6 0.8 I

0.2 0.1 0.05 O.QJ 0 0 ).) 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0.4 0.3 0.2 0.1 0 I 0.99 0.95 0.9 0.8 0.7

CENTER 0.6 0.7 0.8 0.9 ).) 1.2 1.3 1.4 1.5 1.6

<-TOWARD GENERA TOR 2 I

4 5 JO 20 00

10 15:):)

2.5 4 5 JOoo

0.6 0.5 0.4 0.3 0.2 0.1 0

1.7 1.8 1.9

NAME ................................ .

STUDENT# .......................... .

,!>~ S:'1-c '/ if> ,x,}()()40 20 10

~/4'1'%Q'O's, ~40 30 20 10 r:{04'&- 0

~~~.,, l 0.9 0.8 0.7 0.6 05 0.4 0.3 ,!'

o,/ 0.9 0.8 0.7 0.6 0.5

0.0 0.l 0.2 0.3 0.4 0.5

ORIGIN

The Complete Smith Chart Black Magic Design

RADIALLY SCALED PARAMETERS

TOWARD LOAD-> 2.5 2 l.8 1.6 l.4 12 ].] l 15 10 7 5

6 5 ].] l.2 l.3 l.4 l.6 l.8 2

7 8 9 10 12 14 0.l 0.4 0.6 0.8 l 15

0.2 0.1 0.05 0.01 0 0 ].] 1.2 1.3 1.4 1.5 1.6 l.71.81.92

0.4 0.3 0.2 0.l 0 l 0.99 0.95 0.9 0.8 0.7

CENTER 0.6 0.7 0.8 0.9 l.l 1.2 1.3 l.4 15 l.6

<-TOW ARD GENERA TOR 2 l

4 5 10 20 ~

4 5 6 10 150Q

2.5 4 5 10~

0.6 0.5 0.4 0.3 0.2 0.l 0

l.7 l.8 l.9