Cover Page. Solution. James Clerk Maxwell ( )

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1 Cover Page Final Exam (Total 45 points) Professor: Sungsik Lee Subject: Electromagnetics (EM-), Fall Semester in 08 epartment of Electronics Engineering, Pusan National University ate: 5 ecember 08, uration:.5 hours (4:00 ~ 5:0) Name in Korean ( 한글이름 ) Student Number ( 학번 ) Student Information Solution Total Marks Good Luck! James Clerk Maxwell (8-89) *Notice : Write your name and student number in the Table above. And please do NOT open other pages yet until the exam instructor gives you an allowance. *Notice : Write your answers in the grey square blanks only. And other spaces are for your hand calculations. *Notice : There is no chance to ask a question during the exam. But if there is a critical error in a problem, please write down it on that page, then we will give you all full marks to that problem.

2 Part -. Smith Chart and TML (8 points) Problem ( points): Choose correctly drawn Smith-Chart which the most relevant to the circuit diagram shown below. Here, we try to get the impedance matching for the distance (d) from the load of the main transmission line (lossless), connecting the shortest short-circuit Stub: Main transmission line d Toward Characteristic impedance: = 50 Ω Load impedance: Z L = 00 Ω + = =4 Signal s wavelength: λ d/λ h/λ d/λ 4 d/λ Short-circuit Stub with the length (h) h/λ r= r= r= r= h/λ h/λ d/λ Problem (6 points): With the lossless TML circuit diagram shown below, points (A to E) on the Smith-Chart are marked depending on the situation given from the load (Z L ). Here, the signal s wave-length (λ) is 4 m. Now, answer the questions (a to d) listed below: Inductor (x>0) Smith-Chart +x 한바퀴 =0.5=m z=-0.5 L = >0 Toward Lossless Transmission line (TML) λ.5 0 Z L Load z=0 E (a) Find the point which doesn t exist x=0 0=<r< in the TML position of.5 0 for a resistor load Z L < : (*Resistor load includes a zero Ohm as a short circuit) r<, x=0 z [m] capacitor (x<0) = <0 x c A r= B z=-.5 (.5 points) *Note that the grey area is for r C z=- Smallest Smallest VSWR (b) What is the smallest VSWR point among points on the Smith-Chart: (*VSWR: Voltage Standing Wave Ratio) B (.5 points) (c) What is the point for the impedance load with a resistor and inductor in series: A (.5 points) (d) What is the point for the capacitor load: (.5 points)

3 Part -. Smith Chart and TML ( points) Problem (6.5 points): With the lossless TML circuit diagram shown below, points (A to E) on the Smith-Chart are marked depending on the situation given from the load (Z L ). Here, the signal s wave-length (λ) is 8 m, and is 50 Ω. Now, answer the questions (a to d) listed below: Toward Lossless Transmission line (TML) λ 0 한바퀴 0.54m Z L Load z [m] z=0 (a) Find the point which exist r< in the TML position of for the finite resistor load (< 50Ω): x=0 Smith-Chart z=- A B r<, x=0 E z=- C r>, x=0 r= Upper half x>0 *Note that the grey area is for r z=- Lower half x<0 ( points) (b) What is the point when the resistor load Z L > 50 Ω: x=0 r> C (.5 points) (c) What is the point for the impedance load with a 5 Ω resistor and capacitor in series : r< x<0 (d) What is the point for the impedance load with a 00 Ω resistor and inductor in series : r> x>0 E (.5 points) (.5 points) Problem 4 (4.5 points): With the lossless TML circuit diagram shown below, points (A to E) on the Smith-Chart are marked depending on the situation given from the impedance load (only composed of a 50Ω resistor and capacitor) and generator s frequency (f). Now, find the three correct sentences from the list below:, 5,5 중하나맞으면 : 점둘다맞으면 : 4.5 점 V G V G = V p cos(ωt ) ω = πf (.5 points each) ~ lim! # $% Lossless Transmission line (TML) λ 0 Impedance Load z [m] C E B Smith-Chart The point C is closer to the generator than the point A for a fixed frequency. ( 로드가 C 와 A 사이이면, C 가더멀다.) When we increase the generator s frequency of the sinusoidal input signal, a point at the load is getting closer to B or E. When we increase the generator s frequency of the sinusoidal input signal, a point at the load is getting closer to E. 4 When we decrease increase the generator s frequency of the sinusoidal input signal, the point A at the load is getting closer to. 5 We can arrive at the point E with replacing the capacitor by the inductor for a fixed frequency. if a frequency is around 0 Hz, we can get closer to the point E. 0 0 '()*)+,,-./ Same r r= A F *Note that the grey area is for r

4 Part -. Uniform Plane Wave (Reflection, Power, Polarization) (4 points) Problem 5 ( point): Choose Most Incorrect Sentence from the list below:, 4 둘중하나맞으면만점 The reflection of EM-wave at the interface between two media can happen when the intrinsic impedance of each medium is different. When the EM-wave is not reflected at the interface between two media, the wave s power is transmitted 00%. The reflection coefficient at the interface between the first and second media, where the EM-wave is propagating from the first to the second media, can be negative positive when the intrinsic impedance of the first media is smaller than that of the second media. 4 The intrinsic impedance of the vacuum is the smallest largest. Problem 6 ( points): Choose Most Incorrect Sentences from the list below: 5 ( point each) Wave number of the electromagnetic (EM) waves is generally expressed as a complex value. In lossy media, the wave number is a complex number. Existence of conduction current in a space means that the permittivity of that space is a real complex value. 4 Poynting vector is in the dimension of power density. 5 Average power of the EM-wave, which is given from the surface integral of the Poynting vector, is a function of time space. 6 Skin depth is decreased with increasing frequency and conductivity of the medium where the EM-wave is incident Problem ( point): Choose Most Incorrect Sentence from the list below: The linear polarization of the E-wave propagates toward the z-direction happens when the E-wave components of x and y are out of in phase (phase difference between them is non-zero). If the phase difference between x and y components of E-wave is zero or +/- 80 o, the linear polarization happens. If the phase difference between x and y components of E-wave is +/- 90 o, the circular polarization happens. 4 The wave polarization can happens when the wave is passing through a polarizer where the phase difference is made. 5 If the phase (φ) of the y-component of the E-wave is negative in reference to the x-component of the E-wave, the E-wave is being polarized with rotating in counter-clockwise on the fixed plane where the observer is looking at the E-wave s approaching along the z-axis. 4

5 Part -. Uniform Plane Wave (Multi-interfaces and Reflection) (0 points) Problem 8 ( points): Choose Most Correct Sentence from the list below: 4 *Note: the intrinsic impedances of a plastic and a glass are tunable while they are smaller than the vacuum. 0 >0 45,, 0 64 For the three lossless media system (air/a plastic/a glass), the 00% transmission of the EM-wave can be made by the half-wave matching method for the second medium which is a plastic. impossible since the first and last medium are different For the three lossless media system (a plastic/vacuum/a glass), the 00% transmission of the EM-wave can be made by the quarter-wave matching method for the second medium which is vacuum. impossible case due to For the three lossless media system (a glass/a plastic/air), the 00% transmission of the EM-wave can be made by the half-wave matching method for the second medium which is a plastic. impossible since the first and last medium are different 4 For the three lossless media system (air/a plastic/a glass), the 00% transmission of the EM-wave can be made by the quarter-wave matching method for the second medium which is a plastic. possible since it can make 0 45 = Problem 9 (8 points): Find the values of the parameters (a) to (d) listed below, which are relevant to the statement and diagram shown below: Statement () Assuming that no magnetic polarization and lossless in any medium in the figure below (thin enough to be lossless). () EM-wave is orthogonally incident onto the interface between the medium- and the medium-. () Medium- is a plastic (intrinsic impedance η = 5 Ω and reflective index n = ). (4) Wavelength (λ) of the EM-wave in the medium- is.5 µm. (5) Intrinsic impedances of the media,, and 4 (η, η, η 4 ) are all different. (6) Magnitude relationship of the intrinsic impedance of each medium is η > η > η > η 4 = 5 Ω. () We assume that η = η. (8) The incident wave from the medium- is 00% transmitted into the medium-4 without any reflection at any interface. 0 > >0 45,, 0 64 (+:-;<.() *Note: we assume that the intrinsic impedances of the vacuum (η 0 ) is approximately to be 5 Ω for its reflective index n 0 =. QW+ HW HW + QW η = 5 Ω η η η 4 = 5 Ω η 5 5/ η 5 5 L (=/) L (=/8) EM-wave n = n n λ λ λ =.5 µm 00% transmission HW + QW ( 원인 ) thinnest L L (HW+QW 의결과 ) 원인 (a) Find the intrinsic impedance of the medium, η = 5/ Ω (b) Find the intrinsic impedance of the medium, η = 5 Ω Here, we assumed that η = η. (.5 points) (0.5 points) 최종결과 (c) Find the thinnest thickness of the medium (thinnest L ) = / = 0.08 µm 5>>? = =0.08<0.5 (d) Find the thinnest thickness of the medium (thinnest L ) = /8 = 0.5 µm 5>>? B =0.5<0.5 ( points) ( points) 즉, 최소두께 c,d 를얻을수있는조건 ( 원인 ) 은유일하다. 즉 a,b 의답이 QW+HW 로얻은것일수없다. 이경우에 c,d 는 HW+QW 로얻은값보다크므로 Thinnest ( 모든경우를아울러최소 ) 가아니다. 5

6 Part -. Uniform Plane Wave (TE, TM Waves) ( points) Problem 0 ( point): Choose Most Incorrect Sentence from the list below: 없음 : 모두정답처리 TE wave means that E-wave is incident in parallel with the surface. 0 EF =0G,)H 5, =0 Ih(- H 5, =90 TEM wave means that both E- and H-waves are incident in parallel with the surface. Effective intrinsic impedance of TM wave (η ΤΜ ) can be zero by changing the incident or transmitted angle. 4 A total reflection can happen as long as the reflection coefficient is positive, regardless of the type of waves. 5 Snell s law of reflection at the surface of x-y plane is derived from the equivalence between the wave numbers of the x-components (k x ) of incident and reflected waves in the same medium. 6 Effective intrinsic impedance of TE wave (η ΤΕ ) can be infinity. <0 E-wave should be inverted Problem ( points): Choose Most Correct Figure from the list below: 4 And H-wave shouldn t be inverted While getting H *Note: η and η in the figures below are the intrinsic impedance of each medium. H 5 So, this is η <η η >η η <η 4 η >η correct η η η Hr η η η >0 H >H 5 <0 E-wave should be inverted And H-wave shouldn t be inverted While getting H >H 5 Problem ( points): Based on the statement and the figure below, find the values of the parameters (a) to (d) listed below: η η >0 E-wave shouldn t be inverted And H-wave should be inverted Statement and Figure () No magnetic polarization, No charge, and No current in any medium in the figure on the right-hand-side. () Medium- is unknown () Medium- is vacuum (η = η 0 = Ω and n = ). (4) TE-wave is incident from the medium- to the medium- with an oblique angle (0 o < θ i < 90 o ). (5) Characteristic impedances magnitude relationship: η <η. (6) Snell s laws of reflection and transmission are satisfied at the surface between the medium- and the medium-. o () sin 0 o =, sin 60 =, o o cos 0 =, cos60 = η η θ i θ t Er (a) If θ i = 0 o and θ t = 60 o, find the reflective index of the medium, n = (b) If the total reflection with θ t = 90 o is made, find the value of sin θ i = *Note: At this total reflection, the θ i is specially called the critical angle (θ c ). ( points) ( points) TE (c) With this total reflection, find the effective intrinsic impedance of the medium,η = Ω 0 EM L = N O 0 = BRR/ B = BRR L = = - P Q = / B = L (d) With this total reflection, find the effective reflection coefficient TE = unitless ( point) 0 EM = 0 = G,)90 = 0 = = 0 L EM +0 = L EM / - L - =)*-H )*-H 5 = )*-H 5 = - = =/ - L ( points) G,)H 5 = 6