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Instructor: Natalia K. Nikolova

COURSE ELECTRICAL ENGINEERING 4FJ4 Duration of Examination: 3 hours McMaster University Final Examination Dec. 5, 2012 This examination paper includes a total of 7 PAGES AND 5 QUESTIONS. Instructions: 1. You can use only a standard calculator (Casio-FX991) 2. Write your name and ID on each page, the exam booklet included. 3. You are allowed to bring 1 sheet of letter-size paper with any writing on both sides. 4. Answer ALL questions. Provide the solutions in the exam booklet. 5. Submit the Smith chart and the exam booklet with your solutions.

1. [30 points] A transmitter of internal impedance 40GZ can deliver 20 W of power to a matched load. It is connected to a coaxial cable of characteristic impedance 01 50Z and length of 1 / 4L . This coaxial cable is in turn connected to a TV cable of characteristic impedance 02 300Z Ω terminated with a folded dipole antenna of impedance 300LZ Ω. The network is shown in Figure 1 below. Both cables can be assumed to be loss-free.

(a) Find the Thevenin equivalent of the transmitter at terminals 1 1 . Set the phase of the voltage source in the Thevenin equivalent to zero.

(b) Find the input impedance 22Z at terminals 2 2 (looking to the right) at the junction between the coaxial and the TV cable. See Figure 1 below.

(c) Find the following quantities at the transmitter’s terminals 1 1 : incident power 1iP , reflected power 1rP , transmitted power 1tP .

(d) Find the Thevenin equivalent of the transmitter plus the coaxial cable at terminals 2 2 .

(e) Using the Thevenin equivalent found in (d), find the following quantities at the terminals 2 2 : incident power 2iP , reflected power 2rP , transmitted power 2tP .

(f) Find the power LP delivered to the folded dipole antenna (the load LZ ). What would LP be if the antenna was properly impedance-matched to the transmitter?

01 50 Z

1

1

2

2

1 / 4L

02 300 Z 300 LZ

40 GZ

GV

1iP1tP

1rP2iP

2tP2rP

22Z

2. [15 points] Solve for the transmitted power 1tP in Problem 1 using a signal flow graph of the network shown in Figure 1 where the network portion to the right of terminals 2 2 is replaced by the equivalent load 22Z found in part (b) of Problem 1.

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3. [30 points] Use the Smith chart to design a double-stub tuner (see schematic below) to match a load of 100 30LZ j Ω to a 50-Ω line. Set the distance between the two stubs to / 8D and the distance from the load to terminals 2 2 to / 8L . Clearly mark every step on the Smith chart, briefly note each step down in your solution together with the respective numeric value.

LY0,Y

0Y 0Y

D L

1L2L

open orshort

0,Y0,Y

1

1

2

2

3

3

4. [15 points] In a system of system impedance 50 Ω, the S-parameters of a 2-port network are: 11 0.5 0.5S j , 21 0.95 0.25S j , 12 0.15 0.05S j , 22 0.5 0.5S j .

(a) Is the 2-port reciprocal? (b) Is the 2-port loss-free? If not, does it have loss or gain and in which direction? (c) If Port 2 is matched, find the input impedance at port 1. (d) If Port 1 is loaded with 1 0 50LZ Z Ω, what would be the internal impedance of the

generator connected to Port 2 in order to deliver maximum power? (e) Find the S-parameters of this same device if the reference planes at both ports are moved

away from the device by 1 2 / 4L L . [10 points] A plane wave at 1 GHz is normally incident on a thin copper sheet of thickness t.

The specific conductivity of copper is 75.8 10 S/m. a) Compute the transmission loss in dB at the copper-to-air interface. Repeat for the air-to-

copper interface. b) If the sheet is used as a shield to reduce the level of the transmitted wave by 150 dB, find

the minimum required sheet thickness mint .

END OF QUESTION SHEET A Smith chart and mathematical formulas follow (5 more pages)

TOTAL MARKS FOR THIS EXAM = 100