Misr International University Faculty of Engineering Department of Electronics and Communication Course: EET 416 Microwave Engineering Instructors: Prof. Fawzy Ibrahim
Problem Set #3 Solution Transmission Line Theory Question #3.1 [Pozar 2.2] The parameters of a certain transmission line operating at a frequency f = 500 MHz are: L = 0.2 µH/m, C = 300 pF/m, G = 0.01 S/m and R = 5 Ω/m. Calculate: (a) The complex propagation constant , the attenuation constant α and phase constant β. (b) The wave parameters: the wavelength λ, the phase velocity Vp and the characteristic impedance Zo. (c) Repeat parts (a) and (b) for lossless transmission line. Answer: (a) = 0.23 + j 24.3, (b) λ = 25.85 cm, Vp = 1.29x108 m/sec and Zo =25.8 + j0.03 (c) = 0 + j 24.3, (b) λ = 25.85 cm, Vp = 1.29x108 m/sec and Zo =25.8 Question #3.2 [Kraus 3.2][HW] The parameters of a certain transmission line operating at a frequency f = 7 kHz are: L = 1.5 µH/m, C = 1.4 nF/m, G = 1.4 S/m and R = 12 mΩ/m. Calculate: (a) The complex propagation constant , the attenuation constant α and phase constant β. (b) The wave parameters: the wavelength λ, the phase velocity Vp and the characteristic impedance Zo. (c) Repeat parts (a) and (b) for lossless transmission line. Answer: (a) Zo =33 – j 2.5 Question #3.3 [Pozar 2.1] The current on a transmission line is given as: i(z, t) = 1.2 cos(1.51 x 1010 t – 80.3 z) mA. If the characteristic impedance is Zo= 50 , determine: (a) The frequency, f. (b) The wavelength λ. (d) The current representation in phasor form. (c) the phase velocity Vp . (e) The voltage on the transmission line v(z, t). Solution
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Question #3.4 [HW] The voltage on a transmission line is given as: V(z, t) = 0.5e-2z cos(2 x 109 t – 50 z) V. If the characteristic impedance is Zo= 75 , determine: (a) The frequency, f. (b) The wavelength λ. (d) The voltage representation in phasor form. (c) the phase velocity Vp . (e) The current on a transmission line i(z, t). Question #3.5 [Pozar 2.18] A generator (Vg = 10 V and Zg = 100 ) is connected to a transmission line (Zo = 100 and length l = 1.5 λ) and having a load impedance ZL = 80 + j 40 as shown in Fig. 3.6. (a) Find the voltage as a function of z along the transmission line. (b) Plot the magnitude of this voltage for –l z 0.
Fig. 3.6.Transmission line with generator and load Solution
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Question #3.6 [Pozar 2.12] [HW] A radio transmitter is connected to an antenna having an impedance 80 + j 40 with 50 coaxial cable. If the 50 transmitter can deliver 30 W when connected to a 50 load, how much power is delivered to the antenna? Solution
Question #3.7 [Pozar 2.14] Calculate SWR, reflection coefficient magnitude and return loss values to complete the entries in the following table.
Solution
Question #3.8 [Pozar 2.14] [HW] Calculate SWR, reflection coefficient magnitude and return loss values to complete the entries in the following table.
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Solution
Question #3.9 [Kraus 3.12] A lossless 75 transmission line is terminated in a load of 350 . Do the following: (a) Calculate the reflection coefficient, . (b) Deduce and plot the magnitude of the voltage along the line, at increments of / 4 for a distance of one from the load. (c) Calculate the Voltage standing Wave Ratio (VSWR). Answers: (a) = 0.647
(c) VSWR = 4.71.
Question #3.10 [Kraus 3.11][HW] A lossless 50 transmission line is terminated in a load of 400 , find the input impedance Zin at a distance of / 8 from the load. Answers: (a) Zin = 12.3 – j48.5 = 50 -75.9o. Question #3.11 [Pozar 2.30] A losslessy 50 transmission line is matched to a 10V source and feeds a load ZL=100 . If the line is 2.3 long and has an attenuation constant = 0.5 dB / , find; (a) The power delivered by the source to the line. (b) The power delivered to the load. (c) The power lost in the line. (d) The generator or source power. (e) The power lost in the source resistance Rg.
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Solution
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Question #3.12 [Pozar 2.15][HW] A lossless transmission line circuit shown in Fig. 3.12 has Vg = 15Vrms, Zg=75 , Zo=75 , ZL=60 – j 40 and l = 0.7 . Compute the power delivered to the load.
Fig. 3.12Transmission line with generator and load Solution
Question #3.13 [Pozar 2.25] Design a quarter-wave matching transformer shown in Fig. 3.13 to match 40 load to a 75 line.
Fig. 3.13 The quarter-wave matching transformer. Solution Question #3.14 [Pozar 2.19] For the transmission line circuit shown in Fig. 3.14, if its length l = 0.4 , its load impedance ZL= 60 + j50 and its characteristic impedance of the Zo=50 , find the following parameters using the Smith chart: (a) The Voltage Standing Wave Ratio (VSWR). (b) The reflection coefficient (). (c) The admittance YL. (d) Then input impedance Zin. MIU
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Fig. 3.14 A transmission line circuit Solution:
Question #3.15 [Pozar 2.20 & 21] [HW] For the transmission line circuit shown in Fig. 3.14, if its length l = 0.4 , its load impedance ZL= 40 – j30 and its characteristic impedance of the Zo=50 , find the following parameters using the Smith chart: (a) The Voltage Standing Wave Ratio (VSWR). (b) The reflection coefficient (). MIU
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(c) The admittance YL. (d) Then input impedance Zin. (e) Repeat parts (a), (b), (c) and d when the transmission line length, l = 1.8 , Answers: (a) VSWR = 2 (b) = 0.333 270o (c) YL = 16 + j12 mS (d) Zin =93.2 - j21.6 (e) VSWR = 2.6, (b) = 0.422 54o (c) YL =9.84 – j8.2 mS (d) Zin =20.8 – 6.7 Question #3.16 [Kraus 3.15] A quarter wave or /4 transformer is inserted to provide matching between a load impedance ZL= 200 - j100 and a transmission line of characteristic impedance Zo =100 as shown in Fig. 3.16 . Find: (a) The nearest point to the load (d) at which the transformer is connected. (b) The characteristic impedance ZI of the T. L. to be used for the transformer. (c) The Voltage Standing Wave Ratio (VSWR) on d. (d) The Voltage Standing Wave Ratio (VSWR) quarter wave or /4 transformer line.
Fig. 3.16 Matching via quarter wave or /4 transformer.
Answers: (a) d = 0.214, (b) ZI =61.6 (c) VSWR on d = 2.6 (d) VSWR on /4 line = 1.62 Question #3.17 [Kraus 3.15] [HW] A quarter wave or /4 transformer is inserted to provide matching between a load impedance ZL= 300 + j200 and a transmission line of characteristic impedance Zo = 100 as shown in Fig. 3.16. Find: (a) The nearest point to the load (d) at which the transformer is connected. (b) The characteristic impedance ZI of the T. L. to be used for the transformer. (c) The Voltage Standing Wave Ratio (VSWR) on d. (d) The Voltage Standing Wave Ratio (VSWR) quarter wave or /4 transformer line. Answers: (a) d = 0.026, (b) ZI =214 (c) VSWR on d = 4.6 (d) VSWR on /4 line = 2.15 Question #3.18 [Kraus 3.14] A uniform transmission line of characteristic impedance Zo = 100 is terminated by a load impedance ZL= 500 + j0 , use a shorted stub tuner to match this load to the line as shown in Fig. 3.18 . Find: (a) The distance (d) from the load to the stub. (b) The distance or length (d1) of the stub (c) The VSWR on d. (d) The VSWR on the stub. MIU
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Answers: (a) d = 0.182, (b) d1 = 0.082, (c) VSWR on d = 5 (d) VSWR d1= Question #3.19 [Kraus 3.13] [HW] A uniform transmission line of characteristic impedance Zo = 100 is terminated by a load impedance ZL= 150 + j50 , use a shorted stub tuner to match this load to the line as shown in Fig. 3.18 . Find: (a) The distance (d) from the load to the stub. (b) The distance or length (d1) of the stub (c) The Voltage Standing Wave Ratio (VSWR) on d. (d) The (VSWR) on the stub. Answers: (a) d = 0.194, (b) d1 = 0.167 (c) VSWR on d = (d) VSWR d1=
Fig. 3.14 Matching using a stub tuner. Answers: (a) d = 0.194, (b) d1 = 0.167 (c) VSWR on d = (d) VSWR d1=
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