Problems 1. Determine the wavelength for electromagnetic waves in free space with the following frequencies: 1 kHz, 100kHz, 1MHz and 1GHz. Solution: a. λ = c/f = (300x10 6 m/s)/ 1 kHz = 300,000 m b. λ = c/f = (300x10 6 m/s)/ 100 kHz = 3,000 m c. λ = c/f = (300x10 6 m/s)/ 1 MHz = 300 m d. λ = c/f = (300x10 6 m/s)/ 1 GHz = 0.3 m 2. Determine the frequencies for electromagnetic waves in free space with the following wavelengths: 1cm, 1m, 100m and 1000m. Solution: a. f = c/λ = (300x106 m/s)/ 0.01m = 3GHz b. f = c/λ = (300x106 m/s)/ 1m = 300MHz c. f = c/λ = (300x106 m/s)/ 100m
= 300kHz d. f = c/λ = (300x106 m/s)/ 100m = 3MHz 3. Determine the characteristic characteristic impedance for an air-dielectric transmission line with D/r ratio of 8.8. Solution: Zo = 276 log D/r = 276 log 8.8 = 260.68 Ω 4. Determine the characteristic characteristic impedance for an air-filled concentric transmission line with D/d ratio of 4. Solution: Zo = 138/ (√E r) log D/d = 138/ √2.23 log 4 = 55.64 Ω 5. Determine the characteristic characteristic impedance for coaxial cable with inductance L = 0.2µH/ft and conductance 16 pF/ft. Solution: Zo = √L/C = √ (0.2µH/ft)/ (16 pF/ft) = 111.80Ω 6. For a given length of coaxial cable with distributed capacitance C = 48.3 pH/ft and distributed inductance L = 241.56 nH/m, determine the velocity factor and velocity of propagation.
10. Determine the SWR for a 50-Ω transmission line that is terminated in a load resistance Z L = 75Ω. Solution:
Vf = Vp/c = (292.76x10 6 m/s)/ (300X10 6m/s) = 0.98 7. Determine the reflection coefficient for transmission line with incident voltage Ei =0.2V and reflected voltage E r = 0.01V.
SWR = Zo/ZL or ZL/Zo = 75 Ω/50 Ω = 1.5 11. Determine the SWR for a 75 Ω transmission line that is terminated in a load resistance Z L = 50 Ω.
Solution:
Solution:
Ґ = Er / Ei or Ir / Ii = 0.01V/0.2V = 0.05V
SWR = Zo/ZL or ZL/Zo = 75 Ω/50 Ω = 1.5
8. Determine the standing wave ratio for the transmission line in #7. Solution: SWR = Vmax/Vmin = (Ei + Er) / (E i - Er) = (0.2V + 0.01V)/ (0.2V – 0.01V) = (0.21V / 0.19V) = 1.105 9. Determine the SWR for a transmission line with maximum voltage standing wave amplitude Vmax = 6V and minimum voltage standing wave amplitude Vmin =0.5V. Solution: SWR = Vmax/Vmin = 6V/ 0.5V = 12
12. Determine the characteristic impedance for a quarter-wavelength transformer that is used to match a section of 75Ω transmission line to a 100Ω resistive load. Solution: Z’o = √ZoZL = √ (75Ω) (100Ω) =86.68Ω 13. Using TDR, a pulse is transmitted down a cable with a velocity of propagation of 0.7c. The reflected signal is received 1.2µS later. How far down the cable is the impairment? Solution: d = (v x t)/2 = (0.7 x 300x10 6m/s x 1.2µS)/2
14. Using TDR, a transmission line is located 2500m from the source. For a velocity propagation of 0.95c. Determine the time from the beginning of the pulse to the reception of the echo. Solution: t = 2d/v = 2d/ kc = 2(2500)/ (0.95 x 300x10 6m/s) = 17.54µS 15. Using TDR, transmission line impairment is located 100m from the source. If the elapsed time from the beginning of the pulse to the reception of the echo is 833nS. Determine the velocity factor. Solution: d = (v x t)/2 v = kxc k =2d/ct = 2(100) / (300x10 6m/s x 833nS) = 200/249.9 = 0.8 16. Determine the wavelength for electromagnetic waves in free space with the following frequencies: 5kHz, 50kHz, and 500kHz and 5Mhz. Solution: a. λ = c/f = (300x10 6 m/s)/ 5 kHz = 60,000 m b.
λ = c/f = (300x10 6 m/s)/ 50 kHz = 6,000 m c. λ = c/f = (300x10 6 m/s)/ 500 kHz = 600 m d. λ = c/f = (300x10 6 m/s)/ 5 MHz = 60 m 17. Determine the frequencies for electromagnetic waves in free space with the following wavelengths: 5cm, 50cm, 5m and 50m. Solution: a. f = c/λ = (300x106 m/s)/ 5cm = 6GHz b. f = c/λ = (300x106 m/s)/ 50cm = 600MHz c. f = c/λ = (300x106 m/s)/ 5m = 60MHz d. f = c/λ = (300x106 m/s)/ 50m = 6MHz 18. Determine the characteristic impedance for an air-dielectric transmission line with D/r ratio of 6.8. Solution:
Zo = 276 log D/r = 276 log 6.8 = 229.78 Ω
Ei =0.4V and reflected voltage E r = 0.002V. Solution:
19. Determine the characteristic impedance for an air-filled concentric transmission line with D/d ratio of 6.
Ґ = Er / Ei or Ir / Ii = 0.002V/0.4V = 0.005V
Solution: Zo = 138/ (√E r) log D/d = 138/ √2.23 log 6 = 71.9 Ω 20. Determine the characteristic impedance for coaxial cable with inductance L = 0.15 µH/ft and conductance 20 pF/ft. Solution: Zo = √L/C = √ (0.15 µH/ft)/ (20 pF/ft) = 86.60Ω 21. For a given length of coaxial cable with distributed capacitance C = 24.15pH/ft and distributed inductance L = 483.12 nH/m, determine the velocity factor and velocity of propagation. Solution: Vp =1/√LC = 1/√ (24.15 pH/ft) (483.12 nH/m) = 292.76x10 6 m/s Vf = Vp/c = (292.76x10 6 m/s)/ (300X10 6m/s) = 0.98 22. Determine the reflection coefficient for transmission line with incident voltage
23. Determine the standing wave ratio for the transmission line in #22. Solution: SWR = Vmax/Vmin = (Ei + Er) / (E i - Er) = (0.4V + 0.002V)/ (0.4V – 0.002V) = (0.402V / 0.398V) = 1.01 24. Determine the SWR for a transmission line with maximum voltage standing wave amplitude Vmax = 8V and minimum voltage standing wave amplitude Vmin =0.8V. Solution: SWR = Vmax/Vmin = 8V/ 0.8V = 10 25. Determine the SWR for a 50-Ω transmission line that is terminated in a load resistance Z L = 60Ω. Solution: SWR = Zo/ZL or ZL/Zo = 60 Ω/50 Ω = 1.2
26. Determine the SWR for a 60-Ω transmission line that is terminated in a load resistance Z L = 50Ω. Solution: SWR = Zo/ZL or ZL/Zo = 60 Ω/50 Ω = 1.2 27. Determine the characteristic impedance for a quarter-wavelength transformer that is used to match a section of 50Ω transmission line to a 60Ω resistive load. Solution: Z’o = √ZoZL = √ (50Ω) (60Ω) =54.77Ω
