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Homework 5 - Smith Chart Computations
Problem 1.
A 50ohm lossless transmission line is terminated in a complex load impedance
of 40 + j70ohms. Using a Smith Chart find the following. Plot the normalized
impedance, determine the normalized admittance, determine the actual admittance,
calculate VSWR, and the reflection coefficient.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 2.
A 50ohm lossless transmission line is terminated in a real load impedance
RL = 10ohms. Use a smith chart to find the input impedance Zin
for the following lengths, 0.1 wavelengths, 0.25 wavelengths, 0.375 wavelengths,
0.5 wavelengths, and 7.6 wavelengths.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 3.
A 75ohm loss-less transmission line is terminated in a load impedance equal to
105 - j150. The length of the line is 0.35 lambdas. Using a Smith Chart calculate
the input impedance, the VSWR, and the reflection coefficient at the load.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 4.
An antenna is modeled by a 40ohm resistor in parallel with a capacitor with
Xc = -j25. The antenna is fed by a lossless 50ohm signal line 0.15 wavelengths
long. Use a Smith Chart to find the input impedance to the transmission line.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 5.
A 50ohm lossless T-Line has a voltage standing wave ratio, VSWR = 3. A voltage
minimum is located 3.15 wavelengths from the load. Use a Smith Chart to
find the load impedance.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 6.
A 50ohm coax line is terminated in a short circuit. Using a Smith Chart find the
line length in wavelengths (lambda) to produce an input impedance, Zin = 0 - j36ohms.
Repeat for Zin = j100ohms. Determine the length of an open circuit line for Zin = j100ohms.
{ Solution } -
{ Smith Chart Plot }
Problem 7.
An antenna has an input impedance equal to 25 + j40ohms and a frequency at 100MHz.
The antenna is connected to a 50ohm coaxial cable with a short circuit stub to match
the antenna. The velocity of propagation for the cable is 200 million meters per
second. Find the lengths of the stub, and the distance of the stub from the antenna.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 8.
A 50ohm transmission line has an electrical length of 45 degrees and is terminated
in a load, ZL = 50 + j50. Using impedances, find the input impedance,
the load reflection coefficient, and the VSWR.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 9.
Repeat problem 8 using admittances. Convert ZL to an admittance on a Smith Chart and find Yin.
{ Solution } -
{ Smith Chart Plot }
Problem 10.
Determine the input admittance of a short-circuited transmission line having a length
of 1/8 wavelength and characteristic impedance equal to 50ohms. Determine the input
admittance when the line is open-circuited.
{ Solution } -
{ Smith Chart Plot }
Problem 11.
On an impedance smith chart, plot the following as a function of frequency. That is,
frequency varies from zero to infinite hertz, show f = 0 and f = infinite.
a) A resister Rs = 1ohm normalized, and an inductor = +XL in series
b) A resister Rs = 1ohm normalized, and a capacitor = -XL in series
a) A resister Rs = 1ohm normalized, and an inductor and capacitor in series
a) An inductor and capacitor in series
{ Solution } -
{ Smith Chart Plot }
Problem 12.
On an admittance smith chart, plot the following as a function of frequency.
a) A conductance Gp = 1S normalized, and an inductor in parallel
b) A conductance Gp = 1S normalized, and a capacitor in parallel
c) A conductance Gp = 1S normalized, and a capacitor in parallel and an inductor in parallel
d) An inductor in parallel with a capacitor
{ Solution } -
{ Smith Chart Plot }
Problem 13.
On an impedance smith chart, plot six curves corresponding to a value
for Q equal to 1, 2, and 5. Assume a series circuit.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 14.
A transmission line is used to match two impedances, a source and a load. At the
source the reflection coefficient is equal to 0.66 angle - 80deg. At the load the
reflection coefficient is equal to 0.66 at an angle of minus 110 degrees. What is
the electrical length of the transmission line? Note, match the source to the
complex conjugate of the load.
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Problem 15.
A transmission line has a reflection coefficient at the load of 0.71 angle -125
degrees. The frequency is 500MHz. Find the following: Z load, normalized z load,
return loss, mismatch loss, VSWR, and an equivalent circuit of the load. Determine
values using the Smith Chart and verify with calculations. Note, Zo = 50ohms.
{ Smith Chart Plot } -
{ Corrections }
Problem 16.
A 50ohm transmission line has a load impedance Z = 10 - j25. Find the following:
a) z normalized, y normalized, and Y un-normalized
b) The return loss, mismatch loss, and VSWR
c) A series equivalent circuit at 1GHz
d) A parallel equivalent circuit at 1GHz
e) Find the Q for the series equivalent circuit and the parallel equivalent circuit
{ Solution } -
{ Smith Chart Plot } -
{ Corrections }
Engineering key words: smith chart plotting, graphs, impedance chart, admittance chart,
lossless transmission line, line-length, wavelength, lambda, inverse frequency, normalize,
voltage standing wave ratio, VSWR, T-Line, reflection coefficient, computations, calculated,
modeling and simulation, antenna, coaxial cable, twin pair, twisted pair, open circuit,
shorted, short circuit, load, stub, electrical length, hertz, time, conductance, complex
notation, equivalent circuit.
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