PHYS/ENGS 495 Microwave Engineering and Devices Physics, Homework Problems
May 2011
1. Electrical length is the length of a transmission line expressed as a number of wavelengths.
Clearly, electrical length depends on wavelength. Wavelength is related to propagation speed
through v=f, and is reduced by the presence of a dielectric by v  c /  r , where r is the
dielectric constant which you can look up. c=2.998x108 m/s is the speed of light in vacuum,
called the phase velocity. Calculate the electrical length of
a. A 48 inch long PTFE filled coaxial cable, at 1 GHz
b. The same cable with the Teflon removed, at 1 GHz
c. A 12 inch long metal rod, at 1 GHz
d. The same 12 inch long metal rod, at 1 THz
e. A 20 meter tall aerial at 580 KHz
f. Compare the phase velocity for the air filled cable of step b with the drift velocity of
electrons in a 1 Amp current. Any surprises? The center conductor is a copper wire with
a radius of 0.5 mm. See a General Physics textbook for a discussion of drift velocity.
2. Resonator. A 2.00 inch long copper rod is attached to a grounded copper wall of a
rectangular copper cavity. The interior of the cavity is filled with PTFE. What are the 1st three
resonant frequencies? You build this resonant cavity to specification and find that the resonant
frequency is lower than expected. Why?
3. Resonant Transfer Function. Graph [Vout]2
and the phase of Vout versus frequency, f, for 4
GHz < f < 6 GHz. Use Maple or equivalent to
your full advantage. Adjust the sweep width if 2
GHz doesn’t give a good view of the peak.
4. Matrix Methods. Repeat the transmission line with capacitive shunt example from class
using an electrical line length of 180o. What rule and limits do you find?
PHYS/ENGS 495 Microwave Engineering and Devices Physics, Homework Problems
May 2011
5. Decibel Exercises.
a. Convert the following decibels into Watts: -43 dBm, +3 dBW.
b. Convert the following powers into dBm: 4.30x10-3 mW, 12.0 Watts.
c. A power amplifier has a maximum linear input of +20 dBm. The amplifier has a gain
of 28 dB. What is the maximum linear output of this amplifier in Watts?
d. The 3rd order intermodulation distortion (IMD) spur of the previous amplifier is found
to be -3 dBm when a +20 dBm signal is incident on the amplifier. What is the IMD
relative to the output signal in dBc?
6. A test technician is measuring the S-parameters of a directional coupler. The specification
limit on the VSWR of the through port is 1.3. What is the specified limit on the return loss, S11,
in decibels and the magnitude |S11|. If 1 Watt enters this port, how much is reflected?
7. Q measurement. A particular resonator is swept by a scalar network analyzer giving the
following results: center frequency=5.82 GHz, bandwidth=12.1 MHz. S11=-1.4 dB, S22=-2.4
dB. What is the unloaded Q? If 1 Watt is incident on Port 1, how much power is dissipated in
the resonator? (There is a “plugger-chugger” for this one.) Use Origin to make a graph of S21
(dB) versus frequency around the resonance.
8. Transmission line input impedance. A lossless line of impedance Z0 is terminated in a load
impedance ZL at a distance x=ℓ from the line input. The usual convention is to say that the load
is at x=0 and the input to the line is at x=-ℓ. The impedance seen by looking down such a line
varies with the location within the line as (White, Equation 4.14-2)
Z ( x)  Z 0
Z L  jZ 0 tan x
Z 0  jZ L tan x
a. If the line is commensurate with half a wavelength (or a multiple thereof) what
impedance is seen by a wave entering the line (e.g. the input impedance)?
b. If the line is commensurate with a quarter wavelength (or n/4 where n=1,3,5…) what
is the input impedance?
c. If the line is commensurate with an eighth wavelength, what is the input impedance?
d. Now consider a line terminated in a short circuit (ZL=0). Determine the nature of the
input impedance of the line (capacitive, inductive, real, infinite or zero) if the line length
is (i) </4, (ii) =/4, (iii) between /4 and /2, (iv) = /2.
PHYS/ENGS 495 Microwave Engineering and Devices Physics, Homework Problems
May 2011
9. Transmission line design. What is the trace width of a 50  transmission line on 1 oz.,
0.010” thick Rogers 3003 Duroid? How many millimeters long is the line if it is 2 long at 10
GHz? For what types of devices would this particular board prove to be a good choice?
10. Coaxial Cable.
a. Find the capacitance per unit length for a coaxial cable with inner conductor radius R 1,
outer conductor radius, R2, and non-magnetic dielectric constant, r.
b. Find the inductance per unit length for the same cable.
c. Use these results to find the characteristic impedance of 0.141 semi-rigid cable.
11. Reflection problems from Rizzi, Microwave Engineering, Passive Circuits.
a. A reflection coefficient can be computed when a reflected wave is measured. But the
reflection coefficient does not yield unique values for the load or line impedance. To see
this, consider a transmission line with impedance Z0. If the line is terminated in a 50 
load and exactly one ninth of the incident power is reflected, what are all the possible
values of Z0? (Remember power goes as voltage squared.) (Adapted from Rizzi Problem
3-2)
b. A load impedance of ZL=30  - j75  is connected to a 75  lossless line. Calculate
L and in if the line is 0.15 long. Calculate the VSWR. (Rizzi, Problem 3-11)
c. The reflection coefficient at the input of a 50  air-insulated lossless line is 0.6+j0.8 at
1000 MHz. What type of circuit element is connected at the load end if the line is 8.0 cm
long? What is its value? (Rizzi, Problem 3-13)
d. (For practice only) A 300  line is terminated in a pure inductance. The load
reflection coefficient is 1.050o at 1500 MHz. Calculate the inductance in nH. (Rizzi,
Problem 3-16)
12. Low Pass Chebyshev Filter Design. Design a 4 element low pass Chebyshev filter with
f1=1 GHz and Lr=0.1 dB using tabulated g-values. Enter the schematic into Genesys and
simulate it using lossless elements. Does the simulation produce Lr=0.1 dB? What is the
rejection at 2 GHz? Do S11 and S22 appear to satisfy unitary gain?
13. Passive Device Noise Figure.
a. Use Maple to graph the noise figure and the noise factor of a 10 dB attenuator from 0
K up to 500 K.
b. What is the noise figure of the resonator in Problem 7 at the resonant frequency at
room temperature?
c. How much noise power does a 100  resistor at 290 K generate over a bandwidth of 1
MHz?
PHYS/ENGS 495 Microwave Engineering and Devices Physics, Homework Problems
May 2011
14. Cascaded Noise Figure. The subsystem below is operated at room temperature. Make a
graph of the subsystem noise figure using the Friis equation for attenuation values ranging from
0 dB to 40 dB. If the subsystem noise figure is not to exceed 3.0 dB, what is the largest
attenuator value that can be used?
15. Cascaded IP3: the effect of passives. In the subsystem of the previous problem, A=5 dB,
and a -30 dBm signal is incident on the filter. If the IP3i of the LNA is -20 dBm, what are the
IP3i and IP3o of the cascade?
16. Cascaded amplifier IP3. Two single stage LNAs with the following specs are cascaded.
LNA #1: 12 dB gain, 0 dBm IP3i. LNA#2: 16 dB gain, -15 dBm IP3i. Which LNA should be
positioned first in order to have the highest IP3i? What are the IP3i and IP3o?