Laboratory 2: Standing Waves and Waveguides

ECE320H1F: Fields and Waves
Laboratory 2: Standing Waves and Waveguides
1 Objective
Two waveguiding structures implemented using planar transmission lines are introduced. The structures
are characterized using practical field measurements. Using measured standing wave patterns, properties for both the structures, as well as the loads used to terminated the structures, are experimentally
determined.
2 References
[1] F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics, 7th ed. Upper Saddle
River, NJ: Pearson, 2015.
[2] ——, Fundamentals of Applied Electromagnetics, 7th ed. Upper Saddle River, NJ: Pearson, 2015,
ch. 1, pp. 60–65.
[3] Agilent Technologies. (2005) Network analyzer basics. [Online]. Available: http://www.keysight.
com/upload/cmc_upload/All/BTB_Network_2005-1.pdf
3 Background
3.1 Experiment Setup
The following equipment and components are required to complete this lab:
∙ Vector network analyzer (VNA)
∙ Calibration kit
∙ Two SMA cables
∙ Linear translator station
∙ Ruler
∙ Computer running MATLAB or Microsoft Excel, for recording data
ECE320H1F: Fields and Waves Laboratory 2
∙ One short circuit load
∙ One unknown load
The main component in this lab is a linear translator, which is shown in Fig. 1. It consists of a
moveable stage, which is suspended on two rods above a printed circuit board (PCB). On the PCB,
there are two waveguiding structures. The first is a microstrip line [2], and the second is a substrate
integrated waveguide (SIW), which is a specialized implementation of the rectangular waveguide (used
in a later experiment). The translation stage is moved back and forth manually. On the stage, there
are two coaxial probes (one over each of the two waveguiding structures), which independently measure
the electric fields produced by their respective structures. By measuring these fields as a function of
position, useful information about the structures themselves, as well as the loads terminating them,
can be determined.
Figure 1 Linear translator.
In this laboratory, a vector network analyzer (VNA) is used to measure the transmission coefficient from
port 1 to port 2. The transmission coefficient is directly proportional to the voltage, and can thus be
used to reconstruct the voltage standing wave pattern produced by each of the waveguiding structures.
Effectively, the VNA is used as a tuned receiver, whereby signals spanning a specified frequency range
are transmitted from port 1 and received via port 2. That is, port 1 is used to excite the structure of
interest, while port 2 is connected to the coaxial probe in order to measure the fields picked up by the
probe.
3.2 Transmission Line Measurement and Voltage Standing Waves
For both coaxial probes mounted above the PCB, there is a short segment of wire extending from
the coaxial connector. The short wire effectively forms a small monopole antenna, which is sensitive
to the electric field oriented on the axis of the antenna (here it is vertically-oriented). When placed
in proximity to the microstrip line or the SIW, this probe senses the vertically-oriented fringing field
produced by the respective structures in the air region above them. Note that if the microstrip line did
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ECE320H1F: Fields and Waves Laboratory 2
not naturally produce any fringing fields, or the SIW did not have a slot cut along its length, this type
of measurement would not be possible. Since the electric field is proportional to the voltage produced
within the waveguiding structures, the measurement of this field can be used to determine the voltage
standing wave (VSW) pattern along the line.
The voltage on a transmission line is a combination of a forward traveling wave 𝑉
+(𝑧) = 𝑉
+
0
e
−𝑗𝛽𝑧 and
a backward traveling wave 𝑉
−(𝑧) = 𝑉
−
0
e
+𝑗𝛽𝑧. Using the relations 𝑉
+
0 = Γ𝑉
−
0
and Γ = |Γ|e
𝑗𝜃, the
amplitude of the voltage on the line can be written as:
|𝑣(𝑧)| = |𝑉
+(𝑧) + 𝑉
−(𝑧)|
= |𝑉
+
0
e
−𝑗𝛽𝑧 + 𝑉
−
0
e
+𝑗𝛽𝑧|
= |𝑉
+
0
||1 + Γe+𝑗2𝛽𝑧|
= |𝑉
+
0
||1 + |Γ|e
𝑗(2𝛽𝑧+𝜃)
| . (1)
z
V (z)
V1(0)
0
1
|Γ|
|Γ|
−dmin
Figure 2 Voltage standing wave on a transmission line.
As shown in Fig. 2 the amplitude of 𝑣(𝑧) varies between a maximum value of:
|𝑉max| = |𝑉
+
0
|
(︁
1 + |Γ|
)︁
, (2)
located at 2𝛽𝑧max + 𝜃 = 2𝑛𝜋 (where e
𝑗(2𝛽𝑧+𝜃) = +1), and a minimum value of:
|𝑉min| = |𝑉
+
0
|
(︁
1 − |Γ|
)︁
, (3)
located at 2𝛽𝑧min + 𝜃 = (2𝑛 + 1) 𝜋 (where e
𝑗(2𝛽𝑧+𝜃) = −1). The voltage standing wave ratio (VSWR)
is defined as:
VSWR = |𝑉max|
|𝑉min|
=
1 + |Γ|
1 − |Γ|
. (4)
Rearranging (4) for |Γ|, it found that:
|Γ| =
VSWR − 1
VSWR + 1 . (5)
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ECE320H1F: Fields and Waves Laboratory 2
That is, the magnitude of Γ is easily determined using the VSWR. To determine 𝜃 (the phase of Γ) at
the load, as shown in Fig. 2, let the first voltage minimum (i.e. 𝑛 = 0) be located at 𝑧min = −𝑑min.
Then:
𝜃 = 𝜋 − 2𝛽𝑧min
= 𝜋 + 2𝛽𝑑min
= 𝜋 +
4𝜋𝑑min
𝜆
. (6)
Thus, the complex reflection coefficient Γ = |Γ|e
𝑗𝜃 at the input port of a transmission line can be determined by measuring the VSWR, as well as the position of a voltage minimum. The input impedance
can then be computed from the Γ(𝑧) to 𝑍in(𝑧) relationship.
3.3 Microstrip Line
A microstrip line consists of a narrow strip of conductor suspended over a large conducting ground plane,
usually by a dielectric substrate, as shown in Fig. 3a. The geometry of the microstrip line determines
its characteristic impedance. By varying the strip width 𝑤, substrate height 𝑕, and substrate dielectric
constant 𝜀𝑟, a wide range of characteristic impedances can be synthesized.
z
y
x
h
w
Conducting
Strip (µc, σc)
Dielectric
Insulator
(ε, µ, σ)
Conducting ground plane (µc, σc)
(a)
y
x
z z
E
B
(b)
Figure 3 Microstrip line: (a) longitudinal view, and (b) cross-sectional view with 𝐸 and 𝐵 field lines.
When the line is excited by placing the strip at an electric potential relative the ground plane, the
electric and magnetic field lines shown in Fig. 3b result. In addition to the transverse electromagnetic
(TEM) (or “parallel plate” type) field distribution created directly beneath the strip, fringing fields
are also created, most of which propagate within the air region above the transmission line. Since the
fields are not entirely confined to the air region nor the dielectric region, the wave velocity of the line
is somewhere between that of the speed of light in air (𝑐0 = 3 × 108 m/s) and that in the substrate
(𝑐0/
√
𝜀𝑟). As a result, the wave velocity is generally defined using an effective dielectric constant 𝜀eff,
and the corresponding speed of light along the microstrip line is 𝑐0/
√
𝜀eff. The effective dielectric
constant is dependent on the geometry of the line as well as the substrate parameters. It has been the
focus of many studies over the years, and has been found to be best described using a combination
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ECE320H1F: Fields and Waves Laboratory 2
of conformal mapping and empirical formulas [2]. A common expression for the effective dielectric
constant is:
𝜀eff =
𝜀𝑟 + 1
2
+
(︂
𝜀𝑟 − 1
2
)︂(︂1 +
10
𝑠
)︂−𝑥𝑦
(7)
where 𝑠 = 𝑤/𝑕 and:
𝑥 = 0.56 (︂
𝜀𝑟 − 0.9
𝜀𝑟 + 3 )︂0.05
(8a)
𝑦 = 1 + 0.02 ln (︂
𝑠
4 + 3.7 × 10−4
𝑠
2
𝑠
4 + 0.43 )︂
+ 0.05 ln (︀
1 + 1.7 × 10−4
𝑠
3
)︀
(8b)
The characteristic impedance of a microstrip line can be found using empirical formulas as well, such
as:
𝑍0 =
60
√
𝜀eff
ln (︃
6 + (2𝜋 − 6)e−𝑡
𝑠
+
√︂
1 +
4
𝑠
2
)︃
(9)
where
𝑡 =
(︂
30.67
𝑠
)︂0.75
(10)
The microstrip line for this laboratory exercise has been fabricated on a substrate known as FR-4,
which is a commonly used material for making printed circuit boards. As an interesting historical note,
FR-4 was originally developed for aerospace applications, and is so-called because it is flame retardant
(FR). FR-4 has a nominal dielectric constant of 𝜀𝑟 = 4.4. The line on the PCB in this lab has been
designed on a substrate with a thickness of 𝑕 = 1.5 mm, so that it is matched to the system impedance
of the VNA, which is 𝑍0 = 50 Ω.
4 Microstrip Line Standing Wave Measurements
It is very important to wear a static bracelet when operating the VNA. The VNA is a very
sensitive piece of equipment, and can easily be damaged by electrostatic discharge (ESD).
When connecting loads and/or calibration standards, use the provided torque wrench.
If no torque wrench is provided, tighten the loads or standards finger tight. Do NOT
overtighten the loads or standards. The calibration kit (seen in Fig. 4) has protective caps
on all standards; please ensure that the caps are replaced after the calibration procedure is
completed. If any caps are missing, notify the laboratory teaching assistant (TA) and/or
the laboratory manager.
Measurements in this experiment are conducted using a vector network analyzer, which is an important
piece of test equipment for microwave measurements. As the name suggests, it is used for making
measurements of microwave networks, which are circuits with multiple ports (often two). For example,
a network analyzer can measure the impedance matrix of a two-port network by injecting suitable test
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ECE320H1F: Fields and Waves Laboratory 2
Figure 4 The calibration kit used for ECE320 laboratories. Note the protective caps on all the
standards. Please replace the caps after the calibration procedure is completed. If any caps are
missing, please notify the teaching assistant and/or laboratory manager.
signals into each port. Usually, however, we use network analyzers to measure scattering parameters,
or more generally, reflection and transmission coefficients of a microwave network. All measurements
of microwave networks are based on measuring waves reflected from or travelling through microwave
networks [3]. In this experiment, we will use the transmission coefficient measured from the input port
on the translator to the probe as a way of measuring the standing wave pattern produced along the
microstrip line.
The design frequency for the microstrip line is 𝑓 = 1 GHz. Recall that a free-space wavelength 𝜆0
cannot be assumed for the microstrip line configuration.
Notation: In the following instructions, [Command] refers to a softkey on the VNA front panel while
Command refers to an option appearing on the VNA touchscreen (or monitor if it is hooked up to the
VNA), and < Command > refers to something that is typed in using the VNA keypad (or keyboard if
it is hooked up to the VNA).
Before measurements are conducted, the VNA must be properly calibrated. In this way, when the two
cables from the network analyzer are connected together, a transmission coefficient of unity with zero
phase (𝑇 = 𝑆21 = 1∠0
∘ will be measured (|𝑆21| = 0 dB; ∠𝑆21 = 0∘
).
4.1 VNA Setup and Calibration Procedure
1. Set the frequency span over which the measurements and calibration will be performed. On the
VNA front panel, press [Start] → < 800 MHz >, and [Stop] → < 1.2 GHz >.
2. Set the number of points to be measured. Press [Sweep Setup] → Points → < 801 > → Return.
3. Select the calibration kit. Press [Cal] → Cal Kit → 85521A.
4. Initialize the calibration procedure. Press [Cal] → Calibrate → 2-Port Cal.
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ECE320H1F: Fields and Waves Laboratory 2
5. Select Reflection in the menu. Begin with Port 1 by connecting the cable attached to port 1 of
the VNA to the ‘Open’ calibration standard. Then press the corresponding Port1 Open option
on the VNA calibration menu. Repeat for the ‘Short’ and ‘Load’ standards. Then repeat the
three standard measurements on Port 2. After the six reflection measurements (three standards
on each of Ports 1 and 2) are complete, select Return in the VNA menu.
6. Now select Transmission. Connect Port 1 of the VNA to Port 2 using the ‘Through’ calibration
standard. Select Port 1-2 Thru in the VNA menu, then select Return.
7. Make sure you select Done at the bottom of the calibration menu after completing the
six reflection measurements and the thru measurement. Otherwise the calibration
will not be complete, and hence will not be applied. If the Done menu item is grayed
out, it is because one or more of the calibration steps has been skipped.
8. Check your calibration by leaving the SMA cables connected end-to-end with the through, and
ensure that:
(a) The input reflection coefficient is very low (less than −20 dB) across the entire frequency
band.
(b) The transmission coefficient is nearly unity (0 dB) across the entire frequency band.
4.2 Measurement of Microstrip Line Characteristics
In this portion of the lab, the linear positioner will be used to measure the fringing fields produced
above the microstrip line, the magnitude of which are directly proportional to the voltage along the
line. That is, the positioner will be used to plot the voltage standing wave as a function of position.
1. Measure the width 𝑤 of the transmission line. Determine the effective dielectric constant 𝜀eff of
the line using (7) (given in Section 3.3). Then calculate the characteristic impedance 𝑍0 using (9)
to confirm it is near 50 Ω. Finally, determine the phase velocity 𝑣p of the line, assuming it is
constant for all frequencies.
2. Attach the SMA cables and connect port 1 of the network analyzer to the input port of the
microstrip line, feeding the SMA cable through the small hole at the end of the linear positioner.
3. Connect port 2 of the network analyzer to the coaxial probe on the microstrip side of the probe
stage.
4. Connect a short circuit to the end of the microstrip line.
5. Manually move the stage so that the coaxial probe used to measure the microstrip line is positioned
above the load end of the microstrip line.
6. Press [Meas] → 𝑆21 → [Format] → Log Mag.
7. Press [MARKER] → < 1 GHz >. Use [Scale] → Autoscale to see the curve.
8. At this point, the transmission coefficient (𝑆21) measured by the network analyzer should be very
small (less than −60 dB).
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ECE320H1F: Fields and Waves Laboratory 2
9. Move the stage away from the load until the first voltage minima is encountered. This is where
the measurements for this part of the laboratory begin.
10. Manually move the stage in 5 mm increments away from the load, and plot the standing wave
pattern over a minimum one one full wavelength at 1 GHz. Use the ruler supplied to make precise
measurements of the probe location relative to the short-circuited end of the PCB.
11. Locate the maxima and minima to obtain more precise locations for the field readings at these
positions.
12. Compute the VSWR and compare to that of a short circuit, recalling that the field values reported
by the network analyzer are on a power-dB scale.
13. Compute the guide wavelength and corresponding effective dielectric constant of the microstrip
line using the measurement data. Compare these to theoretical expectations, and comment on
any differences.
4.3 Using Standing Wave Patterns for Load Calculations
In this portion of the lab, the standing wave pattern produced when the microstrip line is terminated
in an unknown load is measured. The value of the unknown load can then be determined using
analytical/Smith Chart techniques.
1. Disconnect the short circuit used in the previous experiment and connect the unknown load to
the end of the microstrip line.
2. Re-position the probe so that it is measuring the field directly above the PCB edge where the
load is located.
3. Move the stage in 5 mm increments away from the load, and plot the standing wave pattern over
a minimum one full wavelength at 1 GHz. Use the ruler supplied to make precise measurements
of the probe location relative to the short-circuited end of the PCB.
4. Locate the maxima and minima to obtain more precise locations for the field readings at these
positions.
5. Using the techniques of Section 3.2, compute the impedance of the load based on the standing
wave pattern that has been measured. Recall that the field values reported by the network
analyzer are on a power-dB scale. The maximum and minimum magnitudes are used to find
the VSWR (and hence, the magnitude of the impedance) and 𝑑min yields the phase of the load
impedance.
6. Disconnect the SMA connectors from the microstrip portion of the translator.
7. Connect the unknown load to port 1 of the network analyzer and measure its impedance.
(a) With the unknown load attached, press [Meas] → 𝑆11 → [Format] → Smith → 𝑅 + 𝑗𝑋.
(b) Press [Marker] → < 1 GHz > and record the impedance from by the network analyzer. Compare to your computed result and discuss any discrepancies.
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ECE320H1F: Fields and Waves Laboratory 2
Make sure you speak to you TA before leaving the lab, so that you
can be assessed your oral grade for this experiment!
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