UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
The Edward S. Rogers Sr. Department of Electrical and Computer Engineering
ECE320H1F / ECE357H1F
EXPERIMENT 3: STANDING WAVES AND WAVEGUIDES
1. Introduction
The purpose of this laboratory is to introduce you to a few types of planar transmission lines and
waveguides and to illustrate how practical measurements of the fields produced by these
transmission lines can be carried out. Using knowledge of transmission lines, you will physically
measure the standing wave pattern produced by a terminated transmission line and waveguide.
The lab will also introduce wave propagation characteristics within waveguides, which are
discussed later in the course.
You will need the following equipment and components to complete this lab:
• Agilent vector network analyzer
• Calibration kit (short, open, load, through)
• 2 SMA cables
• Lab 3 linear translator station
• Ruler
• Computer running MATLAB or Microsoft Excel, for recording data
• One unknown load
The main component in this lab is a linear translator, which is shown in Figure 1. It consists of a
moveable stage, which is suspended on three rods above a printed circuit board (PCB). The stage
is moved back and forth using a stepper motor, with controls situated on a small board at one end
of the positioner. On the PCB, there are two circuits. The first is a microstrip line[1], and the
second is a special structure called substrate integrated waveguide, which is a special form of
rectangular waveguide [2]. On the translation stage, there are two coaxial probes, which measure
the electric fields produced by these two structures independently. 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.
In this laboratory, the network analyzer is used as a tuned receiver whereby signals are
transmitted out port 1 and received via port 2, for a set of different frequency points. Hence,
port 1 is used to feed a signal into the input of the structure of interest, while port 2 is connected
to the coaxial probe in order to measure the field picked up by the probe. Essentially, the
network analyzer is used to measure the transmission coefficient from port 1 to port 2, which will
be directly proportional to the voltage standing wave pattern produced with each of the PCB
structures.
Fig. 1 Linear Translator
2. References
1. F. T. Ulaby, E. Michielssen, and U. Ravaioli, Fundamentals of Applied
Electromagnetics, 6th ed. Upper Saddle River, NJ: Prentice-Hall, 2010, pp. 61-64.
2. ——, Fundamentals of Applied Electromagnetics, 6th ed. Upper Saddle River, NJ:
Prentice-Hall, 2010, ch. 8, pp. 381–390.
3. H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a “laminated
waveguide”,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 12, pp. 2438–2443,
Dec. 1998.
4. K. Wu, “Integration and interconnect techniques of planar and non-planar structures
for microwave and millimeter-wave circuits–current status and future trend,” in AsiaPacific Microwave Conf., Taipei, Taiwan, R.O.C., Dec. 3–6, 2001, pp. 411–415.
5. F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated
waveguide,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 1, pp. 66–73, Jan.
2005.
3. Background
3.1 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 Figure 2(a). The geometry of the
microstrip line impacts its characteristic impedance. By varying the strip width w, substrate
height h, and substrate dielectric constant ε r , one can realize a wide range of characteristic
impedances for the application at hand.
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 Figure 2(b) result. Clearly, in addition to the “parallel
plate” type field structure created directly beneath the strip, fringing fields are also created, the
bulk of which propagate within the air region above the transmission line. Since the fields are not
confined to purely the air region or the dielectric region, the wave velocity of the line is
somewhere between that of the speed of light in air ( c = 3×108 m/s) and that in the substrate (
c / ε r ). The wave velocity is usually described using an effective dielectric constant ε eff , so that
the speed of
Fig. 2 Microstrip line (courtesy of Ulaby, Fig. 2-10)
light along the microstrip line is c / ε eff . The effective dielectric constant is dependent on the
geometry of the line and the substrate parameters, just as the characteristic impedance is. There
relationships are well known and best captured in empirical formulas [1]. The effective dielectric
constant is found using
ε eff =
ε r + 1 ⎛ ε r − 1 ⎞⎛
+ ⎜
⎝
2
2
10 ⎞
⎟⎜1 + ⎟
s ⎠
⎠⎝
− xy
where s=w/h and
⎡ ε − 0.9 ⎤
x = 0.56⎢ r
⎥
⎣ ε r + 3 ⎦
0.05
⎛ s 4 + 3.7 − 10 −4 s 2
y = 1 + 0.02 ln⎜⎜
s 4 + 0.43
⎝
⎞
⎟ + 0.05 ln(1 + 1.7 + 10 −4 s 3 )
⎟
⎠
The characteristic impedance of a microstrip line can be found using empircal formulas as well,
given by
Z0 =
60
ε eff
⎛ 30.67 ⎞
t = ⎜
⎟
⎝ s ⎠
⎛ 6 + (2π − 6)e −t
4
ln⎜
+ 1+ 2
⎜
s
s
⎝
⎞
⎟
⎟
⎠
2
The line in this lab has been fabricated on a substrate called FR-4, which is a commonly
used material for making printed circuit boards. FR-4 has a dielectric constant of 4:4 nominally.
The line on the PCB in this lab has been designed on a h = 1.5 mm substrate so that it is matched
to the system impedance of the network analyzer instrument, which is Z0 = 50 .
If you look carefully at the coaxial probe above the microstrip line, you will see that it consists of
a short segment of wire extending from the coaxial connector. This little wire forms a small
dipole antenna which is sensitive to a vertically-oriented electric field. When placed in proximity
to the microstrip line, this probe senses the vertically-oriented fringing field produced by the
microstrip line in the air region above the line. Note that if the microstrip line did not produce
any fringing fields, this type of measurement would not be possible. Since the electric field is
proportional to the voltage produced between the two conductors, the measurement of this field
can be used to determine the voltage standing wave pattern along the line.
3.1.1 Transmission Line Measurement and Voltage Standing Waves The voltage on a transmission line is a combination of a forward travelling wave e V1 (0) and a backward travelling wave e jβzV2 (0) . The amplitude of the voltage on the line is then given by − jβz
v( z ) = V1 ( z ) + V2 ( z ) = e − jβzV1 (0) + e jβzV2 (0) = V1 (0) 1 + e + j 2 βz Γ(0)
As shown in Fig. 3 the amplitude v(z) varies between a maximum value of V
max
= V1 (0) {1 + Γ(0) }
located at ∠Γ(0) + 2βz max = 2nπ
and a minimum value of
V
min
= V1 (0) {1 − Γ(0) }
located at ∠Γ(0) + 2βz min = (2n + 1)π
The ratio
V
V
max
min
=
1+ Γ
1− Γ
= VSWR
is known as the voltage standing wave ratio. Note that Γ is a constant along the line and we find Γ=
VSWR − 1
VSWR + 1
To determine the phase of Γ(0) at the load as shown in Fig. 7, let the first voltage minimum be located at zmin = −d min . Thus ∠Γ(0) = π − 2βz min = π + 2βd min = π +
4πd min
λ
Thus by measuring the VSWR and the position of a voltage minimum, the complex reflection coefficient Γ(0) at the input port of a transmission line can be determined. The input impedance can be derived from the Γ(z ) to Zin (z) transformation. Fig. 3 Voltage standing wave on a transmission line 3.2 Waveguides and Substrate Integrated Waveguide (SIW)
3.2.1 Rectangular Waveguides
The rectangular waveguide is a metallic ’pipe’ of rectangular cross section that can guide
electromagnetic waves along the longitudinal z-axis, as shown in Figure 4. The advantages of the
rectangular waveguide (compared to say planar transmission lines) are that it is a low-loss
medium and can propagate high power signals. Due to these advantages, it is still in use for
applications such as satellite communications and for making base-station channel filters.
Fig. 4 Rectangular waveguide
A rectangular waveguide supports an infinite number of transverse electric (TE) and transverse
magnetic (TM) modes for which the longitudinal electric and magnetic fields vanish,
respectively. In this laboratory, we will experiment with the fundamental mode (lowest cutoff
frequency) which is the TE10 mode (assuming a > b). Normally this is the mode that the
rectangular waveguide is used in practice. The TE10 mode can be thought of as a superposition
of two plane waves bouncing on the two vertical walls and with this understanding, we can write
the electric field in the guide as
⌢
⎛ πy ⎞
E = xE 0 sin ⎜ ⎟e − jkz z
⎝ a ⎠
Indeed this electric field expression satisfies the boundary conditions: it vanishes on the two
vertical walls y = 0 and y = a as it should. Now we can calculate the corresponding magnetic
field by the application of Faraday’s law yielding
kz
⎛ πy ⎞
E0 sin ⎜ ⎟e − jkz z
⎝ a ⎠
j
⎛ π ⎞ ⎛ πy ⎞
Hz =
E0 ⎜ ⎟ sin ⎜ ⎟e − jkz z
ωµ ⎝ a ⎠ ⎝ a ⎠
Hy =
ωµ
To gain further understanding of the fundamental mode, consider the bouncing diagram shown in
Figure 5.
Fig. 5 Reflections within a rectangular waveguide.
As shown, the plane waves are directed along k where k = ω µε and where ε and µ are the
permittivity and permeability of the medium that fills the waveguide. From the bouncing
diagram we can write k 2 = k z 2 + k y 2 (consistency condition). At a given frequency, the bouncing
angle that the plane waves make with the z-axis is not arbitrary and is determined by the fact that
ky =
π
a
(to satisfy the boundary condition E x = 0 at y = a). Therefore, we can determine the
propagation constant of the TE10 mode along the z direction as
2
2π
⎛ π ⎞
k z = k 2 − ⎜ ⎟ =
a
λg
⎝ ⎠
where λ g is the guided wavelength (along z). From the above expression, we see that the propagation
constant k z vanishes when k = π / a . This defines the cutoff frequency:
fc =
c
1
=
2a 2a µε
where c = 1 / µε is the speed of light in the material that fills the waveguide. Note that for
frequencies smaller than the cutoff frequency f < f c the propagation constant becomes purely
imaginary thus leading to evanescent waves (attenuating) along the z direction e − kz z . Therefore,
the waveguide only propagates energy along the z axis for frequencies larger (or equal) than the
cutoff frequency f ≥ f c .
This is understandable since low-frequency currents cannot be guided by a single conductor as
there is no return path. Equivalently, the larger dimension of the guide must be greater than half
λ
2
a dielectric wavelength to ensure operation above the cutoff frequency, i.e. a ≥ . The
propagation constant can be written in terms of the cutoff frequency as
2
⎛ f ⎞
2π
k z = k 1 − ⎜⎜ c ⎟⎟ =
f
λg
⎝ ⎠
This equation reveals more clearly that at cutoff f = f c , the propagation constant is zero, i.e.
k z = 0 . When f < f c the propagation constant becomes imaginary leading to an evanescent wave
along z. It can also be noted that as the frequency tends to infinity k z → k but in practice the
upper
operating frequency is limited due to the excitation of higher-order modes. This equation also
reveals that unlike a transmission line, the rectangular waveguide is inherently dispersive since
k z is a non-linear function of the frequency.
If the rectangular waveguide is terminated to a load (e.g. shorted or left open) then there will be
reflections traveling along the negative z axis. In this case, standing waves will form along the
z axis which will be characterized by the guided wavelength λ g (e.g. the distance between two
successive maxima will be λg / 2 ).
3.2.2 Substrate Integrated Waveguide (SIW)
Waveguides offer a low-loss, shielded transmission medium. In traditional sense, they can be
thought of as hollow metallic pipes having rectangular, circular or elliptical cross-section. These
conventional waveguides are bulky; hence, they are not suitable for board-level integration.
Recently, a substrate integrated waveguide (SIW) structure that is fully compatible with
electronic system substrate has been introduced [3, 4]. The adaptation of waveguides to PCB
technology along with its inherent low-loss bandpass characteristic has made them attractive
structures for microwave and millimeter-wave applications.
Fig. 6 Substrate integrated waveguide.
An SIW is a rectangular waveguide configured within the circuit substrate, as depicted in Fig. 6.
The dielectric layer is sandwiched between two solid conductor planes. Signal propagation is
confined between the waveguide sidewalls, which are formed by rows of plated via holes.
The first step in the design process is to determine the operating frequency range of the SIW. Its
TE10 cutoff frequency ( f c ) can be expressed as
fc =
1
2Weff µ r ε r µ 0ε 0
ε r is the dielectric constant of the substrate material. The TE10 cutoff is usually chosen to be
around 15–20% below the lowest frequency of interest. Weff is the effective width of the SIW,
which is related to the width of the waveguide, W (measured between via centers), by [5]
Weff = W − 1.08
D2
D2
+ 0.1
s
W
s is the center-to-center spacing between the rows of vias, and D is the diameter of the via. Since
the sidewalls are formed by metallic vias, energy leakage can potentially occur if the vias are
placed too far apart. As a rule of thumb, it is recommended to keep s < λg ,min / 10 and D should
be as small as possible. λ g is the guided wavelength. Generally, s and D will be limited by
the PCB fabrication tolerances.
We will now use the above design guideline to design an SIW operating at 1 GHz. The substrate
material is FR-4 ( ε r = 4.4). The TE10 cutoff of the SIW is then set at 0.85 GHz, which gives the
initial value of Weff to be 84.1 mm.
Subsequently, s and D are arbitrarily chosen to be 10 mm and 1 mm, respectively. From (13) and
(14), f c and Weff are calculated to be 0.842 GHz and 84.9 mm, respectively. At 1 GHz, it is found
that λ g = 265.4 mm, which gives s = 0.038 λ g .
4. Microstrip Line StandingWave Measurements
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.
Before we conduct measurements, we must properly calibrate the network analyzer. In
this way, when the two cables from the network analyzer are connected together, we will
measure a transmission coefficient of unity (T = S21 = 1 = 0 dB).
4.1 Network Analyzer Setup and Calibration Procedure
1. Press [Start] and enter 800 MHz. Choose [Stop] to be 1.2 GHz.
2. Press [Sweep Setup] and select Points to be 801.
3. Press [Cal]àCorrection ONàCalibrateàEnhanced ResponseàSelect Ports 2-1.
4. Attach the open, short, load, and through to the network analyzer ports until the calibration is
complete.
5. Check your calibration by connecting the SMA cables end-to-end, and ensuring that:
(a) The input reflection coefficient is very low (< -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, you will be using the linear positioner to measure the fringing fields
produced above the microstrip line, the magnitude of which are directly proportional to the
voltage along the line. Hence, the positioner can be used to plot the voltage standing wave as a
function of position.
The stage is moved back and forth using the buttons on the control board. Pressing one of the
two buttons moves the stage in a corresponding direction until the button is released. This can be
used for fine control of the stage position. Conversely, depressing the buttons momentarily will
result in the stage moving approximately 5 mm in the selected direction.
1. Measure the width of the transmission line w. Then use the formulas given in
Section 2.1 to compute the characteristic impedance to confirm it is near 50Ω. Determine
the effective dielectric constant of the line and the phase velocity 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 a short circuit to the end of the microstrip line.
4. Using the control buttons, move the stage so that the coaxial probe used to measure the
microstrip line is positioned above the load end of the microstrip line.
5. Press [Meas] à S21.
6. Press [Format] and select Log Mag.
7. Press [MARKER] and set the marker to f = 1 GHz. Use [Scale]à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).
9. 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.
10. Using fine positioning of the probe, locate the maxima and minima to obtain more
precise locations for the field readings at these positions.
11. Compute the VSWR and compare to that of a short circuit, bearing in mind that the
field values reported by the network analyzer are on a power-dB scale.
12. Compute the wavelength and corresponding effective dielectric constant of the
microstrip line. Compare the theoretical expectations, and comment on any differences.
4.3 Using Standing Wave Patterns for Load Calculations
In this portion of the lab, you will measure the standing wave pattern produced when the
microstrip line is terminated in an unknown load, and using analytical / Smith Chart techniques
to determine the value of the unknown load.
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. Using fine positioning of the probe, locate the maxima and minima to obtain more
precise locations for the field readings at these positions.
5. Using the techniques of Section 2.1.1, compute the impedance of the load based on the
standing wave pattern you have measured. Bear in mind 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. As in previous labs, the extra length of SMA cable must first be accounted
for using the Port Extension process.
(a) Press [CAL] à Port Extension à “Extension ON” à Auto Port Extension à
Select PortsàPort 1 ON. Now attach an Open and Short connection and press
the corresponding screen buttons.
(b) With the unknown load attached, press
[Meas]àS11à[Format]àSmithàR+jX and select Reflection.
(c) Finally, press [Marker] and set the marker to 1 GHz and record the impedance
from by the network analyzer. Compare to your computed result and discuss any
discrepancies.
4.4 SIW Standing Wave Measurements
In this part of the lab, you will use the field measurement apparatus to measure the electric fields
produced with a type of rectangular waveguide, which has been realized practically in the form
of a substrate integrated waveguide (SIW). Sampling of the electric field is made possible by
introducing a slot in the top (and bottom) of the waveguide, which is clearly visible on the PCB,
and inserting a coaxial probe in the vicinity of the slot. Fields normally contained within the
rectangular waveguide produce fringing effects in the vicinity of the slot, which can be detected
in a similar way as the microstrip line fringing fields. However, structurally, the field
configuration with the SIW is fundamentally different from the microstrip line, as will be made
apparent in your measurements.
One end of the SIW line is probe-fed by a coaxial line. The other end of the line is shorted out
with a short circuit (ZL = 0). Hence, a standing wave is produced with a very high standing wave
ratio along the length of the line, which is visible by measuring the fringing fields from the slot.
The fringing fields are proportion to the values of the transverse electric fields within the SIW.
1. Connect port 1 of the network analyzer to the input port on the SIW. This is the only
port on the SIW side of the PCB, which is a flange-mounted connector on top of the
PCB.
2. Connect port 2 of the network analyzer to the coaxial probe on the SIW side of the
probe stage.
3. Switch the network analyzer back to a log-magnitude view of the transmission
coefficient, and move the marker to a frequency of 900 MHz.
4. Move the stage to the end of the slot farthest from the input connector. Note that the
end of the slot is not at the load (short-circuit) point, but rather, some arbitrary point
along the SIW.
5. Move the stage in 5 mm increments away from the starting position, and plot the
standing wave pattern over a minimum one one full wavelength at 900 MHz. Use the
ruler supplied to make precise measurements of the probe location relative to the shortcircuited end of the PCB.
6. Using fine positioning of the probe, locate the maxima and minima to obtain more
precise locations for the field readings at these positions.
7. Measure the standing wave ratio and determine the phase velocity v p and guide
wavelength λ g of the wave inside the waveguide. How does its value compare to the
speed of light for a TEM wave travelling within the substrate? Explain.