EECS 230
APPLIED ELECTROMAGNETICS I
LABORATORY MANUAL
Cognizant Faculty:
• Professor Fawwaz T. Ulaby
• Professor Anthony Grbic
Contributing Experiment Designers:
• Brian Tierney
• Ahmet Seyit
• Chao Zhang
• Carl Pfeiffer
The University of Michigan
Ann Arbor, MI 48109-2122
Fall 2012
© Ulaby et al. 2012
TABLE OF CONTENTS
Introduction
Laboratory Rules and Procedures
The Lab Notebook
Grading the Lab Notebook
Exercises
Lab Exercise 1:
Lab Exercise 2:
Lab Exercise 3:
Lab Exercise 4:
Lab Exercise 5:
Lab Exercise 6:
Appendices
Appendix A:
Appendix B:
Appendix C:
Appendix D:
Appendix E:
Transmission Line Basics
The Smith Chart
Overview of Magnetic Circuits
Shielded Loop Resonators
Coupled Resonators and Voltage Rectification
Impedance Matching and Frequency Tuning
HP 8712C Network Analyzer
HP 54645A 100-MHz Digital Oscilloscope
HP 8648B RF Signal Source
Adapters, Cables, Connectors, and Components
Error Analysis
2
3
4
6
28
42
64
89
110
A.1
B.1
C.1
D.1
E.1
Laboratory Rules And Procedures
I.
S AFETY
1. Do not touch equipment until instructed.
2. NO student may use the lab equipment alone at any time. A student must be
accompanied at all times by a laboratory partner and/or a lab instructor. This rule
is observed for regular as well as make-up lab sessions.
3. Report all dangerous conditions (stripped AC lines, sparks in equipment, loose wall
socket, etc.) to your lab instructor.
4. If a piece of equipment does not turn on or stops functioning properly, report it to
your lab instructor immediately.
II.
H ONOR C ODE
The Honor Code applies to all laboratory work, which means the pre-lab preparation, the
experiments and the final lab report. You are expected to include the following Honor Code
statement on all your work:
“I have neither given nor received aid on this report, nor have I concealed any violation
of the Honor Code”
III.
C OLLABORATION P OLICY
You are encouraged to work with your laboratory partners and/or with a group of friends
on the pre-lab, lab-reports and homework problems. However, you may not copy the work
of anyone else (enrolled in the class or not) and you should write your own work without
looking at other peoples’. You are allowed to help (or receive help from) your colleagues in
the form of discussing the work, explaining the problem and even describing (or learning)
in detail how to solve a problem (verbally with few sketches), but the work you turn in
should be yours and no one else’s.
IV.
G RADING P OLICY
The lab grade constitutes 20% of the total grade for the course. For each experiment, the
lab grade is distributed as follows:
Pre-lab
Lab Report
Participation
15%
75%
10%
Any student missing a lab experiment (not present in the lab) with no proper or reasonable
excuse will get a zero “0” grade for that lab experiment. If a student misses a lab experiment
with an acceptable excuse, he/she will be allowed to make it up by the laboratory instructor.
2
V.
V.
LABORATORY PROCEDURE
3
L ABORATORY P ROCEDURE
Each lab experiment is designed to be performed within the 2-hour laboratory time.
During each laboratory period, you will be expected to carry out one experiment, to
record the experimental data, to make a few computations to determine how it agrees with
expectations, perhaps plot some graphs, and then answer some questions concerning the
experimental work. The laboratory instructor will then sign on your notebook before you
leave the lab. You will then prepare the final lab report which is due one week after your
lab session. To successfully complete the experiments in one lab period, you must come
prepared to the laboratory. You should read the experiment in advance and answer the prelab questions. The pre-lab counts for 15% of the lab grade. Always bring the following
items to each lab session:
1. Your lab notebook (with numbered pages).
2. An INK pen (color is a plus), and a pencil.
3. A calculator capable of sin, cos, log, ln, e x functions.
4. A ruler, a compass, and a protractor.
Always use an ink pen when you enter data and other information in your notebook. If
you make a mistake, neatly cross it out and continue on. If a graph is incorrect, cross it out
too with a large X, draw a line and continue. No points will be deducted for crossed-out
material, as long as it is neat. However, lab reports with hash marks all over the place, data
presented in haphazard fashion and general sloppiness will suffer as much as 50% penalty
(in severe cases) on the final grade.
The following guidelines will result in a good lab report:
1. Always put a Unit (V, A, Hz, ...) after every quantity or answer. Underline your
answer or place a box around it.
2. Always label the axis on a graph and give the units too.
3. When using an equation, write it out first in analytical form (Ex.:
R1 R2 R1 R2 ) and then substitute the data.
Rp
4. Sketch a circuit diagram or a system diagram neatly and label the component values.
5. Sketch an experimental diagram and label the equipment used.
6. Comment on the agreement or lack of agreement with the calculated values. Also
comment if the measured results are unexpectantly too high or too low. It is rather
OK to have done an unsuccessful experiment but it is NOT OK to accept the data as
it comes without thinking about it, questioning if it makes sense, and knowing finally
what went wrong!
In the lab report (due one week later), you are asked to further analyze your results,
answer a few questions and calculate a few things. The lab report should be written in your
lab notebook. Please write down any special difficulties you encountered in the lab and
your suggestions for any improvements (highly appreciated).
*Start Lab Exercise 1 on page 2. Leave page 1 blank so you may use it later to generate
a table of contents for your lab notebook.
The Lab Notebook
In industry, the lab notebook is not just where you doodle down things related to ideas or
measurements. The notebook is a legal document, which can be used to prove that you
discovered some phenomenon and that you are the person that deserves the credit (or, put
a slightly different way, that your company deserves the profit). Not only can the notebook
be used to protect you from someone trying to steal your ideas, it can also be used to prove
that you didn’t steal an idea, either.
In order to be a legal document, the lab notebook must be bound, so that no pages can
be added or removed. The cover may be hard or soft, but the pages must be numbered, as
proof that no pages were added or removed. Numbered pages also make it easy to index
your notebook, so that it can be used effectively as a reference. You can number the pages
of your own bound notebook, but it is easier to purchase a bound notebook with the pages
already numbered.
As you fill in the pages of your notebook, record the date on each page as you start to
fill it in. Start a new page every new day, regardless of how much space was left on the last
page.
Everything you write in the notebook should be written in non-erasable ink. The only
exception is for calculations. Once made, observations cannot be undone, so they should
be recorded in a permanent fashion. If for some reason you suspect a measurement or
observation to be in error, simply cross out the bad data and put a new (hopefully better)
data down next to it. Somewhere on the same page write down an explanation of why you
suspected the data you crossed out to be incorrect. Put only a single line thru the bad data.
Do not cross out bad data such that it cannot be read: you might be wrong twice and the
original data might have been good!
Calculations may be done in pencil, as an error in calculations can be corrected. Pen is
OK, too.
Sometimes things get on paper that is not in your bound notebook and you wanted to
include them in your notebook. An example might be a plot generated on a computer and
sent to a printer. Make a copy, or, preferably, take the original, and staple, glue or tape it
into your notebook at the page you are working on. There are techniques, like signing your
name across the edge of an included document and onto the notebook page, that insures
that the inclusion is genuine, but that won’t be required in this class. (If you make any new
discoveries, we’ll take credit. Thanks.)
4
5
Grading the Lab Notebook
For the EECS 230 and 330 labs, buy a bound notebook with page numbers. Write
your name on the firs page and leeave the rest of it blank, so that it can be used
later for a table of contents when the notebook is full.
Because we need to grade your weekly lab reports, we’re going to use the
following procedure:
• Write the answers to the pre-lab in your notebook before the lab.
• At the beginning of the lab, the GSI will initial at the bottom of the pages
with pre-lab assignment.
• Record the data taken during the lab directly into your notebook.
• When you are done getting the data, get the GSI to initial the bottom of pages
with the data.
• Write up the lab report using the data in your notebook. Use a computer
to type and print your lab report. A photocopy of the raw data page with
the GSI’s initials should be included in your submitted report. Attach the
photocopied pages with tape, staples, or glue to your report.
• One week after the data was taken, a copy of the lab report must be handed
in to the GSI.
• Two weeks after the data was taken, the GSI will return your graded report.
• Attach the report to the back end of your notebook.
Lab Exercise 1: Transmission Line Basics
Contents
1-1
1-2
1-3
1-4
1-5
1-6
P RE - LAB A SSIGNM ENT . . . . . . . . . .
I NT ROD UC TION . . . . . . . . . . . . . . .
U SEF UL EQUATIONS . . . . . . . . . . . .
E Q UIP M ENT . . . . . . . . . . . . . . . . .
E X P ERIM E NT . . . . . . . . . . . . . . . .
1-5.1 Role of Wavelength . . . . . . . . . .
1-5.2 Standing Waves On The Slotted Line
1-5.3 Network Analyzer . . . . . . . . . .
L AB WRITE - UP . . . . . . . . . . . . . . .
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26
Objective
To examine basic transmission-line concepts. In addition, the network
analyzer will be introduced as a measurement tool, and will be used to
study the reflections caused by various load terminations.
General concepts to be covered:
• Network analyzer
• Role of wavelength
• Standing waves on transmission lines
6
1-1
1-1
1.
2.
3.
1-2
PRE-LAB ASSIGNMENT
7
P RE -L AB A SSIGNMENT
Read the supporting material in the textbook.
Read through the lab exercise.
Prior to the lab, summarize the experimental procedures in your lab notebook (1 paragraph per section):
(a) Section 1-5.1
(b) Section 1-5.2
(c) Section 1-5.3
I NTRODUCTION
With the ever increasing role of communications in our daily lives, it has become important
to connect remote sites together to share data and information. One method for providing
links between remote sites is to use transmission lines. Some everyday examples include
telephone lines, electrical lines, and cable television. In order to provide the best link
possible, it is necessary to understand how signals propagate on transmission lines.
In basic electronic circuits, it is assumed that if a voltage is applied at the input of a
circuit, the output voltage appears instantaneously at the output of the circuit. For circuits
where the line lengths are much smaller than the wavelength of the signal, this assumption
is acceptable with only negligible consequences. However, when the length of the wires or
transmission lines are an appreciable fraction of the wavelength or longer, the output signal
changes phase compared to the input signal, and at impedance discontinuities, reflections
can occur.
In this lab, you will explore experimentally the role of wavelength, standing waves on
transmission lines, and the reflections caused by different load terminations. In addition,
you will be introduced to a new tool for analyzing transmission line systems (or networks),
the network analyzer.
1-3
U SEFUL E QUATI ON S
up
λ= f
(m)
1 + |Γ|
1 - |Γ|
S- 1
|Γ| =
S+ 1
1+ Γ
|ZL | = Z0
1- Γ
SW R = S =
1-4
E QUIPME NT
(Ω)
Item
Part #
Cables & connectors
——
Calibration Kit
HP 85032E
Network Analyzer
HP 8712C
Oscilloscope
Scanner Antenna
Signal source
Slotted Line
Agilent DSO-X 2012A
——
HP 8648B
HP 805A
8
LAB EXERCISE 1: TRANSMISSION LINE BASICS
Station
X
1
2
3
4
5
R1 (Ω)
1689
1484
997
2212
1004
1488
R2 (Ω)
1887
1527
1495
1808
2196
2682
Table 1-1: Voltage divider resistor values.
1-5
E X PERI ME NT
1-5.1
Role Of Wavelength
In this experiment, you will investigate the role of wavelength in circuits. Recall that it
is the ratio of the line length to the wavelength that determines whether transmission-line
analysis is required. If
l
> 0.01
(1.1)
λ
then the system must be analyzed as a transmission-line system. To demonstrate this, you
will compare the output of a voltage divider that is fed by a wire of length l to the input
signal at several frequencies.
Setup
This experiment uses the signal source, oscilloscope, and voltage divider.
Set up the experiment:
• Connect the BNC “Y” to channel 1 of the oscilloscope.
• Connect the input of the voltage divider to one of the arms of the BNC “Y” connector
using a BNC plug to BNC plug adapter.
• Connect the BNC jack to N plug adapter to the RF output of the RF signal source.
• Use a 24" piece of BNC co-axial cable to connect the signal source to the open arm
of the “Y”.
• Use a 180° piece of BNC co-axial cable to connect the output of the voltage
divider to channel 2 of the oscilloscope.
The setup is shown in Fig. 1-1. Block diagrams of the setup and voltage divider
configuration are shown in Fig. 1-2.
180° BNC Cable
1. Record the two resistor values (R1,R2) given in Table 1-1.
Turn the signal generator on and set the frequency to f 1 = 100 kHz.
1-5
EXPERIMENT
9
(a) General setup
BNC ’Y’
24" BNC Cable
(from Signa Source)
(b) Close-up of oscilloscope
Figure 1-1: Setup for Section 1-5.1
10
LAB EXERCISE 1: TRANSMISSION LINE BASICS
Signal Source
Osc il lo sc o pe
Cable 1
Cable 2
Voltage Divider
(a) General setup block diagram
Outer conductor
Inner conductor
R1
+
+
vIn
R2
-
vO u t
-
BNC Connector
(b) Voltage divider block diagram
Signal
Source
+
-
Cable 2
Cable 1 R1
+
+
V1 R2 Vout
24" BNC
Cable
-
+
V2
180" BNC Cable
or 12" BNC Cable
(c) v1 and v 2 are measured by the oscilloscope
Figure 1-2: Block diagrams of (a) the setup for Section 1-5.1, (b) the voltage divider, and
(c) the overall circuit configuration.
1-5
EXPERIMENT
11
• Press Power to turn on the RF signal source. Wait for the RF signal
source to perform its self check.
• Press Mod On/Off until the modulation is turned off
• Press Amplitude followed by 0
MHz
dB(m)
• Press Frequency followed by 1
frequency to 100 kHz.
to set the amplitude to 0 dBm.
0
0
kHz
mV
to set the output
• Press RF On/Off until the RF output is On
If at any point you make a mistake entering a number, you can use the ←
key to delete the last number entered.
Set the oscilloscope to show both channel 1 and channel 2. Adjust the oscilloscope
display so that two periods are shown on each of the channels.
• Press Autoscale
2. Record the signal amplitudes v 1 (channel 1) and v 2 (channel 2).
To measure voltage:
• Press Measure
• Press the Source softkey until channel 1 is selected.
• Press the Type softkey until “Peak-Peak” is selected.
• Press the Add Measurement softkey.
• Repeat for channel 2.
If the signals displayed are noisy and the measured values are jumping around, you
can use averaging to reduce the effects of the noise.
12
LAB EXERCISE 1: TRANSMISSION LINE BASICS
To turn averaging on:
• Press Acq Mode until you choose Averaging.
• Choose the number of averaging as 4 by pushing the # Avgs button.
To turn averaging off:
• Press Acq Mode and choose Normal
Record the delays between 2 signals and find the phase
difference.
• Press Measure
• Press the Type softkey until “Delay” is selected.
• Press the Source softkey until channel 1 is selected.
• Press the Add Measurement softkey.
Note: So far you have measured the time delay between channel 1
and channel 2, which you need to obtain the phase difference in
degrees for the given frequency.
Capture the signal display on your scope and attach it to your
report.
To capture the scope display:
*Run the “IntuiLink Data Capture” program either through the shortcut in the desktop or start
menu as shown below
12a
*Click on instrument and choose the right instrument. It is the second one from the top and can be seen as
either “Agilent 2000/3000 Series” or “DSO-X 2012A”
*Setup the Add-In properties, choose number of points as 1000 and select the channels. The off channel on
the scope is not selectable, as may see below.
* After hitting the OK button, the scope will acquire the measurement data and display the picture.
Save your captured image.
12b
3. Change the frequency of the signal source to f 2 = 100 MHz
.
• Press Frequency followed by 1 0 0
MHz
mV
Adjust the oscilloscope display so that two periods are shown for both the input and
output signals. Record the signal amplitudes v 1, v2 and delay. Attach the display
capture.
12" BNC Cable
4. Replace the 180° BNC co-axial cable with a 12" BNC co-axial cable.
5. Change the frequency of the signal source to f 1 = 100 kHz. Adjust the oscilloscope
display so that two periods are shown for both the input and output signals. Record
the signal amplitudes v1 and v2 .
6. Change the frequency of the signal source to f 2 = 100 MHz. Adjust the oscilloscope
display so that two periods are shown for both the input and output signals. Record
the signal amplitudes v1 and v2 .
Measured Data
Copy the following chart into your lab book and fill in the measured data. If you are missing
any data, please repeat the necessary parts of this experiment before proceeding.
R1
R2
=
=
(Ω)
(Ω)
Frequency
f 1 =100 kHz
f 2 =100 MHz
Cable
12
180
12
180
v1 (V)
v2 (V)
ΔT
Δφ (◦ )
1-5
EXPERIMENT
13
Analysis
1. Compute the wavelength for frequencies f 1 and f 2 . Assume εr of the cable is 1.9.
Arrange the results in tabular form. Record these values.
2. Compute the ratio of l to λ for the 12" BNC cable and the 180° BNC cable
for frequencies f1 and f 2 . Arrange the results in tabular form.
3. If we were to ignore the cables in Fig. 3-2(c), then vout would be the same as v 2
measured by the scope. The output of the voltage divider is given by:
vout = v 1
R2
(V)
R1 + R 2
(1.2)
Calculate this value for each combination.
4. Compare the computed output voltage vout from the previous step to the recorded
output voltage v 2 (channel 2) for both the 12" and 180° BNC cables at each of
the frequencies: f 1 and f 2 . Comment on your results.
5. Comment on the role of wavelength in circuits using the data collected in this
experiment. This can be two or three sentences saying: As X increases, A analysis
fails and we need B. This happens because . . . 1) . . . 2) . . .
Questions
1. Would you expect the output voltage of the RF signal source to be constant over
the frequency range examined? Is this what you observed? If not, explain why the
voltage on channel 1 was not constant over the frequency range.
2. Which of these systems needs to be treated as a transmission line system? Why?
Justify your answer quantitatively. Indicate any assumptions that you are making.
(a) Integrated circuit at high frequencies (500 MHz→1 GHz)
(b) Electrical lines running through your house
(c) Electrical lines connecting cities separated by hundreds of kilometers
(d) VHF antenna leads from a rabbit ear antenna to your television
3. Why is it necessary to treat lines that have a length to wavelength ratio greater than
0.01 as transmission-line systems? Explain in terms of the phase of a co-sinusoidal
signal given by A cos(ωt - βz), where A is the peak amplitude of the signal, z is the
position on the line, and β = 2π
λ . (hint: what are the major assumptions of a wire
in DC analysis and how are those violated on a system requiring transmission line
analysis?)
2-5.2
Standing Waves on the Slotted Line
In this experiment, you will use the oscilloscope and the slotted line to examine the
standing-wave pattern caused by various loads. The measurements you will be making
are uncalibrated measurements. In order to correct for this, the first step is to measure the
standing-wave pattern for a known load to use as a reference.
14
LAB EXERCISE 1: TRANSMISSION LINE BASICS
The reference load that will be used is the short termination since we know the standingwave pattern that should be produced by the short termination. By measuring the slotted
line terminated with various loads and calibrating against the reference measurement, the
standing wave pattern of the loads can be determined.
The slotted line consists of a piece of metal tubing (which is an unshielded transmission
line), a probe, and a detector. The probe measures the electric field present in the line and
uses a detector to convert the measured field to a voltage. The probe and the detector are
housed on a mount which can slide down the line. On the side of the mount is a scale that
can be used to measure the position of the probe. The slotted line is shown in Fig. 1-3.
Detector mount
BNC cable
Load
RF IN
Position adjustment knob
Figure 1-3: The slotted line.
Setup
This experiment uses the signal source, oscilloscope, slotted line, and various loads and
cables.
Setup the experiment as shown in Fig. 1-4:
• Attach the source end of the slotted line to the signal source using a patch cord.
• Attach the output of the probe to channel 1 of the oscilloscope using the 180 piece
of BNC co-axial cable.
• Configure the signal source to output a 10 dBm, 1 GHz, 1 kHz 50% AM modulated
RF signal.
– Press Amplitude followed by 1 0
MHz
dB(m)
– Press Frequency followed by 1 0 0 0
MHz
dB(m)
– Press RF On/Off until the RF output is On
– Press
INT
1 k Hz
– Press Mod On/Off until modulation is On
– Press AM followed by 5 0
%
μV
1-5
EXPERIMENT
15
• Adjust the settings on the oscilloscope so at least 2 periods of the wave-form are
shown.
• Configure the oscilloscope to measure the voltage amplitude.
You may want to use AC coupling to keep the waveform from shifting position on the
screen. Be sure to keep adjusting the voltage scale on the oscilloscope to best fit the
wave on the screen.
Oscilloscope
RF Signal Source
BNC cable
Detector Mount
Slotted line
Patch cord
Position adjustment knob
(a) Setup for Section 1-5.2
Signal Source
O sc illo sc o pe
1
2
BNC Cable
Slotted Line
Load
Patch Cord
(b) Block diagram of setup
Figure 1-4: Setup and block diagram for Section 1-5.2
Station
1
2
3
4
5
R (Ω)
22
100
10
68
39
C (pF)
39
10
39
15
1
Table 1-2: Resistor and capacitor values for the resistive and capacitive loads.
16
LAB EXERCISE 1: TRANSMISSION LINE BASICS
BNC connection (to oscilloscope channel 1)
Tuning Knob
Position indicator
Probe height adjust knob
Figure 1-5: Slotted line probe mount
Procedure
Tuning the slotted line
The two knobs (black and silver) on top of the probe shuttle allow
you to tune the position and electrical characteristics of the probe
needle. The goal of tuning is to have the needle adequately sample
the standing wave without distorting the measured signal.
0.6
0.4
0.2
The figure to the left shows two needle positions. The first signal
(solid) is larger, but suffers from distortion. Therefore the second
location is a better choice despite the lower signal amplitude
0
Time
1. Connect the short termination to the end of the slotted line. Move the detector mount
to a position where a strong signal is shown on the oscilloscope. To move the probe,
push in and turn the black knob (see Fig. 1-4) on the side of the slotted line.
2. Tune the slotted line detector and probe. Turn the silver probe knob on the detector
mount to the right until the probe is as close to the line as possible (see Fig. 1-5).
Adjust the selectivity by turning the black knob on the probe mount to the right and
left until a peak is observed on the oscilloscope.
Short Termination
3. Locate the first minimum nearest the load. Record the voltage amplitude reading on
the oscilloscope and the position on the slotted line. Use the position indicator on
the detector mount to read the value in millimeters from the scale that runs along the
slotted line. Use the 0 marker on the position indicator when reading the position.
1-5
EXPERIMENT
You will use this position as the load plane reference for the remainder of this
λ
experiment. In other words, each of the other terminations will begin their 20
steps
at this measurement point, and not their first zero location. This will allow us to
determine the phase between the standing waves.
4. Locate the first maximum by moving the probe toward the generator (away from the
load). Record the voltage amplitude reading on the oscilloscope and the position on
the slotted line.
5. Locate the second minimum from the load. Record the voltage amplitude reading on
the oscilloscope and the position on the slotted line.
6. Return the probe to the load plane reference. Measure the voltage amplitude on the
λ
line over a half-wavelength interval in steps of 20
. You should be able to calculate
λ from knowledge of the signal frequency and εr =1. If you’re unsure of your
calculation, please check your result with your GSI. Record the voltage amplitude
and position of each measurement.
Open Termination
7. Connect the open termination to the end of the slotted line. Place the probe at the load
reference position (i.e. the same starting point as the first short measurement) and
record the voltage amplitude. Measure the voltage on the line over a half-wavelength
λ
interval in steps of 20
. Record the voltage and position of each measurement.
8. Locate the first minimum nearest the load reference position. Make sure that you
move the probe toward the generator. Record the voltage amplitude and position of
the first minimum.
Matched-load Termination
9. Connect the matched termination to the end of the slotted line. Place the probe at
the load reference position and record the voltage amplitude. Measure the voltage
λ
amplitude on the line over a half-wavelength interval in steps of 20
. Record the
voltage amplitude and position of each measurement.
Resistive Termination
10. Connect the resistive termination to the slotted line. Record the resistor value listed
in Table 1-2. Place the probe at the load reference position and record the voltage
amplitude . Measure the voltage amplitude on the line over a half-wavelength interval
λ
in steps of 20
. Record the voltage amplitude and position of each measurement.
11. Locate the first minimum nearest the load reference position. Make sure that you
move the probe toward the generator. Record the voltage amplitude and position of
the first minimum.
Capacitive Termination
12. Connect the capacitive termination to the slotted line. Record the capacitor value
listed in Table 1-2. Place the probe at the load reference position and record the
17
18
LAB EXERCISE 1: TRANSMISSION LINE BASICS
voltage. Measure the voltage amplitude on the line over a half-wavelength interval in
λ
steps of 20
. Record the voltage amplitude and position of each measurement.
13. Locate the first minimum nearest the load reference position. Make sure that you
move the probe toward the generator. Record the voltage amplitude and position of
the first minimum.
Measured Data
Copy the following charts into your lab book and fill in the measured data. If you are
missing any data, please repeat the necessary parts of this experiment before proceeding to
the analysis section.
Short Termination
Location
1st minimum
1st maximum
2nd minimum
Position (mm)
|v| (V)
Loads
Resistive termination value
Capacitive termination value
=
=
(Ω)
(pF)
Probe Position [λ]
Load
0
λ
20
2λ
20
3λ
20
4λ
20
5λ
20
6λ
20
Minimum
7λ
20
8λ
20
9λ
20
10λ
20
Probe position [mm]
Short
Open
Matched
Resistor
Capacitor
Notes:
• Position 0 refers to the load reference position
• Minimum refers to the exact location of the first minimum unless specified
otherwise. Record both the voltage amplitude and position with greater precision
λ
than 20
steps.
Pos.
X
Vol.
X
1-5
EXPERIMENT
Analysis
1. Using the measured minima and maxima positions for the short termination, compute
the distance between minima (in wavelengths, use εr =1). Record this distance.
Compare to the expected theoretical value.
2. Using the measured minima and maxima positions for the short termination, compute
the distance between the first minimum and maximum (in wavelengths). Record this
distance. Compare to the expected theoretical value.
3. Normalize all of the measured data by the maximum value recorded for the short
termination (i.e. divide all measurements by max([Vshort ]))
Plot the standing wave pattern for each of the measured loads using the normalized
data. Compare the patterns and comment on the results.
Questions
1. What would you expect the standing-wave pattern to look like if the slotted line was
1
)?
terminated in an inductor (assume ωL = ωC
2. Why was it acceptable to define the load reference position to be at a location other
than the load?
1-5.3
Network Analyzer
In this experiment, you will familiarize yourself with the network analyzer and measure the
reflection coefficient of three standard impedances.
The most vulnerable parts of the network analyzer are the RF connectors and the
calibration kit components. Ask the lab instructor to demonstrate how to make connections
and handle the calibration standards. As far as pushing the knobs and keys on the
instrument, no special care is necessary. The worst thing that can happen is the instrument
will lock up. If that should occur, press PRESET or shut the instrument off and turn it back
on.
The network analyzer is a measurement tool for making phase and magnitude
measurements. The network analyzer you will use is capable of making phaser (magnitude
and phase) measurements in the frequency range from 0.0003→1.3 GHz (109 Hz).
Setup
This experiment uses the network analyzer, calibration kit, patch cord, scanner antenna, and
printer.
Setup the network analyzer as follows:
19
20
LAB EXERCISE 1: TRANSMISSION LINE BASICS
• Press PRESET
• Turn channel 2 Off.
– Press MEAS 2 followed by the Meas Off softkey
• Set the display to show only channel 1.
– Press DISPLAY
– Press the More Display softkey
– Press the SPLIT Disp full SPLIT softkey until only channel
1 is displayed (FULL will be displayed in all caps and split
will be displayed in all lowercase letters).
• Set the start frequency to 100 MHz.
– Press FREQ
– Press the Start softkey.
– Enter 1 0 0 using the key pad and press the MHz softkey.
• Set the stop frequency to 1000 MHz.
– Press FREQ .
– Press the Stop softkey.
– Enter 1 0 0 0 using the numeric key pad and press the
MHz softkey.
• Recall the uncalibrated instrument state.
– Press
SAVE
RECALL
.
– Use the ↑ and ↓ keys to select the file nocal.cal from the
root directory.
– Press the Recall State softkey
1-5
EXPERIMENT
Configure the Display of the network analyzer as follows:
• Set the network analyzer to display magnitude data in logarithmic
format.
– Press FORMAT
– Press the Log Mag softkey
• Set the network analyzer to measure the reflections on channel 1.
– Press MEAS 1
– Press the Reflection softkey
• Autoscale the display.
– Press SCALE
– Press the Autoscale softkey
21
22
LAB EXERCISE 1: TRANSMISSION LINE BASICS
Procedure
Note: For the short, open, and matched terminations, you will need to attach the N jack
to N jack adapter to the end of the patch cord. Remove the adapter before attaching the
antenna (i.e. don’t use a male-to-male adaptor to a female-to-female adaptor, when neither
is necessary).
Calibration
1. Connect the patch cord to the Reflection RF Out port on the network analyzer
(channel 1). Attach the short termination to the end of the patch cord. Be sure to
Autoscale the display. Print the magnitude response (see next page for instructions).
Place a marker at 500 MHz:
• Press MARKER
• Press the Marker 1 softkey
• Enter the desired frequency (500) in MHz using the numeric key
pad and press the MHz softkey
The marker should be located at the entered frequency and the value at
that point displayed in the upper corner of the screen.
• Press Hard Copy
• Press the Start softkey.
Note: Please write down your file name and its description while
saving the screen to a floppy disk. Otherwise, you may not know
which figure is which.
1-5
EXPERIMENT
Record the magnitude of the reflection coefficient at 500 MHz. Change the display
to show the phase of the reflection coefficient.
• Press FORMAT
• Press the Phase softkey
Record the phase of the reflection coefficient at 500 MHz.
You should have seen that the response was not at all what you expected. This is
attributed to the fact that the network analyzer had not been calibrated. To make
accurate reflection measurements, you must first calibrate the network analyzer by
performing a single-channel calibration.
• Press FORMAT followed by the Log Mag softkey
• Press CAL
• Press the One Port softkey
• Connect the open standard from the calibration kit to the end of the
patch cord. Press the Measure Standard softkey
• Remove the open standard and connect the short standard from the
calibration kit. Press the Measure Standard softkey
• Remove the short standard and connect the matched (50 Ω)
standard from the calibration kit. Press the Measure Standard
softkey
23
24
LAB EXERCISE 1: TRANSMISSION LINE BASIC
Measurements
2. Now make the following measurements:
Note: After changing a load, press SCALE followed by the Autoscale softkey. Be
sure to change the descriptive title before printing.
(a) Connect the short termination to the end of the patch cord. Sketch, save or
print the magnitude response. Record the magnitude and phase of the reflection
coefficient at 500 MHz.
Note: If this still does not look close to the expected response, repeat the
calibration.
1-5
EXPERIMENT
25
(b) Connect the open termination to the end of the patch cord. Sketch, save or
print the magnitude response. Record the magnitude and phase of the reflection
coefficient at 500 MHz.
(c) Connect the 50 Ω (matched) termination to the end of the patch cord. Sketch,
save or print the magnitude response. Record the magnitude and phase of the
reflection coefficient at 500 MHz.
(d) Remove the matched termination from the patch cord. Connect the scanner
antenna to the end of the patch cord. Be sure to completely extend the antenna
before making measurements.
(e) Change the display format of the network analyzer to SWR.
• Press FORMAT
• Press the SWR softkey
(f) Sketch, save or print the resulting display. Place a marker at 100 MHz. Using
the position knob on the front panel of the network analyzer, move the marker
to the frequency where the SWR is a minimum. Record this frequency and the
corresponding SWR. Using the marker, locate the two frequencies nearest the
minimum where the SWR becomes 2.5. Record these two frequencies.
Measured Data
Fill in the measured data. If you are missing any data, please repeat the necessary parts of
this experiment before proceeding.
Reflection Coefficients
Load
Short (uncal)
Short (cal)
Open
50 Ω (matched)
|Γ|
ΔΓ
Scanner Antenna SWR
Frequency of SWR minimum
Minimum SWR value
(Lower) Frequency of SWR = 2.5
(Higher) Frequency of SWR = 2.5
(MHz)
(MHz)
(MHz)
Analysis
1. Compare the printouts of the short termination before and after calibration. What was
the effect of the calibration? There are two types of error, systematic and random.
How does calibration affect these two types of error?
26
LAB EXERCISE 1: TRANSMISSION LINE BASICS
2. For each of the loads measured (short, open, matched), compute the theoretical
reflection coefficient and compare it to the measured reflection coefficient. Since
the network analyzer performs a power measurement, the conversion to a linear scale
is:
(1.3)
|Γ|linear = 10( |Γ|dB /20 )
Comment on your results.
3. Compute the magnitude of the antenna input impedance at the frequency where the
SWR minimum occurred. Record this value.
Note: For your calculations, assume that that Γ is essentially real.
Questions
1. Is the antenna that you measured good for broad-band communication systems
(20 MHz → 2 GHz)? Why or why not? (Hint: recall the connection between the
reflection coefficient and the SWR)
2. Using the computed input impedance for the antenna at the frequency of minimum
SWR, which of these systems would the antenna work well with?
(a) 50 Ω transmission line system
(b) 75 Ω transmission line system
(c) 300 Ω transmission line system
Use the definition that in order for the antenna to work well, the SWR must be ≤ 2.5.
1-6
L AB W RITE - UP
For each section of the lab, include the following items in your write-up:
(a) Overview of the procedure and analysis.
(b) Measured data where asked for.
(c) Calculations (show your work!).
(d) Any tables and printouts.
(e) Comparisons and comments on results.
(f) A summary paragraph describing what you learned from this lab.
Lab Exercise 2: The Smith Chart
Contents
2-1
2-2
2-3
2-4
2-5
2-6
P RE - LAB
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
U SEF UL EQUATIONS . . . . . . . . . . . . . . . . . . .
E Q UIP M ENT . . . . . . . . . . . . . . . . . . . . . . . .
E X P ERIM E NT . . . . . . . . . . . . . . . . . . . . . . .
2-5.1 Line Parameters of a Lossless Line . . . . . . . .
2-5.2 Lossy and Lossless Lines on the Smith Chart . .
2-5.3 Impedance Measurements Using the Smith Chart
2-5.4 Impedance Matching Using a Single Stub Tuner
.
L AB WRITE - UP . . . . . . . . . . . . . . . . . . . . . .
ASSIGNMENT
IN T RO DU CTIO N
29
29
29
29
30
30
33
34
37
40
Objective
To examine the utility of the Smith Chart and the concept of matching.
The concepts of lossy versus lossless lines and impedance matching will
be explored.
General concepts to be covered:
• Lossy versus lossless lines on the Smith Chart
• Relative dielectric constant and its relationship to u p
• Single-stub matching techniques
• Smith Chart display on the network analyzer
• Transmission line parameters
• Using the Smith Chart to measure load impedances
28
2-1
PRE-LAB ASSIGNMENT
2-1
1.
2.
3.
2-2
P RE-LAB
29
ASSIGNMENT
Read the supporting material in the textbook.
Read through the lab exercise.
Prior to the lab, summarize the experimental procedures in your lab notebook (1 paragraph per section): (a) Section 2-5.1; (b) Section 2-5.2; (c) Section 2-5.3; (d) Section 2-5.4.
I NTRODUCTION
You have seen in class that the wave propagation properties of transmission lines are
governed by several parameters. One of these parameters is the relative dielectric constant,
εr . In this lab, you will measure εr for a lossless transmission line. Using the measured
value of εr , you will determine the phase velocity u p and phase constant β of the line.
There are many formats used to display the reflections caused by mismatched loads.
Some of these formats include magnitude and phase, VSWR, and the Smith Chart. In
this lab, we will focus on the Smith Chart. The Smith Chart is a very useful tool in
microwave engineering. It allows the microwave engineer to represent the complicated
equations governing the reflections at load mismatches in a very compact graphical format.
The network analyzer has the Smith Chart as one of the display options. In this lab,
several different phenomena will be examined using the Smith Chart. In addition, the
concept of matching will be introduced.
There are several techniques that are widely used to match loads to transmission lines.
Among these techniques, three of the most common distributed-element techniques are:
quarter-wave transformers, single stub tuners, and double stub tuners. The first two methods
are covered in this course. The last method, double stub tuning, is covered in more advanced
courses.
2-3
U SEFUL E QUATI ON S
λ = up/ f
(m)
u p = ω/β
(m/s)
εr = (c/u p )
2
2-4
E QUIPME NT
Z0 = √L/C
(Ω)
β = ω √με
(rad/m)
ω√με = ω √L C (m-1 )
Item
Part #
Cables & connectors
——
Calibration Kit
HP 85032E
Lossy line simulator
——
Network Analyzer
HP 8712C
Computer and Printer
Single stub tuner
——
Various Loads
——
30
LAB EXERCISE 2: THE SMITH CHART
2-5
E X PERI ME NT
2-5.1
Line Parameters of a Lossless Line
In this experiment, you will use the network analyzer to compute the line parameters of a
lossless line (β (rad/s), C (F/m), L (H/m), u p (m/s), and εr ) by making measurements to
determine ε r .
Although the cable that you will use is a coaxial cable which does have some inherent
loss, we will make the assumption for this experiment that the attenuation constant α ≈0.
We are justified in making this assumption since the loss for this particular cable is 4.1
dB/100 ft. Since the piece of cable you are using is only 2 in length, the loss is ≈0.082
dB which is less than 1%.
The relative permittivity ε r for the transmission line is determined by measuring u p for
the transmission line and comparing it to u p for free space (a vacuum). Phase velocity can
be determined by measuring the time (Δt) that a wave takes to travel down a transmission
line of length l and return to the network analyzer. The value of ε r can be found from u p
computed using the relation:
up =
2l
Δt
(m/s),
(2.1)
To measure the time delay of the cable, you will use the electrical delay feature of the
network analyzer and terminate the patch cord with a load of known reflection coefficient.
In this experiment, the transmission line will be terminated in a short. Recall that Γ = -1
for a short-circuit termination.
When a transmission line is placed between the network analyzer and a load
termination, an electrical delay is added to the load response. The term electrical delay
means that the signal has an added phase shift from traversing the transmission line. In
particular, if you look at a short termination, you will see that the response is a set of
concentric circles wrapping around the outside edge (|Γ| = 1 circle) of the Smith Chart.
When you add electrical delay, you are actually adding a phase correction to the
received signal. The phase correction is given in terms of time. When you have added
the correct amount of electrical delay, the response of the short termination will collapse to
a point.
In reality, the response will collapse to what is termed a “point-like” response (as
demonstrated in Fig. 2-1). This means that the response is not a true point, but is more
spread out. This is due to the finite loss in the cable and connectors.
2-5
EXPERIMENT
Figure 2-1: Example of a “point-like” response on the Smith Chart
Setup
This experiment requires the network analyzer, patch cord, open-circuit termination, and
short-circuit termination.
Setup the experiment:
• Configure the network analyzer.
– Press Preset .
– Set the start frequency to 500 MHz.
– Set the stop frequency to 1 GHz.
– Set the network analyzer to measure reflection on channel 1.
– Display the data on channel 1 in Smith Chart format:
- Press Format
- Press the Smith Chart softkey
– Perform a 1 channel calibration (no cable present).
Procedure
1. Measure the physical length of the patch cord (measure from the center of the
connectors and add 2.6 cm for the N jack to N jack adaptor). Record this value.
31
32
LAB EXERCISE 2: THE SMITH CHART
2. Attach the patch cord to channel 1 as shown in Fig. 2-2.
Network Analyzer
1
2
Load
Patch Cord
Figure 2-2: Block diagram for Section 2-5.1
3. Connect the short termination to the end of the patch cord. Place a marker at
800 MHz.
4. Add electrical delay until the response is point like.
• Press Scale
• Press the Electrical Delay softkey
• Rotate the knob on the network analyzer to the right until the
response is point like. The amount of added delay is displayed on
the network analyzer.
5. Record the added electrical delay. Print or save the display.
6. Set the electrical delay to 0 ns.
Note: After pressing the Electrical Delay softkey, you can enter 0 on the keypad and
press Enter to set the electrical delay to 0 ns.
Measured Data
Copy the following chart into your lab book and fill in the measured data. If you are
missing any data, please repeat the necessary parts of this experiment before proceeding to
the analysis section.
Patch Cord Length
Added Electrical Delay
=
=
(m)
(ns)
2-5
EXPERIMENT
33
Analysis
1. Using Eq. 2.1, compute u p for the patch cord using the experimentally determined
electrical delay. Record the computed value of u p .
2. Using c = 3×108 (m/s), and the value of u p computed in step 1, compute εr for the
patch cord. Record the computed ε r value.
3. Using the computed value of εr , compute β for this transmission line (assume μ = μ 0
and recall f = 800 MHz). Record the computed value of β.
4. Using Z0 = 50 Ω and the relationship με = L C , compute L and C of the transmission
line. Record the computed values.
Questions
1. No measurements were made to compute R and G . Why?
2. What is the difference between a lossless line and a dispersionless (distortionless)
line?
Would this technique for measuring the line parameters work for a
dispersionless line? Why or why not?
3. When performing the electrical delay measurements, why did we choose the short to
terminate the cables instead of the matched load?
4. When computing u p using the electrical delay technique, why did we have to use
twice the length of the cable?
2-5.2
Lossy And Lossless Lines On The Smith Chart
In this experiment, you will use the network analyzer in the Smith Chart display format to
show graphically how a wave propagates on a lossy and a lossless line.
At low frequencies, it is very difficult to make a lossy line. In this lab, you will be
using a lossy line simulator (see Fig. 2-3) that has been made to demonstrate the response
of a lossy line on a network analyzer. The lossy line simulator consists of a piece of wire
soldered to a BNC connector and is housed in a metal box. Carbon absorber has been placed
in the box to make the system more lossy. The wire acts essentially as a small antenna
element. The impedance of the antenna element changes with frequency. In addition, the
line radiates energy into the carbon absorber where the energy is converted to heat (a tiny
amount). These two effects give the lossy line simulator the same characteristic response
as a lossy line.
Figure 2-3: The lossy line simulator.
34
LAB EXERCISE 2: THE SMITH CHART
Setup
This experiment uses the network analyzer, patch cord, and the lossy line simulator.
Procedure
1. Connect the patch cord (lossless line) to channel 1 of the network analyzer. Print or
save the resulting display.
2. Connect the lossy line simulator to the end of the patch cord using a N jack to BNC
plug adaptor. Print or save the display.
Measured Data
There were no measurements made for this section, only the two printouts.
Analysis
1. Qualitatively comment on the display produced by the lossless patch cord. Why do
you see a series of circles for your response? Are the circles relatively stationary in
their position on the Smith Chart, that is, do they overlap each other or is there a
‘spiral’ pattern?
2. Qualitatively comment on the display produced by the lossy line simulator. Are
the circles relatively stationary in their position on the Smith Chart, that is, do they
overlap each other or is there a ‘spiral’ pattern? Why?
Questions
1. If the lossy transmission line was nearly infinite in length, where would the response
spiral to? Why? Explain in terms of reflected power.
2-5.3
Impedance Measurements Using the Smith Chart
In this experiment, you will use the network analyzer in the Smith Chart display format to
make load impedance measurements.
Setup
This experiment uses the network analyzer and various loads.
Setup the experiment:
• Connect the patch cord to channel 1 of the network analyzer as shown in Fig. 2-4.
• Configure the network analyzer.
– Press Preset
– Set the start frequency to 200 MHz
– Set the stop frequency to 1.2 GHz
– Set the network analyzer to measure reflection on channel 1
2-5
EXPERIMENT
35
– Set the display format to Smith Chart
– Perform a 1 channel calibration
– Save the instrument state
Network Analyzer
1
2
Load
Patch Cord
Figure 2-4: Setup for Section 2-5.3
Station
1
2
3
4
5
R (Ω)
22
100
10
68
39
C (pF)
39
10
39
15
1
Table 2-1: Resistor and capacitor values for the resistive, capacitive, series, and parallel
loads.
Procedure
1. Connect the short termination to the end of the patch cord. Print or save the resulting
display. Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and
1.0 GHz. To speed up the measurement process, you can place a marker at each of
the frequencies by using the Marker function and using markers #1, #2, #3, and #4.
The values of the markers will be displayed on the side of the display when you press
Marker .
2. Connect the open termination to the end of the patch cord. Print or save the display.
Record the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz.
3. Connect the resistive termination to the end of the patch cord. Record the resistor
value listed in Table 2-1. Print or save the display. Record the values of the
impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz.
4. Connect the capacitive termination to the end of the patch cord. Record the capacitor
value listed in Table 2-1. Print or save the display. Record the values of the
impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz.
36
LAB EXERCISE 2: THE SMITH CHART
5. Connect the series termination to the end of the patch cord. Print or save the display.
Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz.
6. Connect the parallel termination to the end of the patch cord. Print or save the display.
Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz.
Measured Data
Copy the following charts into your lab book and fill in the measured data. If you are
missing any data, please repeat the necessary parts of this experiment before proceeding to
the analysis section.
Resistor
Capacitor
=
=
(Ω)
(pF)
Load Impedance
The first line shows the frequency (MHz). The second and third lines show the real and
imaginary components of impedance (Ohm), respectively. Your recording for impedance
should be a complex number.
Load
Short
Open
Resistive
Capacitive
Series
Parallel
400 MHz
600 MHz
800 MHz
1.0 GHz
Analysis
1. For each of the loads, compute the theoretical impedance∗ at each frequency
(400 MHz, 600 MHz, 800 MHz, and 1.0 GHz). Assume that the length of the
microstrip line is 2 cm and εr =7. For the short and open terminations, assume 0.0
cm between the calibration point and the measurement point. Record the theoretical
impedances.
2. Enter each of the theoretical impedances on a Smith Chart. Use a different Smith
Chart for each load, and represent each frequency using a different color. Draw the
associated constant Γ circles.
3. Compare the theoretical and measured values of the load impedance for each of the
loads at each frequency. Comment on your results. If the measured and theoretical
values do not agree, determine if the measured values make sense and explain. (Hint:
what does the constant Γ circle represent?)
4. For each of the loads, comment on the impedance response as a function of frequency
(using the printouts). Is this what you expect to see? Why or Why not?
∗
The capacitor and resistor values used in the series and parallel loads are the same as the values used in the
resistive and capacitive loads.
2-5
EXPERIMENT
Questions
1. Assuming that you did not have a short termination, explain using the Smith Chart
how you could make a short circuit from an open circuit and a transmission line of
appropriate length. How long does transmission line need to be? How frequency
dependent is your solution?
2. Based on your observations, is the microstrip line used in the resistive termination
a wideband or a narrowband transmission line? Why? You may assume that all
connectors and loads are ideal.
2-5.4
Impedance Matching Using a Single Stub Tuner
In this experiment, you will use the single stub tuner to match an unknown load to the patch
cord. You will do it first experimentally, and then confirm analytically that you did achieve
the best match possible.
Setup
This experiment uses the network analyzer, patch cord, single stub tuner, and the unknown
load.
Setup the experiment:
• Attach the patch cord to channel 1 of the network analyzer.
• Set the start frequency to 450 MHz
• Set the stop frequency to 550 MHz
Procedure
1. Connect the unknown load to the patch cord. Print or save the display and record the
value of the impedance at 500 MHz.
2. Insert the single-stub tuner between the patch cord and the unknown load as shown
in Fig. 2-5. Record the distance between the load and the stub tuner.
37
38
LAB EXERCISE 2: THE SMITH CHART
Adjustable Short
Network Analyzer
Unknown Load
Patch Cord
Stub Tuner
(a) Setup for Section 2-5.4 with the single stub tuner
Network Analyzer
Single Stub Tuner
1
2
Load
Patch Cord
(b) Block diagram of setup
Figure 2-5: Setup for Section 2-5.4
3. Adjust the position of the sliding short on the single stub tuner (see below) until the
reflection coefficient is as close to that of a matched load (Γ=0) at 500 MHz as you
can make it. The resulting stub length is measured from the base of the sing stub
tuner (the ‘Tee’) to the center of the sliding short. Record the resulting stub length.
Print or save the resulting display and record the value of the resulting impedance at
500 MHz.
2-5
EXPERIMENT
39
To adjust the length of the stub:
The length of the stub is adjusted by sliding the short up and down the
line. Twist the short to the left to unlock it. Slide the position of the short
to the bottom of the stub (near the connector). Slowly move the short up
the stub until a match is achieved. To lock the short in place, twist the
short to the right.
Measured Data
Copy the following chart into your lab book and fill in the measured data. If you are
missing any data, please repeat the necessary parts of this experiment before proceeding to
the analysis section.
Impedance before matching
Distance between load and stub tuner
Length of stub
Impedance after matching
=
=
=
=
(Ω)
(cm)
(cm)
(Ω)
Note: All measurements made at 500 MHz
Analysis
1. In this experiment, you only had control of the length of the stub tuner, that is, the
distance from the load was fixed. In general, if you were to design a matching system
using a single stub tuner, you would be able to choose both the length of the stub and
the distance from the load. Compute the theoretical value of the stub length by using
the following procedure:
(a) Compute the normalized impedance zl of the unknown load (Z0 = 50 (Ω)).
(b) Enter this point on a Smith Chart. Label this point as A.
(c) Draw the constant VSWR (S) circle that passes through point A.
(d) Mark the point on the Smith Chart that corresponds to the normalized load
admittance y l = 1/zl . Label this point as B.
(e) Compute the recorded distance, d, between the load and the tuner in terms of
wavelengths (f = 500 MHz, εr = 1). Record this value.
(f) To find the input admittance of the load branch, move a distance d toward
the generator along the constant S circle. Remember that the Smith Chart
repeatsλ2 every , so express d as:
λ
d = d + n , n an integer
2
and move a distance d toward the generator. Label this point as D. Record the
admittance y d at point D.
40
LAB EXERCISE 2: THE SMITH CHART
(g) Determine the needed normalized input admittance (ystub ) of the stub to cancel
out the imaginary part of y d (y stub = -i · Im(yd )). Mark this point on the Smith
Chart. Label this point E.
(h) Mark the point on the Smith Chart that corresponds to the admittance of a short
circuit. Label this point F.
(i) Determine the length of the stub in wavelengths by subtracting the position of F
from E on the Wavelengths Toward Generator (WTG) scale. If the stub length
is negative, add λ2 to the length.
(j) Using the wavelength for a 500 MHz signal (assume free space (ε r = 1)),
compute the physical length of the stub. Record the computed length of the
stub. Is this length uniquely determined? Why or why not?
2. Compare the experimentally determined stub length to the theoretical stub length.
Comment on the results.
3. Compute the input impedance of the resulting load impedance and record this value.
Compare the computed load impedance to the measured impedance after matching.
Questions
1. Were you able to achieve a perfect match with the single stub tuner? If not, why not?
2. Would a quarter wave transformer have achieved a better match for the unknown
load? Why or why not?
3. Suggest a way that you could make use of both a single stub tuner and a quarter wave
transformer to achieve a better match than with either of them alone.
2-6
L AB W RITE -U P
For each section of the lab, include the following items in your write-up:
(a) Overview of the procedure and analysis
(b) Measured data where asked for
(c) Calculations (show your work!)
(d) Any tables, printouts, and Smith Charts
(e) Comparisons and comments on results
11 Magnetically Coupled
Circuits
Magnetic flux
Contents
11-1
11-2
11-3
11-4
11-5
Overview
Magnetic Coupling
Transformers
Energy Considerations
Ideal Transformers
Three-Phase Transformers
Problems
i1
i2
+
υ1
_
N1
+
N2
υ2
_
Primary port
Secondary port
Objectives
Learn to:
■ Incorporate mutual coupling in magnetically
coupled circuits.
■ Analyze circuits containing magnetically coupled
coils.
■ Relate input to output voltages, currents, and
impedances for magnetically coupled transformers,
including ideal transformers and three-phase
transformers.
When two physically unconnected inductors are in
close proximity to one another, current flow through
one of them induces a magnetically coupled voltage
across the other one. Magnetic coupling may be
intentional or not. Highly coupled voltage transformers used in power distribution networks are an
example of intentional coupling. If the coupling
between two coils in a circuit is unintentional but
significant, its effects should be incorporated into the
analysis of the circuit.
Overview
i1(t)
Voltage transformers are used in many electrical
systems, including power supply circuits (Section 7-9)
and power distribution networks (Chapter 10). Whereas
resistors, capacitors, and inductors are one-port, twoterminal devices, a transformer is a two-port device
with a primary port and a secondary port. Coupling
of energy between the two ports is realized through
a shared magnetic f eld, without the need for direct
contact between them. Transformers are part of a family
of devices and circuits called magnetically coupled
circuits, whose operation relies on magnetic coupling
rather than current conduction.
We begin this chapter by examining the voltage and
current relationships between the primary and secondary
ports of a coupled two-coil system. We did so previously
in Section 7-9.1, but then our treatment was limited to
the special case of the ideal transformer with perfect
coupling. In this more comprehensive examination, we
introduce the concepts of mutual inductance, equivalent
circuits, and impedance transformations, and we learn
how three-phase transformers are conf gured to stepup or step-down voltage levels in three-phase power
circuits.
11-1
~+_
Φ12
+ Φ11
+
υ1
υ2
_
_
N1 turns N2 turns
(a) Current i1 induces υ2
Φ21
+
i2(t)
Φ22
υ1
+
υ2
_
_
~+_
N1 turns N2 turns
(b) Current i2 induces υ1
Figure 11-1: Magnetically coupled coils.
stepping down voltage levels. On the other hand, mutual
inductance between two inductors in a certain circuit
may be totally unintentional, as well as unavoidable.
In that case, we should learn how to account for the
voltages induced by the mutual inductance and how to
incorporate them in the analysis of the circuit.
The two magnetically coupled coils in Fig. 11-1(a)
have N1 turns on port 1 and N2 turns on port 2. Port 1
is connected to a source that causes current i1 (t) to
f ow through coil 1, which generates magnetic flux Φ11
linking coil 1 alone and f ux Φ12 linking both coils. The
total flux linking coil 1 is
Magnetic Coupling
Magnetic coupling can occur between any two inductors
in close proximity of one another. Current f ow through
the coils of one of the inductors induces a mutual
inductance voltage across the other inductor, and vice
versa. The induction process is described in terms
of a mutual inductance, measured in henrys (H), that
depends on the degree of magnetic coupling between the
two inductors, which in turn depends on their physical
shapes, orientations relative to one another, spacing
between them, and the magnetic permeability µ of
the medium between them. Mutual inductance may
be intentional or not. It is key to the operation of
highly coupled transformers used for stepping up and
Φ1 = Φ11 + Φ12 .
(11.1)
Magnetic flux linkage Λ1 is def ned as the total f ux
linking all N1 turns of coil 1,
Λ1 = N1 Φ1 .
(11.2a)
For coil 2, the f ux is limited to Φ12 and the
corresponding Λ2 is
Λ2 = N2 Φ12 .
43
(11.2b)
44
August 22, 2012
CHAPTER 11
Self inductance L1 of coil 1 is def ned as the ratio of the
magnetic f ux linkage Λ1 to the current i1 responsible for
inducing Λ1 ,
Λ1
L1 =
,
(11.3)
i1
If we were to reverse the roles of coils 1 and 2, by
connecting the source to coil 2 instead of to coil 1,
thereby causing current i2 to f ow through coil 2, as
depicted in Fig. 11-1(b), we would end up with the
following expressions for υ1 and υ2 :
and the voltage induced across inductor L1 is
di1
dΛ1
= L1
.
υ1 =
dt
dt
(11.4a)
dΛ2
dΦ12
dΦ12 di1
×
= N2
= N2
.
dt
dt
di1
dt
(11.4b)
Both υ1 and υ2 are induced by di1 /dt. In the case of υ1 ,
the link is self inductance L1 , as given by Eq. (11.4a).
To establish an analogous relationship between di1 /dt
and υ2 , we rewrite Eq. (11.4b) as
υ2 = N2
dΦ12 di1
×
.
di1
dt
(11.4c)
Next, we def ne the mutual inductance M21 , as
M21 = N2
dΦ12
,
di1
(11.5)
and the expression for υ2 becomes
υ2 = ±M21
di1
.
dt
and
υ1 = ±M12
υ2 = L2
By analogy, Λ2 in coil 2 induces voltage υ2 , with
υ2 =
MAGNETICALLY COUPLED CIRCUITS
(11.6)
◮ Subscripts 21 refer to the fact that M21 is the
inductance of coil 2 due to the magnetic f eld
induced by current i1 . ◭
Mutual inductance M21 is a positive quantity measured
in henrys (H), but υ2 may be positive or negative,
depending on the direction of the winding in coil 2
relative to the direction of the winding in coil 1.
di2
dt
di2
.
dt
(11.7a)
(11.7b)
◮ Because the coupled coils constitute a linear
system, energy considerations (Section 11-3)
require that M12 = M21 = M, where M is now called
the mutual inductance between the two coils. ◭
The ambiguity between the (+) and (−) signs in
Eqs. (11.6) and (11.7a) is resolved through the use of
a standard dot convention based on the directions of the
two windings. For a specif c direction of i1 (left-hand
side Fig. 11-2), the polarity of υ2 depends on whether
the dots are on the same or opposite terminals of the
windings and whether i1 enters coil 1 at its dotted or
undotted terminal.
◮ In a two-coil magnetically coupled system, if
current enters the f rst one at its dotted terminal, the
polarity of the mutual-inductance voltage induced
across the second coil is positive at its dotted
terminal. The polarity of the induced voltage
is reversed if the current in the f rst coil enters
at the undotted terminal. Moreover, reciprocity
applies: current in the second coil induces a mutualinductance voltage across the f rst one in accordance
with the same dot convention. ◭
This dot convention covers all combinations of current
directions and dot locations outlined in Fig. 11-2.
Finally, if we generalize to the conf guration shown
in Fig. 11-3(a) in which currents f ow through both coils
11-1
MAGNETIC COUPLING
i1
August 22, 2012
L1
L2
+
di1
υ2 = M dt
_
+
di2
υ1 = M dt
_
L1
(a)
i1
L2
(e)
M
L1
i2
M
L2
+
di1
υ2 = −M dt
_
+
di2
υ1 = −M dt L1
_
(b)
L2
(f)
M
L1
i2
M
M
45
M
L2
+
di1
υ2 = −M dt
_
+
di2
υ1 = −M dt L1
_
L2
i2
i1
(c)
(g)
M
L1
M
L2
+
di1
υ2 = M dt
_
+
di2
υ1 = M dt
_
L1
L2
i2
i1
(d)
(h)
Figure 11-2: Dot convention for the mutual-inductance voltage induced in coil 2 by current i1 in coil 1, and vice versa.
simultaneously, voltage υ1 will contain two components,
one due to self-inductance of coil 1 and another due
to the mutual inductance between the two coils. That
is, υ1 will be the sum of Eqs. (11.4a) and (11.7a),
and similarly, υ2 becomes the sum of Eqs. (11.6) and
(11.7b). Specif cally:
Dots on same ends and currents entering coils at
same ends [Fig. 11-3(a)]:
υ1 = L1
di1
di2
+M
dt
dt
(11.8a)
di2
di1
+M
.
dt
dt
(11.8b)
and
υ2 = L2
46
August 22, 2012
CHAPTER 11
i1
R1
υs1
i2
M
+
+
_
di1
di2
υ1 = L1 dt + M dt
_
Change + to −
if i1 is CCW
MAGNETICALLY COUPLED CIRCUITS
R2
+
L1
L2
Change + to −
if i2 is CW
di2
di1
υ2 = L2 dt + M dt
_
Change + to −
if i2 is CW
+
_ υs2
Change + to −
if ii is CCW
(a) Dots on same ends
i1
R1
υs1
+
_
i2
M
+
di1
di2
υ1 = L1 dt − M dt
_
Change + to −
if i1 is CCW
+
L1
Change − to +
if i2 is CW
L2
R2
di2
di1
υ2 = L2 dt − M dt
_
Change + to −
if i2 is CW
+
_ υs2
Change − to +
if ii is CCW
(b) Dots on opposite ends
Figure 11-3: Polarities of voltage components for clockwise (CW) and counterclockwise (CCW) current directions.
Dots on opposite ends but current entering coils at
same ends [Fig. 11-3(b)]:
υ1 = L1
di2
di1
−M
dt
dt
(11.9a)
di2
di1
−M
.
dt
dt
(11.9b)
We start by transforming the ac circuit from the time
domain to the phasor domain [Fig. 11-4(b)]. The angular
frequency is
ω = 2π f = 2π × 103 rad/s.
and
υ2 = L2
Solution:
Example 11-1: 1-kHz Circuit
Determine load current iL (t) in the circuit of
Fig. 11-4(a), given that υs (t) = 10 cos(2π × 103t) (V),
R1 = 5 Ω, C1 = C2 = 10 µ F, L1 = 1 mH, L2 = 3 mH,
M = 0.5 mH, and RL = 20 Ω.
Denoting I1 and I2 as the mesh currents in the two loops,
both def ned with clockwise directions, the mesh-current
equations are
µ
¶
j
−Vs + R1 −
+ jω L1 I1 − jω MI2 = 0 (11.10a)
ωC
and
µ
¶
j
− jω MI1 + jω L2 −
+ RL I2 = 0,
ωC
(11.10b)
11-1
MAGNETIC COUPLING
C1
R1
υs(t)
August 22, 2012
C2
M
R1
iL
+
_
L1
L2
RL
+
_
Vs
47
jωL1
+ V1 __
V2
I1
jωM
jωL2
+
I2
RL
(a) Time domain
Figure 11-5: Circuit of Example 11-2.
R1 −j/ωC1
Vs
+
_
I1
−j/ωC2
jωM
IL
jωL1
jωL2
I2
RL
(b) Phasor domain
Figure 11-4: Circuit of Example 11-1.
where C = C1 = C2 . Note that the polarity of the last
term in Eq. (11.10a) is negative because, in accordance
with the convention shown in Fig. 11-2(g), the winding
dots are on the same end in Fig. 11-4(b) but I2 enters the
undotted terminal of coil 2. Simultaneous solution of the
two equations for IL = I2 gives
jω MVs
´³
´
.
R1 + jω L1 − ωjC RL + jω L2 − ωjC + ω 2 M 2
(11.11)
Substitution of the specif ed values leads to
IL = ³
IL = 139.5e j142.2 mA,
◦
iL (t) = Re[IL e
3
Solution:
Before we apply mesh analysis, let us determine V1
and V2 across the two inductors. Voltage V1 consists
of two terms, jω L1 I1 due to current I1 entering at the
(+) terminal of V1 , and jω M(I2 − I1 ) due to current
(I2 − I1 ) through L2 . The polarity of the second term is
governed by the dot convention: if current enters a coil at
its dotted terminal, the polarity of the mutual-inductance
voltage induced across the second coil is positive at its
dotted terminal. In the present case, (I2 − I1 ) enters L2
at its dotted terminal, so the voltage it induces across L1
is positive at the dotted terminal of L1 . Hence,
V1 = jω L1 I1 + jω M(I2 − I1 ).
and its time-domain equivalent is
jω t
The circuit in Fig. 11-5 has the following element values:
◦
Vs = 30e j60 V, L1 = 10 mH, L2 = 30 mH, R1 = 5 Ω,
RL = 10 Ω, and the ac source operates at 60 Hz. The
circuit layout is such that inductors L1 and L2 experience
a relatively small mutual inductance M. Determine
the average power delivered to the load RL for (a)
M = 4 mH, (b) M = 1 mH, and (c) M = 0.
] = 139.5 cos(2π ×10 t +142.2 ) mA.
◦
Application of the same rule to L2 gives
V2 = jω L2 (I2 − I1 ) + jω MI1 .
The f rst mesh equation is
Example 11-2: Coupled Inductors
(11.12a)
−Vs + R1 I1 + V1 − V2 = 0,
(11.12b)
48
August 22, 2012
CHAPTER 11
Error
or equivalently,
(R1 + jω L1 + jω L2 − j2ω M)I1
− ( jω L2 − jω M)I2 = Vs .
Ignoring M altogether would incur an error in PL of
(11.13a)
Similarly, for the second mesh,
V2 + RL I2 = 0,
% error =
PL (@M = 4 mH) − PL (@M = 0)
× 100
PL (@M = 4 mH)
= 2.61%,
when true M is 4 mH,
% error = 0.61%,
when true M is 1 mH.
and
or equivalently,
−( jω L2 − jω M)I1 + (RL + jω L2 )I2 = 0.
(11.13b)
(a) M = 4 mH
Example 11-3: Equivalent Inductance
Upon replacing R1 , RL , L1 , L2 with their specif ed
values, setting M = 4 mH, and multiplying inductances
by ω = 2π f = 2π × 60 = 377 rad/s, matrix solution of
the two equations gives
I2 = 1.657e j63.1 A,
◦
(11.14)
and according to Eq. (8.3), the corresponding average
power absorbed by RL is
PL =
MAGNETICALLY COUPLED CIRCUITS
1
1
|I2 |2 RL = (1.657)2 × 10 = 13.73 W. (11.15)
2
2
For the circuit in Fig. 11-6(a), obtain an expression for
the equivalent inductance, Leq , def ned such that it would
exhibit the same i-υ characteristic at nodes (a, b) as the
actual circuit.
Solution:
Equivalency means that circuits 11-6(b) and (c) will
have the same current I f owing through both loops when
connected to the same voltage source Vs . For the twoinductor circuit in Fig. 11-6(b),
(b) M = 1 mH
I1 = I2 = I,
Repetition of the process with M = 1 mH gives
I2 = 1.64e
and
j62.15◦
A
PL = 13.45 W.
(11.16)
(11.17)
(c) M = 0
and
In the absence of mutual coupling between the two coils,
I2 = 1.635e
and
and while I1 enters L1 at its dotted terminal, I2
enters L2 at its undotted terminal. While guided by
Fig. 11-3(b), application of the dot convention to the
loop in Fig. 11-6(b) gives
j62.03◦
PL = 13.372 W.
A
(11.18)
(11.19)
V1 = jω L1 I1 − jω MI2 = jω (L1 − M)I
V2 = jω L2 I2 − jω MI1 . = jω (L2 − M)I.
At terminals (a, b),
Vs = V1 + V2 = jω (L1 + L2 − 2M)I.
TRANSFORMERS
August 22, 2012
L1
a
(a)
Leq
49
Exercise 11-1: Repeat Example 11-1 after moving
L2
M
b
the dot location on the side of L2 from the top end
of the coil to the bottom.
Answer: iL (t) = 139.5 cos(2π × 103t − 37.8◦ ) mA.
O
DR
M
(See
C
11-2
)
Exercise 11-2: Repeat Example 11-3 for the two
Vs
+
+
_
M
I2
+
V2
_
(c)
Vs
a
+
_
O
DR
)
L2
b
I
Answer: Leq = L1 + L2 + 2M. (See
C
V1 _
M
(b)
L1
a I1
I
in-series inductors in Fig. 11-6(a), but with the dot
location on L2 being on the top end.
Leq
b
Figure 11-6: Finding Leq of two series-coupled inductors
(Example 11-3).
For the circuit in circuit Fig. 11-6(c),
Vs = jω Leq I.
Equivalency leads to
Leq = L1 + L2 − 2M.
Concept Question 11-1: What determines the polarity of the mutual inductance voltage? Summarize
the rules of the dot convention.
Concept Question 11-2: What factors determine
how strong or weak the magnetic coupling is
between two coils?
11-2
Transformers
11-2.1
Coupling Coefficien
To couple magnetic f ux between two coils, the coils
may be wound around a common core [Fig. 11-7(a)], on
two separate arms of a rectangular core [Fig. 11-7(b)],
or in any other arrangement conducive to having a
signif cant fraction of the magnetic f ux generated
by each coil shared with the other. The coupling
coefficient k def nes the degree of magnetic coupling
between the coils, with 0 ≤ k ≤ 1. For a loosely coupled
pair of coils, k < 0.5; for tightly coupled coils, k > 0.5;
and for perfectly coupled coils, k = 1. The magnitude
of k depends on the physical geometry of the two-coil
conf guration and the magnetic permeability µ of the
core material.
◮ A transformer is said to be linear if µ of its core
material is a constant, independent of the magnitude
of the currents f owing through the coils (and hence,
the strength of the induced magnetic f eld). ◭
Most core materials, including air, wood, and ceramics,
are non-ferromagnetic, and their µ is approximately
equal to µ0 , the permeability of free space. When
non-ferromagnetic materials are used for the common
50
August 22, 2012
CHAPTER 11
Transformer core
i1
+
_
N1
υ1
i2
be designed to exhibit coupling coefficient approaching
unity.
As was noted earlier in connection with Fig. 11-1(a),
current i1 , through coil 1 generates magnetic f uxes Φ11
through coil 1 and Φ12 through both coils 1 and 2. The
coupling coeff cient is given by
k=
+
N2
MAGNETICALLY COUPLED CIRCUITS
υ2
_
(11.20a)
The perfectly coupled case corresponds to when the f ux
coupled to coil 2, namely Φ12 , is equal to the selfcoupled f ux Φ1 . Similarly, from the standpoint of coil 2,
Magnetic flux
(a) Cylindrical core
k=
Magnetic flux
i1
Φ12
Φ12
=
.
Φ11 + Φ12
Φ1
Φ21
Φ21
=
.
Φ22 + Φ21
Φ2
(11.20b)
Through energy considerations, k can be related to L1 ,
L2 , and M as
i2
+
υ1
N1
_
k= √
+
N2
υ2
_
M
.
L1 L2
(11.21)
The mutual inductance M is a maximum when k = 1
(perfectly coupled transformer),
Primary port
Secondary port
(b) Rectangular core
M(max) =
√
L1 L2 .
(11.22)
(perfectly coupled transformer with k = 1)
Figure 11-7: Magnetically coupled coils.
11-2.2
core around which the coils are wound, the magnitude
of k depends entirely on how tightly coupled the two
windings are. Such transformers are indeed linear,
but the magnitude of k is seldom greater than 0.4.
Increasing k requires the use of ferromagnetic cores, but
the transformer behavior becomes nonlinear. The degree
of nonlinearity depends on the choice of materials.
With certain types of purifie iron, transformers can
Input Impedance
In addition to the two coupled coils, a realistic
transformer circuit should include two resistors, R1
and R2 , to account for ohmic losses in the coils.
The circuit shown in Fig. 11-8 ref ects this reality by
including resistor R1 on the side of coil 1 and resistor R2
on the side of coil 2. The circuit is driven by a voltage
source Vs on the primary side and terminated in a
complex load ZL on the secondary side.
11-2
TRANSFORMERS
Vs
+
_
M
R1
a
August 22, 2012
L1
I1
We note that
R2
L2
I2
ZL
b
ZR =
Zin (M = 0) = R1 + jω L1 .
a
+
_
I1
Zin
b
(b) Equivalent circuit
Figure 11-8: (a) Transformer circuit with coil resistors R1
and R2 , and (b) in terms of an equivalent input impedance Zin .
In terms of the designated mesh currents I1 and I2 , the
KVL mesh equations are
and
−Vs + (R1 + jω L1 )I1 − jω MI2 = 0
(11.23a)
− jω MI1 + (R2 + jω L2 + ZL )I2 = 0.
(11.23b)
From the standpoint of source Vs , the circuit to the right
of terminals (a, b) can be represented by an equivalent
input impedance Zin , as depicted in Fig. 11-8(b). By
manipulating Eqs. (11.23) to eliminate I2 , we can
generate the following expression for Zin :
Zin =
ω 2M2
Vs
= (R1 + jω L1 ) +
I1
R2 + jω L2 + ZL
= (R1 + jω L1 ) + ZR ,
(11.24)
where we def ne the second term as the reflected
impedance ZR , namely
ZR =
ω 2M2
.
R2 + jω L2 + ZL
ω 2M2
.
impedance of secondary loop
In the absence of coupling between the two windings of
the transformer (i.e., M = 0), Zin reduces to
(a) Original circuit
Vs
51
(11.25)
This is exactly what we expect for a series RL circuit
connected to a load Vs . When M is not zero, the
impedance of the secondary circuit, (R2 + jω L2 + ZL ),
becomes part of the input impedance of the primary
circuit, enabled by the magnetic coupling represented
by M. This dependence is akin to reflectin the
impedance of the secondary circuit onto the primary
circuit. The input and ref ected impedances are related
by
Zin = Zin (M = 0) + ZR .
(11.26)
The expressions given by Eqs. (11.24) and (11.25)
were derived for a transformer circuit in which the
windings have dots on the same ends. Repetition of the
process for windings whose dots are on opposite ends
leads to the same results.
◮ Zin depends on the degree of magnetic coupling,
but not on whether the coupling is additive or
subtractive. ◭
Example 11-4: Input Impedance
Determine current I1 in the circuit of Fig. 11-9.
Solution:
From the given circuit, we deduce that ω M = 2 Ω. By
analogy with Eq. (11.24), Zin is given by
Zin = (3 − j2 + j5) +
◦
22
= 4.2e j42.2 Ω.
6 + 4 − j4 + j20
52
August 22, 2012
CHAPTER 11
I1
120
30o
+
_
V
a
3Ω
Zin
−j2 Ω
j5 Ω
j2 Ω
MAGNETICALLY COUPLED CIRCUITS
−j4 Ω
j20 Ω
6Ω
4Ω
b
Figure 11-9: Circuit of Example 11-4.
Hence,
Vs
120e j30
− j12.2◦
I1 =
=
A.
◦ = 28.6e
j42.2
Zin 4.2e
◦
11-2.3 Equivalent Circuits
A circuit is said to be electrically equivalent to another if
both exhibit the same I-V relationships at a specif ed set
of terminals. For the transformer in Fig. 11-10(a), phase
voltages V1 and V2 are related to I1 and I2 by
V1 = jω L1 I1 + jω MI2
(11.27a)
V2 = jω L2 I2 + jω MI1 ,
(11.27b)
and
· ¸ ·
¸· ¸
jω (Lx + Lz )
V1
j ω Lz
I1
=
.
V2
j ω Lz
jω (Ly + Lz ) I2
(T-equivalent circuit)
(11.27c)
(transformer)
T-Equivalent Circuit
In anticipation of next steps, we have joined in
Fig. 11-10(a) the negative terminals of V1 and V2
together, which imparts no impact on the operation
of the transformer. Part (b) of the f gure displays a
proposed T-equivalent circuit whose element values
Lx , Ly and Lz automatically incorporate the magnetic
(11.28)
The transformer and its T-equivalent circuit exhibit the
same I-V relationships if the four terms in the matrix of
Eq. (11.27) are identical with their corresponding terms
in the matrix of Eq. (11.28). Equalization of the two
matrices leads to
Lx = L1 − M,
which can be cast in matrix form as
¸· ¸
· ¸ ·
V1
j ω L1 j ω M I 1
=
.
V2
j ω M j ω L2 I 2
coupling present in the transformer coils, thereby
avoiding the need to account for the mutual-inductance
terms when writing KVL equations. The I-V matrix
equation for the T-circuit (also called a Y-circuit) is
(11.29a)
Ly = L2 − M,
(11.29b)
Lz = M.
(11.29c)
and
(transformer dots on same ends)
Had the transformer dots been located on opposite
ends, the two terms involving M in Eq. (11.27) would
have been preceded by minus signs. Consequently, the
11-2
TRANSFORMERS
August 22, 2012
I1
+
L1
V1
_
Lx
Ly
+
Lz
V1
_
I2
+
(a) Transformer
I1
M
53
L2
V2
_
Lc
I1
I2
+
+
V2
V1
+
Lb
_
_
(b) T-equivalent
circuit
I2
(c)
La
V2
_
∏-equivalent
circuit
Figure 11-10: The transformer can be modeled in terms of T- or Π-equivalent circuits.
element values of inductors Lx , Ly , and Lz would be
Lx = L1 + M,
(11.30a)
Ly = L2 + M,
(11.30b)
and
Lz = −M.
(11.30c)
(transformer dots on opposite ends)
Even though a negative value for inductance Lz
is not realistic, the mathematical equivalency holds
nonetheless and the equivalent circuit is perfectly
applicable.
Π-Equivalent Circuit
In some situations, it may be easier to analyze the larger
circuit within which the transformer resides by replacing
the transformer with a Π-equivalent circuit instead of
the T-equivalent circuit. In such cases, we can use
the model shown in Fig. 11-10(c). The expressions
for La , Lb , and Lc can be obtained either by repeating
the procedure we used for the T-equivalent circuit or
by applying the Y-∆ transformation equations given in
Section 7-4.2. Either route leads to:
La =
Lb =
and
L1 L2 − M 2
,
L1 − M
L1 L2 − M 2
,
L2 − M
L1 L2 − M 2
.
M
(transformer with dots on same ends)
Lc =
(11.31a)
(11.31b)
(11.31c)
If the transformer dots are located on opposite ends,
M in Eq. (11.31) should be replaced with −M.
Example 11-5: Equivalent Circuit
54
August 22, 2012
CHAPTER 11
I1
120
30o
V
a
−j2 Ω
3Ω
+
_
MAGNETICALLY COUPLED CIRCUITS
−j4 Ω
j2 Ω
j5 Ω
6Ω
4Ω
j20 Ω
b
(a) Original circuit
I1
120
30o
V
a
−j2 Ω
3Ω
+
_
I1
j18 Ω −j4 Ω
j3 Ω
jωLx
jωLz
jωLy
j2 Ω
6Ω
I2
4Ω
b
(b) Equivalent circuit
Figure 11-11: (a) Original circuit and (b) after replacing transformer with T-equivalent circuit.
Use the T-equivalent circuit model to determine I1 in the
circuit of Fig. 11-11.
Solution:
Concept Question 11-3: What does the coupling
coeff cient represent? What is its range?
jω Lx = jω L1 − jω M = j5 − j2 = j3 Ω,
jω Ly = jω L2 − jω M = j20 − j2 = j18 Ω,
jω Lz = jω M = j2 Ω.
Its solution is
I1 = 28.6e− j12.2 A,
◦
How is the mutual
inductance M related to L1 and L2 for a perfectly
coupled transformer?
Concept Question 11-5: Why does the ref ected
impedance ZR bear that name?
Exercise 11-3: The expression for Zin given by
Eq. (11.25) was derived for the circuit in Fig. 11-8,
in which both dots are on the upper end of the coils.
What would the expression look like were the two
dots located on opposite ends?
Answer: The expression remains the same. (See
O
DR
M
The T-equivalent circuit is shown in Fig. 11-11(b).
Application of the mesh analysis by-inspection method
leads to the matrix equation
·
¸· ¸
(3 − j2 + j3 + j2)
− j2
I1
− j2
( j2 + j18 − j4 + 6 + 4) I2
·
◦¸
120e j30
=
.
0
Concept Question 11-4:
C
Use of Eq. (11.29) gives
and
which is identical with the answer obtained in Example
11-4 using the input impedance method.
)
11-3
ENERGY CONSIDERATIONS
August 22, 2012
M
i1
What are the element values
of the Π-equivalent circuit of the transformer in
Fig. 11-11(a)?
Exercise 11-4:
i2
L1
jω La = j32 Ω, jω Lb = j5.33 Ω,
jω Lc = j48 Ω. (See )
55
L2
Answer:
C
11-3
M
O
DR
(a) Transformer
Energy Considerations
i1
Given that the transformer in Fig. 11-12(a)—with
inductance L1 on the primary side, L2 on the secondary
side, and mutual inductance M coupling the two coils—
is equivalent to the T-equivalent circuit in Fig. 11-12(b),
we can use the latter to compute the total amount of
energy stored in the transformer for any specif ed values
of currents i1 and i2 . According to Eq. (5.58), the
magnetic energy stored in an inductor L due to the f ow
of current i through it is
w(t) =
1 2
L i (t)
2
(J).
Ly = L2 − M
i2
i1 + i2
M
(b) T-equivalent
Figure 11-12: Transformer and its T-equivalent circuit.
Reversing the direction of either current or if dots are on
opposite ends, M should be replaced with −M.
(11.32)
For the circuit in Fig. 11-12(b),
1
1
1
Lx i21 + Ly i22 + Lz (i1 + i2 )2
2
2
2
1
1
1
= (L1 − M)i22 + (L2 − M)i22 + M(i1 + i2 )2
2
2
2
1
1
= L1 i21 + L2 i22 + Mi1 i2 .
(11.33)
2
2
w(t) =
Equation (11.33) applies to transformers in which i1
and i2 both enter or both leave the dotted terminals,
and both dotted terminals are on the same end (as in
Fig. 11-12). Reversing the direction of either current or
reversing the locations of the dots requires replacing M
with −M.
Example 11-6: Magnetic Energy
Lx = L1 − M
In the circuit in Fig. 11-13, determine the magnetic
energy stored in the transformer at t = 0, given that
υs (t) = 12 cos(377t + 60◦ ) V.
Solution:
We start by replacing the transformer with its
T-equivalent circuit and then transforming the new
circuit to the phasor domain [Fig. 11-13(b)]. Per
Eq. (11.30), the values of Lx , Ly , and Lz are
Lx = L1 + M = (10 + 6) mH = 16 mH,
Ly = L2 + M = (30 + 6) mH = 36 mH,
and
Lz = −M = −6 mH.
Transforming the inductors to the phasor domain entails
multiplying the inductance values by jω = j377 rad/s,
56
August 22, 2012
CHAPTER 11
υs
+
_
The time-domain equivalents are
6 mH
5Ω
10 mH
30 mH
10 Ω
i1 (t) = 1.91 cos(377t + 26.06◦ ) A
and
i2 (t) = 0.29 cos(377t − 112.5◦ ) A.
(a) Original circuit
j6 Ω
5Ω
12
60o
V
+
_
I1
j13.57 Ω
10 Ω
−j2.26 Ω
I2
(b) Equivalent circuit in phasor domain
Figure 11-13: Circuits of Example 11-6.
which leads to
The magnetic energy stored in the three inductors of the
circuit in Fig. 11-13(b) at t = 0 is
¸¯
·
¯
1
1
1
2
2
2 ¯
Lx i1 + Ly i2 + Lz (i1 − i2 ) ¯
w(0) =
2
2
2
t=0
=
1
× 16 × 10−3 × (1.91 cos 26.06◦ )2
2
1
+ × 36 × 10−3 × [0.29 cos(−112.5◦ )]2
2
1
+ × (−6 × 10−3 )
2
· [1.91 cos 26.06◦ − 0.29 cos(−112.5◦ )]2
= 13.7 mJ.
jω Lx = j377 × 16 × 10−3 = j6 Ω,
and
MAGNETICALLY COUPLED CIRCUITS
jω Ly = j377 × 36 × 10−3 = j13.57 Ω,
jω Lz = j377 × (−6 × 10−3 ) = − j2.26 Ω.
Mesh analysis by inspection gives
·
¸· ¸
(5 + j6 − j2.26)
+ j2.26
I1
+ j2.26
(10 + j13.57 − j2.26) I2
·
◦¸
12e j60
=
.
0
Solution of the matrix equation for I1 and I2 leads to
I1 = 1.91e j26.06 A
◦
and
I2 = 0.29e− j112.5 A.
◦
11-3
ENERGY CONSIDERATIONS
August 22, 2012
57
Chapter 11 Summary
Concepts
Π-equivalent circuits.
• The coupling coeff cient of an ideal transformer
is k = 1. Its secondary-to-primary voltage
and current ratios are def ned by the turns
ratio n = N2 /N1 .
• Three-phase transformers are used to couple any
combination of Y- and ∆-conf gurations on the
primary and secondary sides.
• Current f ow through a coil in close proximity
of a second coil induces a mutual inductance
voltage across the second coil through a shared
magnetic f eld.
• The dot convention, which accounts for the
directions of the windings in the two coupled
coils, def nes the polarities of the induced
mutual-inductance voltages.
• A transformer can be modeled in terms of T- or
Mathematical and Physical Models
Magnetic Coupling
i1
R1
υs1
i2
M
+
+
_
di1
di2
υ1 = L1 dt + M dt
_
Change + to −
if i1 is CCW
R2
+
L1
L2
Change + to −
if i2 is CW
di2
di1
υ2 = L2 dt + M dt
_
Change + to −
if i2 is CW
+
_ υs2
Change + to −
if ii is CCW
(a) Dots on same ends
i1
R1
υs1
+
_
i2
M
+
di1
di2
υ1 = L1 dt − M dt
_
Change + to −
if i1 is CCW
+
L1
Change − to +
if i2 is CW
L2
di2
di1
υ2 = L2 dt − M dt
_
Change + to −
if i2 is CW
(b) Dots on opposite ends
M
k= √
L1 L2
R2
+
_ υs2
Change − to +
if ii is CCW
58
August 22, 2012
CHAPTER 11
MAGNETICALLY COUPLED CIRCUITS
Mathematical and Physical Models (continued)
Equivalent Inductance
Equivalent Circuits
I1
(a) Transformer
I2
M
+
+
L1
V1
_
I1
(b) T-equivalent
circuit
L2
Lx
V2
_
I2
+
+
Lz
V1
M
_
L1
a
Leq
Lc
I1
∏-equivalent
+
circuit
V1
_
La
M
L2
b
I2
+
Lb
L2
Leq = L1 + L2 + 2M
V2
Lx = L1 − M
Ly = L2 − M
Lz = M
V2
_
Leq = L1 + L2 − 2M
Ideal Transformer
V2
=n
V1
L1 L2 − M 2
L1 − M
L1 L2 − M 2
Lb =
L2 − M
L1 L2 − M 2
Lc =
M
La =
I2 1
=
I1 n
Zin =
Replace M with −M if dots are on opposite ends.
Glossary of Important Terms
Π-equivalent circuit
autotransformer
coupling coeff cient
dot convention
input impedance
Leq
b
Ly
_
(c)
L1
a
1 V2 ZL
V1
= 2
= 2
I1
n I2
n
Provide def nitions or explain the meaning of the following terms:
magnetic f ux
magnetic f ux linkage
mutual inductance
mutual voltage
primary side
ref ected impedance
secondary side
step-down transformer
step-up transformer
T-equivalent circuit
three-phase transformer
transformer
turns ratio
PROBLEMS
August 26, 2012
PROBLEMS
+
20 0o V _
f = 60 Hz
*11.1 For the circuit shown in Fig. P11.1, determine (a)
i(t) and (b) the average power absorbed by RL .
0.2 H
10 Ω
+
30 0o V _
f = 60 Hz
j6 Ω
2Ω
Section 11-1: Magnetic Coupling
10 mH
j2 Ω
j4 Ω
4Ω
Figure P11.5: Circuit for Problem 11.5.
i
1H
59
Determine Ix in the circuit of Fig. P11.6.
11.6
RL
200 Ω
j2 Ω
4Ω
Figure P11.1: Circuit for Problem 11.1.
11.2 For the circuit in Fig. P11.2, determine (a) iL (t)
and (b) the average power dissipated in RL .
30
0o
V
j6 Ω
+
_
j4 Ω
−j10 Ω
Ix
8Ω
Figure P11.6: Circuit for Problem 11.6.
14 Ω
100 μF
6 mH
+
_ 12 cos 377t (V) 10 mH
*11.7
30 Ω
30 mH
Determine Ix in the circuit of Fig. P11.7.
iL
j6 Ω
RL 10 Ω
j10 Ω
5Ω
j30 Ω
Ix
Figure P11.2: Circuit for Problem 11.2.
60
0o
V
+
_
j20 Ω
j8 Ω
j5 Ω
20 Ω
11.3 For the circuit in Fig. P11.3, determine Vout .
Figure P11.7: Circuit for Problem 11.7.
4Ω
+
10 0o V _
j3 Ω
j2 Ω
2Ω
+
j1 Ω
j4 Ω
j6 Ω −j2 Ω
11.8 Determine the average power dissipated in the
4-Ω resistor of the circuit in Fig. P11.8.
5Ω
Vout
j10 Ω
j3 Ω
j6 Ω
_
2Ω
Figure P11.3: Circuit for Problem 11.3.
*11.4 Determine Vout in the circuit shown in Fig. P11.4.
11.5 Determine the average power dissipated in the
4-Ω resistor of the circuit in Fig. P11.5.
+
20 0o V _
f = 60 Hz
j2 Ω
j3 Ω
j4 Ω
Figure P11.8: Circuit for Problem 11.8.
4Ω
60
August 22, 2012
CHAPTER 11
8Ω
96
0o
+
V _
2Ω
j1 Ω
MAGNETICALLY COUPLED CIRCUITS
4Ω
+
j2 Ω
−j4 Ω
j6 Ω
−j6 Ω
2Ω
Vout
8Ω
Vout
_
Figure P11.4: Circuit for Problem 11.4.
j12 Ω
6Ω
10
0o
j6 Ω
j18 Ω
+
+
V _
4Ω
8Ω
−j4 Ω
_
Figure P11.9: Circuit for Problem 11.9.
11.9 Determine Vout in the circuit of Fig. P11.9.
2Ω
11.10 The circuit shown in Fig. P11.10 uses a 12-V
ac source to deliver power to an 8-Ω speaker. If the
average power delivered to the speaker is 1.8 W at an
audio frequency f = 1 kHz, what is the value of the
coupling coeff cient k?
4Ω
+
12 cos 2πft (V) _
3 mH
M
12 mH
8Ω
Figure P11.10: Circuit for Problem 11.10.
*11.11 Determine Vout in the circuit in Fig. P11.11.
11.12 Determine Ix in the circuit of Fig. P11.12, given
that Vs = 20∠30◦ (V).
4Ω
−j2 Ω
6Ω
Ix
Vs
+
_
j2 Ω
j2 Ω
j6 Ω
j8 Ω
Figure P11.12: Circuit for Problem 11.12.
11.13 Determine:
(a) Leq at terminals (a, b) in Fig. P11.13(a).
(b) Leq at terminals (a, b) in Fig. P11.13(b).
*(c) Leq at terminals (a, b) in Fig. P11.13(c).
(d) Leq at terminals (a, b) in Fig. P11.13(d).
(e) Leq at terminals (a, b) in Fig. P11.13(e).
(f) Leq at terminals (a, b) in Fig. P11.13(f).
Sections 11-2 and 11-3: Transformers and Energy
11.14 Determine (a) the input impedance and (b) the
ref ected impedance, both at terminals (a, b) in the
circuit of Fig. P11.14.
PROBLEMS
August 22, 2012
4Ω
100
0o
−j2 Ω
j4 Ω
8Ω
−j4 Ω
+
+
V _
j6 Ω
j12 Ω
6Ω
Vout
_
Figure P11.11: Circuit for Problem 11.11.
5 mH
1 mH
a
a
Leq
0.5 mH
5 mH
20 mH
4 mH
Leq
b
6 mH
2 mH
10 mH
b
(a)
(b)
1H
2H
a
a
1H
5H
2H
5H
4H
b
b
6H
(c)
2H
(d)
a
a
20 mH
20 mH
1H
5H
5H
b
40 mH
10 mH
b
(e)
(f)
Figure P11.13: Circuits for Problem 11.13.
61
62
August 22, 2012
10 Ω
a
−j6 Ω
CHAPTER 11
MAGNETICALLY COUPLED CIRCUITS
2H
j2 Ω
j2 Ω
2H
j4 Ω
j6 Ω
b
Leq
4H
11.15 Determine (a) the input impedance and (b) the
ref ected impedance, both at terminals (c, d) in the
circuit of Fig. P11.15.
11.16 For the circuit in Fig. P11.16 (a) determine the
Thévenin equivalent to the left of ZL , (b) choose ZL for
maximum power transfer, and (c) compute the average
power absorbed by ZL .
−j2 Ω
j6 Ω
j4 Ω
11.20 For the circuit in Fig. P11.20, determine the
complex powers: (a) supplied by the source, (b) stored
by the two inductors, and (c) dissipated by the source
and load resistors.
j8 Ω
11.18 In the circuit of Fig. P11.18, what should
the value of the coupling coeff cent k be so that
Vout /Vin = 0.49?
10 Ω
j5 Ω
−j2 Ω
Zin
1Ω
j1 Ω −j4 Ω
j15 Ω RL
j1 Ω
j5 Ω
j1 Ω
12 Ω
+
jωM
10 Ω
j5 Ω
a
Figure P11.17: Circuit for Problem 11.17.
+
_
+
V _
*11.21 Determine input impedance Zin at terminals
(a, b) for the circuit in Fig. P11.21.
10 Ω
1Ω
0o
Figure P11.20: Circuit for Problem 11.20.
20 Ω
j1 Ω
j4 Ω
Rs
10
Zin
6H
Figure P11.19: Circuit for Problem 11.19.
*11.17 Determine the input impedance Zin of the circuit
in Fig. P11.17.
4Ω
8H
2H
Figure P11.14: Circuit for Problem 11.14.
Vin
4H
8Ω
Vout
j5 Ω
j5 Ω
b
_
Figure P11.21: Circuit for Problem 11.21.
Figure P11.18: Circuit for Problem 11.18.
11.19 Apply T- and Π-transformations to determine
Leq in the circuit of Fig. P11.19.
11.22 Determine the average power dissipated in the
10-Ω load in the circuit of Fig. P11.22, given that
Vs = 10 ∠0◦ V (rms).
PROBLEMS
August 22, 2012
j2 Ω
6Ω
4Ω
j6 Ω
j2 Ω
−j2 Ω
12 Ω
c
−j4 Ω
d
Figure P11.15: Circuit for Problem 11.15.
j10 Ω
20 Ω
40 Ω
j20 Ω
20 Ω
j30 Ω
+
_ 120
a
−j10 Ω
0o
V
b
Figure P11.16: Circuit for Problem 11.16.
2Ω
j2 Ω
4Ω
+
j4 Ω
Vs
j6 Ω
+
_
j1 Ω
j5 Ω
10 Ω
j5 Ω
Figure P11.22: Circuit for Problem 11.22.
Vo
_
ZL
63
Lab Exercise 4: Shielded-Loop Resonators
Contents
4-1
4-2
4-3
4-4
4-5
. . . . . . . . . . . . . . . . . .
IN T RO DU CTIO N . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Wireless Power Transfer: A Brief History
4.2.2 The Details of Inductive Coupling
4.2.3 Resonance
4.2.4 Resonant Shielded Loops
4.2.4 Lab Project Overview: Labs 4-6
EQ UIP M EN T
. . . . . . . . . . . . . . . . . . . . . .
EX P E RIM ENT . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Finding the Resonant Frequency
4.4.2 De-Embedding the Feedline
4.4.3 Finding R, L, and C
L AB WRIT E-UP . . . . . . . . . . . . . . . . . . . . . .
P RE - LAB
ASSIGNMENT
65
65
65
69
70
72
76
78
79
79
81
85
88
4
Lab Exercise 4: Shielded-Loop
Resonators
4.1
Pre-Lab Assignment
1. Read the supporting material in Section 4.2.
2. Read through the lab.
3. Prior to the lab, summarize the experimental procedures in your lab notebook (1 paragraph per section):
(a) Section 4.4.1
(b) Section 4.4.2
(c) Section 4.4.3
4.2
4.2.1
Introduction
Wireless Power Transfer: A Brief History
Wireless power transfer has a rich history, which dates back to over a century ago.
Its inception can be traced back to the renowned inventor Nikola Tesla, who was
65
first to demonstrate the transfer of energy without wires in the 1890’s. Wireless
power transfer is useful and often even indispensable when physical interconnects
are inconvenient, dangerous, or impossible. Today, wirelessly-powered systems
are used in implantable medical devices, wireless filtration systems, electric toothbrushes, and a variety of other commercial products. Let’s take some time to
review a brief history of wireless power transfer.
Electrostatic Induction
At the 1893 World’s Columbian Exposition in Chicago, Nikola Tesla demonstrated that he could illuminate phosphorescent lamps without the aid of connecting wires. He positioned the lamps between a pair of rubber plates, which were
suspended in air and covered in tin foil. The plates were fed with high-voltage,
high-frequency alternating current using wires. Tesla had constructed a large capacitor. The lamps between the plates were able to harvest the energy of the electric field between the plates via electrostatic induction. Electrostatic induction
is the redistribution of electrical charge induced by nearby charges. This same
apparatus was also demonstrated two years earlier by Tesla in London, where
it “produced so much wonder and astonishment.” These were the first repeated
demonstrations of wireless power transfer.
Resonant Inductive Coupling
In 1894, Maurice Hutin and Maurice LeBlanc filed U.S. Patent #527,857, in which
they described a system to power railway vehicles wirelessly, posing that it would
be “very difficult to establish a constant communication between the vehicle and
[an] electric conductor under proper conditions.” Hutin and LeBlanc proposed a
system of coils like those used in a transformer coupled by magnetic induction, a
phenomenon discovered 70 years previously by Michael Faraday. However, they
noted the inefficiency of such a system where “the magnetic conductivity [of air]
is about two thousand five hundred times inferior to that of iron”, which is “ordinarily employed in transformers for converting the energy from a primary circuit
to a secondary circuit.” To compenstate for this, the two engineers employed a capacitor to compensate for the self inductance and form a resonant system. Like a
wine glass shattered by a sound wave of prescribed frequency, there is a resonance
for which the wireless power transfer would be more efficient.
66
In that same year, 1894, Tesla lit up incandescent lamps at two different laboratories in New York City using this resonant inductive coupling technique.
Tesla would later express that he favored his ”disturbed charge of ground and
air method”, which is discussed next, over the magnetic induction method.
The Conductive Earth
In 1899, Tesla relocated from New York to Colorado Springs, where he would
have added space for several new, high-voltage experiments. Atop his laboratory
roof, he built a 142-foot mast topped with a copper sphere. He would use it to
generate man-made lightning and test theories about the electrical properties of
the Earth and its atmosphere. Using a large Tesla coil, Tesla was able to produce
a large potential difference (voltage) between the copper sphere and the ground
below, ionizing the air in between and producing lightning. During his first experiment, he strained the local generator and cut power to every residence in Colorado
Springs. Still, he was able to show that the Earth was a good conductor and proposed that it could be used to transfer power wirelessly over unprecedented distances. He was also able to show that the ionosphere was conductive. Separated
by a dielectric, i.e. air, these two layers form a natural capacitor, and, in fact, the
atmosphere of the Earth itself has a fundamental resonant frequency. Tesla would
later speculate about how he might transmit wireless power over the entire planet
via terrestrial propagation using the Earth’s natural conductive properties.
It was at his Colorado Springs laboratory that Tesla famously illuminated vacuum
tubes planted in the ground, but the details of the experiment are unclear. In
1901, construction began on Tesla’s Wardenclyffe Tower in New York, backed by
finacier J.P. Morgan. Here Tesla performed similar demonstrations. Although the
distances in these experiments were much greater than those involving magnetic
induction, they were more hazardous and expensive.
Radiative Methods
In the 1960’s, William C. Brown demonstrated microwave power transmission
by powering a model helicopter for Walter Cronkite and CBS News using an
antenna. This method is radiative, meaning that power is broadcast from the
transmitter and collected by the receiver. Up to this point, the methods of wireless
power transfer had been non-radiative: the power was not radiated, but rather
67
confined to distances less than the wavelength of operation. Radiative methods
using antennas can achieve far greater power transfer distances, but they suffer
from spatial interference, poor efficiency, and cause heating.
Revival of Resonant Inductive Coupling
Recently, interest has been revived in wireless power transfer using resonant inductive coupling: the same phenomenon used by Hutin, LeBlanc, and Tesla in the
late 1890’s. In 2007, an MIT research team led by Professor Marin Soljac̆ić wirelessly powered a 60W light bulb using resonant inductive coupling at a distance of
2 meters. They showed that EM resonators with minimal loss can allow mid-range
power transfer, as opposed to the short-range results studied previously.
Since its revival in 2007 at MIT, non-radiative wireless power transfer (resonant
inductive coupling) has gained strong scientific and commercial interest. In fact,
wireless charging is expected to grow to a 23 billion dollar industry by 2015.
Projections are that it will be used to efficiently charge mobile devices such as
laptop computers, PDA’s, digital cameras, cell phones and even electric vehicles
wirelessly. Devices would gradually charge throughout the course of the day and
have no need for a power cord connection.
Magnetic induction is attractive for several reasons. It is:
• Safe. Magnetic fields interact weakly with dielectric objects and living organisms, compared to electrostatic induction methods, which interact strongly.
• Cheap.
• Simple to implement.
During the past few years, shielded-loop resonators have been developed at the
University of Michigan. These loops are low-loss, self-resonant structures tailored for efficient wireless non-radiative power transfer systems using magnetic
induction. They offer several advantages:
• Low loss.
• High-power capability.
• Confined electric fields, which could otherwise couple to nearby dielectrics.
• Simple to feed at the input.
68
In this lab, we will study the shielded-loop resonator and use it to design our own
wireless power transfer system.
4.2.2
The Details of Inductive Coupling
From Faraday’s Law of Induction, we know that a time-varying magnetic field can
induce a current in a conducting loop of wire. Such a magnetic field could easily
be generated using a second loop of alternating current. In this manner, electromagnetic energy can be transferred wirelessly between two or more conductive
loops. Such loops are said to be inductively coupled. Inductive coupling is a nonradiative mechanism, which is less susceptible to spatial interference and does
not wastefully broadcast power. The circuit representation for magnetic coupling
between two loops/coils is shown in Fig. 4.1.
Figure 4.1: Circuit representation for two magnetically-coupled loops. A
current on one loop will induce a voltage on the other loop.
The symbol M denotes the mutual inductance between the two loops, whereas L1
and L2 are the self inductances of the individual loops. The mutual inductance is
simply a ratio of the voltage induced across a load loop to time-rate-of-change in
current on the source loop.
δi2
δt
δi1
= −M
δt
v1 = −M
v2
In the next section, we will briefly review mutual inductance in more detail.
69
Review: Mutual Inductance
Consider a pair of inductively-coupled loops: loop 1 and loop 2. Suppose loop
1 is fed with a current i1 . We know from Faraday’s Law of Induction that the
voltage induced across loop 2 is given by:
v2 =
dΛ
dt
(4.1)
The term Λ is the total magnetic flux through the loop. Note that (4.1) holds
regardless of the source of Λ. We can decompose Λ into two parts, according to
the source of the flux:
v2 =
d(Λ22 + Λ21 )
dΛ
=
dt
dt
=⇒ v2 = L
di2
di1
− M21
dt
dt
M21 , −
Λ21
I
(4.2)
where Λ22 is the magnetic flux generated by loop 2 through loop 2 and Λ21 is the
magnetic flux generated by loop 1 through loop 2. From energy considerations, it
can be shown that M = M12 = M21 (the coupling is reciprocal).
4.2.3
Resonance
The self inductances of the individual loops result in reactances that store reactive
energy and degrade power transmission. The positive reactance of each loop can
be canceled by a negative reactance (capacitance), as shown in Fig. 4.2, to increase power transfer between the two loops. Such a coupling scheme is referred
to as resonant inductive coupling. When the input reactance of the capacitivelyloaded loop is made to be zero, we say that it is in resonance. However, in order
to maximize power transfer, the input terminal (on the source loop) and output
terminal (on the load loop) must also be simultaneously impedance matched, as
will be discussed in Lab 6. For now, we wish to simply bring each loop to resonance.
70
Figure 4.2: A circuit representation of magnetically-coupled resonators.
The capacitances C1 and C2 create a resonant system at some presribed
frequency ω0 .
Figure 4.3: A schematic of magnetically-coupled resonators with parasitic
resistances R1 and R2 . All real systems will have some loss, modeled by
resistances.
In reality, parasitic resistances (R1 , R2 ) will also be present, as shown in Figure
4.3. The resistances represent ohmic (heating) loss and some inevitable radiation
loss. We can write a pair of loop equations for the circuit shown in Fig. 4.3 using
Kirchoff’s Voltage Law:
1
) − i2 (jωM )
jωC1
1
i1 (jωM12 ) = i2 R2 + i2 (jωL2 ) + i2 (
) + i2 RL
jωC2
Vx = i1 R1 + i1 (jωL1 ) + i1 (
Solving for i2 in terms of i1 , Vx can be written only in terms of i1 :
Vx = i1 R1 + i1 (jωL1 ) − i1 (
j
(ωM )2
) + i1
ωC1
R2 + RL + jωL2 −
71
j
ωC2
The input impedance becomes:
Vx
1
(ωM )2
Zin =
= R1 + j ωL1 −
+
i1
ωC1
R2 + RL + j(ωL2 −
1
)
ωC2
Employing symmetry, the output impedance can be written as:
1
(ωM )2
Zout = R2 + j ωL2 −
+
ωC2
R1 + RS + j(ωL1 −
1
)
ωC1
4.2.4
(4.3)
(4.4)
Resonant Shielded Loops
Although simple wire loops with series capacitors can realize the resonant system shown in Figure 4.3, we will use alternative loops: shielded-loop resonators.
Figure 4.4 shows an example of shielded-loop resonator with rectangular (planar)
cross section. Shielded-loop resonators possess a number of desirable characteristics: they are simple to feed, exhibit high quality (Q) factors (i.e. low loss), tolerate
high input power, and have confined electric fields. Fringing electric fields should
be avoided in a wireless power system since they couple with nearby dielectric
objects.
Figure 4.4: A planar shielded-loop resonator like those used in our experiments. This resonator is basically a looped, planar transmission line with a
cut in the outer conductor halfway around the loop.
Fig. 4.5 shows the inner conductor for the same planar shielded-loop resonator
from Fig. 4.4, revealing its termination, which forms an open stub.
72
Figure 4.5: Inner conductor of the planar shielded-loop resonator from Figure 4.4. Here, we see that the inner conductor is terminated before forming
a complete loop. The result is an open-circuited stub.
Fabricating a Shielded Loop
A shielded-loop resonator is an electrically-small loop that can be constructed
from a length of coaxial cable. One end of the coaxial cable is looped back onto
itself and connected near the opposite end of the cable. The inner conductor of the
coaxial cable is left open-circuited at the termination point (see Fig. 4.5) while the
outer conductors are connected together (see Fig. 4.4) to form a loop. Halfway
around the loop, a small portion of the outer conductor (Fig. 4.4) is removed to
form a slit.
The current flow on the shielded-loop resonator is depicted in Fig. 4.6. The current enters the input terminal of the shielded-loop resonator and propagates down
the interior of the coaxial cable to the slit in the outer conductor. It then traverses
the exterior of the loop to the opposite side of the slit and finally enters the opencircuited stub. Therefore, the loop and open-circuited stub appear in series. The
open-circuited stub provides a capacitance which resonates with the loop inductance. In this lab, the shielded loops are constructed from coaxial transmission
lines that have a planar cross section.
Modeling the Shielded Loop
The shielded-loop resonators can be modeled as an inductor in series with an
open-circuited stub (capacitance), but there is also a feedline that must be included, as shown in Fig. 4.7.
73
Figure 4.6: The behavior of currents on a shielded-loop resonator. The
current at the input propagates down the interior of the coaxial feedline.
Upon encountering the split, the current (red) on the interior of the outer
conductor wraps around to the exterior of the loop. This current (blue),
loops back around to the split, where it returns to the interior, ending at the
open-circuited stub.
Calculating the Capacitance
From transmission-line theory, we know that an open-circuited stub shorter than
a quarter wavelength acts as a capacitance:
−j
= Zin = −jZ0 cot βl
ωC
(4.5)
For an electrically-small (βl << 1) open-circuited transmission line:
jZ0
−j
−j
= −jZ0 cot βl ≈
=
=⇒ C ≈ C 0 l
ωC
βl
ωC 0 l
(4.6)
Here, C 0 is the capacitance per-unit-length of the transmission line. For the resonators in this lab, C ≈ C 0 l is an excellent approximation for the loop capacitance.
74
Figure 4.7: A model of the shielded-loop resonator. The open-circuited stub
acts as a capacitance, forming a resonant system.
The electromagnetic waves supported by the shielded-loop resonators are TEM
waves, and therefore the capacitance per-unit-length C 0 can be found using:
√
0
C =
R 1
c Z0
(4.7)
where c is the speed of light in a vacuum and R is the relative permittivity of the
dielectric.
Calculating the Inductance
For loops of circular cross section, the inductance can be approximated as:
8r
L = µr ln ( ) − 1.75
a0
(4.8)
• µ is the permeability of the surrounding medium
• r is the radius of the loop
• a0 is the cross-sectional radius of the conductor
Equation (4.8) applies only to loops with a circular cross-section, but our loops
have rectangular cross-section. However it can be shown that a wire of thin,
rectangular cross-section and a wire of circular cross-section are approximately
equivalent when:
πa20 = dh
75
(4.9)
Here, d and h are the width and thickness of the rectangular wire, respectively.
By combining (4.8) and (4.9), we can find the loop inductance of the rectangular
shielded-loop resonators, given their cross-sectional dimensions.
Calculating the Resonant Frequency
If we know all of the appropriate parameters for the shielded loop, then using (4.6)
and (4.8), we can find the resonant frequency for the shielded loop:
jω0 L +
1
1
= 0 =⇒ ω0 = √
jω0 C
LC
ω0
(Hz)
f0 =
2π
(rad/sec)
(4.10)
(4.11)
Coupled Shielded-Loop Resonators
The full circuit model for a system of coupled shielded-loop resonators is shown
in Fig. 4.8 and includes the effect of the feedlines and losses.
Figure 4.8: A circuit model for coupled shielded-loop resonators (assuming
identical loops). The model is the same as Fig. 4.3 with the addition of
feedlines.
4.2.5
Lab Project Overview: Labs 4-6
Labs 4-6 discuss the topic of wireless non-radiative power transfer (WNPT) using
a system of inductively-coupled shielded-loop resonators. You will completely
characterize a system of two resonators and use them to design a wireless power
76
transfer system. The system will be used to power a light emitting diode (LED)
wirelessly with a signal generator source.
In Lab 4, you will:
• Measure isolated (uncoupled) shielded-loop resonators using a vector network analyzer (VNA).
• De-embed the feedline of the shielded-loop resonators.
• Model the resonator as an RLC circuit.
In Lab 5, you will:
• Measure the mutual inductance between two magnetically-coupled loops as
a function of distance.
• Observe the effect of misalignment between the shielded loops.
• Study the phenomena of weak and strong coupling.
• Determine the critical coupling distance for a system of unmatched, coupled
loops.
• Measure the input impedance of a full-wave bridge rectifier.
In Lab 6, you will:
• Analyze a frequency-tuned, wireless non-radiative power transfer system.
• Choose a coupling distance at which to match the WNPT system.
• Build matching networks to maximize power transfer efficiency.
• Light up a light-emitting diode (LED) using the WNPT system.
• Use frequency tuning to maintain the brightness of the LED.
Loop Parameters
The properties of the shielded-loop resonators are given here:
77
Dielectric Properties:
R
• Material: Rogers RT/Duroid 5880
• Relative Permittivity: R = 2.2
• Loss Tangent: tan δ = 0.009
Conductor Properties:
• Material: Copper
• Conductivity: 5.8x107 Siemens
• Copper Thickness: 70 µm.
Shielded-Loop Resonator Geometry:
• Loop Radii: 5 cm. and 9 cm.
• Cross-Sectional Width: 15 mm.
• Cross-Sectional Thickness: 3.32 mm.
Rectangular Coaxial Transmission Line:
• Characteristic Impedance: 50 Ω
4.3
Equipment
• 5 cm shielded-loop resonator
• 9 cm shielded-loop resonator
• HP8712C vector network analyzer
• N-type coaxial cable (at least 2 feet long) (male/male)
• N/Female to SMA/Male Adapter
78
4.4
4.4.1
Experiment
Finding the Resonant Frequency
In this section, the resonant frequency of the shielded-loop resonators will be
measured using a vector network analyzer (VNA). From transmission-line theory, we know that the feedline of the shielded-loop resonator will affect its input
impedance. Therefore, the resonant frequency of the shielded-loop is not simply
the frequency at which the input reactance is zero. However, for an RLC resonator, the magnitude of the input reflection coefficient is minimum at resonance
(just look at the Smith chart). An example plot is shown in Fig. 4.9. Knowing
this, we can easily find the resonant frequency.
Figure 4.9: An example of input reflection coefficient (in dB). At the resonant frequency ω0 , the input reflection coefficient is at its lowest.
Setup
1. Attach the male-to-male N-type coaxial cable (at least 2 feet long) to the
reflection port of the VNA.
2. Perform a 1-port calibration of the VNA:
79
•
•
•
•
•
•
•
•
•
•
•
•
Press FREQ
Select the Start Freq softkey
Input 1 MHz and press ENTER
Select the Stop Frequency softkey
Input 150 MHz and press ENTER
Press MENU
Select the Number of Points softkey
Input 801 and press ENTER
Press MEAS 1
Press CAL
Select the One Port softkey
Complete the 1-port calibration, following the steps provided by the VNA
user interface
Procedure
1. Connect the 9 cm shielded-loop resonator to the cable at the reflection port
of the VNA using the N-to-SMA adapter.
2. View the resonance behavior on the Smith chart. If the Smith chart doesn’t
show the proper RLC resonant behavior, you may have a connection or calibration problem:
• Press MEAS 1 to work with the reflection port measurements
• Press FORMAT
• Select the Smith Chart softkey
3. View a log-magnitude plot of the input reflection coefficient:
•
•
•
•
Press FORMAT
Select the Log Mag softkey
Press SCALE
Select the Autoscale softkey
4. Find the magnitude minimum of input reflection coefficient:
• Press MARKER
• Move the marker to the minimum of the plot
5. Record the resonant frequency f0 (in MHz) and corresponding magnitude
80
of input reflection coefficient |Γ0in | into your lab notebook.
6. Repeat the steps 1-4 for the 5 cm shielded-loop resonator.
Measured Data
1. Copy the following chart into your lab notebook and fill in the measured
data:
Loop f0
5 cm
9 cm
|Γ0in |
Analysis
There are no analysis questions for this section of the lab.
Questions
1. From (4.6), which loop should have a higher capacitance?
Hint: The transmission line characteristics are identical for the two loops.
2. From (4.8), which loop should have a higher inductance?
3. From (4.10), which loop should have a higher resonant frequency? Do your
experimental results support your hypothesis?
4.4.2
De-embedding the Feedline
In this section, the electrical length of the shielded-loop feedlines will be determined using the VNA. The feedline changes the input impedance of the shieldedloop resonator. We will remove the effect of the feedline to find the input impedance
of the RLC circuit itself. This process is called “de-embedding”.
81
The insertion of a transmission line before a load will change the phase of the
input reflection coefficient of the load. Referring to Figure 4.10, the phase of the
reflection coefficient before and after the feedline are related as follows:
Γ0in = Γin e−2jβl
√
√
ω R
0
0
β = ω LC =
c
(4.12)
(4.13)
Figure 4.10: Effect of transmission line on reflection coefficient. The transmission line will alter the phase of the input reflection coefficient. A lossless
transmission line will not affect the magnitude.
Thus, the phase of the input reflection coefficient (Γ0in ) should be increased by 2βl
to find the reflection coefficient of the RLC portion of the loop (Γin ), where l is
the physical length of the feedline. Note that β is frequency dependent, and for
de-embedding each frequency point must be adjusted by a different phase value.
Figure 4.11 shows the effect of the transmission line on the Smith chart.
Setup
1. If the VNA is not already calibrated for 1-port measurements, calibrate it
following the procedure described in Section 4.4.1.
Procedure
The goal here is to remove the effect of the feedline and characterize the resonator
as an RLC circuit. To find R, we must measure the input reflection coefficient at
ω = ω0 :
82
Figure 4.11: Behavior of an example RLC resonator for the shielded loop
without (left) and with (right) feedline. The phase will be reduced by 2βl
(see (4.12)).
1. Connect the 5 cm loop to the cable at the reflection port of the VNA using
the N-to-SMA adapter.
2. Change to polar format:
• Press FORMAT
• Press the Polar softkey
3. Move the marker to the resonant frequency omega0 found earlier. This
frequency identifies the frequency at which the reflection coefficient magnitude is minimized.
• Press MARKER
• Move the marker to the loop’s resonant frequency, found in the previous
section
4. Record the magnitude and phase of S11 = Γ0in into your lab notebook.
To find L and C, we must collect information about the input reflection coefficient
at two separate frequencies close to ω0 :
1. Move the marker to the frequency 2 MHz below the resonant frequency:
• Press MARKER
• Move the marker to the frequency 2 MHz below the resonant frequency
83
2. Record the magnitude and phase of S11 = Γ0in into your lab notebook.
3. Move the marker to the frequency 2 MHz above the resonant frequency:
• Press MARKER
• Move the marker to the frequency 2 MHz above the resonant frequency
4. Record the magnitude and phase of S11 = Γ0in into your lab notebook.
• Repeat all of the steps above for the 9 cm shielded-loop resonator.
Measured Data
1. Copy the following chart into your lab notebook and fill in the measured
data:
Loop Γ0in at f0
5 cm
9 cm
Γ0in at f0 − 2M Hz
Γ0in at f0 + 2M Hz
Analysis
1. Calculate the electrical length βl of the feedline.
Hint: See Figure 4.11 and use your measurements.
2. Using (4.13) and the result of the previous question, calculate the electrical
length of the feedline at the frequency 2 MHz below resonance and a frequency 2 MHz above resonance.
Hint: Electrical length is proportional to frequency. To calculate the electrical length at the frequency 2 MHz below resonance, multiply βl by (f0 −
2M Hz)/f0 .
3. Calculate the theoretical β for both loops at their respective resonant frequencies ω0 using (4.13) and the loop parameters provided in Section 4.2.5.
4. Measure the physical length l of the feedline for the two loops.
Hint: The feedline is comprised of half the loop’s circumference and the
small section feeding the loop.
84
5. Calculate the theoretical value for the electrical length using β and l for the
two loops at their respective resonant frequencies ω0 .
Questions
1. Which loop (5 cm or 9 cm) has a feedline with larger electrical length?
Given (4.6), (4.8), and (4.10), does this make sense?
2. Which loop has a larger reflection coefficient magnitude at resonance? Should
we desire a large magnitude or a small magnitude of input reflection coefficient? What would |S11 | be for a lossless resonator?
Hint: Observe the circles of constant resistance on the Smith chart.
4.4.3
Finding R, L, and C
Using the data from the previous section, the loops can be fully characterized by
the circuit model shown in Fig. 4.8. In this section, we develop methods for
finding R, L, and C experimentally.
Finding R
For an RLC circuit at resonance, the input impedance Zin is purely real (Im{Zin } =
Xin = 0). For Zin = Rin < Z0 , the input reflection coefficient Γin is purely real
and negative:
Zin = Rin = Z0
1 − |Γin |
1 + Γin
= Z0
1 − Γin
1 + |Γin |
If we assume the feedline to be lossless (β is purely real), then |Γ0in | = |Γin | from
(4.12). So, at resonance (for Rin < Z0 ):
Rin = Z0
1 − |Γin |
1 − |Γ0in |
= Z0
1 + |Γin |
1 + |Γ0in |
(4.14)
Using (4.14), R can easily be determined using data that we have already collected
at the resonant frequency (Rin = R at resonance).
85
Finding L and C
In the previous section, we also gathered data at two other frequencies: one above
resonance and one below resonance. This is because, to find L and C, we need
measurements at two distinct frequency points. Let us label these frequency points
as ωa and ωb , where:
ωa < ω 0
ωb > ω 0
We have:
1 + Γa
= Xa = Im Z0
1 − Γa
1 + Γb
= Xb = Im Z0
1 − Γb
1
ωa L −
ωa C
1
ωb L −
ωb C
(4.15)
(4.16)
(4.17)
Here, Γa and Γb are the de-embedded input reflection coefficients (Γin ) measured
at ωa and ωb , respectively:
Γa
Γb
Xa
Xb
=
=
,
,
Γ0a e2jβa l
Γ0b e2jβb l
Im {Za }
Im {Zb }
All impedances used here are de-embedded (the effect of the feedline is removed).
Solving (4.15) and (4.16) simultaneously, we can find equations for L and C:
C =
ωb2 − ωa2
ωa2 ωb Xb − ωa ωb2 Xa
ωb X b +
L =
ωb2
1
C
ωb X b + (
=
86
(4.18)
ωa2 ωb Xb −ωa ωb2 Xa
)
ωb2 −ωa2
wb2
(4.19)
Finally, we can solve for the Q factor of our resonators. The Q factor is a measure
of the resonator efficiency. At resonance, the Q factor is defined as:
r
√
1
LC
1 L
ω0 L
=
=
=
(4.20)
Q(ω0 ) ,
R
ω0 CR
CR
R C
Setup
There is no setup for this section.
Procedure
For both the 5 cm loop and 9 cm loop:
1. Calculate R using (4.14).
2. Calculate L using (4.19).
3. Calculate C using (4.18).
4. Calculate Q using (4.20).
Measured Data
1. Copy the following chart into your lab notebook and fill in the measured
data:
Loop R
5 cm
9 cm
L
C
Analysis
For both shielded-loop resonators:
1. Approximate the inductance L using (4.8) and (4.9).
87
2. Calculate the theoretical capacitance C using (4.6), (4.7), and the parameters provided in Sec. 4.2.5.
Hint: The length l used in (4.6) is half of the loop’s circumference.
Questions
1. Which loop has the higher Q factor?
2. How did the approximations for the inductance L and C compare with the
measured values for L and C.
3. Consider the frequency of a wireless power transfer system using magnetic
induction. What are some advantages of operating at a high frequency?
What might be some disadvantages?
Hint: How does a higher frequency impact Faraday’s Law of Induction,
voltage rectification, and resistive losses?
4.5
Lab Write-Up
For each section of the lab, include the following items in your write-up:
• Overview of the procedure and analysis.
• Measured data.
• Calculations (show your work!).
• Any tables and printouts.
• Comparisons and comments on results.
• Answers to all questions.
• A summary paragraph describing what you learned from this lab.
88
Lab Exercise 5: Coupled Resonators and Voltage Rectification
Contents
5-1
5-2
5-3
5-4
5-5
P RE - LAB
. . . . . . . . . . . . . .
IN T RO DU CTIO N . . . . . . . . . . . . . . . . . .
5.2.1 Calculating Mutual Inductance
5.2.2 Input Impedance of Coupled Shielded-Loop
Resonators
5.2.3 Feedline Effects
5.2.4 Weak, Critical, and Strong Coupling
5.2.5 Power Transfer
5.2.6 Introduction to Frequency Tuning
5.2.7 Voltage Rectification
E Q UIP M ENT . . . . . . . . . . . . . . . . . . . .
E X P ERIM E NT . . . . . . . . . . . . . . . . . . .
5.4.1 Measuring the Mutual Inductance
5.4.2 Strong and Weak Coupling
5.4.3 Measuring the Rectifier
L AB W RI TE- UP . . . . . . . . . . . . . . . . . .
ASSIGNMENT
. . . .
. . . .
90
90
91
95
95
96
98
98
99
. . . . 99
. . . 100
100
104
106
. . . . 100
5
Lab Exercise 5: Coupled Resonators
and Voltage Rectification
5.1
Pre-Lab Assignment
1. Read the supporting material in the textbook.
2. Read through the lab.
3. Prior to the lab, summarize the experimental procedures in your lab notebook (1 paragraph per section):
(a) Section 5.4.1
(b) Section 5.4.2
(c) Section 5.4.3
5.2
Introduction
In the previous lab, an isolated (uncoupled) shielded-loop resonator was characterized. In this lab, we introduce a second (load) loop that will inductively couple
to the first (source) loop. Recall the circuit model for the system of coupled loops,
shown in Fig. 5.1.
90
Figure 5.1: A circuit model for shielded loop resonator (assuming identical
loops).
Using coupled resonators, we can efficiently deliver power wirelessly via magnetic induction. As a demonstration, a light-emitting diode (LED) will be illuminated wirelessly in the next lab. In this lab, we will introduce voltage rectification,
which will convert the high-frequency signal from the generator source to a DC
value to power the LED.
5.2.1
Calculating Mutual Inductance
Figure 5.2: Illustration of two coaxially-aligned, coupled loops. For this
case, the equation for mutual inductance M is simplified and given in (5.1)
A well-known formula for mutual inductance between two coaxially-aligned, filamentary loops, like in Fig. 5.2, is given by (5.1):
91
√
M = µ r1 r2
2
2
−k K − E
k
k
(5.1)
Here, K(k) is the complete elliptic integral of the first kind, and E(k) is the
complete elliptic integral of the second kind.
Z
π
2
K(k) =
0
Z
E(k) =
dβ
p
π
2
1 − k 2 sin2 β
q
dβ 1 − k 2 sin2 β
(5.2)
(5.3)
0
s
k ,
4r1 r2
(r1 + r2 )2 + d2
(5.4)
The mutual inductance vs. distance for 5 cm loops and 9 cm loops are shown in
5.3 and Fig. 5.4, respectively.
92
Figure 5.3: Plot of mutual inductance M for 5 cm loops
93
Figure 5.4: Plot of mutual inductance M for 9 cm loops
94
5.2.2
Input Impedance of Coupled Shielded-Loop Resonators
Figure 5.5: A circuit representation of magnetically-coupled resonators.
The capacitances C1 and C2 create a resonant system at some presribed
frequency ω0 .
Recall the input impedance for the RLC circuit shown in Fig. 5.5:
Vs
1
(ωM )2
Zin =
= R1 + j ωL1 −
+
i1
ωC1
R2 + RL + j(ωL2 −
1
)
ωC2
Consider two loops with the same resonant frequency ω0 . At ω0 , the input impedance
reduces to:
Zin |ω=ω0 = Rin |ω=ω0 = R1 +
(ω0 M )2
R2 + RL
(5.5)
In this lab, (5.5) will be used to experimentally determine the mutual inductance
M between the shielded-loop resonators at various coupling distances. The input
impedance Rin at ω = ω0 is required.
5.2.3
Feedline Effects
For our shielded-loop resonators, we must include the effect of the feedlines
shown in Figure 5.1.
These particular shielded loops were designed using Z0 = 50Ω coaxial transmission lines with a rectangular (planar) cross section. If the load impedance ZL is
95
Figure 5.6: Impedance simplification at load loop for coupled shielded-loop
resonators (assuming identical loops). The load is Z0 = 50Ω, then the
feedline of the load loop will not affect the input impedance at the source
loop.
equal to Z0 = 50Ω, the feedline of the secondary loop will have no effect on the
input impedance, reducing the system to the circuit shown in Fig. 5.6.
As in the previous lab, we can de-embed the feedline of the source loop to obtain
the simplified circuit model shown in Fig. 5.5.
5.2.4
Weak, Critical and Strong Coupling
Consider the system of magnetically-coupled RLC circuits shown in Fig. 5.5. We
know that an isolated loop resonates when:
ω0 = √
1
LC
(5.6)
In general, resonance occurs in the coupled system when the the input impedance
becomes purely real (Im{Zin } = Xin = 0). For a symmetric system (two identical loops), we can show that resonance occurs when:
1
) = 0 =⇒ ω = ω0
ωC
1 2
(ωL −
) = (ωM )2 − (R + RL )2
ωC
(ωL −
(5.7)
(5.8)
Equation (5.7) yields the familiar resonance ω0 of an isolated loop. The quadratic
equation (5.8) yields two solutions for ω when:
96
(ωM ) > (R + RL )
(5.9)
When the equation (5.9) is satisfied, there are three different resonant frequencies
and we say that the loops are in strong coupling. One of the frequencies given
by (5.8) is lower than the resonant frequency given by (5.7) and is referred to as
the odd mode. The second frequency given by (5.8) is higher than the resonant
frequency given by (5.7) and is referred to as the even mode. The terms “even”
and “odd” refer to the direction of the currents in the two loops. At the even
mode frequency, the currents in the coupled loops are in-phase. At the odd mode
frequency, the currents in the coupled loops are 180◦ out-of-phase. At the resonant
frequency ω = ω0 , the current in the load loop leads that of the source loop by
approximately 90◦ . Figure 5.7 shows the presence of one resonant frequency for
weak coupling (ωM < R + RL ) and two additional resonant frequencies for
strong coupling (ωM > R + RL ).
Critical Coupling
We define critical coupling as the distance at which (5.9) becomes an equality:
ωM = R + RL . At this distance, the even and odd mode frequencies merge to the
resonant frequency ω0 , since the right-hand side of (5.8) becomes zero. When the
loops are far apart, (ωM ) < (R + RL ), the system is said to be weakly coupled.
In weak coupling, there is only one resonant frequency: ω0 .
Figure 5.7: An example of input reflection coefficient: weak coupling (left)
and strong coupling (right). In strong coupling, the odd mode and even
mode resonances appear.
97
5.2.5
Power Transfer
The power transfer efficiency of a wireless power transfer system is the ratio of
power delivered to the load and the power available from the source. When operating at the resonant frequencies described in Section 5.2.4, the equations for power
transfer efficiency of inductively-coupled RLC circuits are greatly simplified. The
power transfer efficiency is equal to |S21 |2 if the source and load impedances are
50Ω.
For a system operating at the resonant frequency ω0 given by (5.7), the power
transfer efficiency is:
4RL2 (ω0 M )2
η=
((R + RL )2 + (ω0 M )2 )2
(5.10)
For a system operating in strong coupling at an even or odd mode frequency given
by (5.8), the power transfer efficiency is:
η=
RL2
(R + RL )2
(5.11)
At critical or weak coupling, there is only one resonant frequency ω0 and the
efficiency is given by (5.10).
5.2.6
Introduction to Frequency Tuning
Lab 6 will show that, in the strong coupling regime, operating at either the even
or odd mode frequency will yield a higher efficiency than operating at the selfresonant frequency ω0 of the loops. Therefore, operating at one of these frequencies is desirable. Equation (5.8) also indicates that, in strong coupling, the even
and odd mode resonant frequencies separate further with decreasing distance (increasing coupling). Tracking the even or odd mode frequency is referred to as
frequency tuning. Frequency-tuned wireless non-radiative power transfer systems
operate in strong coupling. They change the operating frequency with coupling
distance to maintain high efficiency that is constant with distance (see (5.11)). In
this lab, we will observe the shifting resonant frequencies in the strong coupling
regime.
98
5.2.7
Voltage Rectification
The shielded-loop resonators operate at megahertz frequencies, but many applications require DC power. This requires voltage rectification. Two simple rectifiers
are half-wave rectifiers and full-wave bridge rectifiers. In this lab, we will measure
the input impedance of a voltage rectifier.
Half-Wave Rectifiers
Figure 5.8: A simple half-wave rectifier circuit.
Figure 5.8 shows the schematic of a standard half-wave rectifier. The diode blocks
the negative cycle of the input AC signal while the RC filter “smooths” the signal
to a DC value.
Full-Wave Rectifiers
Figure 5.9 shows the schematic of a standard full-wave bridge rectifier. In this
circuit, the negative cycle of the input AC signal is not blocked. Instead it is
converted to a positive value for the RC filter to “smooth”. We will use a fullwave rectifier in this lab.
5.3
Equipment
• Two 5 cm shielded-loop resonators
99
Figure 5.9: A simple full-wave rectifier circuit.
• Two 9 cm shielded-loop resonators
• HP8712C vector network analyzer
• N-type coaxial cable (at least 2 feet long) (male/male)
• N-type coaxial cable (at least 4 feet long) (male/male)
• Two N/male to 3.5mm SMA/female adapters
• Distance-marked PVC pipe
• Two tripods
• Full-wave voltage rectifier
5.4
5.4.1
Experiment
Measuring the Mutual Inductance
In this lab, the resonant frequency ω0 , the resistance R of the shielded-loop resonator, and the input resistance Rin of coupled loops are used to find M using
(5.5).
100
Setup
1. Attach the male-to-male N-type coaxial cable (at least 2 feet long) to the
reflection port of the VNA.
2. Perform a calibration on the reflection port of the VNA:
•
•
•
•
•
•
•
•
•
•
•
•
Press FREQ
Select the Start Freq softkey
Input 1 MHz and press ENTER
Select the Stop Frequency softkey
Input 150 MHz and press ENTER
Press MENU
Select the Number of Points softkey
Input 801 and press ENTER
Press MEAS 1
Press CAL
Select the One Port softkey
Complete the 1-port calibration, following the steps provided by the VNA
user interface
3. Attach the male-to-male N-type coaxial cable (at least 4 feet long) to the
transmission port of the VNA.
4. Perform a calibration on the transmission port of the VNA:
•
•
•
•
Press MEAS 2
Press CAL
Select the Response softkey
Complete the 2-port calibration, following the steps provided by the VNA
user interface
Procedure
1. Connect a 9 cm loop to the reflection port of the VNA using an N-to-SMA
adapter.
2. Connect the second 9 cm loop to the transmission port of the VNA using an
N-to-SMA adapter.
101
3. Hook the two loops onto the distance-marked PVC pipe. Suspend the pipe
and loops above the table using the two tripods.
4. Measure the magnitude of the input reflection coefficient Γ0in at ω0 for coupling distances from 2 cm to 26 cm (2 cm increments). Align the loops
coaxially for maximum coupling. Do not turn the frequency to the frequency with minimum input reflection magnitude, as this might not be ω0
for the case of coupled loops. To perform the measurements:
•
•
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Polar softkey
Press SCALE
Select the Autoscale softkey
Press MARKER
Move the marker to the resonant frequency, ω0
5. At each distance, use the VNA’s transmission plot to determine S21 at ω =
ω0 . To perform the measurements:
•
•
•
•
•
•
•
Press MEAS 2
Press FORMAT
Select the Log Mag softkey
Press SCALE
Select the Autoscale softkey
Press MARKER
Move the marker to the resonant frequency, ω0
6. At a coupling distance of 10 cm, turn the load loop approximately 30◦ so
that the loops are no longer coaxially-aligned. Measure the magnitude of
the input reflection coefficient at ω0 , as detailed in step 4. Do the same for
an angle of approximately 60◦ . These angles need not be exact: estimate
them visually. We are simply trying to show the effects of misalignment.
Do not hold the loops anywhere except the short input segment.
7. Remove both loops from the VNA.
8. Repeat steps 2-7 using the 5 cm loops.
102
Measured Data
1. Copy the following chart into your lab notebook for both the 5 cm loop and
the 9 cm loop and fill in the measured data:
Distance (cm.)
2
4
6
8
10
12
14
16
18
20
22
24
26
|Γ0in |ω=ω0
|S21 |(dB) at ω = ω0
2. Copy the following chart into your lab notebook for both the 5 cm loop and
the 9 cm loop and fill in the measured data:
Distance (cm.)
10cm (30◦ misalignment)
10cm (60◦ misalignment)
|Γ0in |ω=ω0
Analysis
For both the 5 cm and 9 cm loop:
1. Calculate the de-embedded input resistance for the measured distances at
ω = ω0 . Recall that for ω = ω0 , only the magnitude of the reflection coefficient is needed:
Rin = Z0
1 − |Γ0in |
1 + |Γ0in |
103
(for R < Z0 )
(5.12)
2. Using (5.5), calculate the mutual inductance at these distances.
3. Calculate the mutual inductance for the system of loops misaligned by 30◦ .
Do the same for the 60◦ case.
4. Calculate the power transfer efficiency at each distance using the transmission coefficient S21 .
Note: These efficiency values are for an unmatched system.
Questions
1. How do the measured mutual inductance values compare with the plot in
Fig. 5.3 and Fig. 5.4? Why might the experiment not match exactly with
the plot?
Hint: What assumptions are made in (5.1)?
2. What happened to the mutual inductance for the misaligned loops? Why?
5.4.2
Strong and Weak Coupling
Setup
If the VNA is not already calibrated for both reflection and transmission measurements, perform the calibrations described in Section 5.4.1.
Procedure
1. Connect a 9 cm loop to the reflection port of the VNA using an N-to-SMA
adapter.
2. Connect a 9 cm loop to the transmission port of the VNA using an N-toSMA adapter.
3. Begin with the loops far apart (i.e. in weak coupling), but coaxially-aligned.
Decrease the coupling distance, while maintaining coaxial alignment. Measure the critical coupling distance.
Hint: At critical coupling, a transition occurs from a single resonance to
two resonances: even and odd mode resonances.
104
4. Bring the loops even closer together in order to enter strong coupling. Observe the behavior of the resonant frequencies as the loops are moved closer
together. Does the behavior match the plot in Fig. 5.7?
5. Repeat steps 1-4 using 5 cm loops.
Measured Data
1. Copy the following chart into your lab notebook fill in the critical coupling
distances:
Loop
5 cm
9 cm
Experiment Theory
Analysis
1. Use the plots in Fig. 5.3 and Fig. 5.4 as well as equation (5.9) to determine
the theoretical critical coupling distance for the 9 cm loops.
2. Determine the theoretical critical coupling distance for 5 cm loops.
Questions
1. As the resistance R of the shielded-loop resonators increases, what happens
to the critical coupling distance?
2. As R increases, what happens to the efficiency in strong coupling?
3. Describe the behavior of the resonant frequencies as the loops are moved
closer into strong coupling. What are the implications for frequency tuning?
4. How do the theoretical and experimental critical coupling distances compare?
105
5.4.3
Measuring the Rectifier
The schematic of the provided full-wave bridge rectifier is shown in Fig. 5.10. In
this section, we characterize and test the rectifier. It will be used in Lab 6 when
the complete wireless non-radiatve power transfer system will be assembled. For
the complete system, we will use only the 9 cm loops, so all measurements will
be performed at the resonant frequency of the 9 cm loop.
Figure 5.10: The schematic for the full-wave bridge rectifier circuit used in
these experiments.
Setup
1. Attach the male-to-male N-type coaxial cable (at least 2 feet long) to the
reflection port of the VNA.
2. Perform a 1-port calibration of the VNA:
106
•
•
•
•
•
•
•
•
•
•
•
•
Press FREQ
Select the Start Freq softkey
Input 1 MHz and press ENTER
Select the Stop Frequency softkey
Input 150 MHz and press ENTER
Press MENU
Select the Number of Points softkey
Input 801 and press ENTER
Press MEAS 1
Press CAL
Select the One Port softkey
Complete the 1-port calibration, following the steps provided by the VNA
user interface
Procedure
Measuring the Input Impedance:
1. Connect the rectifier to the reflection port of the VNA using an N-to-SMA
adapter.
2. Change the output power level of the VNA to 18 dBm. This input power
is higher than we will use in the final system, but most of the power will
be reflected for an unmatched rectifier. To get a more accurate impedance
measurement, we use a higher power. Note that the LED will be blinking
strangely because the VNA is sweeping over a range of frequencies during
measurement:
• Press POWER
• Input 18 dBm
3. Measure the input return loss and input impedance of the rectifier at the resonant frequency ω0 of the 9 cm loop. The input impedance should be in the
lower right quadrant of the Smith chart.
To read the input impedance:
107
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Smith Chart softkey
Press MARKER
Move the marker to the resonant frequency ω0 of the 9 cm loop
To measure the return loss:
• Press FORMAT
• Select the Log-Mag softkey
Measured Data
Record the following into your lab notebook:
1. Rectifier input return loss at ω = ω0 (input power: 18 dBm).
2. Rectifier input impedance at ω = ω0 (input power: 18 dBm).
Analysis
There are no analysis questions for this section.
Questions
1. Why must we increase the power level of the VNA to measure the rectifier?
Hint: What do we know about diodes?
5.5
Lab Write-Up
For each section of the lab, include the following items in your write-up:
• Overview of the procedure and analysis.
• Measured data.
• Calculations (show your work!).
108
• Any tables and printouts.
• Comparisons and comments on results.
• Answers to all questions.
• A summary paragraph describing what you learned from this lab.
109
Lab Exercise 6: Impedance Matching and Frequency Tuning
Contents
6-1
6-2
6-3
6-4
6-5
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
6.2.1 Complex-Conjugate Matching
6.2.2 Matching Network Topologies
6.2.3 Solving for the Reactances
6.2.4 Capacitors vs. Inductor: Q Factor
6.2.5 Return Loss
6.2.6 (Optional Section) Matching Networks as
Two-Port Networks
6.2.7 Matching for the Shielded-Loop Resonators
6.2.8 Analysis of Frequency-Tuned System
EQ UIP M ENT . . . . . . . . . . . . . . . . . . . . . .
EX P ERIM EN T . . . . . . . . . . . . . . . . . . . . .
6.4.1 (Pre-Lab) Designing the Matching Networks
6.4.2 Building the Matching Networks
6.4.3 Frequency-Tuned System
LAB WRI T E-UP . . . . . . . . . . . . . . . . . . . . .
P RE- LAB AS SIG NM EN T
INT RO DU CTI ON . . . .
111
112
112
113
115
117
118
119
120
123
126
127
127
128
135
140
6
Lab Exercise 6: Impedance
Matching and Frequency Tuning
6.1
Pre-Lab Assignment
1. Read the supporting material in the textbook.
2. Read through the lab.
3. Prior to the lab, summarize the experimental procedures in your lab notebook (1 paragraph per section):
(a) Section 6.4.2
(b) Section 6.4.3
4. Find the optimal source and load impedances for the loops and design the
matching networks, as described in the Section 6.4.1.
5. Design a matching network for the rectifier, as described in the Section
6.4.1.
111
6.2
Introduction
In the previous two labs, inductively-coupled, shielded-loop resonators were introduced for the purposes of wireless non-radiative power transfer. In this lab,
we will optimize the power transfer efficiency of the system using impedance
matching and deliver power wirelessly to a light-emitting diode. To this end, we
introduce the concept of complex-conjugate matching.
6.2.1
Complex-Conjugate Matching
Consider the simple circuit shown in Fig. 6.1. We can find an expression for the
power delivered to the load using simple circuit analysis.
Figure 6.1: The reference circuit for complex-conjugate matching proof.
The voltage across the load ZL and the current through it are:
ZL
(ZS + ZL )
vS
i =
(ZS + ZL )
vA = vS
The voltages and currents are written in terms of phasors. The power, PD , delivered to the load, ZL , is:
112
1
|VS |2 Re{ZL }
∗
PD = Re{vA i } =
2
2 |ZS + ZL |2
(6.1)
Using (6.1), the source impedance ZS for which maximum power is delivered to
the load can be determined. Decomposing the complex impedances into real and
imaginary components, ZS = RS + jXS and ZL = RL + jXL , we have:
PD =
|VS |2 Re{ZL }
2 ((RS + RL )2 + (XS + XL )2 )
(6.2)
It can be easily shown using (6.2) that maximum power transfer occurs when:
RS = RL
XS = −XS
In other words, maximum power is transferred when:
ZS = ZL∗
(6.3)
Similarly, it can be shown that for a two-port (two-terminal) system, power transfer is maximized for a simultaneous, complex-conjugate match at the two ports.
In Fig. 6.2, the impedance ZIN into the first port of a two-port network is ZS∗ and
the impedance ZOU T into the second port is ZL∗ . The power delivered from the
voltage source v to the load ZL will therefore be maximized.
Using matching networks, these impedance conditions are achievable. Next, we
review how to construct these matching networks.
6.2.2
Matching Network Topologies
Classes of Matching Networks
Matching networks fall into two classes: transmission line and lumped element
networks. We know that transmission lines can be used to match any impedance
113
Figure 6.2: An impedance-matched, two-port network. Here, both the
firs and second port of the matching network present a complex-conjugate
impedance match, resulting in maximum power transfer to the load ZL .
on the Smith chart to another using prescribed lengths of lines and stubs. Our
shielded-loop resonators operate at frequencies lower than 100 MHz, resulting
in wavelengths too large to design a reasonably-sized matching network based on
transmission lines. Instead, we wil use lumped-element matching networks.
Lumped Element Topologies
Figure 6.3: There are two “L” matching types: load-parallel-series (left)
and load-series-parallel (right). Each can match to a finite set of load
impedances ZL .
The simplest lumped element topologies are shown in Fig. 6.3, where the reactance can be realized using capacitors and inductors. Using these topologies, any
impedance on the Smith chart can be matched to another. More complex topologies, such as “π” networks or “T” networks, allow manipulation of bandwidth and
other factors. The matching network topologies shown in Fig. 6.3 are called “L”
matching networks.
114
6.2.3
Solving for the Reactances
The prescribed reactances for “L” matching networks can be found using the
Smith chart. Alternatively, the reactances can be determined analytically by a
set of equations. There are two “L” matching topologies (see Fig. 6.3): loadparallel-series and load-series-parallel. A pair of equations have been derived for
each topology that solves for the prescribed reactances to match to a given load
impedance ZL .
Load-Series-Parallel
Figure 6.4: The load-series-parallel “L” matching network.
Figure 6.4 shows the load-series-parallel “L” topology. Consider the matching
network attached to a load ZL = RL + jXL . Setting the input impedance of
the network to Z0 and solving for Xs and Xp , we derive the required reactances.
There are two solutions to the problem:
Solution 1:
p
RL (Z0 − RL ) − XL
p
(Z0 − RL )/RL
Xp =
Z0
Xs =
(6.4)
(6.5)
Solution 2:
p
RL (Z0 − RL ) − XL
p
(Z0 − RL )/RL
Xp = −
Z0
Xs = −
115
(6.6)
(6.7)
Note that negative reactances are realized using capacitors and positive reactances
are realized using inductors. From (6.5) and (6.7), we see that a real solution
requires:
RL < Z0
(6.8)
Therefore, the load-series-parallel topology is limited to matching load impedances
ZL that lie in the shaded region of the Smith chart in Fig. 6.5.
Figure 6.5: The region of Smith chart for matching with load-series-parallel
“L” topology, according to equation (6.8).
Load-Parallel-Series
Figure 6.6 shows the load-parallel-series “L” topology. Consider the matching
network attached to a load ZL = RL + jXL . Setting the input impedance of
the network to Z0 and solving for Xs and Xp , we derive the required reactances.
Again, there are two solutions to the problem:
116
Figure 6.6: The load-parallel-series “L” matching network.
Solution 1:
Xp =
XL +
p
RL /Z0 RL2 + XL2 − Z0 RL
RL2 + XL2
1
XL Z0
Z0
Xs = ( ) +
−
Xp
RL
Xp RL
p
(6.9)
(6.10)
Solution 2:
Xp =
XL −
p
RL /Z0 RL2 + XL2 − Z0 RL
RL2 + XL2
1
XL Z0
Z0
Xs = ( ) +
−
Xp
RL
Xp RL
p
(6.11)
(6.12)
Note that negative reactances are realized using capacitors and positive reactances
are realized using inductors. To obtain real solutions for (6.9) - (6.12), the load
impedances ZL are restricted to a mirror image of (6.8) on the Smith chart (see
Fig. 6.7):
Re
6.2.4
1
RL
<
1
Z0
(6.13)
Capacitor vs. Inductor: Q Factor
The “L” matching networks shown in Fig. 6.3 can be implemented using inductors
and capacitors. When matching to the characteristic impedance of the system, e.g.
117
Figure 6.7: The region of Smith chart for matching with load-parallel-series
“L” topology, according to equation (6.13).
Z0 = 50Ω, there is always more than one way to realize the match. However, in
terms of resistive loss, it is generally best to use capacitors when possible. This is
because inductors exhibit higher parasitic resistances. In other words, capacitors
have a higher Q factor. The Q factor of a component is defined as the ratio of
reactance to parasitic resistance:
ωL
(Inductor)
R
1
Q ,
(Capacitor)
ωCR
Q ,
6.2.5
(6.14)
(6.15)
Return Loss
When a system is impedance-mismatched, power incident into the system is reflected. In circuit design, a figure of merit for the quality of a match is the return
loss, defined in (6.16). Essentially, the return loss is the magnitude of the input
118
reflection coefficient. A negative sign is applied so that the number is always
positive for a passive system.
Return Loss = −20 log10 (|S11 |)
6.2.6
(6.16)
(Optional Section) Matching Networks as Two-Port Networks
In linear systems theory, two-port (two-terminal) networks are circuits that can be
fully described mathematically using a 2x2 impedance matrix, as shown in (6.17).
This impedance matrix relates the total currents and voltages at the two ports,
shown in Fig. 6.8. “L” matching networks are examples of two-port networks, as
shown in Fig. 6.9.
V1
Z11 Z12 I1
=
(6.17)
V2
Z21 Z22 I2
Figure 6.8: A “black box” representation of a two-port network.
By describing matching networks lossless, two-port networks, it can be shown
that the following two statements are equivalent:
• A load ZL at port 2 is matched to the characteristic impedance Z0 at port 1
(left side of Fig. 6.10).
• If an impedance of Z0 is presented to port 1, the output impedance of the
matching network will be ZL∗ (right side of Fig. 6.10).
It is important to understand this equivalence when designing matching networks.
119
Figure 6.9: An “L” matching network represented as a two-port network.
Figure 6.10: Matching network impedance relationships. For a lossless,
two-port matching network, given an impedance Z0 measured for the network on the left, an impedance ZL∗ will be measured for the network on the
right.
6.2.7
Matching for the Shielded-Loop Resonators
Matching Two-Port Networks: Simultaneous Matching
A system of coupled shielded-loop resonators also form a two-port network. In
Section 6.2.1, it was mentioned that power transfer is maximized for a two-port
network with a simultaneous, complex-conjugate impedance match at both ports,
i.e. the input of the source loop and the output of the load loop. The keyword
here is simultaneous. Suppose we wish to match our two-port system of coupled
loops, and we begin by matching the input port. Having matched the input port,
we match the output port. However, the matching network at the output will
have disturbed the impedance match at the input. Therefore, we must achieve
the impedance matches simultaneously. For our loops, we can do this using the
equations derived in the next section.
120
Figure 6.11: Coupled RLC circuits with matching networks (assuming identical loops). The matching networks are represented by two-port “black
boxes”.
Matching the Loops as a Function of Distance
Figure 6.11 depicts coupled RLC resonators attached to both source and load
matching networks. Recall the input and output impedances of a system of inductivelycoupled RLC circuits (loops):
Zin
Zout
1
(ωM )2
= R1 + j ωL1 −
+
ωC1
R2 + ZL + j(ωL2 −
1
(ωM )2
= R2 + j ωL2 −
+
ωC2
R1 + ZS + j(ωL1 −
1
)
ωC2
1
)
ωC1
(6.18)
(6.19)
For a simultaneous, complex-conjugate match, we must ensure that:
∗
ZS = Zin
∗
ZL = Zout
(6.20)
(6.21)
Using (6.18) - (6.21), it can be show that the required source and load impedances
for a simultaneous, complex-conjugate match are:
ZSOP T
ZLOP T
r
j
R1
R2
=
− jωL1 +
R22 + (ωM )2
ωC1
R2
R1
r
j
R2
=
− jωL2 + R22 + (ωM )2
ωC2
R1
(6.22)
(6.23)
(6.24)
121
Assuming identical loops (R1 = R2 , L1 = L2 , C1 = C2 ), the optimal impedance
values are:
ZLOP T
=
ZSOP T
=
p
j
− jωL + R2 + (ωM )2
ωC
(6.25)
For our shielded-loop resonators, we must include the effects of the feedline. We
must therefore provide a simultaneous conjugate match for the circuit depicted in
Fig. 6.12.
Figure 6.12: Coupled shielded-loop resonators with matching networks.
The matching networks are represented by two-port “black boxes”.
The impedance transformation resulting from the feedline can easily be taken into
account using the following expressions from transmission-line theory:
ZS − jZ0 tan (βl)
Z0 − jZS tan (βl)
ZL − jZ0 tan (βl)
ZL0 = Z0
Z0 − jZL tan (βl)
ZS0 = Z0
(6.26)
(6.27)
Here, ZS , ZS0 , ZL , and ZL0 are the impedances labeled in Fig. 6.12. We combine
(6.25), (6.26) and (6.27) to find the impedance values for a simultaneous complexconjugate match of the coupled shielded-loop resonators:
h
ZLOP T = ZSOP T
i
p
− jωL + R2 + (ωM )2 − jZ0 tan (βl)
h
i
(6.28)
= Z0
p
j
2
2
Z0 − j ωC − jωL + R + (ωM ) tan (βl)
j
ωC
122
At the self-resonant frequency of the loops ω = ω0 :
ZLOP T = ZSOP T
hp
i
R2 + (ωM )2 − jZ0 tan (βl)
hp
i
= Z0
Z0 − j
R2 + (ωM )2 tan (βl)
(6.29)
Note that the conjugately-matched source and load impedances (given by (6.29))
depend on M, which decreases monotonically with increasing distance. Therefore, the conjugately-matched impedances depend on distance, making it difficult
to maximize efficiency over a wide range of distances. Tunable matching are
required to compensate for changes in mutual inductance in order to maintain a
simultaneous complex-conjugate impedance match over all distances. However,
such an impedance-tuned system is rather complex. Oftentimes, frequency tuning
is employed to increase efficiency, since it is a tradeoff between efficiency and
system complexity. Frequency tuning was briefly introduced in Lab 5, but a more
rigorous analysis will now be performed.
6.2.8
Analysis of Frequency-Tuned System
Maximum Power Transfer Efficiency
Recall from Lab 5 that if the mutual coupling between the loops is strong, two
additional resonant frequencies appear in addition to the resonant frequency ω0
of the isolated loops given by (6.30). The frequency above ω0 is called the even
mode and the one below is called the odd mode, given by (6.31).
1
) = 0 =⇒ ω = ω0
ωC
1
(ωL −
) = (ωM )2 − (R + RL )2
ωC
(ωL −
(6.30)
(6.31)
Recall that the power transfer efficiency is equal to the magnitude squared of the
transmission coefficient |S21 | if the source and load impedances are 50Ω:
η , |S21 |2
123
(6.32)
For a system operating at the resonant frequency ω0 given by (6.30), the power
transfer efficiency is:
η=
4RL2 (ω0 M )2
((R + RL )2 + (ω0 M )2 )2
(6.33)
(6.34)
For an impedance-matched system, the magnetically-coupled loops are always in
weak coupling, for a realistic system with R 6= 0. This can be seen by comparing
(6.25) and the strong coupling condition (6.35):
(ωM ) > (R + RL )
RL =
p
R2 + (ωM )2 =⇒ ωM < R + RL
(6.35)
(weakly-coupled)
(6.36)
Substituting (6.25) into (6.33), the maximum possible power transfer for a system operating at ω0 under a simultaneous, conjugate impedance match can be
derived:
ηmax
4 [R2 + (w0 M )2 ] (w0 M )2
= 2
2
p
2
2
2
R + R + (w0 M )
+ (w0 M )
(6.37)
As can be seen from the expression, the maximum possible efficiency increases
with increasing M (decreasing loop separation).
Operation at Critical Coupling
Consider a conjugately-matched wireless non-radiative power transfer system operating at ω0 . If M increases (distance decreases), the system will become slightly
mismatched, but the efficiency still increases until the critical coupling distance
is reached. This can be verified by showing that η is maximum when ω0 M =
R + RL . In other words:
124
δη
=0
δM
when ω0 M = R + RL
(6.38)
The right hand side of (6.38) is the familiar critical coupling condition. For
ω0 M > R + RL (shorter distances), the efficiency begins to drop off, as shown in
Fig. 6.13.
At the critical coupling distance, defined by ω0 M = R + RL , the even mode
frequency, odd mode frequency, and ω0 all coincide. The efficiency η at critical
coupling can be found by substituting ω0 M = R + RL into (6.33) to obtain:
η=
RL2
(R + RL )2
Operation at an Even or Odd Mode: Strong Coupling
Equation (6.2.8) also holds for a system operating in strong coupling at an even or
odd mode. For shorter distances (ω0 M > R + RL ), the efficiency stays constant
when operating at the even and odd mode frequencies but drops off with distance when operating at ω0 . Therefore, operating at either the even or odd mode
will yield higher efficiency than operating at ω0 , when the matching networks are
fixed.
Frequency Tuning
To maintain resonance, one solution is to tune the elements of the system as the
mutual coupling changes. An alternative is to frequency tune, in order to operate
at the even or odd mode frequencies and maintain a constant efficiency. The frequency of operation varies with mutual coupling (distance). A frequency-tuned
system would be designed as follows:
1. Pick a distance for providing an impedance match.
2. As the loops come together and enter strong coupling, the excitation frequency of the system is tuned to either the even or the odd mode in order
to achieve a constant efficiency (assuming R and RL remain constant with
frequency).
125
Figure 6.13: An example of power transfer efficiency for a system of coupled shielded-loop resonators under various operating conditions: constant
frequency, frequency tuning, and maximum efficiency (perfect impedance
match at all distances).
From (6.29), a larger matching distance requires a smaller load resistance RL .
From (6.2.8), a smaller load resistance RL yields lower efficiencies. A tradeoff
must be made between distance and efficiency.
6.3
Equipment
• HP8712C vector network analyzer
• HP8648B signal generator
• Two 9 cm shielded-loop resonators
• N-type coaxial cable (at least 4 feet long) (male/male)
• Three N-type coaxial cables (at least 2 feet long) (male/male)
• Three N/female to 3.5mm SMA/male adapters
• Distance-marked PVC pipe
• Two tripods
126
• SMA-to-SMA adapter (female-to-female)
• Three PCBs for building matching networks
• Full-wave bridge rectifier
• Assortment of capacitors: 1 pF to 1000 pF (1206 package)
• Assortment of inductors: 3.3 nH to 1.2 uH (1206 package)
• A soldering iron
• Solder
6.4
6.4.1
Experiment
(Pre-Lab) Designing the Matching Networks
In this section, you will be designing “L” matching networks for both the loops
and the rectifier. Two identical matching networks will be designed for the loops,
while a different network will be designed for the rectifier.
Setup
There is no setup for this section.
Procedure
Find the optimal source and load impedance for the loops:
1. After reviewing Section 6.2.8, choose a matching distance. Recommended
distances are anywhere between 5 cm and 20 cm.
2. From Lab 5, determine the mutual coupling, M , at this distance.
3. From Lab 4, recall the resistance R of the loop’s RLC circuit.
4. Setting ω = ω0 , use (6.29) to determine the optimal source and load impedance.
Because the system is symmetric, these impedances are equal: ZS = ZL .
127
5. Take the complex conjugate of your answer. This is the value that must be
matched to Z0 = 50Ω (see Section 6.2.6 and Fig. 6.12).
6. Use Section 6.2.3 to solve for the prescribed reactances. The impedance
should be in the upper half of the Smith chart, and it should be possible to
match using only capacitors!
Designing the matching network for the rectifier:
1. Locate the input impedance of the rectifier, measured in Lab 5, on the Smith
chart. You do not need to take the complex conjugate of this value (why?).
2. Use Section 6.2.3 to solve for the prescribed reactances. The impedance
should be in the lower half of the Smith chart. You will likely need an
inductor to match the rectifier.
Measured Data
There is no measured data for this section.
Analysis
There are no analysis questions for this section.
Questions
There are no questions for this section.
6.4.2
Building the Matching Networks
In this section, you will be building and testing the matching networks. The GSI
will provide PCBs and components to build the matching networks.
128
Testing Matching Networks
There are two ways to test a matching network: measuring the output impedance
of the matching network and measuring the return loss of the load with the matching network attached. In this experiment, we will be doing both, but let’s first
discuss how to measure the output impedance of the matching network.
We know that a matching network should present complex-conjugate impedance
to a given load ZL when the characteristic impedance Z0 is connected to its first
port (see Sections 6.2.1 and 6.2.6). Therefore, we can test the matching network
by connecting an impedance Z0 to its first port and measuring impedance of the
second port. Recall that the second port is that connected to the load ZL .
As an example, consider the load matching network for the loops (see Figure
6.12). In this case, the “first port” is shown on the right side. Disconnecting the
load matching network from the loops, we can measure the impedance. In the
experiment, we will use the VNA transmission port to present Z0 , as in Fig. 6.14,
since its input impedance is Z0 = 50Ω.
Figure 6.14: Test procedure for measuring the output impedance of the
matching network. The VNA transmission port provides an impedance Z0
to the matching network.
Setup
1. Attach the male-to-male N-type coaxial cable (at least 2 feet long) to the
reflection port of the VNA.
2. Perform a 1-port calibration of the VNA:
129
•
•
•
•
•
•
•
•
•
•
•
•
Press FREQ
Select the Start Freq softkey
Input 1 MHz and press ENTER
Select the Stop Frequency softkey
Input 150 MHz and press ENTER
Press MENU
Select the Number of Points softkey
Input 801 and press ENTER
Press MEAS 1
Press CAL
Select the One Port softkey
Complete the 1-port calibration, following the steps provided by the VNA
user interface
Procedure
Building the matching network for the loops:
1. Find circuit components with values closest to what was determined analytically.
2. Solder the circuit components to the appropriate locations on the PCB. The
male SMA connector will be used to connect to the loops. Knowing this,
solder the parallel component on the correct side.
3. Make sure that all solder joints are properly formed.
4. Build a second matching network in an identical manner. One is required
for each loop.
Building the matching network for the rectifier:
1. Find circuit components with values closest to what was determined analytically.
2. Solder the circuit components in the appropriate locations on the PCB. The
male SMA connector will be used to connect to the rectifier. Knowing this,
solder the parallel component on the correct side.
3. Make sure that all solder joints are properly formed.
130
For the rectifier’s matching network, perform the following procedure:
1. Recall Section 6.4.2. Connect the reflection port of the VNA to the port
of the matching network designed for the rectifier input (the side of the
matching network with the male SMA connector). You will need an N-toSMA adapter and the SMA-to-SMA adapter (female-to-female).
2. Connect the transmission port to the other port of the matching network
using an N-to-SMA adapter.
3. Measure the input impedance seen by the VNA at the resonant frequency of
the 9 cm loop. This is the output impedance of the matching network.
To perform the measurement:
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Smith Chart softkey
Press MARKER
Move the marker to the resonant frequency of the 9 cm loop to view the
impedance
4. The measured impedance should be the complex conjugate of the load impedance
ZL for which the match is designed (see Sec. 6.2.6). If the impedance is significantly off, make sure that the matching network was properly soldered.
5. Having tested the impedances presented by the matching network, we will
attach it and measure the input return loss of the matched rectifier. Detach
the connections between the VNA and the matching network.
6. Connect the matching network to the rectifier. The male SMA of the matching network should connect directly to the rectifier. There should be no long
lengths of cable connecting the two (why?).
7. Connect the matched rectifier to the reflection port of the VNA.
8. Change the output power level of the VNA to 13 dBm:
• Press POWER
• Input 13 dBm
9. If the LED doesn’t light up, then there is a connection problem in your
131
system.
10. Measure the input return loss of the matched rectifier at ω = ω0 :
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Log Mag softkey
Press MARKER
Move the marker to the resonant frequency of the 9 cm loop to view the
magnitude of the reflection coefficient
11. A good input return loss is at least 10 dB. If the match is worse than this at
ω = ω0 , check all connections. Note that the rectifier may just be difficult
to match to.
12. Remove the matched rectifier from the VNA.
For both of the designed loop matching networks, perform the following procedure:
1. Recall Section 6.4.2. Connect the reflection port to the port of the matching
network that is designed to be connected to the loop (the side with the male
SMA connector). You will need an N-to-SMA adapter and the SMA-toSMA adapter (female-to-female).
2. Connect the transmission port to the other port of the matching network
using an N-to-SMA adapter.
3. Measure the input impedance seen by the VNA at the resonant frequency of
the 9 cm loop. This is the output impedance of the matching network.
To perform the measurement:
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Smith Chart softkey
Press MARKER
Move the marker to the resonant frequency of the 9 cm loop to view the
impedance
4. The measured impedance should be the complex conjugate of the load impedance
ZL for which the match is designed (see Sec. 6.2.6). If the impedance is sig132
nificantly off, make sure that the matching network was properly soldered.
The measured impedance should be on the bottom half of the Smith chart
and should be the same value that was computed using (6.29).
Measure the return loss of the matched system of loops:
1. Having tested and verified the impedances presented by the matching networks, we can attach them and measure the input return loss of the complete
system at the matching distance. Detach the connections between the VNA
and the matching networks.
2. Connect the matching networks, one to each loop. The male SMA of the
matching networks should connect directly to the loops. A long length of
cable should not be used to connect the two (why?).
3. Connect the input loop (either loop) to the reflection port of the VNA. Connect the load loop (the other loop) to the transmission port of the VNA.
4. Hook the two loops onto the distance-marked PVC pipe. Suspend the pipe
and loops above the table using the two tripods.
5. Measure the input return loss of the system at ω = ω0 for coupling distances
from 2cm to 26cm (2cm increments) and at the matching distance, aligning
the loops coaxially:
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Log Mag softkey
Press MARKER
Move the marker to the resonant frequency of the 9 cm loop to view the
magnitude of the reflection coefficient
6. A good input return loss is at least 10 dB. If the match is worse than this
around the matching distance at ω = ω0 , check all connections.
7. Move the loops away from the matching distance and observe the behavior
of the input reflection coefficient.
133
Measured Data
1. Loop matching network #1: Measured output impedance (input power: 13
dBm)
2. Loop matching network #2: Measured output impedance (input power: 13
dBm)
3. Rectifier matching network: Measured output impedance
4. Input return loss of matched rectifier
5. Input return loss of the matched system of loops at the matching distance
6. Input return loss of the matched system of loops at:
Distance (cm.)
2
4
6
8
10
12
14
16
18
20
22
24
26
Input Return Loss
Recall that return loss is defined as:
Return Loss = −20 log10 (|S11 |)
Analysis
There are no analysis questions for this section
134
(6.39)
Questions
• Why shouldn’t we use a length of cable between the matching networks
and the loops or the rectifier and its matching network? Why doesn’t it
matter whether we have a length of line between the matched rectifier and
the matched loops?
6.4.3
Frequency-Tuned System
Setup
1. Calibrate the reflection port of VNA, as detailed in Section 6.4.2.
2. Attach the male-to-male N-type coaxial cable (at least 4 feet long) to the
transmission port of the VNA.
3. Perform a calibration on the transmission port of the VNA:
•
•
•
•
Press MEAS 2
Press CAL
Select the Response softkey
Complete the 2-port calibration, following the steps provided by the VNA
user interface
4. If the system of matched loops (without the rectifier) is not connected to the
VNA, connect them.
Procedure
Efficiency at Constant Operating Frequency:
1. Determine S21 (in dB) of the system for coupling distances of 2 cm to 26
cm (2 cm increments) at ω = ω0 , aligning the loops coaxially:
135
•
•
•
•
•
•
•
Press MEAS 2
Press FORMAT
Press the Log Mag softkey
Press SCALE
Press the Autoscale softkey
Press MARKER
Move the marker to the resonant frequency of the 9 cm loop
Find the strong coupling resonances:
1. For coupling distances of 2 cm to 26 cm (2 cm increments), find the resonant
frequency of the coupled system, i.e. the frequencies with the lowest input
reflection coefficient. Align the loops coaxially for maximum coupling. For
strong coupling, there will be three resonant frequencies. The even and odd
mode frequencies will have the lowest input reflection coefficient, so record
both of these frequencies. Also note which distances are in strong coupling.
To setup these measurements:
•
•
•
•
•
Press MEAS 1
Press FORMAT
Press the Log Mag softkey
Press SCALE
Press the Autoscale softkey
2. Find the critical coupling distance.
Efficiency of Frequency-Tuned System:
1. For coupling distances of 2 cm to 26 cm (2 cm increments), use the data
collected in the previous steps to tune to the resonant frequency. When
the loops are in strong coupling, tune to the lower frequency (odd mode).
Record S21 (in dB) at all distances for this frequency-tuned system into your
lab notebook:
• Press MEAS 2
• Press FORMAT
• Press the Log Mag softkey
• Press SCALE
• Press the Autoscale softkey
136
Measure the input return loss of the complete system:
1. Disconnect the loops from the VNA.
2. Connect the matched rectifier to the matched load loop using a new maleto-male N-type coaxial cable (why doesn’t the length matter here?) and two
N-to-SMA adapters.
3. Connect the matched source loop to the reflection port of the VNA.
4. Change the output power level of the VNA to 14.5 dBm:
• Press POWER
• Input 14.5 dBm
5. Align the loops coaxially at a coupling distance equal to the matching distance.
6. Measure the input return loss of the complete system at ω = ω0 :
•
•
•
•
•
Press MEAS 1
Press FORMAT
Select the Log Mag softkey
Press MARKER
Move the marker to the resonant frequency of the 9 cm loop to view the
magnitude of the reflection coefficient
Lighting up the LED:
1. Remove the connections from the loops to the VNA.
2. Turn on the signal generator.
3. Turn the RF off on the signal generator:
• Press
RF ON/OFF
so that ”RF OFF” displays
4. Connect the matched input loop to the signal generator. Use another N-type
coaxial cable.
5. With the loops separated by their matching distance, turn on the RF on the
signal generator, with the frequency set to the resonant frequency of the
loops:
137
•
•
•
•
•
Press FREQUENCY
Input the frequency and press ENTER
Press AMPLITUDE
Input the 14.5 dBm and press ENTER
Press RF ON/OFF so that ”RF OFF” does not display
6. Now push the loops farther apart. Observe the behavior of the LED.
7. Move the loops back to their matching distance. Now move them closer
together. Observe the behavior of the LED.
8. With the loops in strong coupling, i.e. when the LED begins to dim as the
loops are moved close together, tune the frequency of the source to achieve
maximum brightness in the LED. Try using the even and odd mode resonant
frequencies determined earlier in this section.
Measured Data
1. For distances in strong coupling, record the even and odd mode frequencies:
Distance (cm.) Odd Mode Frequency
2
4
6
8
10
12
14
16
18
20
22
24
26
Even Mode Frequency
2. Record the efficiency data. For distances in strong coupling, record frequencytuned efficiency at the odd mode frequency:
138
Distance (cm.)
2
4
6
8
10
12
14
16
18
20
22
24
26
|S21 | (Constant Frequency)
|S21 | (Frequency-Tuned)
3. Input return loss of completely matched system at the matching distance
(input power: 14.5 dBm)
4. Critical coupling distance
Analysis
1. Using (6.33), determine the theoretical power transfer efficiency η at ω = ω0
for coupling distances of 2 cm to 26 cm in 2 cm steps.
2. Using (6.2.8), determine the theoretical power transfer efficiency at the even
or odd mode frequency for coupling distances of 2 cm to 26 cm in 2 cm
steps.
3. Using (6.37) and the circuit values from the previous labs, plot the maximum power transfer efficiency at coupling distances of 2 cm to 26 cm in 2
cm steps.
Questions
1. How does the input reflection coefficient of the matched system behave as
the loops are moved away from the matching distance?
139
2. Plot the theoretical efficiency vs measured efficiency for the constant frequency system. How do they compare?
3. Does the coupling distance with minimum return loss correspond to the
coupling distance with maximum S21 for a constant w0 system?
4. Plot the theoretical efficiency vs measured efficiency for the frequencytuned system. How do they compare?
5. Why is the critical coupling distance here different from the unmatched
case?
6.5
Lab Write-Up
For each section of the lab, include the following items in your write-up:
• Overview of the procedure and analysis.
• Measured data.
• Calculations (show your work!).
• Any tables and printouts.
• Comparisons and comments on results.
• Answers to all questions.
• A summary paragraph describing what you learned from this lab.
140
Appendices A - E.
Equipment and component reference guide
EECS 230
TABLE OF CONTENTS
Appendix A: HP 8712C Network Analyze . . . . . . . . . . . . . . . . . . . .
A.1
List Of Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B: Agilent DSO-X 2012A Digital Oscilloscope . . . . . . . . . . . .
B.1
List Of Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix C: HP 8648B RF Signal Sourc . . . . . . . . . . . . . . . . . . . . .
C.1
List Of Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix D: Adapters, Cables, Connectors, And Components . . . . . . . . . .
D.1
Connector Types . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.2
Adaptors And Connectors . . . . . . . . . . . . . . . . . . . . . .
D.3
Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.4
Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix E: Error Analysi . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.1
Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . .
E.2
Derived Quantities . . . . . . . . . . . . . . . . . . . . . . . . . .
E.3
Error Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1
A.1
B.1
B.1
C.1
C.1
D.1
D.2
D.3
D.6
D.8
E.1
E.1
E.2
E.2
HP 8712C Network Analyzer
The HP 8712C network analyzer is used in several of the labs to perform both amplitude
and phase measurements of reflections caused by various loads. The front panel of the
HP 8712C network analyzer is shown in Fig. A.1.
Numeric Key Pad
Display
Disk Drive
Control Knob
System Keys
Softkeys
Configure Keys
Channel 1
Power
Measurement Keys
Channel 2
Source Keys
Figure A.1: The HP 8712C network analyzer front panel.
A.1
L IST O F P ROCEDURES
The procedures used in the processes of making measurements with the HP 8712C network
analyzer in the various labs are listed in this section for easy referencing.
The procedures listed use specific numerical examples. Substitute the desired numerical
value when performing the procedure.
Autoscaling
Press SCALE followed by the Autoscale softkey
Adding Electrical Delay
Press SCALE
Press the Electrical Delay softkey
Rotate the knob on the network analyzer to add the desired amount of electrical delay
A.1
A.2
HP 8712C NETWORK ANALYZER
Calibration - Cable
Press CAL
Press the Calibrate Cable softkey
Press the Specify Length softkey
Enter the length of the patch cord in feet and press ENTER
Press the Measure Cable softkey
Calibration - Fault Location
Press CAL
Press the Full Band Cal softkey
Connect the open termination from the calibration kit and press the Measure
Standard softkey
Remove the open and connect the short termination from the calibration kit and press
the Measure Standard softkey
Remove the short and connect the matched termination from the calibration kit and
press the Measure Standard softkey
Note: It will take a few seconds for the network analyzer to measure the calibration
loads due to the large bandwidth being used.
Calibration - Single Channel
Press CAL
Press the One Port softkey
Connect the N jack-jack adapter to the end of the N patch cord
Connect the open standard from the calibration kit to the end of the patch cord and press
the Measure Standard softkey
Remove the open standard and connect the short standard from the calibration kit.
Press the Measure Standard softkey
Remove the short standard and connect the matched (50 %) standard from the
calibration kit. Press the Measure Standard softkey
Changing The Display Format
Press FORMAT
Press the desired softkey (e.g. Phase)
A.1 LIST OF PROCEDURES
Configuring The Printer
Press
HARD
COPY
Select the parallel output port and printer type
– Press the Select Copy Port softkey
– Use the and keys to select the configuration which corresponds to:
Device Type: HP Printer
Language: PCL
HardCopy Port: Parallel Port
– Press the Select softkey
– Press the Prior Menu softkey
Configure the graph and printer settings
– Press the Define Hardcopy softkey
– Press the Graph Only softkey
– Press the Prior Menu softkey
– Press the Define Printer softkey
– Press the Monochrome softkey
– Press the Landscape softkey
Set the printer resolution to 150 dpi.
– Press the More Printer softkey
– Press the Printer Resolution softkey
– Enter 1 5 0 and press the Enter softkey
Set the printer switch box to output channel B.
Customizing Titles
Press the Enter Line 1 softkey
Enter desired title
– Use the knob on the front panel to highlight the desired character
– Press the Sel Char softkey
– Repeat for all characters. To enter a space, press the Space softkey. To move
back a character, press the Backspace softkey.
– Press the Enter softkey when done
Displaying data and memory on the screen
Press DISPLAY
Press the Data and Memory softkey
A.3
A.4
HP 8712C NETWORK ANALYZER
Displaying data only on the screen
Press DISPLAY
Press the Data softkey
Displaying the date and time on the screen
Display the date and time on the screen.
– Press DISPLAY
– Press the More Display softkey
– Press the Title and Clock softkey
– Press the Show Clock on Line 2 softkey
Fault Location Measurements
Press BEGIN
Press the Cable softkey
Press the Fault Location softkey
Press the Feet softkey
Markers
Press MARKER
Press the Marker X softkey, where X is the desired marker number
Enter the desired frequency in MHz using the numeric key pad and press the MHz
softkey
The marker should be located at the entered frequency and the value at that point displayed
in the upper corner of the screen.
Measuring Cable Loss
After performing a cable calibration
Press CAL
Press the Cable Loss softkey
Measuring Cable Velocity Factor
After performing a cable calibration
Press CAL
Press the Velocity Factor softkey
A.1 LIST OF PROCEDURES
Measuring Reflections
Press MEAS X , where X is the desired channel (1 or 2)
Press the Reflection softkey
Number of Points
Press MENU
Press the Number of Points softkey
Enter the desired number of points
Press ENTER
Printing The Display
Add a descriptive title (in addition to the title you added previously)
– Press DISPLAY
– Press the More Display softkey
– Press the Title and Clock softkey
– Press the Enter Line 1 softkey
– Add the descriptive title (i.e. Short Uncal)
* Use the knob on the front panel to highlight the desired character
* Press the Sel Char softkey
* Repeat for all characters. To enter a space, press the Space softkey. To
move back a character, press the Backspace softkey.
* Press the Enter softkey when done
Press
HARD
COPY
Press the Start softkey
Saving Calibration Coefficients And Instrument State
Press
SAVE
RECALL
Press the Define Save softkey
Press the Inst State softkey until On is selected
Press the Cal softkey until On is selected
Press the Data softkey until Off is selected
Press the Prior Menu softkey
Press the File Utilities softkey
Press the Directory Utilities softkey
A.5
A.6
HP 8712C NETWORK ANALYZER
keys to select the EECS230 directory
Press the Change Directory softkey
Use the and keys to select the SECxxx directory, where xxx is your three digit
lab section number.
Press the Change Directory softkey
Press the Prior Menu softkey
Press the Save State softkey. A new file name of the form STATEy.STA, where y is
an incremental counter starting at 0 will appear in the current directory. This file
contains the state of the network analyzer, including the calculated calibration
coefficients.
Press the File Utilities softkey
Press the Rename File softkey
Press the Clear Entry softkey
Enter the desired file name. The file name must be a legal DOS file name.
Press the Enter softkey
Saving Data In Memory
Press DISPLAY
Press the Data
Memory softkey
Setting the Bandpass Maximum Span (Fault Location)
Press FREQ
Press the Fault Loc Freq softkey
Press the Bandpass softkey
Press the Bandpass Max Span softkey
Enter the desired span using the numeric keypad
Press ENTER
Setting the Start Distance
Press MENU
Press the Distance softkey
Press the Start Distance softkey
Enter the desired distance
Press ENTER
A.1 LIST OF PROCEDURES
Setting the Start Frequency
Press FREQ
Press the Start softkey
Enter the desired frequency using the key pad and press the MHz softkey
Setting the Stop Distance
Press MENU
Press the Distance softkey
Press the Stop Distance softkey
Enter the desired distance
Press ENTER
Setting the Stop Frequency
Press FREQ
Press the Stop softkey
Enter the desired frequency using the numeric key pad and press the MHz softkey
Setup
Turn the network analyzer on and wait for the instrument to perform its self test.
Press PRESET
Turn channel 2 off.
– Press MEAS 2
– Press the Meas OFF softkey
Set the display to show only channel 1
– Press DISPLAY
– Press the More Display softkey
– Press the SPLIT Disp full SPLIT softkey until only channel 1 is displayed
(FULL will be displayed in all caps and split will be displayed in all
lowercase letters)
A.7
AGILENT DSO-X 2012A DIGITAL OSCILLOSCOPE
Measurement Keys
Soft Keys
Channel 2
Channel 1
Fig B.1: Agilent DSO-X 2012A Digital Oscilloscope front panel
B.1 LIST OF PROCEDURES
The procedures used in the processes of making measurements with the Agilent DSO-X 2012A
oscilloscope in the various labs are listed in this section for easy referencing.
The procedures listed use specific numerical examples. Substitute the desired numerical value
when performing the procedure.
Adjusting Oscilloscope Display
•
Press Autoscale
OR
•
•
Set V/div knob right above active channel button to scale the signal in vertical direction.
Set the horizontal knob at the top left side of the keys to scale time axis. At least 2 period
of the signal should be displayed.
Averaging
•
•
•
•
Press Acquire button under measurement Keys
Press Acq Mode soft key until you choose Averaging
Push the # Avgs button, then choose desired averaging number
To turn off averaging, choose Normal in Acq Mode soft key
B.1
Saving the Data and Capturing Display of Oscilloscope to Computer
•
•
•
•
Run the “Intuilink Data Capture” program on your computer
Under “Instrument,” choose the second one from the top, which is either “Agilent
2000/3000 Series” or “DSO-X 2012A”
Setup Add-In properties, choose Number of Points as 1000 and select the channels. If any
channel on the scope is not active, it is not selectable in this setup.
After hitting the “OK” button, the scope display picture and data is acquired into the
program. Both the picture and the data can be saved in an appropriate file format.
Using Cursor to Find Voltage and Time Differences
•
•
•
Push the “Cursors” button under measurement keys.
For measuring the voltage difference (ΔV or ΔY ), choose cursor Y1 in soft key and level
it to the voltage level of the first point by using cursor knob. Choose Y2 and do the same
thing for the second point. Record ΔY.
For measuring the time difference (ΔX or Δt ), choose cursor X1 in soft key and set it to
the first point by using cursor knob. Choose X2 and do the same thing for the second
point. Record ΔX.
Measuring Signal Amplitudes
•
•
•
•
Push “Meas” button under Measure keys
Push “Type” soft key until desired voltage measurement selected (peak-peak, peak,
amplitude etc.)
Push “Add Measurement” soft key
Use cursors to find voltage and time difference between any two points you choose
Measuring the Phase Shifts or Time Delays Between Channels
•
•
•
•
Push “Meas” button under Measure keys
Push “Type” soft key until Phase or Delay is selected
Push “Add Measurement” soft key
Use cursors to find voltage and time difference between any two points you choose
B.2
HP 8648B RF Signal Source
The HP 8648B RF signal source is used in several of the labs to provide the test signal.
The output of the HP 8648B RF signal source is a sine wave at the selected frequency and
amplitude. In addition, the signal can be modulated by either an FM or AM signal. The
front panel of the HP 8648B RF signal source is shown in Fig. C.1.
Increment control
Function Keys
Display
Data entry key pad
Frequency control
Amplitude control
RF On/Off
Memory
Power
HP-IB
Modulation
RF output
input/output Attenuation
hold on/off
Modulation control
Figure C.1: HP 8648B RF Signal Source front panel
C.1
L IST O F P ROCEDURES
The procedures used in the processes of making measurements using the HP 8648B RF
signal source in the various labs are listed in this section for easy referencing.
The procedures listed use specific numerical examples. Substitute the desired numerical
value when performing the procedure.
Adding A 1 kHz, 50% AM Modulation
Press
INT
1 kHz
Press Mod On/Off until modulation is On
Press AM followed by 5 0
%
µV
C.1
C.2
HP 8648B RF SIGNAL SOURCE
Changing The frequency
Press Frequency followed by the desired frequency (e.g. 5 0 0 ) followed by
kHz
mV
Configuring The HP 8648B To Output A 0 dBm, Non-Modulated, 50 kHz Signal
Press Mod On/Off until the modulation is turned Off.
Press Amplitude followed by 0
MHz
dB(m)
Press Frequency followed by 5 0
to set the amplitude to 0 dBm.
kHz
mV
to set the output frequency to 50 kHz.
Press RF On/Off until the RF output is On.
If at any point you make a mistake entering a number, you can use the
last number entered.
key to delete the
Adapters, Cables, Connectors, And Components
This section contains photographs for most of the cables, connectors, adapters, and
components used in the labs for quick and easy reference.
D.1 C ONNECTOR T YPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2
D.2 A DAPTERS AND C ONNECTORS . . . . . . . . . . . . . . . . . . . . . . D.3
BNC Jack To BNC Jack . . . . . . . . . . . . . . . . . . . . . . . D.3
BNC Plug To BNC Plug . . . . . . . . . . . . . . . . . . . . . . . D.3
BNC Jack To N Plug . . . . . . . . . . . . . . . . . . . . . . . . . D.3
BNC Plug To N Jack . . . . . . . . . . . . . . . . . . . . . . . . . D.3
BNC ‘Y’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4
BNC ‘Tee’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4
N Jack To N Jack . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4
N Plug To N Plug . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4
N ‘Tee’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5
D.3 C ABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.6
Alligator To Alligator Plugs (Alligator Cable) . . . . . . . . . . . . D.6
Banana To Alligator Plugs . . . . . . . . . . . . . . . . . . . . . . D.6
Banana To Banana Plugs (Banana Cable) . . . . . . . . . . . . . . D.6
BNC To BNC Cable (BNC Cable) . . . . . . . . . . . . . . . . . . D.7
BNC To Banana Cable . . . . . . . . . . . . . . . . . . . . . . . . D.7
N To N Cable (N Cable) . . . . . . . . . . . . . . . . . . . . . . . D.7
D.4 C OMPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.8
Button Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.8
Digital Function Generator . . . . . . . . . . . . . . . . . . . . . . D.8
Halogen Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.8
Lossy Line Simulator . . . . . . . . . . . . . . . . . . . . . . . . . D.9
Matched Termination . . . . . . . . . . . . . . . . . . . . . . . . . D.9
Microphone Pre-amp Box . . . . . . . . . . . . . . . . . . . . . . D.9
Mounting Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.10
Microphone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.10
Multi-meter (HP 973A) . . . . . . . . . . . . . . . . . . . . . . . . D.11
Open/Short Termination . . . . . . . . . . . . . . . . . . . . . . . D.11
Printer Switch Box . . . . . . . . . . . . . . . . . . . . . . . . . . D.12
Ripple Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . D.12
Ripple Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.12
Scanner Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . D.13
Short Termination (see Open/Short Termination) . . . . . . . . . . D.11
Single Stub Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . D.14
Slotted Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.14
Speaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.14
Speaker Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.15
Stop Watch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.15
Strobe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.16
Unknown Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.16
Voltage Divider . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.16
D.1
D.2
ADAPTERS, CABLES, CONNECTORS, AND COMPONENTS
D.1
C ONNECTOR
TYPES
There are several types of connectors available on the commercial market. The main
parameters that determine which type of connector is used include: existing connector
type, size, loss, and frequency. Some typical types of connectors include: BNC, N, F,
RCA, SMA, and GR.
In this lab course, you will encounter three different types of connectors, BNC and N.
Each of these connector types are pictured in Fig. D.1.
(a) BNC
(b) N
Figure D.1: BNC and N connectors
D.2 ADAPTERS AND CONNECTORS
D.2
A DAPTERS
AND
BNC Jack To BNC Jack
Quantity per station:
1
BNC Plug To BNC Plug
Quantity per station:
2
BNC Jack To N Plug
Quantity per station:
1
BNC Plug To N Jack
Quantity per station:
1
C ONNECTORS
D.3
D.4
ADAPTERS, CABLES, CONNECTORS, AND COMPONENTS
BNC ‘Y’
Quantity per station:
1
BNC ‘Tee’
Quantity per station:
1
N Jack To N Jack
Quantity per station:
3
N Plug To N Plug
Quantity per station:
2
D.2 ADAPTERS AND CONNECTORS
N ‘Tee’
Quantity per station:
2
We also have the SMA female-to-female adapters and the N-female to SMA-male adapters.
Each station will require 1 of the first and 3 of the second.
D.5
D.6
ADAPTERS, CABLES, CONNECTORS, AND COMPONENTS
D.3
C ABLES
Alligator To Alligator Plugs (Alligator Cable)
Quantity per station:
5
Banana To Alligator Plugs
Quantity per station:
2 red, 2 black
Banana To Banana Plugs (Banana Cable)
Quantity per station:
5 red, 5 black
D.3 CABLES
D.7
BNC To BNC Cable (BNC Cable)
Quantity per station:
1-12 , 2-24 , 1-180
BNC To Banana Cable
Quantity per station:
1
N To N Cable (N Cable)
Quantity per station:
4
D.8
ADAPTERS, CABLES, CONNECTORS, AND COMPONENTS
D.4 C OMPONENTS
Lossy Line Simulator
Quantity per station: 1
Used in Lab:
2
Matched Termination
Quantity per station: 1
Used in Lab:
1
Multi-meter (HP 973A)
Quantity per station: 1
Used in Lab:
6
D.9
Open/Short Termination
Quantity per station: 1
Used in Lab:
1, 2
Note: The short termination and open termination are contained on the same load. The arrows on the load
indicate which end is the short termination and which end is the open termination.
Printer Switch Box
Quantity per station: 1
Used in Lab:
1, 2
Short Termination
(See Section D.4: Open/Short Termination)
Single Stub Tuner
Quantity per station: 1
Used in Lab:
1, 2
Slotted Line
Quantity per station: 1
Used in Lab:
1
D.10
ADAPTERS, CABLES, CONNECTORS, AND COMPONENTS
Unknown Load
Quantity per station: 1
Used in Lab:
2
Voltage Divider
Quantity per station: 1
Used in lab:
1
E.1. MEASUREMENT ERRORS
E.1
Error Analysis
Errors are not the same thing as mistakes. An error is the difference between what a
quantity actually is and what it was measured to be. For example, if a rod were exactly 341.078 mm long but measured to be 341.0 mm, then the error is −0.078 mm,
or −0.023%. No one can measure anything exactly: all measurements have some
level of uncertainty. Knowing the errors of a measurement means you know the
quality of the measurement. If a second measurement of the rod reported the same
length of 341.0 mm with an estimated error of 2 mm, the second measurement
would not be as good as the first A good scientist or engineer always reports the
quality of the numbers they are working with, so that they, and others, know how
good the results are. Also, knowledge of the errors of two measurements of the
same thing allows the two measurements to be compared to see if they agree. A
third measurement of the rod with a reported length of 342.0 mm and an estimated
error of 2 mm wouldn’t be as good a measurement as the first but it would agree
with both of the other measurements. A fourth measurement with a reported length
of 346 mm and an estimated error of 1 mm wouldn’t agree with any of the other
measurements. So, unless the rod has physically changed, there is at least one
mistake in the measurements.
E.1
Measurement Errors
Obviously, if nothing can be measured exactly, errors can’t be quantifie exactly.
Errors must be estimated. There are two primary ways of estimating the error of a
measurement:
• Knowing in advance of the measurement how good it could be
• Repeating the measurement many times to see how much it varies.
The measurement of the rod is an example of a measurement for which the error can be estimated in advance. Rulers have a smallest unit that can be measured, e.g., 1 mm, so the error of all measurements made with that ruler must be
within ±0.5 mm. A voltmeter that reads 3.89 V on the display has an error of
at least ±0.005 V, since the smallest unit measured by the voltmeter is 0.01 V.
Sometimes it is easier to repeat a measurement a number of times, and report “the
measurement” as the average of the measurements and the estimate of the error
as the standard deviation of the measurements. The errors in the measurements
of time are most easily estimated by calculating the standard deviation of a set of
measurements. The formula for the standard deviation of a set of measurements
{t1 , t2 , . . . , tN } is given by
v
u
u
std dev = t
1
N −1
ÃÃ N
X
t=1
where the mean of t is given by
N
1X
t=
ti .
n i=1
t2i
!
− Nt
2
!
(E.1)
E.2
ERROR ANALYSIS
E.2
D ERIVED QUANTITIES
Often we aren’t interested in the measurements themselves but in something derived from
them. For example, the velocity of an object can be derived from measurements of the
time it takes to travel a measured distance. Suppose the measured distance is l 2 m with
an error of -l 0 1 m and the measured time is t 1 s with an error of -t 0 15 s. Then
the velocity is v l t 2 1 2 m/s. A "quick and dirty" estimate for the error of the
velocity measurement can be found from
l
t
-l
-t
v
(E.2)
-v
There are actually four equations in (E.2): each on the left side of the equation must be
evaluated separately. The largest resulting -v is the estimate of the error. The four
calculations for the error of the velocity are
2
1
2
1
2
1
2
1
01
0 15
01
0 15
01
0 15
01
0 15
2
0 17
(E.3)
2
0 35
(E.4)
2
0 47
(E.5)
2
0 24
(E.6)
and so -v 0 47 m/s. The estimate of the error is always a positive quantity, because it is
understood that it can be added to or subtracted from the reported value. In this example,
the true velocity can be anywhere in the range 2 0 47 m/s. This technique is crude, and it
overestimates the error of derived quantities, but that is better than underestimatingthe error.
E.3
E RROR
BARS
When plotting results of measurements or derivations, error bars are used to graphically
represent the errors. If, for example, the velocities in the following table were calculated
for some object as a function of starting position,
Starting Position
110 cm
130 cm
150 cm
170 cm
190 cm
Error
0 5 cm
0 7 cm
0 5 cm
1 6 cm
2 8 cm
Velocity
2 4 m/s
2 2 m/s
2 1 m/s
2 1 m/s
2 0 m/s
Error
0 2 m/s
0 3 m/s
0 4 m/s
0 2 m/s
0 5 m/s
the plot would look like that in Figure E.1. The measured or calculated data points are
represented by the little circles, while the errors are represented by vertical or horizontal
lines. For the first three points, the errors in the starting position are so small that they are
not visible as horizontal error bars for those points.
Knowing the error and plotting it can be of great assistance in preventing incorrect
conclusions. Note that from the table, the mean starting position and mean velocity appear
E.3 ERROR BARS
E.3
3
2.5
Velocity (m/s)
2
1.5
1
0.5
0
100
110
120
130
140
150
160
Starting position (cm)
170
180
Figure E.1: Derived Velocity vs. Starting Position
190
200
E.4
ERROR ANALYSIS
to have an inverse relationship. However, from the graph, which incorporates the errors
into it, it is equally plausible that the velocity does not depend at all on the starting
position.