
Searching for the Q


Sunshine Design Engineering Services
Joe Cahak, RF engineer and owner of Sunshine Design Engineering Services, just sent me another
paper he has authored on the subject of determining the quality factor ("Q") of rectangular
and cylindrical cavities. Before you decide to skip reading this article because the topic might
seem too esoteric, Joe does much more than present formulas and tables. He also includes a tale
of the research needed to cull and vet the many, often conflicting, opinions of which formulas
are correct. Ultimately, a decision was made to select formulas that most closely matched empirically
determined (measured) values from published data. If Mr. John T. Frye was still writing
Carl &
Jerry adventures for Popular Electronics,, this might make a good story line.
See list of all of Joe's articles at bottom of page.
Searching for the QBy Joseph L. Cahak Copyright 2014 Sunshine Design Engineering Services
What is Q? Q, or Quality factor, is the measure of the ability of a cavity or other resonant
structure to have and hold energy. Another description is the center frequency of a resonant
circuit divided by the 3 dB Bandwidth.
Figure 1  Bandwidth vs Q of resonant circuit courtesy
AllAboutCircuits.com
The search for the Q began after I got an email from an engineer at a filter manufacturer.
He had purchased my RF Calculator
and desired to see if I could add the calculation for rectangular and cylindrical cavities with
their Q. His company had something they used internally and he sent me their code to look at.
No documentation or details about how to use it was included. So, it wasn’t much use, other
than to review and see if it matched anything I found; it did somewhat. His calculation was
one of the five for the rectangular prism case where the mode orders l, m & n are all non
zero.
With this article I would like to tell you a story of one of my recent development
projects for an addition to my RF
Calculator product. I hope you will forgive my indulgence in telling it. I am a consulting
RF/Microwave Test Engineer with process automation and application development as specialties.
I design, build and program automated test systems. I also do some application development for
myself and for sale. I have been in the RF/Microwave business for 30 years collecting reference
literature all along. I have a rather large collection of engineering books, notes, articles
and digital content. The formulas cover the gamut from RF, microwave, astronomy, physics, math,
chemistry, electronics, programming and lots more. Having the formulas at hand and easy to use
and understand is key to making them useful. Lugging all the books around is not an option,
nor is trying to correctly recalling them all by memory.
Over the years I worked in BASIC,
Pascal, Fortran, HP Basic, HT Basic, QBasic then Visual Basic 3 thru 6, Borland C and Microsoft
C, VB.net, C# and now LabVIEW. The question with software libraries for sale is what language
makes it marketable? There are so many different programming environments out there today that
targeting a specific language or programming environment is a difficult decision. The programming
environments of choice have gone thru many changes over the years. My choice for these calculator
applications was a light weight Windows program in Visual Basic. It is easy to program and install
and the Windows executable still works on Windows 7 and 8. For more modern RF Test development
I program more with LabVIEW now and compile libraries (DLL) and as LabVIEW Libraries. I am considering
going to the mobile platform for the applications, but haven’t been able to afford the development
tools yet.
About 15 years ago I started collating the best formulas in my
RF Calculator program. I also
developed a rather useful astronomy calculation product called
AstroCalculator and an astrophotography
and optics calculator called OptiCalculator.
The calculators were initially developed for my use. I am always on the hunt to find new formulas
and features to add to my calculators.
RFCalculator had a cavity resonance section, but
it was incomplete. It had a rectangular cavity with partial dielectric fill (PCB) and cylindrical
cases that needed work and did not perform the Q calculations. The more general solutions for
the cylindrical and rectangular prism cases was always preferred, but I saw a lot of work and
time to resolve and understand all the information and glean a solution. Life’s conditions had
to be ripe for the effort. I had some employment uncertainties for a couple of months, so I
used the time off to do some special projects while looking for more employment. It was good
challenging mental work and kept me occupied and purposed. I had to complete the complex and
Sparameter library and functions first. I was able to complete all that sizable effort and
finish a salable LabVIEW and C language DLL library to sell for the libraries. I could now tackle
the Q.
Research for the effort began with a reference search to see what material I could
come up with. In the old days searching meant at the book stores, libraries with index cards,
books, magazines and microfilm. All of them were very time consuming and book purchases were
very expensive. Today, a literature search with the Internet takes only seconds. This is a tremendous
resource for the paperchallenged (remote locations w/o books). Going to the library or bookstore
used to be easy and cheap, but now with living in the backcountry, distance costs a lot. So,
I now order new or used books from Amazon for pennies to dimes on a dollar for some very good
used reference books. With Amazon, I found if I added the expensive ones to my wish list, I
would see the occasional used deals that allowed me to purchase the odd book for my library
for less. I had to be quick for these short flash gems of bound paper. They usually go fast.
Just yesterday I was able to finally purchase the Microwave Filters, ImpedanceMatching Networks,
and Coupling Structures book by Matthaei, Jones and Young for $49.99 instead of the usual $119.
My Internet searches not only scan for book or article titles, but also content in them.
eBay, Google, Amazon and a slew of companies now deal in used books and a lot of classic technical
books that are coming available for very cheap prices as libraries dump their old books. I use
this extensively now to grow my engineering and science reference library. The public loss is
my gain.
I purchased what reference books I could afford. Many of my engineering friend’s
libraries were scoured and book loans were secured. The search was on to find the general solution
for Q of rectangular prisms and cylinders and almost no avenue was left unturned. Initial searches
thru the microwave reference books were fair, but showed many poor and incomplete examples for
Q formulas that did not cover all the cases for TE and TM modes, so the search continued.
For the rectangular prism resonant cavity the coordinates for x, y and z is alength, bwidth
and dheight. Other authors use different dimensional symbols and this can confuse novices (like
me) on the topic due to many using mode orders (l, m, n) or (m, n, L) and then l for height
(Pozar). It quickly gets very confusing. In addition to the dimensions of the cavity a few other
things must be determined to perform all the calculations leading up to the Q formulas. We need
the type of metal or other surface (like the earth for instance in the spherical case). I also
use the metric system of measurements. Matthaei, Jones and Young’s book or MJY, as I will refer
to it, uses English inch units and also fixes a constant instead of showing the full formulation.
Pozar’s book has one formula each for the rectangular and cylindrical case. They did not compute
well at many instances outside his singular examples. Other books showed a blithering array
of variations of the coordinate systems and formulas. Most authors’ examples were singular and
poorly expounded in most cases, almost as if offhanded, disappointing to be sure.
I
kept gathering my reference materials and looking it over occasionally when I was bored, usually
when the weather was bad or it got dark early in winter. I kept trying Google searches to see
what I could find and discovered that search phrases could bring big differences in the search
results. I was also finding a lot of protected content on the MTT, Journal of Applied Physics
and many other more obscure sites and other references too numerous to mention. They only showed
partial content if any.
One day while Web searching, I finally stumbled on the first
of the Bell System Technical Journal or BSTJ notes in the references. I had been using Cavity
Resonance Q and Microwave Cavity Q in my searches to no avail. I then tried a different variation
on search string and used Microwave Cavity Testing and there was a paper from “BSTJ” and had
volume numbers on them. The two papers had the same Applied Physics Journal table of formulas,
see figure 1. I found out later that these papers were from Bell System Technical Journal and
that they had the tables, formulas and charts that helped a lot. I now had at least three independent
references for the full general Q methods to use for the various mode cases. Several of the
links to this and other referenced papers are only available to subscribers. I happen to find
a site or so that had the PDF copies available. I also found several related references from
the same Bell Systems Technical Journal which I was also able to find copies of.
These
references had the same copy of the table of cavities and there reference formulas. Rectangular
had 5 total cases, the cylindrical had three cases and the coaxial has three cases. I was seeing
cross agreement with MJY for the rectangular and Collin’s formulas for cylindrical. I also found
a reference to the full tables in the Microwave Engineer’s Handbook Volume 1. The coaxial case
was where the only difference occurred. Also the BSTJ had the full formulation and in metric
units, so the formulas had no constants other than expected physical constants.
Figure 2  Cavity Formulas
The next step was to work up the formulas
and test them. I used
Mathematica for the formulation and testing. It was easy to enter the formulas and test
them. I made some changes to the variables to help me cross over to several authors’ formulas
simultaneously. They needed to be tested over the range of typical values. I was successful
at this stage and was starting to notice trends in the values calculated and whether they were
consistent over TM/TE modes and modal states. All the most significant resonant cases were run
for testing the Q formulas against each other, and results were gathered. I still had the problem
of determining how could I better test my values to know if the ranges I was seeing were reasonable?
I had just a few examples from the books and articles to go by, but wanted better data. The
MJY reference gives the Q as a compound function. The charts are difficult to interpret. Rereading
thru all the references again I found some statements in Collin’s book “Foundations of Microwave
Engineering”. He gave the expected range for cylindrical cavity as 10000 to 40000 with wavelengths
in the 3cm range as having optimal Q. He also gave a 3x3x3cm cube in copper Q as 12700. My calculations
produced 12734.3 (see Figure 2). I was beginning to have some level of confidence in the calculations.
Figure 3  3x3x3 cm Rectangular Cavity
After all the formulas had been
tested in Mathematica, I next performed the coding and testing in my RF Calculator program.
I had to work out an interface for the mode options and present them in a easy to understand
and use format. This also required looking further into the TE and TM modes and the most commonly
used modal orders so I could bring some containment to the interface, yet provide a wide selection.
I also had to allow the user to select between the surface material and the tangent loss and
dielectric constant.
Figure 4 – Rectangular Cavity screen
Figure 5 – Cylindrical Cavity screen
As can be seen in the screenshots
answers are provided to a number of cavity values like the wavelength, sheet resistivity, characteristic
impedance and the skin depth along with the loaded and unloaded cavity Q, resonant frequency
and both the Applied Physics value for Q from BSTJ and the Pozar value for Q. Loaded Q uses
the formula
for
the Q of the dielectric in the cavity. Adding this in parallel with the cavity Q produces the
total Q.
Rectangular Prism Cavity
Figure 6  Rectangular Prism Cavity
The formulas for the power are complex
integrals of three dimensions. For the electric field the formula is
Or for the magnetic fields
For the rate of loss or just “loss” we use the skin depth calculated below
Resistivity
of copper is ρ=5813000 mhos/meter or S/m.
μ_{o} =4π10^{7} H/meter
Characteristic impedance Z or
Ohms
Sheet resistivity or
Ohms/Square
Skin depth or
cm
We calculate by defining the coordinates, dimensions, modes and mode orders x=a;
y=b and z=d. The corresponding mode orders are l, m and n for the x, y and z axis and a, b and
d length. We are calculating for a half cycle or half wavelength.
The definitions used
for the calculation
Wavelength or
cm
Resonant Frequency
Hz
For TE mode order l, m > 0
For TE mode order l = 0
For TE mode order m = 0
For TM mode order n > 0
For TM mode order n = 0
Rectangular Modes Table
The table shows the various rectangular
cavity modes and mode orders as functions of the dimensions and frequency.
Figure 7  Rectangular Cavity Models Chart
Cylindrical Cavity
Figure 8  Cylindrical Cavity
The formula for the cylindrical cylinder
power is a complex integral of 3 dimensions.
For the rate of loss or “loss” we use the skin depth calculated below
Resistivity
of copper is ρ=5813000 mhos/meter or S/m
μ_{o} =4π10^{7} H/meter
Characteristic impedance Z or
Ohms
Sheet resistivity or
Ohms/Square
Skin depth or
cm
We again calculate using the coordinate, dimensional mode and mode order definitions.
The diameter=a and the height or z=d.
The corresponding mode orders are l, m and n for
the x, y and z axis and a diameter and d length. The l and m mode orders are not addressed the
same for cylindrical cavities. For these, the zero order Bessel J0 for the TM modes and first
derivative (prime) Bessel Jp0 are used for the TE modes. These Bessel zero values can be looked
up in mathematical tables. See table in figure 8. In this case the J0 and Jp0 values are interspersed
and marked by the E or M mode and the mode order from lowest order out to 180th order.
For TE mode
The definitions used for the calculation
Wavelength
or
cm
Resonant frequency
Hz
For TM mode
The definitions used for the calculation
Wavelength or
Resonant frequency
when mode order n>0
when mode order n=0
Cylindrical Modes Table of Bessel Functions
Figure 9  Cylindrical Modes Table of Bessel Functions
Corrections for Q
Q is
a complex thing and is comprised not only of the power storage/loss mechanism Qe , but also
by other effects. Dielectric materials used for cavity fill can cause a power loss resulting
in
is
the dielectric tangent loss. Another Q factor is the surface roughness Qs and finally
Qp is any perturbations to the geometry such as a small aperture and electric probe wire.
These addition Q factors are summed by the parallel sum method.
Coaxial, Spherical and other Cases
Figure 10 – Coaxial Cylinder
The coaxial cylinder case can be seen in
the BSTJ documents and is quite lengthy. I have not yet tackled that or the spherical and elliptical
cases. In the article “Redefining
the Kelvin” an elliptical cavity was used to get a very accurate value for the Kelvin and
Boltzmann’s constant (k). For the spherical cavity case, in
Lecture Notes 10.5 they discuss the Earth’s spherical resonance with Lightening stimulation.
The reader can pursue further studies in these areas if they desire. Maybe the next long bout
of extended time off and bad weather I will tackle the coaxial and spherical Q cases. For now
I am satisfied having resolved the rectangular and cylindrical cases to add to my calculator
functions. I would like to thank you the reader for allowing my indulgence with “Searching for
the Q”.
References
Links
Books
 Fields and Waves in Communication Electronics; Ramo, Whinnery and Van Duzer; Wiley; iSBN
 Foundations of Microwave Engineering; Collin;Wiley; ISBN: 9788126515288
 Microwave Engineering; Ishi; Harcourt, Brace and Jovanovich; ISBN: 0155586580
 Microwave Engineering; David M Pozar; Wiley; ISBN: 9971512637
 Microwave Engineers Handbook; Artech House; ISBN:0890060029
 Microwave Engineering and Applications; Om P. Gandhi;Pergamon Press; ISBN: 0080255884
 ARRL UHF/Microwave Experimenters Manual; ARRL; 0872593126
 Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, Matthaei, Jones
and Young
Papers and/or Articles
 Package Resonance and Field Leakage, R. N. Simons
 Techniques Engineers the Cavity Resonance in Microstrip housing design; Edwin F. Johnson;
Pacific Monolithics; MSN & CT Feb 1987
 Predict Resonance of Shielded PCBs, Vinash Sharma; Microwaves & RF July 2007 and Aug
2007
 Design Guidelines for Microwave Cavities; Triquint Product Application Note March 2002
 EEstimation of Qfactors and Resonant Frequencies; IEEE Trans on Microwave Theory and
Tech. Vol51 No. 3 March 2003
Sunshine
Design Engineering Services is located in the sunny San Vicente Valley near San Diego, CA, gateway to the
mountains and skies. Are you looking for new things to design, program or create and need assistance? I offer design
services with specialties in electronic hardware, CAD and software engineering, and 25 years of experience with Test
Engineering services in RF/microwave, transceiver and semiconductor parametric test, test application program development,
automation programs, database programming, graphics and analysis, and mathematical algorithms. 
See also: 
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with Vector Analysis 
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Scattered Thoughts on Scattering Parameters 
Sunshine Design Engineering Services 23517 Carmena Rd Ramona, CA 92065 7606851126
Featuring: Test Automation Services, RF Calculator and SParameter Library (DLL & LLB)
www.AstroCalculator.com
SunshineDesign@cox.net 
Posted March 13, 2014 


