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The
Two Faces of Q

Wes Hayward, w7zoi, November 4, updated
November 8, 2010

Updates:  14Dec10,

Abstract
Most
home-lab measurements of Q only evaluate an LC
resonator.   The resulting Q is usually then attributed to inductor
loss while the capacitor Q is usually assumed to be quite high.   This
assumption is now in greater doubt, especially with SMT components.  In
an effort to isolate capacitor Q from inductor Q, some well specified high Q
mica capacitors were purchased and used in measurements.   These
allowed evaluation of some inductors that then become "standards" that can
be used to evaluate capacitors.    Several available capacitor
types were investigated and some inductor Q measurements were
extended.

Introduction

I've always
been interested in the design of frequency selective filters and impedance
transforming networks.   Intimately connected to this has been a long
standing interest in the measurement of Q with some of this work having appeared
on this web site.   This report is an update (and summary) of parts of that
work.

This figure shows an ideal LC tuned
circuit, or resonator.   Click here to see some of the analytic formality.

Measuring Resonator Q

Three methods for Q
measurement are briefly discussed here.   The details can all be derived
from a simple model for resonator Q where all losses are
represented by a single resistor.    The resistor value is

A subscript "R" appears on both the resistor
and the Q term, indicating that these are resonator related terms.
For details, see Introduction to RF Design,
ARRL, 1994, p58, or many other classic engineering text books.   The three
measurement methods presented are all  resonator measurements and are not
necessarily determinations of component Q.   All analysis done in this
discussion uses series resistors to model loss, but parallel resistors can also
be used with no change in results.

1.  The
Q of a tuned circuit is equal to the center frequency divided by the 3 dB
bandwidth of that resonator.    This is not a definition or a rule of
thumb, but is a derived result.    This resonator
characteristic can be used to measure the Q of a LC tuned circuit embedded in
the following circuit:
The basic circuit to be measured is the tuned
circuit built from L and C0.   These components form a parallel tuned
circuit and are then coupled to a 50 Ohm source and a 50 Ohm load with coupling
capacitors Cc.    The two capacitors should be nearly identical in
value, should themselves have high Q, and should be much
smaller in capacitance than C0.    Typically, Cc=C0/100.   The
following experimental procedure is used:
a)  The source is tuned to
produce a maximum output response.    The center frequency, F0,
and the insertion loss is noted.   The loss is just the power
available from the generator to the load divided by the power
delivered to the load.    The available power from the
generator is measured by removing the filter (the part of the circuit between
the dotted lines) and replacing it with a "through" connection, attaching the
generator directly to the load.
b)   The insertion loss should
be 30 dB or more.   If the loss is less than 30 dB, the values of Cc should
be decreased and the procedure should be repeated. Remember to keep the Cc
values approximately equal to each other.
c)    Assuming IL> 30
dB,  the exact value of IL is carefully measured and recorded.   The
generator is then tuned on either side of the peak to find the places where the
loss is (IL+3 dB.)    This requires careful experimental
procedure.
d)   The difference between the two 3 dB frequencies is B,
the bandwidth.   Once this number is determined, the bandwidth
is known and the Q can be calculated as

Q=F0/B

The reason for picking an insertion loss of
30 dB or more is that this produces a Q number that represents the intrinsic
loss in the resonator.   This is called the unloaded Q, or Qu, and is the
value we seek.   An IL well under 30 dB would generate a lower Q value that
is partially determined by tuned circuit loading by the
external source and load.    The loaded BW would be wider than the
unloaded value.   The center frequency for the composite filter will be
slightly lower than the raw resonator formed by L and C0 owing to the two extra
capacitors Cc.

A variation of this method uses larger values of Cc, which
yield lower loss through the measurement filter.   A mathematical
correction is then applied to extract the unloaded Q information.   It is
now important to obtain a better measurement of the actual insertion loss.
The information in IRFD, ARRL 1994, p58, can be used for this
variation.

Some workers do a similar measurement where a source is weakly
coupled to the resonator and the output is sampled with an oscilloscope and 10X
probe.   This method can provide reasonable results if done with great
care, but can also be compromised by the loss in the scope probe.   I
prefer to avoid this uncertainty by using a measurement scheme where all loads
are well defined.

2.  A second scheme for measuring the Q of a
resonator is to configure the L and C as a series tuned circuit.   This
series tuned circuit is then attached as a parallel connected trap as shown
here.

The source generator is tuned to produce the lowest
output in the load, which occurs as a narrow notch.   The following
procedure is used:
a)   The generator is tuned to produce the notch
response.   The frequency is carefully noted.
b)   The attenuation
of this notch filter is carefully determined.    This can be read
directly if a network analyzer is used as the source and load.   If a
signal generator is used with a spectrum analyzer, power meter, or 50 Ohm
terminated oscilloscope as the load, the attenuation can be obtained by removing
the trap filter and inserting a step attenuator in its place.   The
variable attenuator, which should have 1 dB or finer steps, is adjusted to
produce the same response that the filter produced.    It may be
necessary to interpolate to obtain resolution of about 0.1 dB for attenuation.

c)   The trap is disassembled and the capacitance of C0 is
measured.   I usually do this with an LC meter from AADE.   (See
"Almost All Digital Electronics" on the web.)
d)   The measured data is
then used to calculate the Q.    Additional detail is given on page
7.36 of EMRFD.   (Experimental Methods in RF Design, ARRL,
2003.)  The equation is

It is important that the source impedance used for
this measurement be well known at Z0, usually 50 Ohms.   If there is
uncertainty, it is best to place a pad right at the coax connector where the
trap is applied.  I usually use a 14 dB pad, even when using a network
analyzer for the measurements.     Source impedance is much more
important than load impedance for this measurement.

This method has the
advantage over the direct bandwidth method that only one careful level
measurement is needed.   This is the determination of the attenuation
value.    I tend to use method 2 for routine measurements, and use the
bandwidth scheme (method 1) to confirm the experimental results.

A
variation of this method uses a parallel tuned circuit connected as a series
trap.   This is not as handy, for neither component is attached to ground.
It is usually handy to have C0 attached to ground.

3.
The fundamental circuit used in commercial Q-Meters is shown below and
this is a third method.

There are two critical features to this circuit.
The first is a very low source impedance.
This is realized with a carefully built transformer in the instruments
found on the surplus market.   It may be possible to get good low Z results
with modern wide bandwidth operational amplifiers.  (That's an experiment
on my list.)   The other difficult, but important element in building a Q
meter is the capacitor, C, usually a variable.   The cap should have the
highest possible Q.   Some workers have reported good results with Jennings
Vacuum Variable capacitors.    If the capacitor is good enough, the
measured results will faithfully describe the Q of the inductor.
The manual for the HP-4342A Q Meter is often sold on the web and offers
interesting reading.    I believe this is the last commercially built
Q meter and know of nothing on the market at the present time.
Commercial Q meters have largely been replaced by network
analyzers.

Modeling the Two
Faces

Our home-lab measurements are all done on
resonators, usually fabricated from discrete inductors and capacitors.
Traditionally, we have assumed that the capacitor Q has been sufficiently high
that it can be ignored, attributing all loss to the inductors.   That is a
reasonable viewpoint in some, but not all situations.    In
particular, many surface mount capacitors are sufficiently poor that they
severely compromise resonator Q.    We really must be able to measure
both inductors and capacitors.

The ideal LC is modeled, as a resonator, with a single
resistor, Rr.  This resistance is the value that would generate the
observed Q if the L and C were otherwise ideal.     The resistor
value is just the ratio of the inductive reactance to
resonator Q, as presented in an earlier equation.     The resonator
resistor is drawn in series with the inductor in the above figure, but it is
clearly in series with both the L and C.

The Q for the inductor and the
capacitors alone is modeled with individual resistors.   Again, these
resistors are just the reactance of the elements divided by the Q of that
element.    We should emphasize that the these resistors are just
models, usually frequency dependent.

The
resonator R is a sum of the two component related resistors.
(A mesh equation is written for the analysis.)   This leads to
a formula for resonator Q in terms of inductor and capacitor Q values,

Although not immediately obvious, this equation is
the familiar hyperbolic form.    The equation can be solved for
inductor Q as a function of capacitor Q, treating resonator Q as a parameter.
The result is the following curve:
In this example, the resonator Q was set to 200.   Even when
the capacitor Q is 10 times the resonator Q, there is still a 10% difference
between inductor and resonator Q.

Direct Measurement of Inductors and
Capacitors

It is, in concept, possible to measure a capacitor or
inductor directly with a Vector Network Analyzer, VNA.   This is shown
below

In this measurement, the "DUT" to be measured is
placed in the signal path between the source and the load.   The loss
resistance of the L will alter the response at the output.     In
concept, measuring the magnitude and the angle of the voltage at the output will
allow both L and R-L to be calculated for any applied frequency.   But it
is an extremely difficult measurement requiring exacting calibration of the VNA.
The reactance is so large that it almost completely dominates the
overall impedance.   After all, this is what we desire -- we are
predominantly interested in high Q inductors and/or capacitors that have low R
components.

The above figure uses a transmission
measurement.   It is also possible to measure an L or a C attached as a
load on a bridge attached to the VNA.   This is termed a "reflection"
measurement.      The results are similar and remain equally
difficult.     The severe errors of this direct method are discussed
in Agilent Applications note 1369-6.    Much better measurements are
obtained when one uses a scheme called RF I-V where a radio frequency source is
applied to an unknown impedance.   Then the current through the impedance
and the voltage across it are both measured.   The vector ratio of the
values is calculated to obtain a better complex impedance value.   This
method is discussed in Agilent Applications Note 1369-2.   (Thanks to N2PK
for both references.)

An excellent reference I've found for this subject
is the collected information presented on the web by Paul Kiciak, N2PK.
Google "N2PK VNA" and you will immediately get to Paul's site where he describes
his homebrew vector network analyzer.    There is a Yahoo Group
devoted to this design.   The information on the Yahoo site provides links
to numerous pertinent HP/Agilent documents and application notes having to do
with network analysis.

Another interesting treatment of the VNA
problem is the discussion by Thomas Baier, DG8SAQ.   His first article
appeared in QEX for March/April 2007, p46.   A later, more refined
instrument was described in January/February 2009 and in May/June
2009.

I've done experiments with a version of the N2PK VNA and get
reasonable results for low and modest Q elements.    However, the
results are far from the "warm and fuzzy" ones that we would like to have,
especially when measuring higher Q parts.    N2PK has built his own
version of the Agilent RF I-V scheme and has obtained much better
data.

Using Capacitors
with Known Q

There is an alternative to a
direct measurement and that is the method we have used in this study.
Rather than trying to do the complete measurement, we
merely looked for capacitors that were of moderately high Q and had well defined
and published Q specifications.   These capacitors would then become the
basis for resonator measurements that could be extended to provide inductor
results.

We searched the specifications in the
data sheets for some readily available capacitors and found some good ones at
Mouser.    The parts we selected are mica capacitors manufactured by
Cornell-Dubilier (CDE).   These parts are in the CDE MC line.
The data sheets can be downloaded from the Mouser web site.   The MC
capacitors are SMT parts, but they are physically large enough to be quite easy
to handle, even for those folks uncomfortable with chip parts.    The
data is sparse with little more than typical curves for a few representative
capacitors.   However, the Q values are high enough at almost 4000 at lower
frequencies.   Q data is only given for a few samples, but they all seem to
converge to a constant value at low frequency.   Our procedure was to
measure resonator Q with a selection of several of the MC mica caps, which would
then give us a sampling over frequency.  We used the highest Q toroid
inductor we could build when doing these measurements, forcing a measurement
that would emphasize capacitor Q.    More data will be presented
regarding the inductor.   The Q data from the CDE MC data sheets, or
estimates of it, were then used to characterize the inductors.

Very high
quality Porcelain capacitors are also available from American Technical Ceramics
and from Johanson, both well established component vendors.   See
www.atceramics.com and www.johansontechnology.com on the web.

This is a close-up view of a loose CDE mica type MC
capacitor and another mounted on a piece of single sided PC board.
I used single sided board to hold these capacitors.   (Later, we will
present some data on the Q of circuit board.)   The parts shown are 100
volt, 200 pF MC12FA201J-F.    This part is kept on the small board, so
it is only soldered once.  When additional measurements are to be done, the
original board is used.

This photo shows a bandwidth test
fixture using method #1 from above where the toroid is measured with a 100 pF
100 volt Mica MC capacitor.   1 pF ceramic "dog bone" capacitors couple the
resonator to the outside world.   The inductor is 20 turns of #18 wire on a
T68-6A core.    Although the T68-6 core is readily available, this "A"
part is not. The "A" designator indicates a shape with a greater cross section
of powdered iron material than is available from the usual T68 sized cores.
This measurement yielded a Q value of 399 at 10.7 MHz, assuming a
capacitor Qc of 2500.   The resonator Q was 344 for this measurements.

This photo shows the same board with the same
inductor and and the same MC capacitor, but now configured for measurement with
the trap method, scheme #2 from above.   This resulted in an inductor Q of
420 at 10.8 MHz, again assuming Qc=2500.   Resonator Q was 360.   The
two measurement methods produce nearly identical results.

I modified the
computer program that I normally use with the Notch type Q measurement to
include the effect of an assumed capacitor Q.   The program screen is shown
below.  (Eventually, I'll make the program available via the EMRFD errata
page on this web site.)
The high resolution with the Attenuation results
from the VNA output.  The usual cautions having to do with excessive
resolution must be applied!   That is, just because we calculate or measure
things to a thousandth of 1 dB does not mean that this extreme resolution is
meaningful.

Some Measurement
Results

1.  "Standard" inductor "L2",
which is 20 turns of #18 on a T68-6A, tightly wound on the core.
This was measured with the variable capacitor mentioned in item 3 below this
entry.   The Q of the inductor, the Q of the resonator during measurement,
and the value of Qc assumed for the measurement are all plotted on the curve.

The blue trace shows the Q of the capacitor that we
assumed for calculations.   The Qc plot is actually of Q/10, so the actual
low-frequency Q is 3000.   Capacitor Q drops with increasing
frequency.

This inductor was initially measured with three or
four values of CDE MC fixed capacitor, providing the first points that set the
inductor Q.   More detail is given in item 3 below.   Note that the
resonator Q data above is a firm measurement that does not depend upon any
assumptions.   Inductor Q is then extracted from an assumed capacitor Q and
the previously discussed tradeoff
equation.

2.  "Standard Inductor L3."
This inductor also uses a T68-6A core, but has only 10 turns of #18,
evenly spaced along the core.   The Q is not as high as with "L2," but the
smaller inductance allows operation to higher frequency.
The blue trace shows the Q of the
capacitor that we assumed for calculations.   The Qc plot is actually of
Q/10, so the low frequency Q is 2500.

3.   Variable
Capacitor.
The initial experiments with the CDE MC capacitors used values
of 100, 200, and 470 pF with two different inductors.      The
inductor Q for our L2 "standard" (20 t #18 on T68-6A) was 353 at 5 MHz as
determined with a 470 pF MC capacitor with an assumed Qc value of 3600.
Having a few measurements with this and the other MC capacitors
provided a base for the inductors over a modest frequency range.
This was used to measure the Q of the variable capacitor that was the base
element in our Q measurement test fixture.   This capacitor, which I think
was manufactured in the UK by Jackson Brothers, is a dual section variable with
almost 500 pF capacitance per section.    We had
determined it to be about the best variable in our junk box in early Q
experiments a few years ago.  The present results indicated the following
capacitor Q values:
6 MHz         Qc=2855
8 MHz
Qc=2423
11 MHz
Qc=2801

Based upon this data, the default capacitor Q in the program
shown above has been set at 2500.  The capacitor Q can be edited by the
program user.

4.   Porcelain Capacitors.
My junk
box included a capacitor kit with a large variety of capacitors made by
Tansitor.   These were very expensive capacitors, even in their day and are
no longer available.   A 200 pF unit was tested with the previously
mentioned standard inductor (20t #18, T68-6A).   Assuming a 7 MHz inductor
Q of 400 with a measured resonator Q of 363 yielded a capacitor Q of 3900.
Several other measurements were done with capacitors from this kit, all
producing values of several thousand.    When the capacitor Q values
become this high, the measurements become more difficult.    Some of
the other capacitors from this kit were used to fill in gaps and to extend the
measurements.

5.  Leaded Silver Mica
A 190 pF
SM capacitor from the junk box was determined to have Qc=1600 pF at 7.85 MHz.
This is not as good as the variable or the MC capacitor, but is
good enough for many communications applications.

6.  Leaded 100
pF Ceramic Capacitors.
I was able to put my hands on quite a variety of
ceramic capacitors.   They were all measured at about 10 MHz.  QL of
400 was assumed, again for the T68-6A core.      Three capacitors
tested had Q of 2900, 4400, and 5900.    All of these parts were
pretty good and would be fine for homebrew LC filters, even though some were low
priced parts.

7.  SMT Capacitors
Several SMT
caps were evaluated.   All were elements from the junk box and all used my
"standard L2" inductor.
1206
220 pF      7.3 MHz    Qc=472.
(unknown origin from junk box)   This is the worst single capacitor I've found so far except for PC
Board material.
0805
330 pF       6.0 MHz   Qc=1800
(Panasonic-ECG PCC 331 CGCT-NT from Digi-Key)
1206
120 pF
9.9            Qc= 1070  (PCC
121 CCT-ND from DigiKey).
1206        100
pF        10.6
Qc=660.  (junk box unknown)

8.  Double Sided Circuit Board Capacitors.
Two
different pieces of PCB material were evaluated.   The first was some
standard FR-4, 3.5 x 3.7 inches, 234 pF.   Qc=47 at 7.1 MHz.   The
second piece was 429 pF with a piece that measured 4.7 x 6 inches, with
Qc=1368 at 5.2 MHz.    This second piece was a material called Duroid
and is used for microwave applications.

9.  Polystyrene Film
Cap.
This was a junk box cap with C=220 pF.   At 7 MHz, we measured
Qc=1180.      More measurements with Polystyrene caps are in
order.

10.  Common Toroid Inductor ("L1", 17 turns #24, evenly
spaced, on T50-6.)
This is a toroid that many of us have used in filters.
The measured inductor Q values obtained for this core are:
14
MHz        QL=273
10 MHz
QL=291
6.7 MHz
QL=281
This part would be satisfactory for the main inductor in these
experiments, if I didn't have the higher Q parts
available.

11.  Solenoid at MF with 660/46 Litz wire.
This high inductance "crystal set special" 44 turn coil
was wound on a 4.5 inch diameter styrene from with a coil length of 3.2 inches.
The coil was resonated with a 200 pF MC capacitor at 786 kHz.
Measured resonator Q was QR=1164.   Assuming Qc=3700, the inductor
QL was almost 1700.   A higher capacitor Qc would produce a more realistic,
but nonetheless stellar value of QL around 1500.   The same inductor was
measured with a 200 pF Tansitor porcelain capacitor.    The resonator
Q was higher at 1286, indicating that the porcelain capacitor is measurably
higher in Q than the MC mica capacitor.    Our main interest in
measuring this inductor was to extend the measurements down to the 1 MHz area
where the MC mica capacitors have their highest Q.   Incidentally, this is
by far the highest inductor Q we have even seen.    Some UHF helical
resonators were getting close though.

12.  Film Trimmer.
A capacitor that I've used in many filter designs is a 4.5 to
65 pF film trimmer manufactured by Sprague-Goodman.    One was
measured as 75 pF fully meshed.   It produced a Q of 2500 at 12 MHz, a very
respectable number.

13.  More SMT Ceramic Capacitors.
(8Dec10 update)  Item #7 in this
list presents some preliminary results with SMT capacitors that were available
in my stock.   After these had  been measured, I remembered that I had
purchased a selection of SMT capacitors just for such a comparison.   A
little digging produced the parts.   They were either 56 or 120 pF in value
and were in either 0805 or 1206 sizes.    These parts were measured
with my "standard" inductor, "L2."   Here are the results, presented in
order of increasing capacitor Q. Only one part from each strip of 10 was
measured.
a)  120 pF 0805  C0G 50V, manufactured by AVX, part #
08055A121JAT2A,   Qc=635 at 10 MHz.
b)
120 pF 1206  C0G 50V, Panasonic-ECG, part#ECU-V1H121JCH,  Qc=880 at 10 MHz.
c)  56 pF, 0805, NP0, 50V, AVX,
part # 08055A560JAT2A, Qc=955 at 14 MHz.
d)
56 pF, 0805, C0G, 50V, Kemet, part # C0805C560J5GACTU, Qc=1100 at 14 MHz.
e)  56 pF, 1206, C0G, 50V,
Panasonic-ECG, part # ECU-V1H560JCM, Qc=1240 at 14
MHz.
f)  56 pF, 1206, C0G, 50V, AVX, Part # 12065A560JAT2A, Qc=1500 at 14 MHz.
g)  56 pF, 0805, C0G, 100V,
Murata, Part # GRM2195C2A560JZ01D, Qc=1840 at 14
MHz.
------------- The next two measurements were done merely to further
validate the measurement scheme. -----------------
h)  130 pF, Porcelain
chip, TC?, V?, Tansitor Corp.   Qc=4600 at 9.7
MHz.
i)  56 pF, Porcelain chip, TC?, V?, Tansitor Corp.
Qc=3900 at 14
MHz.
We should avoid generalizations about these
parts, for the sampling is very small.    The last two in the list
were parts pulled from my stash of "golden" parts and were done merely for
"calibration."   The one valid conclusion we can draw is that none of the
routine SMT parts (in our junk box) are spectacular and some are pretty poor.

Another observation was that this is a tedious
measurement and some sort of a test fixture is needed where a SMT part can be
inserted, measured, and returned to the appropriate envelope to be used at a
later time.

14.  Mica Compression Trimmer
(Tektronix surplus, marked GMA40300)    I have often used
this part, or similar ones, for tuned RF power amplifiers at HF and 50 MHz.
Set C to 104.8 pF with AADE L/C meter and resonated with the "L2"
standard.   Capacitor Q was 2187 at 10.5 MHz, assuming Q-L=400.  See
the tradeoff
curve and the photo below.

The inductor is our "standard, L2" used for many of
these measurements.   The upper trimmer is the mica compression measured in
case 14 while the lower capacitor is the rotary ceramic part measured in case
15.

15.  Ceramic Trimmer.
Nominal 50 pF max C.   This is a classic rotary ceramic design, set for
maximum capacitance and measured with L2.   The Q was only 1000 at 14.7
MHz.

16.  Dog Bone Ceramic, nominal 47 pF.    This
is a part that remains a mainstay of my junk box, even though it is perhaps 40
years old.   It is a very stable NP0 part that I've used in perhaps too
many variable frequency oscillators.    This sample measured 47.3 pF
and had a Q over 3000 at 15.5 MHz.

Conclusions

The
purchase of and measurement with the CDE MC capacitors, although less than
profound, was certainly a worthwhile exercise.   Extracting Q values from
the curve offered in the CDE data sheet has allowed some internal "standard"
inductors to be characterized.     Those inductance standards are then
used to evaluate a variety of other capacitors.     The mica
capacitors have moderately high Q and are readily available with affordable,
although not cheap prices.

The results obtained in this study are
not offered as being high accuracy.      We are still estimating
capacitor Q values when an inductor is being measured.   The results are,
however, consistent and are a step in the right direction.    The Q-L
versus Q-C curve presented in the text illustrates the nature of the tradeoff.

It
is probably not necessary for the casual experimenter to refine his or her
measurements to isolate capacitor Q values.    The most common
application is the fabrication of LC bandpass filters.    For that, it
is perfectly acceptable to measure resonator Q.   Even there, it is rarely
necessary to have highly accurate knowledge of resonator unloaded Q.
Rather, all that is required is to be sure that the Qu values are high enough
that a desired filter can be realized.    This detail always emerges
from a simulation so long as Q is included in the
models.