Abstract
Most home-lab measurements of Q only evaluate an LC resonator.   We then tend to associate the resulting Q with inductor loss while capacitor Q is 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, s

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

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

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

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.  

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.
 

    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.    

17.  Polyvaricon variable capacitor.    A popular, inexpensive variable capacitor is one with plastic sheets between the plates.  It otherwise tunes like a familiar air variable with a rotary motion.   The shaft holds an edge driven knob.   Owing to the close plate spacing filled with dielectric material, we expect higher loss than a similar structured air dielectric capacitor.   But I had no idea how bad or good it might be.     The Sprague-Goodman film trimmer (data in item 12 above) produced quite good Q in a rotary structure with a plastic dielectric.  
The part with the black knob is the one measured here.
The variable that I measured has two sections, each with a capacitance that ranged from 6 to 270 pF.   The measurement used inductor "standard" L2 and the variable capacitor set at 100 pF.   The result was a very poor Qc of 540 at 10 MHz.   This was a major surprise.    The measurement was then extended down to 6.5 MHz where Qc was even worse at 340.   Capacitance was then maximum.   Just to be sure that the equipment was behaving properly, a high quality air variable was then measured; the result there was a Q of 3100 at 6.5 MHz, which is great.   This particular polyvaricon capacitor has now become the lowest Q capacitor that I have measured.    It may still be useful in some applications.  
    The term