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Measuring Distributed Capacitance
Revision History
11 Sept 2009. Original
18 Sept 2009. Revised to include comments from Terman, "Electronics
Measurements"
I've written elsewhere about distributed capacitance of inductors.
Self-resonant frequency of
Inductors being one example.
Table of Contents
Introduction
Distributed_Capacitance_and_Self-Resonant_Frequency
Direct_method
Grid_Dip_Meter
Q-Meter_Direct_Method
SRF_Determined_with_Impedance_Meter
Extrapolated_Capacitance_with_Q-meter_Readings
Extrapolated_Low_Frequency_Inductance_Measurements_with_Impedance_Meter
Comparison_of_Results
Introduction
Inductors, like all non-theoretical electrical parts, are
not perfect. The schematic below shows a commonly used model of a real inductor,
of the type you might buy from a parts supplier or make from wire and a core. In
addition to the inductance L, the real part also has loss (modeled as a simple
resistance, shown in the model as R) and parasitic capacitance, shown as C. (It
is also possible, and desirable under some circumstances, to model the circuit
with parallel resistance. For simplicity, our discussion will consider the
series model only.)
This model has been criticized as being overly simplistic,
with a more detailed model being proposed at
http://www.edn.com/contents/images/159688.pdf.
For the purpose of this discussion, however, the
traditional three-component model suffices. And, although the model shows R, L
and C as "ideal" or theoretical components with no imperfections, in fact
in a real inductor R, L and C are anything but constant, perfect components. For
example, the skin effect causes the winding resistance to increase with
increasing frequency and, in an air core inductor, L will also change with
frequency as the distribution of current within the wires changes due to
proximity effect. If wound on a magnetic core, then the permeability of the core
will also have a frequency dependent component, which means L is a function of
frequency. And, the core will have a loss that is also frequency dependent,
which means another factor that changes R with frequency. |
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Classical model of a real inductor, consisting of
inductance L, series resistance R and parasitic capacitance C. These values
are not constant with frequency in the general case.
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Distributed
Capacitance and Self-Resonant Frequency The classical description of C is that it represents the
turn-to-turn distributed capacitance of the inductor (and turn-to-core, etc.).
At some frequency, the "self-resonant frequency" or SRF, this turn-to-turn
capacitance resonates with the inductance L and the inductor becomes a parallel
resonant tuned circuit.
This traditional viewpoint
has been challenged and the inductor analyzed as a transmission line, with the
SRF frequency being determined as the frequency corresponding to the wire used
in L being a half-wave long. David Knight, G3YNH, makes a very persuasive
argument for this viewpoint at
http://www.g3ynh.info/zdocs/magnetics/appendix/self-res.html and I recommend
it to any interested in the subject. In fact, all of Dr. Knight's web site is
worth detailed consideration. I have made some measurements of air wound
inductors confirming G3YNH's SRF approach, and hope to do more work in this
regard before reaching a conclusion. I believe, however, his analysis is worthy
of careful consideration.
For the purpose of measuring and describing a typical
store bought inductor, it makes little difference whether C is the turn-to-turn
capacitance or whether it is just a fictitious capacitance of a value computed
based on the SRF.
How then do we measure the distributed capacitance?
This page discusses three approaches to measuring Cd:
- Direct method; at what frequency is the inductor self-resonant.
- Extrapolated capacitance when measured with a Q-meter
- Extrapolated inductance when measured with an impedance-measuring
device.
Let's look at each method in more detail. To keep the math more manageable
and in order to focus on the usual way in which inductors are used, our
discussion assumes reasonable inductor Q, in which the series and parallel
inductance are nearly identical and in which case the three definitions of
parallel resonance converge to nearly identical frequencies.
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Direct method
At "phase resonance" our RLC model has a purely resistive
impedance—the L and C elements cancel, leaving a pure resistive impedance. The
impedance Z at phase resonance is: Z=R+ω2L2/R
where ω is the frequency of phase resonance in
radians/sec. ω=2πf where f is in cycles per second or Hz.
The impedance is perhaps more recognizable as Z = (1+Q2)R.
Q is traditional "quality" factor of the inductor. In the series model we're
using, Q=ωL/R. (Our model assumes all losses from whatever source are subsumed
in R.)
Since Z is multiplied by Q2 (for Q>10, 1+Q2
is approximately Q2), a high Q inductor can present high impedance at
resonance. This effect gave rise to Q also being known as the "magnification"
factor.
How might we go about directly measuring the SRF of the
inductor? The answer depends on the available test equipment. I'll go over a few
methods, but undoubtedly many other approaches are possible, such as applying a
fast rise pulse and observing the natural resonant frequency.
The inductor I'll use is a 103A-21, air core, 100μH
inductor, made by Bellaire Electronics, as a clone of the Boonton
103A-series working inductors provided as adjuncts to Boonton's Q-meters. (From
the markings on the name plate, Bellaire made this part under a US military
contract.) |
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103A-21 Inductor. The nameplate says 100uH, 6 pF distributed capacitance and
approximate Q of 200, intended frequency of use 800-2000 KHz.
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Inside the shield is an air core inductor, wound on a ceramic core. It's not
visible from this angle, but the uppermost winding is spaced out a
considerable distance, and at an irregular angle, from the remaining
windings. This is how the inductor was trimmed to the target value.
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Grid Dip Meter The venerable grid dip
meter has been around since the 1930's, and like most older hams with an
interest in building, I have one at the back of the test equipment locker. Mine
is a B&W manufactured (probably built from a kit; I bought mine used at a swap
and shop 25 years ago) in the late 1950's or early 1960's.
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B&W Grid Dip Meter and plug-in coil set
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For those too new to the hobby to be exposed to a grid dip meter, it is exactly
as the name implies. It is a single tube oscillator with a sensitive meter
reading the oscillator grid current. The oscillator frequency is determined by a
combination of the plug-in coil and a variable capacitor. When the plug-in coil
is brought near the tuned circuit to be measured and the tuning capacitor
rotated throughout its range, at the resonant frequency of the circuit being
measured, the grid current meter will exhibit a sharp dip. This occurs because
energy is being coupled from the grid dip oscillator to the circuit under test
and this coupling causes the grid current to drop. It's important to have
sufficient coupling to cause a perceptible dip, but no too much so that the grid
dip meter influences the resonant frequency of the inductor being measured. |
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Grid dip meter oscillator coil coupled to the inductor under test
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Meter reads around 90 when off resonance and dips to perhaps 60 at
resonance. The meter sensitivity is adjustable with a potentiometer control
and I set it to be near the top end of the range.
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The dip occurs at about 6.8 MHz. (White scale corresponds to white
color-coded inductor, usable from 5-14 MHz.
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As the calibration scale should show you, the B&W grid dip meter is not a
precision instrument. (The king of grid dip meters is without question the
Measurements Model 59 and later 159, with honorable second place going to the
James Millen 90651 and its follow-on versions.) How close is the grid dip
meter? Based on the 6pF and 100uH nameplate data, we calculate the SRF as 6.497
MHz. Not all that far off the 6.8 MHz on the grid dip meter scale. (And, yes, it
would be possible to read the grid dip frequency more accurately with a counter
or receiver.)
Working from 6.8 MHz as the SRF, the computed distributed
capacitance of the 103A-21 inductor is 5.5pF.
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Q-Meter Direct Method
It's also
possible to use a Q-meter as the resonance detecting device. The principle
behind a Q-meter is illustrated in the image below. The Q-meter has a variable
frequency oscillator, an RF voltmeter and a calibrated precision variable
capacitor. The oscillator injects a signal into the inductor being tested, which
is brought to resonance with the variable capacitor C. The voltage across the LC
circuit is read by a high impedance RF voltmeter. This voltage is proportional
to Q and hence the voltmeter scale can be calibrated in terms of indicated Q.
(The instrument is constructed so that losses in the variable capacitor,
the voltmeter circuit, the oscillator signal injection circuit, etc. are
negligible compared with typical inductor Qs in the range up to 1,000 or so.)
Of significance in determining the SRF of an inductor is that the Q-meter's
resonating capacitor has external connection points brought out—HI
and GND in the schematic fragment. |
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Conceptual view of Q-meter
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Assume some "working inductor" is brought to resonance at a
frequency near the SRF of the (different) inductor under test. If the inductor
under test (IUT) is connected across the tuning capacitor, one of three
conditions will occur:
- If the working inductor is resonated at a frequency
above the IUT's SRF, it appears as a capacitor and C must be reduced to
return the working coil to resonance.
- If the working inductor is resonated at a frequency
below the IUT's SRF, it appears as an inductor and C must be increased to
return the working coil to resonance.
- If the working inductor is resonated at the IUT's SRF,
it will appear as a pure resistance and no change in C is necessary to
restore the working coil to resonance when the IUT is connected. (Of course,
the indicated Q of the working inductor will drop when the IUT is connected.
But the capacitance required for resonance will not change.)
Through an iterative process of connecting and
disconnecting the IUT whilst observing how C is adjusted to restore resonance,
the Q-meter's frequency is adjusted to the SRF of the IUT.
I've extracted the 4342A's instruction manual description
of this process, which may be read by clicking
here.
(Adobe PDF format.)
I've used a Boonton 103A-5 working inductor. Its a 2.5μH
inductor plugged into the normal inductance jacks. The IUT, the 100μH inductor,
is at right, connected to the Q-meter's capacitance terminals by the two clip
leads.
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Using the process, I determined the zero reactance, resonant frequency of the
IUT was 6650 KHz. For 100uH inductance, this corresponds to a distributed
capacitance of 5.81pF, quite close to the nameplate 6pF.
The added capacitance
of the two vertical alligator clips (measured at 1.78pF) does not enter into
this measurement because the clips are always in place, with the IUT being
connected and disconnected to the clips. Hence, the clip capacitance is not
material.
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SRF Determined with Impedance
Meter It's also possible to use an impedance meter, or vector network
analyzer to determine the self-resonant frequency. In this case, the impedance
of the inductor is measured as a function of frequency. As the SRF is
approached, the impedance switches from inductive to resistive to capacitive.
Or, if the impedance meter is set to display inductance, the apparent inductance
will switch to a negative value above the SRF. At the SRF, the inductance is
zero.
I used an HP 4192A LF Impedance Meter to look at the impedance of the 103A-21
inductor as a function of frequency. |
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103A-21 Inductor connected to HP 4192A LF Impedance
Meter.
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4192A with 103A-21
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The plot below shows the indicated inductance between 100 KHz and 10 MHz. As the
frequency approaches the SRF, the indicated inductance first rises to very high
levels and then abruptly reverses to a negative value. The SRF is the frequency
at which the indicated inductance is zero. |
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It's difficult to read the SRF with any degree of accuracy
from the wide range plot, but the expanded graph below shows the zero crossing
point to be approximately 6094.8 KHz. |
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The capacitance required to resonate 100μH at 6094.8 KHz is
6.82pF. Not visible in the photograph is the fixture used to connect the
inductor to the 4192A. The fixture adds 0.78pF in parallel with the 103A-21
inductor's distributed capacitance. Thus, the distributed capacitance is 6.04pF,
matching the nameplate value of 6μμF. |
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Other Methods Many other techniques are possible to directly measure
the SRF, which I will leave as an exercise for the interested reader.
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Extrapolated
Capacitance with Q-meter Readings In
addition to direct SRF measurement with the Q-meter, it's also possible to
determine the distributed capacitance with lower frequency measurements. This is
convenient if the inductor's SRF is above the upper frequency limit on the
Q-meter.
Two related methods are commonly used.
f1 & f2 Method. This involves making
an inductance measurement at frequency f1 corresponding to a low
capacitance, typically 50pF and then a second inductance measurement at a
lower frequency f2. This is described in the
4342A
manual extract as an "approximate method" of determining the distributed
capacitance.
As an example, I measured the 100uH 103A-21 inductor using
the f2 = f1/2 method with the following results:
|
Parameter |
f1 |
f2 |
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Frequency |
2.146
MHz |
1.073 MHz |
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Resonating Capacitance |
C1
= 50pF |
C2 =
215pF |
Plugging these values into equation 3-10 yields: Cd=
(215-4*50)/3 = (215-200)/3 = 5 pF.
This simple measurement is about 1pF low, somewhat better
than the instruction manual's note that this method has a typical error of at
least ±2pF. (I read the frequency with a digital counter and did not rely upon
the 4342A's frequency calibration.)
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It's possible, and generally desirable, to use more data
points from the Q-meter. The capacitance and associated resonant frequency can
then be plotted and a regression line fitted with both the distributed
capacitance and inductance extracted from the intercept and slope of the
regression line. The plot has capacitance on the Y
axis and 1/ω2 on the X axis. ω is, of course, 2πf. The reason 1/ω2
is plotted is that the standard equation for resonance, ω2=1/LC. C
consists of the external Q-meter capacitance plus the distributed capacitance Cd
associated with the inductor. Hence, ω2=1/L(C+Cd) or, with a bit of
algerbra:
Source: Clifford & Wing, eds., Electronics
Circuits
and Tubes,
McGraw-Hill, NY, NY (1947), Sect. 4.7.
Substituting X for 1/ω2, C+Cd = (1/L)X. This is
a linear equation with a slope of 1/L and intercept of Cd.
The plot below shows data points collected with the 4342A
Q-meter and the 103A-21 inductor, with a linear regression fit to the data. The
intercept is 4.90 pF and the slope is 9.9923 x 1015. Since the Y axis
is in pF, the slope must be multiplied by 1X10-12 to convert to
farads, since the fundamental equation has units of Hz, Henries and Farads.
Hence, L = 1/(9.9923X1015 x 1X10-12) = 100.08X10-6
or 100.08μH.
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The regression fit parameter R is 1.00, indicating a perfect
fit to a straight line. And, the results match the expected inductance about as
well as possible. The distributed capacitance of 4.9
pF, however, is well below the nameplate distributed capacitance of the 103A-21
inductor. It is, more interestingly, very close to the 5pF value obtained from a
simple two point measurement. |
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Extrapolated Low Frequency Inductance Measurements with Impedance Meter
In addition to determining the distributed capacitance
through measuring inductance at various frequency by resonance, as with the
Q-meter, a similar approach can be used with inductance measurements. The data
may come from an automated instrument or from a measurements made with a manual
bridge.
When using direct inductance measurements, it's most
convenient to plot 1/L on the Y axis and ω2 on the X axis. A linear
regression line fitted to the data points yields the inductance and distributed
capacitance.
It can be shown that the indicated or measured inductance
L at angular frequency ω of a coil with distributed capacitance Cd
and low frequency ("DC") inductance Ldc is:
This relationship can be recast into a form that is
linear:
Source: Clifford & Wing, eds., Electronics
Circuits
and Tubes,
McGraw-Hill, NY, NY (1947), Sect. 4.7.
The plotted data is over the frequency range 100 KHz to 2300 KHz. It's
important to only collect data up to a frequency at which the
The inductance is 1/intercept, whilst the distributed capacitance is the
slope. (The negative sign indicates capacitance that must be removed from the
inductor to make it "perfect," i.e., without distributed capacitance.)
The data shows an almost perfect linear fit (R is nearly 1). The computed
parameters are:
Cd=5.92pF net of fixture capacitance
L=100.24μH
Although I used the same fixture to adapt the 103A-21 to the 4192A, I took
the data on different days and measured the fixture capacitance slightly
differently. |
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Comparison of Results We have six
distributed capacitance results. How do they compare?
|
Methodology |
Cd
(pF) |
Comments |
| Grid
Dip Meter |
5.5 |
Crude accuracy of
frequency scale on grid dip meter limits accuracy |
|
Q-meter direct |
5.81 |
|
| HP
4192A direct |
6.04 |
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Q-meter approximate method from two measurements |
5 |
HP 4342A
instruction manual recommends this approach where Cd ≥10pF. Accuracy is
stated as not better than
±2pF |
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Q-meter from regression fit of multiple measurements |
4.9 |
Should be more
accurate than data from two measurements. |
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Regression fit of multiple inductance measurements with HP 4192A |
5.92 |
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Nameplate |
6 |
No significant
figures given. |
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Three of the six measurements are quite close to the nameplate value. Two are
direct measurements of the SRF, taken with a Q-meter and with an impedance
analyzer. Even the venerable grid dip meter with its crude frequency scale is
within 10% of the nameplate value.The outliers are
from the Q-meter, both the two frequency measurement and the multi-frequency
regression fit. While the 4342A Q-meter is less accurate than the 4192A, its
expected error is not so large as to account for the 1 pF difference. And, the
plotted data has an excellent fit to the regression line, which suggests the
Q-meter's main tuning capacitor linearity is quite good.
The conclusion from the measurements might be that a direct
measurement of the SRF is preferred to indirect measurements, but that indirect
methods can produce good results.
In this connection, the view expressed by F.E. Terman in
Electronics
Measurements,
2nd Ed.
p. 102 (McGraw Hill, 1952) is worth noting:
A method sometimes proposed for determining distributed capacitance
consists of measuring the resonant frequency of the coil when tuned only by
the distributed capacitance. This can be done by loosely coupling the coil
to an oscillator and observing the frequency at which the coil reacts
to the oscillator to cause a sudden change in the grid or plate
current. A knowledge of this self-resonant frequency and the true inductance
of the coil will permit a determination of an apparent distributed
capacitance. However, the capacitance obtain in this way will always be
smaller than when the same capacitance is measured by the preceding methods.
This is because when the driving voltage is a distributed induced voltage,
the voltage and current distribution in the coil with no external tuning
condenser is quite different from the distribution when an appreciable
capacitance is shunted across the coil terminals. For this reason the
distributed capacitance of a coil normally should not be determined by the
self-resonant-frequency method. [emphasis in original]
(The "preceding methods" Terman refers to are the Q-meter methods discussed
earlier on this page.)
Consistent with Terman's observation, the distributed capacitance measured by
the "direct" method using the grid dip meter is below the values calculated by
indirect methods as well as below the test inductor's nameplate
distributed capacitance value. However, the direct measurements made with the
Q-meter and with the HP 4192A agree quite closely with the nameplate. This
discrepancy may well be related to Terman's observation on application of
exciting voltage. The grid dip meter excitation encompasses Terman's objection
exactly.
In contrast, when the inductor is placed across the Q-meter or across the
4192A impedance meter, the test voltage is applied across the inductor in much
the same fashion as in the indirect method. Thus, one may expect the voltage and
current distribution to be similar to the indirect methods. I believe,
therefore, Terman's caution may be less important when other than the grid dip
meter method is used to directly determine the self-resonant frequency.
Inherent in many of these calculations is the assumption
that the inductance remains constant over the frequency range up to the SRF.
With an air core coil, this is only approximately true. The inductance of an air
core coil changes with frequency as the current distribution changes. Skin
effect and proximity effect cause small shifts in the current distribution
within the wires comprising the inductor. These small shifts change the
effective coil diameter and hence the inductance. Hence, to compute the
distributed capacitance at 6 or 7 MHz based on the low frequency inductance has
an error contribution of its own.
With a magnetic material core, iron, powdered iron or
ferrite, inductance can change radically with frequency and hence extrapolated
distributed capacitance measurements made of inductance at low frequency will
not necessarily accurately predict the SRF. |
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