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Capacitance Change with Applied Voltage;
or "when is a 0.1uF capacitor not a 0.1uF capacitor
Revision History
Original
Revised 07 July 2007�Added data for 1000
pF, 1 μF ceramic, electrolytic and tantalum capacitors at
page bottom.
Revised 04 Dec 2008�Added distortion data for 0.1μF
Y5U ceramic, monokap and polyester film; also added SPICE simulation for Y5U
distortion.
Revised 05 Dec 2008�Corrected text regarding nonlinear capacitor
simulation in LTspice
Revised 15 September 2010�Added butterfly plot
We usually assume capacitors are ideal passive elements;
at most we might consider dielectric loss, known as dissipation factor, or D.
Or, we might concern ourselves with lead inductance and selfresonance, as
illustrated here. But, for the most part, if we
measure a capacitor as 0.1uF, we assume it always is a 0.1uF part. This is not
necessarily a safe assumption, as some capacitors made with high value ceramic
dielectrics exhibit a significant shift in capacitance with applied voltage.
I measured the capacitance versus applied DC voltage for a
selection of six capacitors, selected for a variety of dielectrics. The
capacitors are pictured below. All, except for F are 0.1uF value. Capacitor B is
a film dielectric, whilst the remaining ones employ various ceramic dielectrics.
ID 
Nominal Value 
Manufacturer 
Form 
Rated Voltage 
Dielectric 
Comments 
A 
0.1μF 
Xicon 
Small Disk 
50V 
Y5U 

B 
0.1μF 

Tubular 
100C 
Mylar? 

C 
0.1μF 
Centralab? 
Disk 
50V 
Z5Z 
Oldfrom junkbox 
D 
0.1μF 

Disk 
100V 
unknown 
Oldfrom junkbox 
E 
0.1μF 

Monokap 
50V 
unknown 
Monokap 
F 
100pF 
Centralab? 
Disk 
unknown 
NP0 
Oldfrom junkbox 
Dielectric codes Y5U, Z5Z, Z5U indicate high dielectric
constant but poor temperature stability. NP0 (now C0G) indicates a temperature
stable capacitor.


I measured the capacitance and dissipation factor of each of these six
capacitors using a General Radio 1650A RLC bridge, whilst applying varying DC
bias voltage with a HP6217A variable voltage power supply, monitored with a
Simpson 467 digital voltmeter. The 1650A bridge operates at 1 KHz and changes in
capacitance of well under 1% can easily be discerned.
To permit a direct comparison, I normalized the measured
capacitance values to be 1.000 with no bias voltage.
The figure below shows essentially zero change in
capacitance with voltage for samples B, E and F. All three 0.1 disk
ceramic capacitors show considerable change in capacitance with applied voltage,
with sample A showing spectacular reduction in capacitance with applied voltage.
Indeed, except for their abysmal stability with temperature changes, one might
consider a Y5U dielectric capacitor as a substitute for a varactor diode in
voltagetuned oscillator design! (In fact, special voltage varying dielectrics
were developed in the 1950's for exactly this purpose, as varicap diodes had yet
to be invented.) 

We also see a change in dissipation factor with applied
voltage. (Dissipation factor is equivalent to inductor Q, except that by
convention D is measured in an inverse, so the smaller the D factor, the less
loss.) Sample E, which exhibits capacitance stability with applied
voltage, shows some change in dissipation factor. 

Although the data is based on DC bias voltage, we would see a
similar change in capacitance and D from an AC voltage applied across these
capacitors. This means that the capacitance (and hence the capacitive reactance)
and D (hence the equivalent series resistance) are functions of the applied
voltage.
The figure below shows the voltage across sample A
(channel 1) and the current through sample A (channel 2) with an applied
frequency of 992 Hz. Channel 2's scale is 50 mA/division. Channel 2 clearly
shows the current through sample A is highly nonlinear. The data is taken with
a Tektronix TDS430A digital oscilloscope and a Tektronix TCP202 Hall effect
current probe.


Examining the spectrum of the current sample, with an applied
400 Hz signal, we see significant even harmonic generation, with the second
harmonic about 20 dB down. These harmonics are generated by the nonlinear
action of the capacitor, as the test data is taken with no active components in
the circuit. The data was taken with the TDS430A
and TCP202 current probe, with the TDS430A executing an FFT spectrum analysis
appliqué.


We normally use high dielectric constant capacitors
such as sample A only for bypassing purposes, where changes in capacitance and D
with applied voltage are unimportant. However, if our design calculations
suggest a 0.1uF bypass capacitor is appropriate for a particular circuit
carrying 25V DC, we would find it useful to know that sample A has only 0.03uF
when biased to 25V. Sample E would make a much better selection.
We may also be tempted to use sample A as a coupling
capacitor, where the exact value may not be important, so long as it is "small"
with respect to the circuit's working impedance. In many cases, sample A may
work, but with large voltage excursions across sample A, harmonic and
intermodulation distortion caused by the coupling capacitor may be important.


Expanded Data.
The data presented above was taken in July 2001. It's now
July 2007, and I've recently acquired a new measuring device, a General Radio
GR1658 Digibridge. It's a digital readout device, as the name suggests, with an
accuracy in the ±0.1% range, so it's a factor of 10 better than the manual
GR1650A bridge I used in 2001. Since it's digital, data is easier and faster to
collect. Unfortunately, the GR1658 I acquired is not equipped with the optional
GPIB interface, so it still requires manually transcribing the values into Excel
and then Origin for plotting. Although the 1658 seems accurate, or at least it
matches all the other instruments I have, it has not been recently calibrated,
but we can have greater confidence in relative readings.
DC bias voltage is supplied with a HP6217A variable voltage
power supply, monitored with a Simpson 467 digital voltmeter. The 1658
Digibridge bridge operates at 1 KHz or 100 Hz for large capacitance values. (The
instrument I have is the European model with 100 and 1000 Hz test frequencies.)
The data below compares the relative change in capacitance
measured in 2001 with the manual GR 1650A bridge and 2007 data taken with the
1658 Digibridge. The capacitor under test is, in both cases, a 0.1 μF Y5U
capacitor from the same lot, but unfortunately not the same unit. The generally
good agreement provides confidence in both the 2001 and 2007 data.
There's little to add to the earlier comments�high
dielectric constant parts with poor temperature coefficients, such as Z5U and
Y5U dielectrics, exhibit major capacitance shifts with applied voltage. This
should be kept in mind when sizing bypass capacitors subject to DC voltage. 

1 μF monolithic capacitor
I've used these parts extensively in the Z90/91 for, amongst other things, DC
charge pump capacitors associated with RS232 level conversion. Mouser's P/N is
581SR215E105MAR. These are manufactured by AVX and the data sheet is at
http://www.avxcorp.com/docs/Catalogs/skycap.pdf. The part has a Z5U
dielectric, 50 WVDC and ±20% tolerance.
These parts show an even faster drop in capacitance with
increasing bias voltage than the similar dielectric 0.1 μF parts in the above
graph. At 10V bias, these are 0.5 μF parts, not 1.0 μF. All three units tested
behave similarly.
I also measured the equivalent series resistance of these
capacitors as 2.80 ohms at 1000 Hz, average of the three devices tested. This
means that above 50 KHz, or so, the effectiveness of these devices as bypass
elements is limited by the series resistance. (At 50 KHz, the capacitive
reactance and series resistance are about equal.) 

Since we've started with high value capacitors, let's look at
two other types commonly found; the aluminum electrolytic and the dipped
tantalum, as illustrated below.

The two high value capacitors tested. At center, an 82 μF,
63V electrolytic. At right, a 33 μF, 10V tantalum.



We'll look at both the absolute capacitance and the dissipation factor D for
both capacitors.The graph below shows both the
electrolytic and tantalum capacitors change capacitance and dissipation factor
very little with applied voltage. The tantalum is somewhat more constant than
the electrolytic, but both show relatively little variation with bias voltage,
unlike the high capacitance ceramics we've looked at so far.
Another point of interest is that the tantalum's
dissipation factor is lower. This is, of course, not a surprise, but it should
be kept in mind. Although more expensive, the tantalum provides lower
dissipation (usually a good thing) and usually (for the same capacitance value)
lower ESR, which is also usually a good thing, although there are rare
circumstances where a higher ESR may be necessary. (Of course, a series resistor
can be added to increase ESR.) 

Let's now look at smaller value parts, 1000 pF, such as may
be used in RF or audio design, where we require the capacitance to be unchanged
with applied voltage. The figure below shows
normalized results for four stable capacitor types:
 C0G/NP0 temperaturestable ceramic
 Polystyrene
 Dipped silvered mica
 Polyester film
The data shows very little change in capacitance versus
voltage. In fact, the full scale graph represents only 0.2% change in
capacitance and the measured data closer to 0.04%. Interestingly, all the
capacitors exhibit the same general C versus E curve, which leads me to suspect
that the majority of the variation, if not all of it, is residual error in the
GR1568 Digibridge. The instrument's accuracy is 0.1% or thereabouts (depends on
a variety of factors, such as the specific capacitance value, D, etc., but 0.1%
is the best case) and the changes seen in the data are a fraction of the
instrument's rated accuracy. Unfortunately, I don't have a 1000 pF vacuum or air
capacitor for a comparison standard. 

Although I didn't plot the data, I also measured a 1000 pF
Z5U capacitor. It shows a fair bit of variation with bias voltage, but not
nearly as much as the 0.1 or 1.0 μF Z5U parts. The
last plot shows the dissipation factor D for these four capacitor types and also
D for the 1000 pF Z5U part. D is a measure of loss in the capacitor, so if you
are looking for a capacitor to resonate a high Q circuit, you want the lowest D
capacitor, assuming, of course, other factors are acceptable, such as
selfresonant frequency.
I knew that polystyrene capacitors were good, but
I did not realize how good the C0G/NP0 ceramic capacitors were. Both beat the
dipped silvered mica. Of course, dielectric performance is a function of
frequency and the data below is taken at 1 KHz. Performance at radio
frequencies will not necessarily be the same.
I would take the dissipation factor data with some caution
for low values of D, as it presses against the 1658's accuracy limits to measure
D values in the range we see for the best capacitor values. Also, even if the
Digibridge were completely accurate in its D values (not the case), D data is
presented with four digits, so the smallest possible values are 0.0001, 0.0002,
etc. The C0G/NP0, polystyrene and some mica readings were all in those low
ranges, where the digital instrument's ±1 digit factor makes a major difference.
Hence, don't read more into this data than you should. It is fair to say, I
believe, that the C0G/NP0 and polystyrene capacitors have an excellent loss
factor but don't hold me to these specific numbers. 

Distortion Analysis (Added
04 December 2008). I thought it would be useful to run several capacitor types
through an audio distortion analyzer, type HP 8903B, to demonstrate how the
change in capacitance versus voltage translates into distortion. I looked at
three capacitors:
 Xicon 0.1µF disk ceramic, dielectric type Y5U
 Monokap, 0.1 µF
 Polyester film, 0.1µF
The plot below shows how the capacitance varies with
applied DC bias. This data is taken with a General Radio 1658 Digibridge with DC
bias provided by an external HP 6217A power supply. 

It's easier to judge the relative change in capacitance with
bias voltage by replotting the normalized capacitance, i.e., the
capacitance at 0 volt bias = 1.00 for each capacitor. 

The data is consistent with the earlier data on this page;
the Y5U has a strong capacitance versus applied voltage dependence; a monocap
has a lesser dependency, but still is far from linear. The polyester shows
almost no change. (The total change from 0 to 50 volts is about 900 PPM, or 14
PPM per volt. This degree of change presses the limits of my test equipment, so
take it as an estimate rather than a guaranteed number.)
To see how this nonlinearity causes imperfections in
waveforms, I built a simple test fixture as shown below. The signal source is 1
V RMS at 10 KHz, generated by the oscillator section of an HP 8903B audio
analyzer, with 50 ohm generator output selected. The RC network output (port
OUT) connects to the 8903B's analyzer input. External DC bias is provided
by an HP 6216B analog power supply.
As should be evident from the test setup, noise on the
bias supply will be read by the 8903B as increased distortion. (The 8903B
employs a notch system to measure distortion.) Even with extra filtering at the
test jig, the lowest possible THD measurable with my setup is around 65 dB with
1V applied test signal. Incidentally, the 6216B provides the lowest noise of any
of my HP power supplies. The digital power supplies are, as might be expected,
worse than the analog supplies. To measure lower distortion levels with a 1V
test signal, either additional noise filtering or a custom lownoise power
supply is necessary.
The capacitor under test is arranged as a high pass RC or
typical coupling arrangement. Configuring it as an RC low pass filter would, of
course, roll off the harmonics and provide an overly optimistic view of waveform
distortion. At 10 KHz, 0.1 µF has an impedance of 159 ohms. 

The figure below plots the measured THD at 10 KHz with 10V DC
bias and with no bias. No real surprises here either�the polyester capacitor
shows essentially instrument noiselimited distortion without DC bias. When DC
bias is applied, the THD measurement floor is limited by noise from the bias
power supply. The Y5U and monokap parts show
inferior performance and increasing THD with increasing drive signal, the
reverse of polyester. This is because the polyester capacitor generates very low
THD levels and the measurements are noise limited (both with and without
bias) so that the larger the exciting signal, the greater the 8903B's dynamic
range and lower measured THD. Both ceramic capacitors have a different mechanism
at work; their distortion is caused by the change in capacitance with
instantaneous voltage of the applied AC waveform. The larger the exciting
voltage, the more nonlinearity the capacitor exhibits. 

The most interesting capacitor is, of course, the Y5U. From
the plot of capacitance versus voltage, we expect the distortion to vary with
applied bias for a constant exciting voltage.
The figure below plots THD (in percent, not dB) as a
function of applied DC bias. Distortion peaks with around 5V bias. 

The plot below is the derivative of the Y5U's capacitance
change with bias voltage, i.e., how much the Y5U's capacitance changes with an
infinitesimal change in bias voltage centered around the bias voltage shown on
the horizontal axis. We should expect maximum distortion to coincide with
maximum dC/dV, as this will represent the maximum change in waveform.
The maximum distortion occurs at 5V bias, whilst the
maximum dC/dV is around 7.5 V, at which voltage the distortion is improved over
the value at 5V. Although I cannot be sure, I suspect the reason maximum
distortion does not occur at maximum dC/dV is that at the maximum dC/dV point,
the derivative is more or less symmetrical for small values of ΔV and rapidly
reduces as the voltage moves from 7.5V. This symmetry likely reduces the
distortion. At 5V, the peak distortion point, the dC/dV plot has a steeper slope
than for bias voltages above 10V. The shallower dC/dV slope above 10V will
translate into lower distortion. 

As an experiment in modeling, I curve fitted a 7th order
polynomial to the Y5U's capacitance versus voltage curve with the results
reflected below. 

This curve fitted equation can then be used in LTSpice to
model the Y5U capacitor, with its capacitance versus voltage relationship.
To model a nonlinear capacitance in LTSpice, it's necessary
to write an equation relating charge (in Coulombs) versus bias voltage. This
mathmatical relationship is written into the value of the capacitor (instead of
so many uF or pF) as Q=f(x) where X is the predefined variable in LTspice representing the
instantaneous voltage across the capacitor.
More generally, the charge Q stored in a capacitor is:
C(V) is the relationship between capacitance and applied
voltage, in this case, as determined by our 7th order polynomial fitted to the
measured C versus V data:
C(V) =0.07099 + 0.00295X 0.00105X^{2}
+8.30549E5X^{3 }3.25626E6 X^{4} +6.9665E8 X^{5}
7.78278E10 X^{6 }+3.558E12 X^{7}
In a theoretically perfect capacitor of constant value, of course, the
relationship between charge and voltage is simple; Q=CV where Q is the charge in
Coulombs, C is the capacitance in Farads and V is the voltage in Volts.
This is the case since the integral from 0 to V of CdV is simply CV. Things get
a bit more complex when C is not constant, of course.
In the case of the Y5U's nonlinear capacitance, it's necessary to
integrate our curvefitted
equation relating C to voltage. The resulting integration defining Q(X) for this
particular Y5U capacitor is thus:
Q(X)=1E6*(4.4475E13 * x* (9.57823 + x) * (5002.74 +
(138.86 + x)*x)*(3188.84 + (92.7184 + x)*x)*(1044.61 + (27.9884 + x)*x))
The 1E6 factor is needed to convert to Coulombs because
the curve is fitted to numerical values stated in µF.
Incidentally, this particular Y5U capacitor has a
different C versus V relationship when the bias voltage is reversed.
There's no + and  polarity on these capacitors, so "normal" and "reverse" don't
have a meaning other than by arbitrarily selecting one terminal and identifying
it as "+" for the purpose of the analysis.
If you don't care to perform the integration manually, you
can use the online symbolic integrator at
http://integrals.wolfram.com/index.jsp, which is what I did. 
The first simulation versus measured data is to simply look
at the 10 KHz voltage level developed into the analyzer section of the 8903B,
i.e., the voltage at <OUT> in the schematic, versus bias for an applied
stimulus signal of 1 V RMS. As the bias voltage increases, C decreases,
which reduces the voltage into the voltmeter. The
figure below shows reasonable agreement between the measured and simulated data,
with a typical error over most of the range of 5% or so, with greater
discrepancy only near zero volts. The reason for greater error around 0 volts
can be seen by looking at the equation fit; it's valid only for V > 0. (Now how
fast it drops with negative voltage.) With 1V AC RMS excitation applied,
the instantaneous voltage across the capacitor has a negative component until
the bias exceeds 1.4V. 

Unfortunately, the SPICE simulation does not match the
measured distortion data very well, as is apparent from the plot below.
Even if we disregard the expected error for bias voltage < 1.5 volts or so,
there is little agreement between the measured and predicted distortion data. At
this time, I don't have a good reason for the divergence, particularly in light
of the fairly good agreement between measured and SPICE simulated voltage across
the load. One possibility may be that the Y5U capacitor requires a more detailed
model to include nonlinearity dielectric soakage. I'll leave
it as an exercise for the interested reader to expand this analysis and
reconcile the SPICE and measured data. 

Butterfly Plots As this
discussion suggests, high dielectric constant (often designated by the Greek
letter κ) ceramic capacitors exhibit less than perfect behavior. As the TV
pitchman says, "but wait�there's more."
In fact, the capacitance versus voltage relationship of
high κ capacitors has a memory; the capacitance at a particular DC bias
voltage depends upon the DC voltage history, a phenomena known as hysteresis.
This relationship between capacitance and DC bias voltage
is usually illustrated in a 'butterfly plot' such as the one I made below of a
0.1uF, 50V Z5U dielectric ceramic capacitor. (The plot is called a 'butterfly
plot' because of its resemblance to the wings of a butterfly.)
To demonstrate the hysteresis effect, I wrote a simple
computer program in EZGPIB to
control an HP 4192A LF Impedance Meter. The 4192A, amongst other things, can
accurately measure capacitance with an impressed DC bias voltage. The 4192A's
internal bias generator has a range of 35V to +35V. The program logic starts
the bias voltage at 0V and increases it to 35V, then decreases to 35V, reverses
direction again and returns to 0V, all in steps of 0.25V.
I ran the program with dwell times at each voltage step of
1 second, 10 seconds and 100 seconds.
The plot well illustrates the complex relationship between
capacitance and DC bias in high κ capacitors. For any particular DC bias, the
part has two possible capacitance values, depending upon the bias voltage
history; in which direction was the bias voltage changing. As an example,
consider the blue trace at +20V bias. Depending upon whether the DC bias was
increasing or decreasing at the time of measurement, the capacitance is either
8.05nF or 8.15nF
Moreover, there's an even more complex relationship
relating to how long the bias voltage steps are held. At 20V DC bias, for
example, the measured capacitance can be between 7.6nF and 8.15nF, depending
upon how long the DC bias voltage has been applied and the direction of bias
voltage change.


There's yet another oddity visible in the plot. Notice how
the starting capacitance at 0 volts is never achieved again when the 0 volt bias
point is crossed, either from the positive or negative direction.
What causes these odd effects? It's related to the
structure of the dielectric. For reasons too complex to go into here, the
relationship between electric field and charge is non linear and also has some
electric domains that are slower to respond than others. 








