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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 self-resonance, 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	Old-from junkbox
D	0.1μF		Disk	100V	unknown	Old-from junkbox
E	0.1μF		Monokap	50V	unknown	Monokap
F	100pF	Centralab?	Disk	unknown	NP0	Old-from junkbox

Dielectric codes Y5U, Z5Z, Z5U indicate high dielectric constant but poor temperature stability. NP0 (now C0G) indicates a temperature stable capacitor.

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 voltage-tuned 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.)

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 non-linear. The data is taken with a Tektronix TDS-430A digital oscilloscope and a Tektronix TCP-202 Hall effect current probe.

The data was taken with the TDS-430A and TCP-202 current probe, with the TDS-430A 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. 

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 GR1650-A 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 1650-A 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.

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 581-SR215E105MAR.  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.)

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.)

The figure below shows normalized results for four stable capacitor types:

• C0G/NP0 temperature-stable 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.

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 self-resonant 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.

(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.

To see how this non-linearity 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 low-noise 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 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 non-linearity the capacitor exhibits.

The figure below plots THD (in percent, not dB) as a function of applied DC bias. Distortion peaks with around  5V bias.

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.

To model a non-linear 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 pre-defined 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 -0.00105X² +8.30549E-5X³ -3.25626E-6 X⁴ +6.9665E-8 X⁵
-7.78278E-10 X⁶ +3.558E-12 X⁷

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 non-linear capacitance, it's necessary to integrate our curve-fitted equation relating C to voltage. The resulting integration defining Q(X) for this particular Y5U capacitor is thus:

Q(X)=1E-6*(4.4475E-13 * x* (9.57823 + x) * (5002.74 + (-138.86 + x)*x)*(3188.84 + (-92.7184 + x)*x)*(1044.61 + (-27.9884 + x)*x))

The 1E-6 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 on-line symbolic integrator at http://integrals.wolfram.com/index.jsp, which is what I did.

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. 

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.

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.