A capacitor measured with a $100 handheld multi-meter can give a substantially different result than the same capacitor measured with a $10,000 LCR meter. That same capacitor measured with two different handheld multi-meters may also give results that vary several percent depending on the dielectric material of the capacitor and on the measurement algorithm used. In order to know the factors that contribute to this variation and, even more importantly, know when to upgrade to the $10,000 LCR meter, it is important to understand the principles behind the measurement algorithms used to make capacitance measurements.

The analysis of capacitance measurements is best understood by examining the way that resistors are measured. When a digital multi-meter measures a resistance, it uses a constant-current source of some known value, to generate a voltage across the resistor under test. This results in a DC voltage that is easily translated into a resistance value by the ADC and signal processing firmware. The error terms inherent in a resistance measurement are easily understood and avoided. Thermal EMF, lead resistance, leakage currents, and self-heating are some of the more significant error sources, and they can be controlled through proper measurement techniques and built-in multi-meter features such as offset compensation.

Even in moderately priced instruments, resistance measurements are accurate to better than 30ppm and can be made without much trouble. Making a sufficiently accurate measurement of a different type of passive component, such as a capacitor, is an entirely different matter.
This article describes various capacitance measurement techniques and compares the effectiveness of their accuracy.

**High-accuracy capacitance measurements**

The obvious extension of the resistance measurement to capacitors is to stimulate the capacitor under test with an AC source. In high-performance LCR meters, one technique used is to find the value of a capacitor does just that. An AC signal of known frequency is applied through an internal low value resistor and the capacitor under test in a series configuration. The AC current flowing into the capacitor must also flow through the resistor, creating an AC voltage across the resistor.

The magnitude and phase of this voltage can be measured and compared to the original AC signal, and the capacitance can be computed. Techniques such as this frequency-domain measurement can be very accurate and can give information about additional parameters such as dissipation factor; however, instruments that implement these techniques are specialized, only measuring passive networks and costing in excess of $3500.

More general-purpose instruments have cost constraints that do not allow them to include an AC signal source; however, they still implement a capacitance function. The way that they do this is by utilization of the same DC current source used to measure resistance.

**Low-cost capacitance measurements**

As discussed previously, digital multimeters contain a precise, internal current source that is used to create a DC voltage across a resistor. This same precise current source can be used to create a voltage across a capacitor. An ideal capacitor being charged by an ideal constant-current source will create a ramp, characterized by the equation *I = C dV/dt*.

Therefore, a value for capacitance C can be computed in the time domain by applying a constant-current source and observing the rate of change on the voltage across the capacitor. Many low-cost bench and handheld multi-meters make capacitance measurements under the assumption that the current source and capacitor are both ideal.

However, there are no ideal capacitors. Capacitors exhibit non-ideal factors such as dielectric absorption, leakage, dissipation factor, and equivalent series resistance (ESR). These terms can introduce substantial error into the time-domain measurement technique described above. Therefore, most of the low-cost instruments that measure capacitance have a footnote stating that their "specifications apply only for film capacitors."

Film capacitors, such as those with polyester and polypropylene dielectrics, have low-enough loss terms that this time-domain technique can give results that are accurate to 1%. The errors introduced by non-film dielectrics, however, do not necessarily demand the use of a high-performance LCR meter. There are other techniques that have recently been introduced in bench multimeters that can reduce the error caused by non-film dielectrics, without the expense of an LCR meter.

**Better low-cost capacitance measurements**

The loss of a capacitor being charged with a DC current source is best modeled as a parallel resistance. This model is shown in **Figure 1**:

*Figure 1: Time-based capacitance measurement model.*
A constant-current source connected to a parallel RC circuit results in a voltage curve that changes with time, and is represented by:

assuming that there is no initial voltage across the capacitor. In this equation, * *is the time constant equal to *R* times *C* and *I *is the value of the constant-current source. Both this curve and the ideal straight-line curve are shown in **Figure 2**.

*(Click to enlarge image)*

* Figure 2: Capacitor voltage, with and without loss.*
Notice that the parallel resistance tends to bend the straight line down by an exponential factor. The area between the straight line and the curve is due to the loss term that creates error in the measurement. Since this is a transcendental equation, it quite difficult to solve without using an iterative technique. The derivative of this equation,

can be solved in closed-form. If the RC time-constant

is known, the capacitance value

*C* can be found by substituting it into this equation, in a method similar to the time-domain capacitance algorithm without loss rejection. The essential measurement enhancement therefore lies in finding the magnitude of the RC time-constant

*. *
To find the RC time-constant, the capacitor under test is first discharged by connecting either a resistor in parallel, or by reversing the polarity of the current source. The constant current is turned ON and high-speed readings are taken by the multimeter's analog/digital converter (ADC). An exponential fit is performed on these readings and, using both the readings themselves and the slope of the line between adjacent readings, the RC time-constant is computed.
This algorithm has a number of stringent requirements that makes it unsuitable for every digital multimeter:

- First and most critical, the ADC in the digital multimeter must sample fast enough to capture multiple points on the charge ramp of the capacitor under test, and without introducing significant noise to the measurement.
- Second, the multimeter's constant-current source must not exhibit non ideal behaviors, such as a thermal tail when turning on.
- Third, the internal capacitance of the multimeter and the lead capacitance of the probes must be calibrated out, which can be as simple as using a "Math Null" function to subtract the current reading from all subsequent readings.
- Finally, the internal capacitance of the multimeter must have a relatively high quality factor to avoid introducing errors due to its own RC time-constant.

If these requirements are all met, users can obtain substantially improved reading accuracy. (The capacitance measurement in the Agilent 34410A is based on a method very similar to the one described above.)

The measurement described above requires a current source of only one polarity, as an internal resistance can be used to discharge the capacitor under test. With a little more cost in the current source, a different method of rejecting loss can be implemented. If a precision current source is available which can both sink and source current, then a square-wave AC current signal can be created by reversing its polarity at a predetermined interval. This AC current source will create a triangle-shaped waveform in voltage when it is connected to a capacitor. If the capacitor exhibits loss, the slopes on the triangle wave will contain exponential terms, shown in Figure 2.

These exponential terms change the magnitude of harmonics in the frequency spectrum of the voltage waveform. By investigating the harmonics, the loss term can be removed. The National Instruments NI 4072 multimeter uses a method similar to this, in which a fast Fourier transform (FFT) is used to determine the frequency spectrum, and the first and third harmonics are compared to remove loss terms.

**Errors in time-based capacitance measurements**

There are a several significant issues with any time-based implementation of a capacitance measurement. The first is that the value of the capacitance can change substantially with frequency. LCR meters such as the Agilent 4263B have the ability to measure capacitance at multiple frequencies with the aid of an internal, variable AC source. For an aluminum electrolytic capacitor, the capacitance can vary as much as a few percent between frequencies of 100 Hz and 1 kHz.

A less-expensive algorithm typically operates at a single frequency, and will therefore not give additional information about the performance at higher frequencies. Although the measurements in multimeters may not be incorrect, they will differ from those that an LCR meter makes, simply because of the difference in measurement frequency.

Another non-ideal behavior characteristic of capacitors which can lead to misinterpreted results by a lower-cost measurement is the equivalent series resistance of the capacitor, or ESR. Assume for the moment that a positive current is used to charge the capacitor under test during the measurement cycle. If a resistor connected between the capacitor and ground is used to discharge the capacitor to prepare it for the next measurement cycle, then the lowest possible voltage on the capacitor will be 0 V.

Since the constant current creates a voltage ramp on the capacitor, the average voltage over multiple measurement cycles will be greater than 0V. This DC bias term does not create significant errors for film and ceramic capacitors; however, for aluminum-electrolytic capacitors, it can have a large impact on the result. This is due to the property that the ESR changes nonlinearly with DC bias.

The simple way to solve this problem is to keep the DC bias on the capacitor as small as possible, which is done by using an AC current source, by discharging below 0 V, or by reducing the amplitude of the voltage swing on the capacitor. Any of these techniques can provide accurate results. In some instances, the value of the capacitor with a DC bias applied may be desired, if the capacitor is to be used in a circuit that will place a DC bias across it in normal operation (such as in power-supply decoupling). Meters that use an AC source, such as LCR meters, generally provide a DC bias option to measure the value under this condition.

This analysis has shown there are a lot of properties to consider when measuring the value of a capacitor. For general bench troubleshooting or for measuring high-quality film capacitors, the simple and inexpensive time-based technique built in to general purpose multi-meters should be more than adequate. For measurements that require extremely high precision and measurements of additional parameters, a high-performance LCR meter should be the instrument of choice. There is, as shown above, some middle ground between these two extremes which reduces loss terms using relatively inexpensive methods. These methods will not allow for computation of properties such as dissipation factor, but will improve the accuracy of measurements on lower quality capacitors.

**About the authors**

*Bill Coley*graduated from the Georgia Institute of Technology (Georgia Tech) in 1999 with BEE and MSEE degrees, specializing in analog-circuit design. He spent six years designing signal-conditioning and analog-to-digital converter circuits for high-performance digital multimeters, including the Agilent 34410A and 34411A. He was also the designer of the capacitance-measurement algorithm in these meters. Bill is named on several patents related to his expertise in designing digital multimeters. (*Bill is currently employed by Linear Technology Corp. *(www.linear.com)* as an analog IC design engineer*.)

*Conrad Proft* has BSEE and MSCS degrees. Conrad has worked for Hewlett-Packard/Agilent (www.agilent.com) for 27 years and has spent about half of that time between R&D and Marketing, specializing in General Purpose instrumentation for bench and system measurements. From Application Engineering and technical writing in Marketing to R&D project manager on function generators and digital multimeters, Conrad's career has spanned a wide range of experience with many successful designs.