# Measuring Capacitor Self-inductance and ESR

Technical Background
The parasitic parameters of a capacitor, that is its equivalent series resistance (ESR) and its inductance, affect the way the capacitor performs in circuits. Some applications are very sensitive to these parameters. For instance, a bypass capacitor used between power and ground in a digital circuit must be able to supply current quickly to nearby active devices. If it has too much inductance it will not be able to do this. Similarly, the transient response of a capacitor used to divert a current pulse due to electrostatic discharge is very important to the ability of the capacitor to do its job.

So how can the parasitic parameters of a capacitor be measured? One could certainly connect the capacitor to a network analyzer and get a very good characterization. Such an instrument can be quite expensive though. Even the less expensive capacitance measuring instruments may not be available when needed. Both instruments may not provide the information in an easily usable form. If you have a pulse generator (preferably with a 50 ohm output impedance) and an oscilloscope, you can easily measure the transient response of a capacitor. From this data, the capacitor’s ESR and inductance can be determined.

First, construct the simple network shown in Figure 1 at the end of a 50 ohm coaxial cable fed from a 50 ohm pulse generator. A 50 ohm resistor is used in Figure 1 to terminate the coax during the rising edge and provide a total of 100 ohms of source impedance. The resistor shown is a 51 ohm 1/2 watt carbon composition resistor with one lead trimmed so that the resistor just seats with the trimmed lead fully inserted into the BNC connector. It may be necessary to put a little solder bump on the resistor lead so that it stays firmly in the BNC connector. The capacitor to be tested is connected between the end of the resistor and the shell of the BNC connector. An oscilloscope is connected directly across the capacitor, using the shortest leads possible to connect the probe. Probes with a resistive input impedance of 500 to 1000 ohms are recommended. Standard 10X HiZ probes often have rising edge effects that will distort the part of the waveform used for the calculations.

Figure 1: Test circuit for measurement of capacitor self-inductance and ESR

For a pulse length that is long with respect to the RC time constant, one will see an exponential rise to the open circuit voltage of the pulse source. For the purposes of this discussion, we will be looking at the first couple of hundred millivolts of a 5 volt exponential rise. An example of this appears in Figure 2.

Figure 2: Initial rise

Figure 2 shows the beginning of the exponential voltage rise across the capacitor when the generator pulse starts. The vertical scale is about
200 mV. The horizontal time is a small fraction of the RC time constant of 100 ohms and the capacitor being measured. Since the capacitor voltage is still very small compared to the 5 V open circuit output of the generator, the current through the capacitor may be presumed to be constant and equal to the generator open circuit voltage divided by 100 ohms, 50 mA in this case.

The risetime of the current will be the same as the generator voltage. If the rise is a ramp with a constant slope and the capacitor had no inductance, the initial rise shown in Figure 2 would follow the dotted line and then the slope would change to the initial slope of the exponential rise determined by:

dv/dt = i /C = 50 mA/C     (1)

where C is the value of the capacitor at this low voltage and the risetime of the current << RC.

The offset between the baseline and the beginning of the exponential rise is just the voltage that the current, 50 mA for this case, develops across the ESR of the capacitor. The ESR can be easily estimated in this case by dividing the voltage offset (labeled ESR in Figure 2) by 50 mA.

Parasitic inductance in the capacitor will cause the spike in the waveform shown in Figure 2, exceeding the value of the dotted line along its length. If the current rise were in fact a ramp with constant slope and very sharp corners (high di2/dt) then the spike would be a square pulse of value:

E = L*di/dt     (2)

where L is the parasitic inductance of the capacitor.

As is the case for most generators I have used, the current rise from the generator used for the data in this article was not a ramp with very sharp corners and constant slope. That characteristic of the generator combined with probe effects led to a peaked shape, shown as the Ldi/dt spike in Figure 2. The inductance of the capacitor can be calculated using Equation 2. Often, one does not need to calculate the inductance or ESR but just choose a capacitor with the lowest inductance and/or ESR from several that are available.

Soldering the components onto a BNC connector, as in Figure 1, works up to 300 MHz. I estimate the inductive reactance of the loop formed by the capacitor and resistor to be about 20 ohms at 300 MHz (estimating the inductance at 10 nH). This is small enough relative to the 100 ohms of resistance in the circuit to not significantly affect the initial current very much. For this frequency range, a generator with a risetime of one to two nanoseconds will do.

If you need to check the capacitor using faster risetimes, it would be best to build the test setup on a small circuit board with a ground plane and controlled impedances. At this point, the parasitic capacitance of the 50 ohm resistor would also be an issue to be taken into account. Fortunately, such accuracy is often not needed. Especially, if one is just comparing the relative performance of several capacitors.

Data

Figure 3 shows the initial rise from the generator. The black square indicates the vertical voltage and horizontal time scales. The open circuit voltage was a little over 4 Volts with about a 5 nanosecond risetime. The data in Figures 3 through 6 were taken with an analog scope some years ago. Figures 4 through 6 show data obtained from several leaded capacitors (as opposed to surface mount). Two traces were taken for each capacitor. The lower trace was measured at the capacitor body where the leads entered and the upper trace included the minimum amount of lead to practically connect the capacitor to a printed wiring board. The upper trace would not be needed for modern surface mount capacitors unless one wanted to model the connection inductance from the capacitor to the point of interest on a printed wiring board.

Figure 3: Input from pulse generator

Figure 4: 4uF capacitor

Figure 4 shows data from a 4 uF electrolytic capacitor. The ESR offset is about 50 mV yielding an estimate of the ESR of just over 1 ohm. Notice that there appear to be some oscillations on the 1/C part of the slope. This could be scope probe resonance or a resonance in the capacitor. The data was taken with a standard 10X HiZ probe, so the probe is suspect. I have seen capacitors with pronounced oscillation from internal resonance. If you are planning to put a large capacitor in parallel with a smaller one, especially if they are constructed from different technologies, it would a good idea to check out the impulse response of the combination using this method. It is possible for the smaller capacitor to resonate with the inductance of the larger one, causing an unexpected result.

Figure 5 shows the result for a 1 uF capacitor of the same construction as the 4 uF capacitor tested in Figure 4. Note that the inductance is similar to the 4 uF capacitor, but the ESR is
slightly lower. Since an analog scope was used, the waveform was repetitive and the slight slope on the left half of the waveform was the end of the exponential fall from 5 V. If a single pulse on a digital scope was used, the slope to the left of the Ldi/dt spike would be zero.

Figure 5: 1uF capacitor

Figure 6 shows the result for a 1 uF radial ceramic capacitor (square case). Note the low inductance and undetectable ESR. Note also that the slope of the 1/C exponential rise is flatter, indicating more capacitance than the 1 uF capacitor of Figure 5. This may be due to the fact that the electrolytic capacitor used for Figure 5 may have lower capacitance near zero voltage than at its operating voltage, whereas the ceramic capacitor has a more constant capacitance with voltage. The inductance corresponding to the lower trace is estimated to be 4.4 nH. It is interesting to note that a 0.1 uF ceramic capacitor in the same size package as the 1 uF of Figure 6 showed a slightly higher inductance in this test setup. I believe this was due to the fact that the smaller capacitor did not fill the package and internal lead inductance caused the effect. In this case, a 1 uF capacitor was a better choice than a 0.1 uF!

Figure 6: 1uF ceramic capacitor

One of the advantages of this test is that the output waveform is the transient response of the capacitor. The voltages developed across the capacitor in this test are directly related to what will happen in a real circuit if the current risetime from the generator is similar to what the capacitor will see in its intended application.

For more Technical Tidbits, please visit Doug’s site, http://emcesd.com.

 Douglas C. Smith Mr. Smith held an FCC First Class Radiotelephone license by age 16 and a General Class amateur radio license at age 12. He received a B.E.E.E. degree from Vanderbilt University in 1969 and an M.S.E.E. degree from the California Institute of Technology in 1970. In 1970, he joined AT&T Bell Laboratories as a Member of Technical Staff. He retired in 1996 as a Distinguished Member of Technical Staff. From February 1996 to April 2000 he was Manager of EMC Development and Test at Auspex Systems in Santa Clara, CA. Mr. Smith currently is an independent consultant specializing in high frequency measurements, circuit/system design and verification, switching power supply noise and specifications, EMC, and immunity to transient noise. He is a Senior Member of the IEEE and a former member of the IEEE EMC Society Board of Directors.His technical interests include high frequency effects in electronic circuits, including topics such as Electromagnetic Compatibility (EMC), Electrostatic Discharge (ESD), Electrical Fast Transients (EFT), and other forms of pulsed electromagnetic interference. He also has been involved with FCC Part 68 testing and design, telephone system analog and digital design, IC design, and computer simulation of circuits. He has been granted over 15 patents, several on measurement apparatus. Mr. Smith has lectured at Oxford University, The University of California Santa Barbara, The University of California Berkeley, Vanderbilt University, AT&T Bell Labs, and internationally at many public and private seminars on high frequency measurements, circuit design, ESD, and EMC. He is author of the book High Frequency Measurements and Noise in Electronic Circuits. His very popular website, http://emcesd.com (www.dsmith.org), draws many thousands of visitors each month to see over 150 technical articles as well as other features. He also provides consulting services in general design, EMC, and transient immunity (such as ESD and EFT), and switching power supply noise. His specialty is solving difficult problems quickly, usually within a couple of days. His work has included digital and analog circuits in everything from large diesel powered machinery to IC chip level circuits. His large client base includes many well known large electronic and industrial companies as well as medium sized companies and start-up companies.