# Measuring capacitance & ESR

*Last Modification: January 20, 2014*

There are various ways to determine the capacitance of capacitors. This article describes a number of measurement methods. Also the ESR (equivalent series resistance) can be measured with most of the presented methods.

A capacitor has beside the most important property; the capacitance, also parasitic properties. The most important of these are the series resistance and the self-inductance who is also connected in series with the capacitance. The model of a capacitor with its parasitic components is shown in the figure on the right. This model doesn't include the in parallel connected leakage resistance. In most cases these are negligible, but if this parameter is necessary, it can be measured with a DC-resistant measurement.

The following measurement methods measure the capacitance and ESR. To determine also the parasitic self-inductance another measurement method must be used described in the article Parasitic properties.

## Measuring with a squarewave

By connecting a capacitor to a squarewave generator a typical voltage shape across the capacitor arises. By analyzing the voltage shape the capacity and internal series resistance can be determined.

### Measurement arrangement

Figure 2 shows the measurement arrangement. The capacitor to test is directly connected to the output terminals of the function generator who's delivering a squarewave voltage. The voltage across the capacitor is measured with an oscilloscope. The function generator is set at the maximum output voltage, and the frequency is adjusted so that the voltage across the capacitor is kept at a low level. In this way almost the entirely voltage is dropped across the internal generator resistance. It is like the capacitor is connected to a current source. The current that flows is calculated as:

The voltage *V _{g}* is the open terminal top value of the generator. In most cases the generator voltage is given as a top-top value at a specific load, often 50 Ω. The unloaded top value is therefore equal to the loaded top-top value.

### Capacitance

The screen shot in figure 3 shows linear increasing and decreasing voltages. The amplitude of the slopes is marked in a blue color and labeled *V _{C}*. This voltage change occurs during the time

*t*, marked in red. With these numbers the capacitance can be calculated:

### Internal resistance (ESR)

The yellow arrow represents the step voltage *V _{R}* which is caused by the capacitor internal series resistance under the influence of current polarity reversal. The current changes here from -0.2 A to +0.2 A, thus a current step of 0.4 A. With this information the ESR, or equivalent series resistance, can be calculated:

### Self-inductance

The self-inductance can't be determined with this method. However, the effects of the inductance are noticeable as spikes in the oscilloscope image. One of these spikes is marked with an asterisk.

## Measuring with a sinewave

With this measurement method the capacitor is inserted in a half bridge configuration which is connected to a sinewave generator. By the measured voltages and phase difference the capacity and ESR can be determined.

#### Limitations

Capacitors can almost be considered as ideal components. The equivalent series resistance is normally very small as well as the capacity in most cases. To measure the capacity and the internal resistance accurate, the measure frequency must be chosen so that the reactance and resistance are approximately the same. The phase difference between the capacitor voltage and the voltage cross the reactance is than circa 45 °. This means that the measure frequency must be very high in some cases, tens to hundreds megahertz. The self-induction at these frequencies is very dominant what makes the measurement worthless. Only electrolytic capacitors who have a relative high capacity in combination with a high ESR, the measure frequency can kept low enough to be usable. When measuring other capacitors the frequency must be chosen lower than desired what means that only the capacitance can be measured.

Two examples are given: The first one is for measuring only the capacitance, and the second one is for measuring the capacity as well as the ESR.

### The measuring arrangement

The measurement arrangement is shown in figure 4. This is a half bridge configuration consisting of the resistor *R _{s}* and the unknown capacitor

*C*. The bridge voltage and the voltage at the junction is measured with a two channel oscilloscope. Also the phase difference between these voltages is measured. To measure the internal resistance accurate the channel 2 probe must be placed as close as possible at the capacitor. The resistor

_{x}*R*must have approximately the same value as the impedance of the capacitor.

_{s}### Method 1: Measuring capacitance

De first method describes the measurement of small capacitors whereof the series resistance is negligible.

#### Mathematical model

Figure 5 shows the capacitor model that is used for the calculation of the capacity *C _{x}*. The associated vector diagram is shown in figure 6. The reactance of the capacitor is relative high in this measurement. The ohmic series resistance breaks down into nothingness and is therefore not included into the model.

Parallel to the capacitor under test is the probe connected represented by the capacity *C _{p}* and the ohmic resistance

*R*. The probe capacity

_{p}*C*and the unknown capacitor

_{p}*C*are taken together as one replacement capacity

_{x}*C*. Because

*C*(and also

_{p}*R*) are known is it easy to figure out the unknown capacity.

_{p}The current is measured with the aid of *R _{s}*. The value of this resistance should be in the proximity of the reactance of the capacitor to be measured and is thus dependent on the measurement frequency, and the capacitance:

For a capacity of 100 pF and a measure frequency of 50 kHz, R

_{s}must be approximately 33 kΩ. For an unknown capacity the resistance

*R*has to be determinate empirical. The reactance is set by manipulating the frequency. The probe for measuring the terminal voltage of the generator

_{s}*V*does load the function generator but has no further influence on the measurement. This probe is represented by

_{g}*C*and

_{p}**R*.

_{p}*#### Processing measurement data

Figure 7 shows the result of a measurement to an 100 pF capacitor. On the basis of this measurement is shown how the results should be processed. The following values are measured: the generator voltage *V _{g}* (5.076 V), the voltage across the capacitor

*V*(3.242 V), the phase angle between these two voltages

_{x}*α*(48.89 °) and the frequency

*f*(50 kHz). The resistor value

*R*is 33 kΩ, and the probe capacitance

_{s}*C*is 12 pF.

_{p}*R*is calculated:

_{s}And the generator current is therefore:

The loss angle φ (with respect to the parallel resistor):

The reactance of the total capacitance:

The total capacitance is:

The probe capacitance is subtracted from this total to calculate the unknown capacitance:

### Method 2: Measuring capacitance and internal resistance (ESR)

The second method describes a measurement that is suitable for measuring on larger capacities and can also determine the internal series resistance (ESR). This method is thus mainly suitable for measuring on electrolytic capacitors.

#### Mathematical model

Figure 8 shows the mathematical model with the associated vector diagram in figure 9. *C _{x}* and

*R*are the capacitive and resistive parts of the measured capacitor. With the use of the resistor

_{x}*R*the current is measured. This resistor must have a low ohmic value to keep the circuit at a low impedance. This is necessary in order to be able to determine the internal resistance.

_{s}The probe capacitance and resistance are ignored.

#### Processing measurement data

On the basis of a measurement on a 100 µF, 16 V electrolytic capacitor is demonstrated how the capacitance and ESR can be calculated with the measured parameter. The scope picture in figure 10 shows the measurement. The measured generator voltage *V _{g}* is 417 mV, the capacitor voltage

*V*is 291 mV, the phase difference between these two voltages

_{x}*α*is 31.18 ° and the frequency

*f*is 700 Hz. A 2.2 Ω resistor is used for

*R*.

_{s}With the measured voltages *V _{g}* and

*V*and the phase shift

_{x}*α*the voltage across the resistor

*R*is calculated:

_{s}And therefore the current though the circuit:

The loss angle is:

The voltage across the capacitive part:

The reactance:

The capacity:

The voltage across the ohmic part:

And the ESR of the capacitor is :

There is a side note with this example: The loss angle *φ* which has a value of 14 ° that makes the difference between the reactance and ohmic resistance relative high. This has the effect that the capacitance is calculated with a greater accuracy than the internal series resistance. If one wants to determine the ESR with a greater accuracy than the measure frequency has to be adjusted to make the loss angle lager.

## Resonance measurement

Another way to measure the capacitance is to include the unknown capacitor in a resonance circuit. The accuracy is directly dependent on the used reference inductor. Inductors with a small tolerance are rare and expensive.

### Measurement arrangement

The unknown capacitor is connected in series with a reference inductor and connected to a sinewave generator. The voltage across this resonance circuit is measured with an AC-voltmeter. The most voltmeters are not suited to measure very low or high frequencies. In most cases a peak detector can fulfill this task.

If the resonance circuit has a high DC-resistance it is possible that the resonance frequency can't be found. In this case a extra resistor *R _{a}* can be placed in series with the generator.

### The measurement

With the sinewave generator the frequency range where the resonance frequency is expected is slowly sweeped. The resonance frequency is noticeable by a sharp voltage drop. If the sinewave generator doesn't produce a clean signal the chance exists that a harmonic causes a weaker dip. It is advisable to continue looking for a more distinctive dip.

With the found resonance frequency *f _{o}* the unknown capacitance can be calculated:

[F]

The internal series resistance can't be determined with a great accuracy. Even if the ohmic resistance of the inductor is well known, the various junction resistances are still unknown.

## Accuracy

### Influence cables and instruments

As noted previously, the measurement is affected because it is connected to measuring equipment. For a more accurate measurement the model of the measurement arrangement has to be added to the capacitor model, where the self-inductions, resistances and capacitances of the measurement instruments and cables is taken into account. This is particularly important at high measuring frequencies.

### Dielectric losses

The dielectric of capacitors is not ideal. It involves losses appear as ohmic losses. The dielectric losses are also frequency and voltage dependent.

### Measuring electrolytic capacitors

Be aware of the polarity when measuring electrolytic capacitors there should be no AC voltage present. Voltages that become negative with respect to the capacitors polarity may cause faulty results and may damage the capacitor. An offset voltage can in most cases be included to the sinewave of a function generator. In this way can be ensured that never the wrong polarity is applied to the capacitor.

### Parasitic self-induction

The described measuring methods cant measure the parasitic self-inductance. The article Parasitic properties how this and the frequency dependent ohmic resistance is measured.