This article explains capacitor losses (ESR, Impedance IMP, Dissipation Factor DF/ tanδ, Quality FactorQ) as the other basic key parameter of capacitors apart of capacitance, insulation resistance and DCL leakage current.

There are two types of losses:

**Resistive real losses**– these are real losses caused by resistance of leads, electrodes, connections etc. During current flow these losses are dissipated by Joule heat. Usually (unless it is intended by designed) the effort is to minimize these losses for maximum efficiency and high power load ratings.**Reactance imagine losses**– these are losses caused by capacitive reactance and inductive reactance “stored” in the component that can be reverted back

A capacitor creates in AC circuits a resistance, the *capacitive reactance*. There is also certain inductance in the capacitor. In AC circuits it produces an ** inductive reactance** that tries to neutralize the capacitive one. Finally the capacitor has

*resistive*

*losses*. Together these three elements produce the impedance, Z.

If we apply an AC voltage over a capacitor its losses release heat. They can be regarded as a resistive part of the impedance, i.e., as resistive elements distributed in different parts of the component, e.g. in accordance with the equivalent circuit in Figure 1.

- C = Capacitance
- IR = Insulation Resistance (IR>>Rs)
- Rs = Series losses
- L = Inductance in lead-in wires

R_{s} consists of resistance in lead-in wires, contact surfaces and metallized electrodes, where such elements occur, as well as dielectric losses. If we apply a DC voltage over the capacitor, the generator ”feels” a purely resistive loss dominated by the IR. But because of the high value of the IR the heat release will be negligible. Should we instead change over to an AC voltage and let the frequency rise the current will increase proportionally and eventually release a considerable heat in the R_{s}. If we transform the IR to a small series resistance and join it with the R_{s} we get a total series resistance called ESR (Equivalent Series Resistance, sometimes called Effective Series Resistance). The series impedance, Zs, in Figure 1. can be written:

As a root mean square value we obtain the formula:

(Ω)……………………… [1]

The capacitive reactance, 1/ωC, in the formula above decreases with frequency to that level where the inductive reactance, L, takes over. It happens at the resonance frequency fo of the capacitor where 1/ωC = L. Above the resonance frequency the capacitor is inductive. *Exactly at the resonance frequency remains of the impedance Z only the resistive ESR *(Figure 2.). By determining the losses at the resonance frequency we gain accuracy. But there is a condition for this accuracy. We need to know the frequency dependence of the ESR which very much is conditioned by the dielectric material. In certain materials it is negligible, in others considerable.

The expression for capacitance in the formula for Z_{s} above can be simplified to a series capacitance C_{s}. If C means the nominal capacitance then we obtain C_{s} as

………………………… [2]

and

………………………… [3]

The equivalent circuit diagram then looks like the one in Figure 3.

**Impedance around the resonance frequency**

Figure 2. shows an example of the impedance diagram around the resonance frequency. We shall evolve the reasoning further.

Because of the approximations used during derivation of formula [2] it applies only far below the resonance frequency, f_{0}. There, however it may cause discernible deviations from the true value. Already at 0.2 x f_{0} C_{s} will be approximately 4% greater than the nominal value C.

Often the expression for C_{s} is used when the frequency dependence of capacitance is shown in diagrams. This means that the capacitance quite contrary to physical and electrical laws starts rising at higher frequencies. The explanation accordingly is to be sought in errors in the measurement method.

Except for electrolytics and other high loss capacitors the *impedance curve* usually has the appearance of the one shown in Figure 4.

The sharp tip at the resonance frequency is typical for capacitors with comparatively small losses. In this frequency range the impedance contribution from the ESR is smaller than those of the capacitive and inductive reactances. When the decreasing capacitive reactance reaches the same magnitudes as those of the rising inductive reactance there will be an increasing influence from the latter. It reduces the capacitive reactance and eventually eliminates it. The curve bends down in a sharp tip. The bottom of the bend is determined by the ESR.

In capacitors with relatively high losses, for example electrolytics, the impedance curves reach and are influenced by these losses long before we get to the resonance frequency. A frequency dependent decrease in capacitance may also play a certain role in the frequency range. The impedance curve will deviate from the initial reactance curve and level out in a pliable bend on the ESR contribution, high above the point of intersection between the capacitive and inductive branch. The phenomenon is illustrated in Figure 5.

**Loss Dependent Derating**

The heat release from AC applications limits the temperature range of for example paper capacitors where the loss raises the internal temperature appreciably. While DC applications for example allow +85 or +100°C, AC applications already at 50 Hz may require limitations to maximum +70°C.

Higher frequencies require further derating because of the current which grows correspondingly. The R.M.S. value of the AC voltage furthermore is derated according to the permitted DC value not only with respect to the peak value and the temperature rise but also because of the additional strain that every repolarization exerts on the dielectric. The higher the rated voltage, the higher the degree of derating.

Example:** AC/DC** = 40/63, 63/100, 125/250, 220/400, 300/630, 500/1000, 660/1600. But please, always check what the relevant data sheets specify.