A capacitor creates in AC circuits a resistance, the *capacitive reactance* (Formula C3-3). 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

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 C3-16.

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

*Figure C3-16. Circuit diagram of a capacitor.*

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 with the help of formula C3-14 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 C3-16 can be written:

As a root mean square value we obtain the formula:

(Ω)……………………… [C3-9]

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 C3-17). 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. Some guidance is obtainable through those frequency diagrams which sometimes are shown in this book for certain dielectric materials where this information is of importance.

* Figure C3-17. The impedance diagram of a capacitor.*

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

………………………… [C3-10]

and

………………………[C3-11]

The equivalent circuit diagram then looks like the one in Figure C3-18.

*Figure C3-18. The equivalent series circuit diagram of a capacitor. Valid at higher frequencies.*

**Impedance around the resonance frequency** **[1]**

Figure C3-17 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 C3-10 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 C3-19.

*Figure C3-19. The appearance of the impedance vs. frequency curve around the resonance frequency in low-loss capacitors.*

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 C3-20.

*Figure C3-20. The appearance of the impedance vs. frequency curve around the resonance frequency of high-loss capacitors.*

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.

The losses in Figure C3-18 are concentrated to the ESR which consequently becomes significant when we leave the low frequency range. For HF chips and high loss components as for example electrolytics often the ESR is stated in the data sheets. If the ESR information is missing you always find, for all component types, a specified *dissipation* *factor *(DF), the** tanδ ** (Figure C3-21).

*Figure C3-21. Definition of Tanδ in a series circuit.*

Hence at **higher** **frequencies** the series circuit according to Figure C3-18 applies. There is

……………….. [C3-12]

Tanδ usually is expressed in %.

If the frequency declines to zero the circuit becomes resistive according to Figure C3-10 and -16, without any capacitance, and the losses are limited to the IR. Also at very **low frequencies** the IR is predominant but here it should be completed with dawning AC dependent losses to an equivalent loss resistance R_{p}. The diagram in Figure C3-16 now can be simplified to a parallel circuit with the capacitance C_{p} (Figure C3-22).

*Figure C3-22. The equivalent parallel circuit of a capacitor. Applies at low frequencies.*

If we describe the impedance in a parallel circuit according to Figure C3-22 it’s easy to show that its dissipation factor – which applies at low frequencies – can be written as

……………………………[C3-13]

The difference between C_{s} and C_{p} usually is negligible. We shall return to the connections in some formulas. If we equate the formula C3-10 with C3-11 we obtain

Let us terminate the discussion about the capacitor losses by distinguishing the different types of losses as in following Figure C3-23.

- R
_{d}= dielectric losses; - R
_{s}= losses in leading-in wires, joints and electrode metallizations; - ESR = R
_{s}+ R_{d}; - C = C
_{1}+ C_{2}.

*Figure C3-23. Equivalent diagram with dielectric losses particularly marked.*

Sometimes we encounter the expression Q or Q value, especially in high frequency applications. Instead of describing the capacitor losses as DF (Tanδ) we rather specify its freedom from losses, its figure of merit, the Q value. It is defined as

…………………….[C3-15]

Typical Q values for ceramic Class 1 dielectrics range from 200 to 2000 at 100 MHz and will vary strongly with frequency.

We shall use the Q value to describe the connection between the quantities in the series and parallel circuits in Figure C3-18 and -20. By depicting the expressions for the impedances and Q values of these circuits and equate the real and imaginary parts of the impedances we can show that

………………………………[C3-16]

………………………………[C3-17]

…………………………………….[C3-18]

……………………………[C3-19]

*Figure C3-24. Dipole losses versus frequency.*

Figure C3-24 illustrates the behavior of different dielectric dipoles when they are affected by an alternating field. They will oscillate at the same frequency as the field’s if allowed by their reaction time. Every rotary motion requires energy and the executed work produces heat. The most inert dipoles will react to the very low frequencies and will here contribute to the losses. But as the frequency increases the different types of dipoles will not be able to respond quickly enough, one after another, as shown in the figure.

Just in the range where the reaction time of the dipoles and the frequency period coincide a kind of resonance occurs which causes the dipole types to react with a loss peak (Figure C3-24).

Observe that Figure C3-24 deals with *dipole* *losses*, nothing else. There are other dielectric materials which have no molecular dipoles. They are called non-polar while the others are called polar. This has nothing to do with the polarity dependence of electrolytics.

The sum of losses in a polar and a non-polar capacitor may look like the ones in Figure C3-25.

*Figure C3-25. Total losses versus frequency in a polar and non-polar dielectric material.*

[1] The one who wants to study the subject on behavior of capacitors around the resonance frequency more closely is recommended the CARTS 99 paper *A Capacitor’s Inductance*, by Dr. Gary Ewell and Bob Stevenson, PE.

rev.1.: P-O.Fagerholt., CLR Passive Components Handbook

© European Passive Components Institute

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