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Equivalent Series Resistnace 1 Engineering Bulletins Capacitors 1. Equivalent Series Resistance (ESR) . 2 2. Charging Capacitors: Polarization and Leakage Currents . 10 3. Metallized Electrode Capacitors for Pulsed Power Applications . 21 1 Equivalent Series Resistance (ESR) Many electrical engineers are General Atomics Energy Prod- familiar with a circuit model (Figure 1) ucts (GAEP) has attempted to avoid in- of a non-ideal capacitor which includes correct application of ESR information a series inductance (the ESL) and a se- by not listing an ESR value in its stan- ries resistance (the ESR). This model dard product sales literature. Each cus- is very misleading because it often re- tomer who is concerned about this pa- sults in the assumption that the equiva- rameter is asked how the ESR value is lent resistance is actually a true resis- to be used in his or her calculations or tance, having essentially constant value models. This allows us to tailor a mea- over a wide range of conditions [7]. In surement method to the requirements of truth, the Equivalent Series Resistance a particular application. Although this is the value of resistance which is equal is more complex and time-consuming, to the total effect of a large and com- it is the only way of assuring that the plex set of energy loss mechanisms quoted ESR values have real mean- occuring under a particular set of ing for the intended purpose. measurement or operating conditions. Capacitor manufacturers often define the ESR as the impedance mea- CS LS RS sured at the resonant frequency of the (ESL) (ESR) capacitor, or the value of the ESR at a V particular AC frequency (e.g. 100 kHz), usually measured with a bridge, LCR meter, or impedance analyzer at about I 1 V. These are definitions which make δ ωCS it easy to measure the ESR but may be θ inadequate to describe the behavior of a high voltage capacitor under actual RSI operating conditions. Figure 1. AC voltage signal applied to circuit below resonant frequency. 2 In this Technical Note, we in- energy stored over all frequencies (con- tend to briefly describe why the ESR stant DF), the ESR decreases with in- does not behave like a simple resistance creasing frequency, as can be seen by as the result of the physics of a capaci- rearranging equation (1): tor. We will then examine several meth- ods of measuring the ESR and how the ESR = DF/ωC ~ 1/ω (2) exact test conditions affect the results of each. Finally, we will discuss how Figure 2 illustrates the hypo- ESR values are often used and suggest thetical constant-DF case. how to specify the ESR and the proper C = 1 µF measurement technique for your appli- DF = Constant = 0.001 cation. 1.6 1.4 Capacitor Physics 1.2 and ESR 1.0 Dielectric Loss 0.8 Metal Loss If we apply an AC voltage sig- Both nal to the circuit of Figure 1 at a fre- ESR (Ohms) 0.6 quency well below the resonant fre- 0.4 quency, we observe a phase shift be- 0.2 tween voltage and current which is less than 90 degrees. The difference be- 0.0 tween the phase angle and 90 degrees is 246810 the defect angle (δ), which can be used Log (Frequency, HZ) to determine the effective resistive im- pedance using: Figure 2. Hypothetical constant/DF case. tangent (δ) = Zr/Zc . C = 1 µF = ESR ω C (1) DF (dielectrric) = 0.001 R (metal) = 0.00001Ω .00002 where Zr and Zc are the resistive and .00018 capacitive impedances, ω is the angular frequency, and C is the capacitance. .00016 Note that tangent δ is the same as the .00014 Skin dissipation factor, or DF [4]. .00012 Effect .00010 The energy loss in the circuit is .00008 ESR (Ohms) ESR proportional to the power factor, PF, .00006 which is given by the cosine of the phase Total ESR Dielectric Loss .00004 Metal Loss angle. The DF is approximately the Both same as the PF for small values (e.g. DF .00002 < 0.1 or 10 %). .00000 24612 810 14 If we assume that the energy Log (Frequency, HZ) dissipated is a constant fraction of the Figure 3. ESR as a function of frequency. 3 The ESR of a 1 µF capacitor has The dielectric losses of a given been modelled assuming that the di- material can be described by its Dissi- electric losses result in a constant DF pation Factor (DF). If the dielectric loss of 0.001 or 0.1% and the ohmic resis- were the only loss mechanism operat- tance is 0.01 milliohm. The resulting ing in a capacitor, then the DF of the ESR is shown as a function of fre- capacitor would be independent of its quency in Figure 3. The ESR falls size and geometry and internal configu- from a value of 1.6 ohm at 100 Hz to ration. Capacitors of any size made with a value of 0.01 milliohm at 5 MHz. the same material would have the same Above this frequency the ESR is DF when measured under identical con- largely determined by the ohmic re- ditions. The ESR could then easily be sistance of the conductors, and re- computed using equation (2). However, mains relatively constant until the skin the capacitor DF and ESR do depend effect begins to reduce the effective on the electrodes and their configura- thickness of the foil electrodes. The tion, as described below. ESR then begins to increase in pro- portion to f 0.5 above about 30 MHz. Ferroelectric hysteresis losses are observed in certain high dielectric When examined more closely, constant materials, most notably ceram- the losses vary as a function of voltage, ics. These losses are a strong function temperature, and other aspects of the of applied voltage. This loss mechanism waveform. This is because there are a arises when the internal polarization variety of energy loss mechanisms field has the same order of magnitude which act within a capacitor. Some of as the applied field. Under these condi- these reside within the dielectric while tions the dielectric response saturates. others involve the conductors carrying Capacitors made with such materials the current. Below we describe some exhibit permanent polarization, variable of these mechanisms and the operating capacitance as a function of voltage, and parameters which strongly effect the extreme sensitivity to reversals of voltage. magnitude of the losses which are asso- ciated with them. Dielectric conduction losses are caused by the actual transport of Dielectric losses are usually the charge across the volume of the dielec- most important losses in a film capaci- tric or across internal dielectric inter- tor. These losses are associated with the faces. These losses are largest at low polarization and relaxation of the dielec- frequencies and higher temperatures. tric material in response to the applica- Because conduction in a dielectric ma- tion and removal of voltage from the terial can be strongly nonlinear (non- capacitor. As such, the magnitude of the Ohmic), conduction losses are often dielectric losses in a capacitor are, in strongly dependent on the voltage ap- general, both frequency and temperature plied to the capacitor. dependent. In this case, the largest losses occur at low temperatures or high Interfacial polarization losses frequencies, where dipole orientation is are closely related to dielectric conduc- most hindered. Dielectric losses (as tion. Many high voltage capacitors con- defined here) are not voltage-dependent. tain two or three different materials 4 within their dielectric systems – film within the metallic electrodes, the inter- and oil or paper, film and oil. Each nal wiring, and the terminals of the ca- material has different conduction prop- pacitor. (In electrolytic capacitors, erties and permittivity. As a result, the ohmic resistance in the electrolyte itself application of a DC voltage over a pe- represents the largest loss mechanism.) riod of time will result in a build-up of The resistance losses in the metal are conducted charge at the internal inter- quite constant as a function of tempera- faces between materials. This polariza- ture and frequency (until the skin depth tion of the dielectric is largely a low- in the electrodes becomes important, frequency phenomenon and the energy usually at several megahertz or higher). stored in this way is not available for Losses in the internal wiring and the ter- discharge at high frequency. Again, minal can be important in high current since the conduction is nonlinear, inter- applications, and should not be ignored. facial polarization will also generally be When high voltage capacitors are inter- nonlinearly voltage dependent. nally configured as a series string of lower voltage capacitor windings or This loss mechanism can be es- units, the ohmic resistance within a pecially important in pulse discharge given container size increases at the applications, where the capacitor is square of the voltage (or as the number charged over a relatively long period of of series elements). time and then discharged much more rapidly. Sparking between conductors or different points on the same conduc- Partial discharge losses can tor during the discharge has been re- occur within gas-filled or defective solid ported to occur in pulse capacitors [3]. capacitors or even in liquid-filled ca- For example, capacitors manufactured pacitors at high voltages. It is also com- with an inserted tab connection to the mon to have external corona on capaci- electrode foil which is only a pressure tor terminals (which can be considered contact have been found to exhibit a capacitor energy loss mechanism). points of localized melting after pulse Partial discharges are most energetic at discharge operation resulting from high rates of change of voltage (high dV/ sparks between the adjacent metallic dt), such as during a capacitor pulse dis- surfaces [3].
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