Other factors contribute additional losses: the bulk metal of leads; methods of lead attachment; capacitor plate material; and general construction. Particularly with larger capacitance values, these factors become a significant percentage of the total loss as their contribution to ESR increases.
There are three types of losses in a film capacitor: metal losses (Rs); leakage or insulation losses at DC or low frequencies (Rin); and dielectric losses (Rda). Leaving out extensive derivation, the following formula shows how these three losses are related to a capacitor's ESR:
ESR = Rs + 1/Rin(2(pi) f C)2 + DF/2(pi) f C
Metal losses include all the losses in the capacitor's leads, termination junctions, and capacitor plates. As shown in the equation above, the insulation resistance (Rin) leakage losses prevail (second term), particularly at low frequencies. Then, as frequency increases, dielectric losses (third term), and then metal losses (first term) become prominent.
Metalized film capacitors utilizing thin conductors or plates have a distinct size advantage over other types - they can be very small. Formed by a vacuum-deposition process that laminates a film substrate with a thin aluminum coating measured in Angstroms, these capacitors are used where small signal levels (low currents/high impedances) and small physical size are primary factors. They are generally inappropriate for large signal AC applications.
The film-and-foil capacitor has much thicker plates (foil) than those made of metalized film; these plates result in lower losses. The thicker foil also helps carry away heat build-up; this means longer life, greater reliability, and minimal effect upon the capacitor's dielectric or DF. film-and-foil construction capacitors may be used in both small and large signal applications, but the higher performance (and higher price) may be unnecessary in some small-signal applications.
Plate resistance increases with its length and so does the capacitor's value. A larger diameter plate can be used to compensate for a shorter length. This short length/large diameter construction further reduces ESR because of the reduced plate resistance and enlarged contact area between the plate and lead and also avoids the inductance increase typically found in long length/small diameter plate construction.
The junction between leads and the capacitor's plate(s) also influences ESR, usually for the worse. in addition to the size and shape of the leads (mentioned above), the following considerations are important: the method of attachment (welding or soldering); the size and shape of the capacitor; and the size of the sprayed metal droplet used to short all the plates together. Long-term reliability and ESR stability also depend on the use of metals that do not form electrolysis action, such as that resulting from the direct contact of aluminum and copper.
Figure 5: Typical capacitor impedance vs. frequency.
For low capacitance values, the inherent dissipation factor of the dielectric material contributes most significantly to ESR. As capacitance value increases, plate resistance, lead resistance, and endcoating resistance become, in turn, the dominant factors affecting total ESR. Careful selection and control of these factors results in uniformly stable and low ESR.
Impedance, Inductance, and Resonant Frequency
An ideal capacitor's reactance decreases as frequency increases, as shown by the formula: Xc = 1/(2(pi) f C). Of course, impedance (Z) also varies with frequency, owing to the ESR and inductance (L) of the capacitor, as shown in Figure 5.
The point of minimum impedance (ESR) marks the frequency at which L and C form a series-resonant circuit, where the inductive reactance equals the capacitive reactance. Above this resonant frequency, the capacitor functions as an inductor. For many applications, the capacitor's series resonant frequency will be a circuit's useful upper frequency limit, especially where the phase angle of the capacitor is expected to maintain a 90-degree (tan ð = 0) or near 90-degree voltage/current relationship. This is a common assumption in filter network design.
The length of the capacitor and its construction determine the capacitor's self-inductance and thus its resonant frequency. The lead length to the capacitor's external circuit load influences the incircuit performance, usually in a quite different manner from that which was calculated based upon ideal (that is, no inductance) conditions. Figure 6 shows the effect of lead length increased from 3/8'/ to 3", a typical length when you add the circuitboard trace length or circuit wiring to the capacitor's lead length. Note that the useful upper frequency limit has gone from 490 KHz to about 290 KHz. the reduction is about 33 KHz per inch for this particular capacitor! for a dielectric loss angle of 10 degrees (tan ð = .176) in the audio pass-band (a common occurrence with larger capacitor values, inductance, and ESR), there is an even more pronounced effect. here the corresponding frequencies for 1/4" and 3" lead lengths are 260KHz and 141KHz (figure 6).
Figure 6: Lead length aiters a capacitor's range of operating frequency. Here a 2 uf capacitor's self-nesonance decreases from 490 kHz to 290 kHz when its leads are lengthened from 3/8 inch to 3 inches. In other words, the capacitor's usable operating range is reduced by almost half.
Figure 7: A cross sectional drawing showing the coax al construction of an MultiCap. For simplicity's sake, only tow of the 10 sections are included.
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