Page 290 - Analog and Digital Filter Design

20
Selecting Components for Analog Filters

reactance vector Xc, and the impedance vector (Xc + ESR), where
the ESR vector is at right angles to the reactance vector.

One of the most notable problems with capacitors is self-resonance. Self-
resonance occurs due to the device construction: leads are inductors (albeit low
value) and wound capacitors can have some inductance because the currents
circulate through the capacitor’s plates. Consider the self-resonant frequency of
capacitors, of various dielectrics, having a lead length of 2.5mm (or 0.1 inch):
a lOnF disc ceramic has a self-resonance of about 20MHz: the same value of
polyester or polycarbonate capacitor also has a self-resonance of about 20 MHz.
Mica capacitors are better, and a lOnF device with this dielectric has a self-
resonance frequency of over I GHz.

A rough idea of the self-resonant frequency can be found by calculating the
inductance of a component lead. For example, a 0.5mm diameter lead that is
5mm long (2.5mm for each end of the component) has an inductance of 2.94nH
in free space. When combined with a 1 nF capacitor, the self-resonant frequency
is calculated to be about 93 MHz. Replacing the 1 nF capacitor in the preced-
ing calculations with a lOnF capacitor, results in the self-resonant frequency
falling to 29MHz. But. wait a moment, I just said that the self-resonant fre-
quency of a lOnF capacitor with 2.5mm leads was about 20MHz. The reason
for the discrepancy between the calculated frequency and the actual frequency
is that inductance in the plates was not taken into account. As the value of the
capacitor increases, the inductance of its plates also increases and so does the
discrepancy.

For small value capacitors of less than 1 nF the self-resonant frequency can be
approximately calculated by the following equations.

1
where L is the lead inductance.
fR =GiE’
{i r31 1
L =0.0002b In - -0.75 pH, where “a” equals the lead radius arid
“b” equals the lead length. All dimensions are in millimeters (mm)
and the inductance is in pH.

Using the formulae, if a = 0.25mm (0.5mm diameter) and b = 5mm (2.5mm
each leg), the inductance is 2.94 x 10-jpH. This is 2.94nH. When substituted
into the frequency equation, with a 1 nF capacitor, the self-resonant frequency
is calculated to be 92.8 MHz.

The formula given for inductance is that for a wire in free space. This should
work for leads that are perpendicular to an earth plane, but not for those