here to go to our main page on capacitors
here to go to our reactance calculator
Below is an index of our mathematical
discussion of capacitors:
of a transmission line (separate page)
charge storage capacitance (separate page, new for March
ESR effects (separate page, new for February 2009!)
Use our reactance
calculator if you are interested in this topic!
is the "imaginary" impedance of a capacitor expressed
in ohms. Note the negative sign, which implies that on a Smith
Chart, adding series capacitance tends to rotate your reflection
coefficient counterclockwise. It's a function of frequency.
Hey, Bill Gates,
before we go any further with this page, we want to tell you that
Microsoft Equation Editor 3.0 completely sucks! Now let's fix that
formula for the more practical units of pico-Farads and GHz:
The quality factor
is a measure of how lossless a capacitor is:
Note that as you increase frequency,
quality factor always gets worse (decreases). Let's put the equation
into GHz and pico-Farads:
Thus a 1 pF cap with one-tenth
ohm resistance would have a Q of 159 at 10 GHz. Dissipation factor
is merely the reciprocal of the quality factor:
of the same 1 pF cap at 10 GHz would be 0.6%.
The well-known formula for parallel-plate
capacitors of "infinite size" is given below. Most parallel
plate capacitors behave close to ideally because there area dimensions
(length and width) are much bigger than their plate separation.
No consideration of the fringing fields is usually necessary unless
you are working with extremely small capacitors (perhaps less than
This is a case where
microwave engineering prefers the metric system, because the permittivity
of free space is always expressed in Farads per meter, not Farads
per inch. Because practical microwave circuits are much smaller
than meters and Farads, we will rewrite the capacitance formula
using millimeters and pico-Farads:
is a useful simplification of the capacitor formula, if you are
always using a given dielectric and thickness to construct capacitors
(such as on a MMIC):
Thus, if you are
working with a MMIC foundry that offers 2000 Angstroms (0.2 micron)
silicon nitride (er=7.5) for thin-film capacitors, the sheet capacitance
7.5 x 0.00885/0.0002=332 pF/mm2
If you construct
a capacitor of 100 x 100 microns (0.1 x 0.1 mm), it will have a
value of 3.3 pF.
The first resonance
of a capacitor is the series resonant frequency. Referring to the
model below, this is the frequency where the capacitive reactance
and inductive reactance due to LS cancel.
The frequency at which the series inductance of a capacitor is equal
but opposite to its capacitance. Click
here for an explanation of series resonance on our filter page.
This is where the capacitor behaves as low-value resistor (equal
to the ESR value).
Most capacitor vendors
tell you the series resonant frequency, not the inductance LS.
To determine LS in nano-Henries, use the following equation
(fixed on February 3, 2006!):
Parallel resonant frequency
This typically occurs at twice the SRF frequency. Click
here for an explanation of parallel resonance on our filter
page. Usually you have no business operating a capacitor at or near
the PRF, because it acts like an open circuit in this case!
That's all for now!