# (OMG F&%k This Math)

Impedance is often made up of the reactance of capacitors and inductors.  Reactance is frequency dependent. This shit is what separates good designs from great designs.  Reactance calculations help you figure out things like coupling caps, RIAA filters, output transformer inductance, power supply filtering, and way more. Now’s a good time to grab a towel in case your head explodes.

## Capacitors

### Output Capacitors & High Pass

Math:

f = 1 / (2 * pi * Cout * Zin)

Cout (in farads) = 1 / (2 * pi * f * Zin)

Why:

If you want to achieve good low-end bandwidth (ie awesome bass), you have to consider the input impedance of whatever you’re connecting your stage to as well as the capacitor you’re using to couple them (Cout).  The lower the input impedance of the following stage, the higher the value of the capacitor you need to maintain a low -3db point (generally, 5-10hz is good enough for hifi).

### Miller Effect, Power Supplies, & Low Pass

Math:

Cmiller = Cg-k + Cg-p * (stage gain + 1)

f = 1 / (2 * pi * Cmiller * Zout)

Why:

The Miller Effect is the capacitance created by the spacing of the electrodes in the tube envelope and the effect of gain.  Along with the output impedance of the previous stage, this creates a high pass filter.  For shimmering, twinkling, sparkling treble, you want to minimize the output impedance of the previous stage, the miller capacitance in the current stage, or both.  Miller Effect capacitance is usually very, very small and sometimes its effects will be outside of the typical hifi bandwidth goal of 20khz.  That said, it is an important consideration, especially where there is high gain.

Capacitors in power supplies also create a low pass filter that helps to roll off high frequency noise/ripple.  In this case, the impedance of the power supply must be taken into account in order to calculate…wait a second…screw that, use PSUDII.

### Cathode Bypass Capacitors

Math:

C (in farads) = 1 / (2 * Pi * f * R’)

where R’ = Rk || ((Rp + Ra) / (Mu +1))

Why:

Cathode resistors create degenerative feedback, reducing gain and linearity in grounded cathode amplifier stages. To get around this, we can bypass the cathode resistor with a capacitor, thus creating a short to ground (in AC terms). But we need to ensure that this capacitor will pass all audio frequencies of interest (usually using an “f” value of 5 or 10 hertz).  If this looks like the high pass equation above, that’s because it basically is.

## Inductors

### Output Transformers & High Pass

Math:

f = Z / 2 * pi * L

L = Z / 2 * pi * f

Why:

As explained in the output transformers section, the low-end bandwidth in a transformer coupled amp can be affected by the inductance (L) of your output transformer.  Higher inductance means a lower -3db point (and usually a higher cost and size).

### Power Supplies & Low Pass

Math:

f = Rt / 2 * pi * L

Why:

Finally, the last one of these awful AC equations.  I’m sick of math.  Are you sick of math?  Screw this one.  Stick a big choke in your power supply so that it rolls off any high frequency ripple and have a nice day.  May you never know the horrors of writing technical explanations to stuff.