Impedance & Reactance

rage flip

Tube Stage I/O Impedance

Think of input and output impedance in your audio equipment like an arrow and target.  Would you rather shoot an arrow at a nickel or the side of a barn?  Unless you’re William fuckin’ Tell, you’d prefer that nice fat barn.  Impedance, when it comes to connecting preamplifiers to power amplifiers or one tube stage to another, works in much the same way.  Ideally you want your output impedance (arrow) to be as small as possible relative to the input impedance (target) that it will be connected to.  Generally, a 1 to 10 ratio is a pretty good “size” difference to design for.  

Here are theoretical maths to calculate the input and output impedance of grounded cathode amplifiers and cathode followers.  There are plenty more tube arrangements that all have their own equations, but fuck that noise because this shit is already way too technical for WTF-level tube hooliganry.  

Input Impedance

Grounded Cathode Amplifier

Math:

Zin = Rg || Zmiller

Why:

If the first stage of your amplifier is a grounded cathode amplifier (and it very often is), it’s going to determine the input impedance that whatever you hook up to it sees.  Due to the Miller Effect (see further down this page), the input impedance is frequency dependent, so it’s good to make sure the output impedance of the previous stage is low.

Cathode Follower

Math:

Zin = Rg / (1 – stage gain)

Why:

Really this is just interesting because it shows that cathode followers have super high input impedance.  Remember that cathode follower stage gain is always close to 1 so you’re dividing an already fairly large grid resistor by a very small number.  Tits.

Output Impedance

Grounded Cathode Amplifier

Math:

If Rk is bypassed, Z = Rp || Rl

If Rk is not bypassed, Z = (Rp + Rk*(Mu + 1)) || Rl

Why:

Grounded cathode amplifiers are the simplest way to get voltage gain with a single tube and so they’re all over the place.  It’s good to consider the output impedance of a grounded cathode gain stage when designing what it will be connected to.  In many cases, the plate resistance (Rp) dominates the output impedance, but the load resistance (Rl), cathode resistor (Rk), and Mu may all play a role.  

Cathode Follower

Math:

Zout = 1 / Gm or Rp / Mu (in a triode)

Why:

Cathode followers are great because they offer a really low output impedance (at the expense of voltage gain).  As should be apparent from the equation, higher transconductance generally means lower output impedance.

Reactance (OMG Fuck This Math)

Impedance (in addition to resistance) 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.  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?  Just fuck 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.