WTF is Impedance

After you read about and understood Ohm’s Law, you learned that when resistors are in series, they share the same current, but have different voltages across them (if their values are different). Current through a series resistance is calculated as input volts divided by total resistance, and voltage across each resistor is calculated as current multiplied by the resistance. So if one resistor is a much larger value than the other, it will have more volts across it. Consider this:

If we only measure the voltage across R2, we will see:

[ V(source) / (R1 + R2) ] * R2

or rearranged as V(source) * R2 / (R1 + R2)

If we want to see as much of the (fixed) source voltage as possible across R2, we need to either maximize R2’s value or minimize the value of R1. Now here comes the meat and taters. Ready? Impedance is the AC equivalent of DC resistance. Sink that into your thinker and replace R1 with the output impedance of your source and R2 with the input impedance of your amp. Higher input impedance (R2) or lower output impedance (R1) means better voltage transfer. Congratulations, you now successfully understand the basics of the most mysterious DIY audio topic: impedance matching and signal transfer. This kind of circuit analysis is know as Thévenin’s Theorem, or alternatively Norton’s Theorem. The next step is understanding frequency dependant impedance, or reactance.

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 Tell, you’d prefer that nice fat barn.  Impedance, when it comes to connecting sources to preamplifiers and preamplifiers to power amplifiers, works in much the same way.  Ideally you want your output impedance to be as small as possible relative to the input impedance that it will be connected to.  Generally, a 1 to 10 ratio is a pretty good minimum difference to design for.  

This general concept can also be applied to power supply filters.  Replace R2 with a capacitor (low AC impedance) and R1 with a choke or resistor (high AC impedance).  As long as R2 is little and R1 is big, very little ripple signal will make it through the filter.

Note that with an audio signal, rather than power supply ripple, we often want the opposite. For example, our R1 may be a coupling capacitor, which we want to have a small impedance at the frequencies of interest. Our R2 may be a resistor referencing a grid to ground, where a large impedance minimizes signal lost in R1.

Impedance and Tubes

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 would involve their own equations, but screw 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 this is the first stage of your amplifier, it’s going to determine the input impedance that whatever you hook up to it sees.  Due to the Miller Effect, it is frequency dependant.  

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.  

Output Impedance

Grounded Cathode Amplifier

Math:

If Rk is bypassed, Z = Rl || Rp

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

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, cathode resistor, and gain may all play a role.  

Cathode Follower

Math:

Zout = 1 / Gm = Rp / Mu

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.