What is impedance? And why does it do those terrible things?
Beginners with electronics get down Ohm's law pretty quickly. The concept that Voltage, Current, and Resistance to electrical flow are related by the simple expression V=IR seems to set pretty easily. They usually learn about capacitors and inductors as "something else", not like resistors at all, but as perhaps a way to store energy or filter some frequencies from others. But sooner or later, they run up against the need to understand the effect of source impedance and/or load impedance, and things get un-simple quickly. Impedance seems to become a vague concept of something like electrical strength or drive capability, not simple at all, especially if the impedance in question involves capacitors or inductors, or (ugh!) both.
It's not that complicated.
Impedance is the generalization of the concept of resistance from DC to AC. That is, it's a way to represent how much current will flow with a specified (AC) voltage across the impedance. That is, if you have one volt AC across an impedance that lets one ampere of AC current flow, the impedance is defined by the AC version of Ohm's law and is one ohm.
Since AC has not only amplitude, like DC, but also frequency and phase, this introduces the possibility that an impedance will not only allow a current to flow, but will change the phase of the signal, and respond with different amplitudes and phases as frequency changes. You can have a resistor, a capacitor, and an inductor that each have an impedance of one ohm (or a Kohm or a Mohm) at any given frequency.
The resistor (ignoring the "imperfections" of parasitic capacitance, lead inductance, etc., which is usually valid at audio frequencies) will have the same impedance at every frequency. The capacitor will have an impedance that goes down with frequency (making the same assumptions as with the resistor, ignoring parasitics) and the inductor will have an impedance that goes up with frequency (ditto).
We can calculate the impedance of any of these, given a frequency to work with.
The cap is Xc = 1/(2*pi*F*C) and the inductor is Xl= 2*pi*F*L, so they vary linearly with frequency. As to phase, the inductor's voltage is always 90 degrees ahead of its current (the current takes some time to change); the capacitor's current leads the voltage across the cap, as the capacitance slows down voltage changes. If you can remember "E" for voltage, I for current, you can keep this straight by the phrase "ELI the ICEman", which is a nonsense way to remember "E leads I in inductors (L's)" and "I leads E in Capacitors (C's).
In AC power line circuitry, where the "signal" on the line is always a fixed frequency, fixed amplitude sine wave, inductors and capacitors make perfectly good current limiters, better than a resistor would be because they cause no heat to be dissipated as a result of their current limiting. In fact, the impedance of inductors and capacitors is so different from a resistance that it's given a special name - "reactance". In engineering schools, impedance is expressed as a "complex" number represented by "Z". This Z is the sum of a resistance and some reactance, Z = R+jX. You won't need to know this, but I thought I'd mention it in case you had run across it and wondered.
Ok, so how do we do some quick calculations of impedance?
There are ways to do it that are analytically correct, but very confusing. There are more intuitive ways to find some approximations; we can ignore some things and simplify. I'm going to skip the correct-but-tedious stuff entirely.
Any series element that has a very small impedance compared to the other things in series with it can be replaced with a wire for the purposes of calculations. Any parallel impedance that is very large can be replaced by an open circuit; these simplifications produce only tiny errors. When we come to things with similar (less than 10:1 differences) values in series we add them, and in parallel, we compute the parallel values.
Using these approximations, we can usually get a pretty good idea of the source impedance (how much impedance is inherent inside a signal source) and the input or loading impedance (how much current it takes to drive a signal into it)
Example: The input to an NPN transistor amplifier, with a pull down resistor of 1M to ground at the input, a capacitor of 0.1uF in series to the base, which also has a resistor of 220K to the positive supply and a resistor of 22K to ground, and a 1K emitter resistor from the transistor emitter to ground.
Unstated but assumed is that the supply voltage is that, a pure voltage source with an impedance that is vanishingly small. We approximate that with a battery, which may have an internal resistance of only a few ohms. Since this is by far the smallest impedance here, we replace it with a short, our first approximation. So now we have the 220K and the 22K in parallel to ground from the base.
Let's see about that input cap. The impedance can be as high as 1/2*pi*82Hz*1E-7 = 19.4K at the low end of the guitar range, and as low as 79.5 ohms at 20KHz. The 1M is much higher than the cap at all frequencies of interest, so we replace it with an open circuit and ignore it.
So now we have a series cap going to the parallel combination of a 220K, a 22K, and a transistor base. Ack!! How do we compute that?
It turns out that it's not that hard. If the transistor has a modest gain, say 100, the impedance seen at its base is the series combination of the current gain times any emitter resistance *including the internal base-emitter resistance*. Since we know (here's about seven pages of stuff I won't type) that the internal base emitter resistance is usually low, we'll assume it's zero and just use the 1K; that gives us an impedance as seen at the transistor base of 100*1K, or 100K, effectively in parallel with the 220K and the 22K. The actual value is somewhat higher, but is dominated by the emitter resistor. (Note - the transistor's internal base-emitter resistance can be computed simply, too - it's Rbe=25mv/Ie).
So we do the calculation for parallel resistors, and get an equivalent resistance of close to 16.7K. That's NOT negligible compared to the capacitor, so we have to figure out what happens with frequency. However, now we have a simple way to calculate the impedance variation with frequency - the impedance will always be the sum of the capacitor's impedance and the equivalent resistance.
We know that at high frequencies, the capacitor will eventually be much less than the 16.7K, so we'll be able to ignore the cap there. We know at lower frequencies the cap will dominate as its impedance gets much higher. We need to calculate the "turnover point" where the cap and the resistor have equal impedances; this will be where the input impedance stops looking like a capacitor and starts looking more like a resistor. That frequency is the one where the cap's impedance is 16.7K, or Xc=16.7K=1/(2*pi*F*C), or F = 1/(2*pi*R*C) = 95Hz.
So while we can calculate the sum of the R and the Xc, we know that above 95 Hz, the input looks more and more like a 16.7K resistor as the capacitor Xc decreases.
We made a lot of assumptions getting there, and we know that the numbers are not exactly correct, but they are very close, probably closer than our ability to get a transistor with Hfe=100. With a 16K +/- a few K load at the input to this amplifier, we know it's gonna suck treble out of guitar pickups because of the loading.
The quick and dirty way of measuring input impedance with a variable resistor to halve the input voltage works, and gives a correct number at every frequency, but it gives you no insight into what is happening with frequency until you do enough points to find the turnover frequency.
Output impedance can be calculated the same way, except that there is nearly always an active source like a transistor emitter or collector, a transformer secondary, etc. that has R's, C's and L's in series/parallel with it. In that case you figure the source impedance of the device that provides the energy, and then do the approximations and calculations toward the output node.
This is only a very brief intro. In most cases, the real trick is to make the input and output impedances not matter: that is, you make input impedances large compared to the output impedances of whatever the signal source is, and make output impedances tiny compared to the load it's driving. A guitar pickup may have an impedance between 8K and 100K; in general, we want an input impedance of 1M or more to keep from loading a guitar down. An amplifer is loaded by 16, 8, 4 or 2 ohms; we want the amplifer to be much less than the load, maybe under 0.2 or less ohms.
This is usually not possible with power outputs on tube amplifiers, but the calculation of the proper impedances and matching in vacuum tube amplifier power stages is a subject that deserves its own explanation.