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This is presented as both a resource for the community and a learning experience for myself. I have just enough knowledge of the subject to get myself into trouble, but I don't have the best grasp of the subject's details. Some helpful responses might be:
Thanks for the help!
To the question "what is Impedance," I would note that impedance is a broad concept of physics in general, of which electrical impedance is only one example.
To get a grasp of what it means and how it works, it's often easier to consider mechanical impedance instead. Think of trying to push (slide) a heavy couch across the floor.
You apply a certain amount of force, and the couch slides at a certain velocity, depending on how hard you push, the weight of the couch, the type of floor surface, the type of feet that the couch has, and so on. For this situation, it's possible to define a mechanical impedance that gives the ratio between how hard you push and how fast the couch goes.
This is actually a lot like a dc electrical circuit, where you apply a certain amount of voltage across a circuit, and current flows at a certain corresponding rate through it.
For the case of both the couch and the circuit, the response to your input may be simple and fairly linear: a resistor that obey's Ohm's Law, where its electrical impedance is just the resistance, and the couch may have friction slider feet that allow it to move with a velocity proportional to your force.*
Circuits and mechanical systems may also be nonlinear. If your circuit consists of a variable voltage placed across a resistor in series with a diode, the current will be near zero until you exceed the forward voltage of the diode, at which point current will begin to flow through the resistor, in accordance with Ohm's law. Likewise, a couch sitting on the floor will usually have some degree of static friction: it won't begin moving until you push with a certain amount of initial force. In neither the mechanical nor electrical system is there a single linear impedance that can be defined. Rather, the best that you can do is to separately define impedances under different conditions. (The real world is much more like this.)
Even when things are very clear and linear, it's important to note that impedance just describes a ratio-- it doesn't describe the limits to the system, and it's not "bad." You can definitely get as much current/velocity as you want (in an ideal system) by adding more voltage/pushing harder.
Mechanical systems also can give a pretty good feel for ac impedance. Imagine that you're riding a bicycle. With each half-cycle of the pedals, you push left, push right. You can also imagine pedaling with just one foot and a toe-clip, such that you push and pull with every cycle of your pedal. This is a lot like applying an ac voltage to a circuit: you push and pull in turn, cyclicly, at some given frequency.
If the frequency is slow enough-- like when you're stopped on the bike, the problem of pushing down on the pedals is just a "dc" problem, like pushing the couch. When you speed up, though, things can act differently.
Now, suppose that you're biking along at a certain speed, and your bike is a three-speed with low, medium, and hi gear ratios. Medium feels natural, hi gear is difficult to apply enough force to make any difference, and at low gear, you just spin the pedals without transferring any energy to the wheels. This is a matter of impedance matching, where you can only effectively transfer power to the wheels when they present a certain amount of physical resistance to your foot-- not too much, not too little. The corresponding electrical phenomenon is very common as well; you need impedance matched lines to transmit RF power effectively from point A to point B, and any time that you connect two transmission lines together, there will be some loss at the interface.
The resistance that the pedals provide to your feet is proportional to how hard you press, which relates it most closely to a simple resistance-- particularly at low speeds. Even in AC circuits, a resistor behaves like a resistor (up to a certain point).
However, unlike a resistor, the impedance of a bicycle is dependent on frequency. Suppose that you put your bike in high gear, starting from a stop. It can be very hard to get started. But, once you do get started, the impedance presented by the pedals goes down as you get going faster, and once you're going very fast, you may find that the pedals present too little impedance to absorb power from your feet. So there's actually a frequency-dependent impedance (a reactance) that starts out high and gets lower as you head to higher frequency.
This is much like the behavior of a capacitor, and a fairly good model for the mechanical impedance of a bicycle would be a resistor in parallel with a capacitor.
At dc (zero velocity), you just see the high, constant resistance as your impedance. As the pedaling frequency increases, the capacitor impedance becomes lower than that of the resistor, and allows current to flow that way.
There are, of course, various other electrical components and their mechanical analogies**, but this discussion should give you some initial intuition on the general concept to stay grounded (pun intended) as you learn about the mathematical aspects of what can at times seem like a very abstract subject.
*A word to the picky: Ohm's law is never exact for a real device, and real-world friction forces never give velocity exactly proportional to force. However, "fairly linear" is easy. I'm trying to be all educational and stuff here. Cut me some slack.
**For example, an inductor is something like a spring-loaded roller on your wheel that adds drag as you get to higher frequency)
The impedance of a circuit element is the ratio between voltage and current in that element.
Constant voltages and currents
For constant voltages and currents, impedance is just resistance. A resistor is a device that maintains the same ratio of voltage to current, even as the voltage changes. They're linear-- double the voltage and the current doubles too. If you drew a graph of voltage vs. current, the slope would be the impedance.
A capacitor, which is like two metal plates, acts like an open circuit for constant currents and voltages. An inductor, which means a curly wire, acts like a short circuit for constant currents and voltages.
(In reality, it's not quite this clean. Resistors tend to let through less current then they should when they get hot. Capacitors let a little bit of current leak through, even when they shouldn't. Inductors have a small amount of resistance, like any normal wire.)
Voltages and currents that change with time
Here's where it gets more interesting. Some circuit elements, like capacitors and inductors, allow more or less current flow depending on the frequency of the voltage they're subjected to. You could think of them as frequency-dependent resistors. The frequency-dependent part of impedance is called reactance. Add reactance and resistance and you get impedance.
Examples of reactance
Suppose you had a box that generated sine waves of amplitude 120 V. You set the box for 60 cycles per second and connect the box's signal across a 0.1 F capacitor. The current that flows will be a sine wave at the same frequency. The current will be:
I = V * 2 * pi * frequency * C
I = 120 * 2 * 3.14 * 60 * 0.1 = 4522 amps.
(In reality, that much current would make the capacitor explode.)
If you doubled the frequency of the sine wave, the current would double. This kind of behavior is useful in RC filters-- you can make circuits that have high resistance at one frequency, but low resistance at another, which lets you pick out a signal from amongst noise, for example.
An inductor behaves similarly, but as you increase frequency, the impedance increases rather than decreases.
The real world
In reality, everything has some resistance as well as some reactance (either a little capacitance or inductance, but not both). In addition, all circuits have non-linearities, like temperature dependence or geometrical effects that make them deviate from the ideal model.
Also, the voltages and currents we deal with are never perfect sine waves-- they're a mix of frequencies.
For example, suppose you're running a solenoid to open a door lock, like the buzzers in apartment buildings. The solenoid is a massive inductor that creates a magnetic field that pulls back a latch against the force of a spring. When you turn off the solenoid, you're making a current change drastically with time. As you try to make the current drop quickly, the inductance of the solenoid makes the voltage rise quickly.
This is why you see what's called a "flyback diode" in parallel with big inductors-- to allow the current to drop more slowly, avoiding the voltage spike caused by a high-frequency change.
The next step
From here, the next step is to learn how to model circuits built of multiple reactive elements (say, a bunch of resistors and capacitors). For that, we have to track not just the amplitudes of voltage and current, but also the phase shift between them-- the peaks of the sine waves don't line up in time.
(Unfortunately, I have to get some work done here, so I'll have to leave you with this link: http://www.usna.edu/MathDept/CDP/ComplexNum/Module_6/ComplexPhasors.htm)
Impedance is an extension of the concept of resistance that includes the effects of capacitance and inductance. Inductors and capacitors have "reactance", and impedance is the combination of the effects of resistance and reactance.
n00b introduction: Essentially, it lets you think about capacitors and inductors as if they were resistors, making calculations simpler and more intuitive. For instance, if you know how to calculate the output of a purely resistive voltage divider:
then you can also calculate the magnitude of the output of an RC filter at a given frequency:
Say R is 1 kΩ and C is 1 uF, for instance, and you want to know the output voltage if you input a sine wave at 160 Hz. The reactance of the capacitor at 160 Hz has a magnitude of about 1 kΩ, so both "resistors" are the same, and the voltage across each will be the same. Each component has 0.707 of the input voltage across it, though, not 0.5, as in the resistive case.
At other frequencies, the magnitude of the reactance of the capacitor would be different, which is why the filter responds differently to different frequencies. You can also work with imaginary numbers to calculate the phase shift in the output, but often the magnitude is the only part you care about.
The mechanical analogy I like for impedance is a vertically-hanging spring with a collection of weights hanging on it. If the system is initially motionless and one gives a brief upward jerk to the weight at the top, returning it quickly to its original position, the disturbance will travel down the spring. Each weight will be pulled upward by the weight above, then push upward on the weight above (and be pushed downward by it) while it pulls upward on the weight below (and is pulled downward by it), and finally be pushed upward by the weight below. Once all these things have happened, the weight will return to its original position and (zero) velocity.
Note that the behavior of the downward-propagating wave does not depend upon anything below it. Once the wave reaches the bottom, however, one of three things may happen depending upon whether the end of the spring is dangling, rigidly fixed to something, or fixed to something that can move with some resistance.
If the end of the spring is dangling, the bottom weight won't have anything below it to pull down on it when it jerks upward. The effect of this will be that the weight will jerk upward more than it otherwise would, and more than the weight above would be expecting to cancel out its energy. This in turn will cause the weight to push upward on the weight above, and generate an upward-traveling wave which will be (absent frictional losses) be equal in magnitude to the initial downward wave. The displacement direction will be the same as the original wave (i.e. upward) but the stress will be opposite (the original wave was a tension wave; the rebound will be compression).
By contrast, if the end of the spring is fixed, the bottom weight will find that the spring below it resists more strongly than expected. The bottom weight will thus not move up as much as the weight above it was expecting, and the net effect will be as though the bottom gave an extra "tug", sending a wave upward. The displacement direction of this wave will be the opposite of the original wave (i.e. downward) but the stress will be the same (compression).
If the bottom of the spring is attached to something which moves somewhat, but not as much as a dangling spring, the two behaviors above may cancel out to some degree. If the bottom of the spring is allowed to move just the right amount, the behaviors will cancel and the wave will disappear. Otherwise one or other type of wave will rebound, but the magnitude will generally be less than it would with a dangling or fixed end. The amount of resistance required is effectively defined by the impedance, which is in turn a function of the mass of the weights and the spring constant of the springs.
Note that many impedance-related behaviors are captured by this model. For example, if all the weights above a certain point weight 100g while those below weigh 200g, and all springs are equal, the transition from the lighter weights to the heavier weights will cause some of the wave energy to be reflected upward (in a manner similar to the fixed bottom end) since the heavier weights won't move as much as expected. The key notion is that for things which are pushed to return to zero velocity, they must transfer both their kinetic energy and their momentum. If the they can transfer their energy and momentum to something with the same characteristics as whatever pushed them, they will accept all the energy and momentum and pass them on. Otherwise they will have to send back some of the energy and/or momentum.
I will limit my answer to the electrical realm. Impedance (Z) is literally just V/I. It is as simple as that. But 'that' is not so simple in all cases. Let's start with simplist and work up.
If the impedance is a simple lumped resistor and V is a DC voltage (frequecy = f = 0), we can rewrite Z=V/I to be R=V/I.
If the impedance is due to a cap or an inductor, then impedance is frequecy dependent.
If frequencies get high enough that components do not appear as lumped elements, then impedance is not only frequency dependent but location dependent. Sometimes these elements are designed to be distributed (e.g, wave guides, antennas and EM waves in free space), and sometimes not.
The general tool which has been developed to portray these higher frequency effects in time and space (1 dimension) is . . . Z=V/I. But 'V' and 'I' are both complex vector quantities of the form (A)(e)^(j(wt+x)), where j=SQRT(-1), 'A' is a constant, 'e' is the base of the natural logarithm, 'w' is frequency in radians/second, 't' is time in seconds, and 'x' is distance along the 1-D path. Since 'Z' is a ratio of these two complex vectors, it, too, is a complex vector which varies in time and space. The electrical engineer manipulates these quantities for the time and location desired, and then takes the real portion of V or I (or Z) to get what is observed in the real world.