What is a general definition of impedance?

Impedance is a concept that shows up in any area of physics concerning waves. In transmission lines, impedance is the ratio of voltage to current. In optics, index of refraction plays a role similar to impedance. Mechanical impedance is the ratio of force to velocity.

What is a general definition of impedance?

What are some examples of "impedance matching" other than in electrical transmission lines?

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I found a general, qualitative answer in David Blackstock's book Physical Acoustics, on page 46:

Impedance is often described as the ratio of a "push" variable $q_p$ (such as voltage or pressure) to a corresponding "flow" variable $q_f$ (such as current or particle velocity).

I also received a nice answer to this question on another Q&A site which expands a bit on this qualitative statement with a quantitative one. In particular, this answer makes the point that impedance is the ratio (transfer function) of a force applied at a particular point to the velocity at that point.

I suppose what I am looking for next is an intuitive explanation of the phenomenon of impedance matching and maximum power transfer.

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Uh... I liked this question. And a quick look at the Google or Google scholar did not give me much. So is written here more what I grasp people understand about impedance. Like Noldorin was saying I take impedance as how much is impeding power to be transferred from one place to another. In a homogeneous medium power is transferred as wave unimpeded but when it find an interface of another medium then power transfer won't be so straight forward. The point is at the two mediums dynamic equations of the two place does no match. Say, electromagnetically one medium has the Maxwell equations of $(\mu_1,\epsilon_1)$ and the other medium has the equations with $(\mu_2,\epsilon_2)$ At the interface the countour conditions needs to be satisfied $$\left(\vec{D_2}-\vec{D_1}\right)\cdot\hat{n}=0$$ $$\left(\vec{B_2}-\vec{B_1}\right)\cdot\hat{n}=0$$ $$\left(\vec{E_2}-\vec{E_1}\right)\times\hat{n}=0$$ $$\left(\vec{H_2}-\vec{H_1}\right)\times\hat{n}=0$$ where $\hat{n}$ is the normal of the interface. Using the above conditions to connect the solutions of wave equations at each medium, will give rise to the Reflections and Transmission Coefficients. In the case of Maxwell's equations $$R=\frac{Z_2-Z_1}{Z_2+Z_1}$$

For a one dimensional wave, with standard solutions $$f_i\left(x,t\right)=A e^{k_i x-\omega t}+B e^{k_i x+\omega t}$$

the conditions that needs to be satisfied are the equality of displacement and the its derivative $$f_2(x_0,t)=f_1(x_0,t)$$ $$f'_2(x_0,t)=f'_1(x_0,t)$$ where $x_0$ is the position of the interface. Using these conditions with the solutions of the wave equations we will get. $$R=\frac{k_2-k_1}{k_2+k_1}$$ So in this case the impedance will be characterized by $k$.

So generalizing what you want is find what is the physical conditions that needs to be matched at the interface and use it to connect the solutions of each side of the interface, and figure how you get the reflections coefficients, which should look something like the reflection coefficients above. The terms should be your impedance for that system. Note that the definition of your impedance in that system is not quite unique. As in the previous example you could normalize the $k_i$ by some arbitrary $k_0$ and call this the impedance. $$z_i=\frac{k_i}{k_0}$$ Check the literature to find what is the exact definition for the system you are interested in. I suppose you already went to Wikipedia for other examples.

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A good example of impedance matching outside the world of transmission lines is acoustic horns, which match acoustic impedance between a sound source (like a vibrating string or reed) and the air.

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Mechanical impedance matching does have an application in electrical transmission lines (or any elastic cable/structure vibration) because it helps describe how much of the wave gets through a discontinuity, and how much is being reflected. Mechanical impedance is force over velocity and along the cable it is equal to tension over wave speed. The discontinuity may be either an elastic support (with damping properties), or a change unit weight of the cable (bigger cross section) which changes the wave speed.

For a cable with tension $T$ and unit weight $w$ (weight per length) the wave speed is $c=\sqrt{T/w}$ and the impedance is $z=\frac{T}{c}=\sqrt{T\,w}$. The practical application of this is in the design of stockbridge dampers that go on cables and structures to absorb aeolian vibrations. There is a theory of matching impedances and measuring the "Inverse Standing Wave Ratio" in testing to check the efficiency of the damper. With a good match most of the wave energy does not get reflected back into the cable once it reaches the end near the tower. There is a IEEE standard covering all this.

In case you missed it, mechanical impedance is force over speed. Maybe someone something similar for the electrical, or acoustical impedance.

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Impedance is general simply refers to the "amount of impeding (or block)". It's specific meanings in the various branches of physics (mechanical, electrical, wave impedance) varies, but are all loosely based on the concept of the everyday word.

A good example of impedance matching I always think of relates to audio systems. When you want to chain a number of amplifiers in stages, you usually want to match the output impedance of one with the input impedance of the other in order to maximise power transfer.

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Impedance matching is not used in audio electronics. Not in microphones/preamps, nor interconnects, nor speaker amplifiers. In all these cases, the source impedance is (much) smaller than the load, to maximize fidelity, not power. Also, you misunderstand the maximum power theorem, which only applies when the source impedance is fixed and you're trying to get the maximum power out of it. If you have control of the source, you just make its impedance as small as possible to increase power to the load. –  endolith Nov 9 '10 at 19:35
@endolith: I'm afraid you're mistaken, at least in part. When you connect power amps in series, you want to impedance match. I was explained this by an electrical engineer, no less. –  Noldorin Nov 9 '10 at 19:55
@Noldorin: What do you mean by "connect power amps in series"? –  endolith Nov 9 '10 at 20:22
@Noldorin: Can you explain further? I don't like to make a lame "argument by credentials", but I'm an electrical engineer, too, and my job is designing audio equipment. If I'm mistaken I'll be pretty embarrassed. :) –  endolith Nov 12 '10 at 5:34
@endolith: Sure, I'll try to look up a reference and get back to. I admit I might be wrong, but if so my memory is playing tricks on me! Disturbing, in any case... –  Noldorin Nov 12 '10 at 19:22