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Ohm’s Law…isn’t *December 29, 2010*

*Posted by mareserinitatis in Uncategorized.*

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One of the first things that every electrical engineer learns in the course of their education is Ohm’s Law:

The problem with Ohm’s Law is that it isn’t a law: it’s an approximation, or at least it is in the form shown above. The real form of Ohm’s Law is:

*What’s the difference?*

The first equation, V=IR, says that there is a relationship between the potential across an object and the current through it. This constant of proportionality is the resistance. The second relationship, however, is between the current and the electric field.

*But potential and the electric field are proportional, aren’t they?*

Yes, this is true. The first equation is a special case or an approximation of the second one, where the electric field is held constant. The first equation holds for two cases. The first is the static case, where you are dealing with DC or constant potential (and hence electric field). The second situation is when you have an alternating current, and the object is much smaller than a wavelength. However, once the object becomes a significant portion of a wavelength, usually 1/10th or greater, one can no longer assume that the approximate form of Ohm’s law, V=IR, holds because the electric field will change over that distance. Therefore, the potential will change as well.

*Why does this matter?*

Most of the time, it doesn’t. Most circuits one makes are very happy with the approximate form of Ohm’s Law. However, one has to consider cases where things may not be so simple: when the object you’re working with is very large or the wavelengths are very small. People who work in power, where AC power is being transmitted over wires that are miles long, and people who work at microwave frequencies need to be concerned and avoid using the approximate form. They can, however, use the telegrapher’s equations, which I’ll discuss in another post.

So the lesson to take away today is that Ohm’s Law isn’t a law.

Futhermore, current through a physical resistance is really a stochastic process. The “I” in Ohm’s law is the expected value of current rather than the actual instantaneous value of current at any specific point in time.

And then there’s self heating…

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I’m beginning to think I ought to have you ‘peer review’ my posts on electrical engineering. In fact, you can be the third reviewer. 😉

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This was basically the second half of EE 14c.

Basically, transmission line circuits are redrawn with distributed capacitance and inductance such that the first form of Ohm’s Law still holds.

Or even more basically: we rewrite the rest of EE to ensure that we don’t make a liar out of Mr. Ohm. 🙂 I think this has actually been beneficial. Anytime Ohm’s law doesn’t appear to hold, it indicates there’s a circuit element you forgot to account for.

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The only EE class I took as an undergrad was CS/EE3 at Tech, which was all digital. They don’t really cover it in physics, so I wasn’t introduced to transmission line theory until grad school. (Unfortunately, it was on a test, no less.) It was the foundation for almost everything we did in microwave engineering, however, so I made up for it later. 🙂

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CS/EE 4? That was one of the classes I TA’ed my senior year (and, since Rod Goodman was out working on his NSF proposals and — ahem — reappropriating funds, did a few lectures for). Dang, that’s a fun class on both sides.

We reworked the problem sets and CS/EE 11 labs to be a bit more challenging and let students explore a bit more. Heck, I remember creating extra portions on the labs, not knowing if it was solvable with the hardware in the lab (and running out of time to test it out). Fortunately, the students were eager to give it a go (and made it work).

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Hmmm…must’ve been CS/EE4. I think CS3 was part of the freshman computer series. I remember that was how I got into using Macs because we used some sort of digital simulator to do problems and check ourselves.

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It’s unfortunate that digital class didn’t at least mention transmission line theory b/c digital is where it tends to come up. In the time domain you can see it as the length of a digital transition being short compared to wire you want to send it down. In the frequency domain you can take the Fourier transform of that edge and find some very high-frequency spectral components.

As you say in your post, if your wire is less than 1/10th of the wavelength (some people say 1/6th of the length of the digital edge) you don’t have to worry about this stuff.

If the driver impedance, transmission line characteristic impedance, and the load impedance are all the same, you can transfer all the power from the source to the load (i.e. transmitter-to-antenna or digital-output-to-input) without reflections. If they’re not all the same, I call an RF engineer and face the fact that a simple application of Ohm’s Law won’t solve my problem.

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Everything a couple GHz behaves this way, so I get to deal with it a lot, and there isn’t much digital involved for me. 🙂

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