Pilfered Power

Some time ago I saw an article about the concept of harvesting power from high-voltage transmission lines. The article reported disturbingly large voltages obtained from the equipment they set up, which led them to stop the experiment for the sake of safety (and legal protection). I was curious if I could figure out a way to predict the voltage we could expect from such a situation.

The electric system transmits power with alternating current (AC) at high voltages, which gets stepped-down by transformers when it is delivered to our houses. To see why the transmission is done at high voltage, we can imagine sending out a power P through lines with resistance R:

The power lost is inversely proportional to the square of the voltage, meaning raising the voltage can quickly decrease the power we lose to resistance.

When current travels through a wire, it produces a magnetic field in circles around the wire. The equation for this field is

where μ is the permeability of free space, I is the current in the wire, and ρ is the radial distance from the wire. For alternating current we can write

where I0 is the amplitude of the current and f is the oscillation frequency, 60 Hz in the US. If we set up our fence directly below the transmission line with the z-axis pointing along the fence and the y-axis pointing up toward the wire, then the B-field at this point will be

where h is the height of the transmission wire, and we've replaced I0 with the power & voltage carried by the line. Now to get the voltage induced in the fence, we can use Ampere's Law:

Plugging in our B and integrating to get E gives

Electric field is measured in volts/meter, so this will tell us the relation between the length of our fence and the voltage we can expect between the two ends. The example case discussed in this Wikipedia article uses P = 1000 MW and V = 765 kV. This page gives the minimum height for wires carrying that voltage as 45 ft. Plugging in those values, along with the 60 Hz frequency I mentioned, gives 330 megavolts/meter, which is pretty insane! Now I did assume best (or maybe worst) case numbers in this calculation, but if you watch the video in the article that started this, the presenter seems a bit concerned with the size of the spark he gets at the end. Good (grounded) fences make good neighbors!

Re: Re: Re: Charge

A topic of much debate when using battery-powered devices, like phones, laptops, or electric toothbrushes is, how often should they be charged? The main schools of thought are, charge whenever you have power available, or charge only when the battery dies or is low. I tend to subscribe to the second theory, on the principle that batteries are typically rated for a fixed number of charge/discharge cycles. My laptop has some built-in protection that limits how often it charges:

Batteries work through a chemical reaction between two reactants suspended in an electrolyte. Electrons pass between the reactants, creating a current and dissolving some of the reactants into the electrolyte. This is a bit oversimplified, but I am, after all a physicist, so that's how I like things. I wanted to see whether this model could back up my charging habits.

The setup I settled on was several electrodes, each surrounded by a block of reactant, and the whole thing surrounded by electrolyte. Then I put the system through several charge/discharge cycles. During each discharge step, we find all the reactant that's in contact with electrolyte, and dissolve it into electrolyte with some probability. During charging steps, we do the reverse: Find electrolyte in contact with either reactant or electrode, and precipitate reactant with some probability. We keep track of how many changes happen each cycle, which corresponds to the amount of current produced, and we vary the number of repeated discharge steps before switching to charging, and vice versa.

There's a lot of parameters to tune here (probability of state change, number of electrodes, amount of electrolyte), so I haven't come close to exploring the full space, but I'm still pretty happy with the results. This case used 200 steps for each discharge/charge cycle:

You can see we use up most of the reactant (yellow) on each cycle. If we switch to only 80 steps, things look a little more ragged:

We can measure how much reactant is around at the start of each discharge, and plot how it changes as we go through cycles for several different cycle lengths:

For all but the extreme 2-step case, these quickly reach an equilibrium max charge (plotted as a fraction of the initial). I was curious what was going on with 2-steps, so I plotted what it looks like at the end of the simulation:

Because of the way I set up the cycling between dissolution and precipitation, the system tends toward holding only as many as it adds on during charging – That's the source of the pattern in the max charge plot above, and not (as I initially thought) my belief about battery health. As result, I think I have to consider this sim inconclusive as far as charging habits. Maybe I'll do a followup later (which will of course be called "Re: Re: Re: Re: Charge")!

The Fault in Our Ground

Yesterday, we took the camper to refill propane and get it cleaned. When we came back to our site, and plugged in to the campground power, we found nothing was flowing! We tried a couple ideas to solve it, but we were exhausted and it was already dark, so we turned up the (newly filled) propane heat and went to sleep. This morning, after much more troubleshooting, we discovered the issue was the ground fault circuit interrupter (GFCI), which we had tried the previous night, but hadn't performed the necessary reset steps correctly. I thought I'd get some good out of the frustrating experience, and write a post on GFCIs.

Humans are essentially big bags of salty water, which makes us good at conducting electricity. Unfortunately, letting anything more than around 100 milliamps flow through us can be fatal. The 120 volts that the US uses for mains power can create this much current under the right conditions, so we need safety mechanisms to prevent it.

A "ground fault" refers to the amount of outgoing current not matching the amount of current returning to the power source, meaning that charge is finding another path to flow down. We can measure the current in each line using a loop of wire, which picks up the magnetic field of the moving charges and generates a voltage in the loop. If the two lines don't match, we break the circuit:

In this diagram, the green loops measure the current, and the red is the GFCI's circuitry to compare and break the connections. Initially, I had wondered how a single outlet could cut power to all outlets, but this diagram makes that clear: Typically houses are wired with a pair of lines going from the source to some endpoint, and outlets are wired across the two in parallel. However, if we can break these lines immediately after the source, none of the downstream plugs will be powered.

That brings us to our issue: Once the GFCI is tripped, what do you do next? Typically, GFCIs have two buttons on them, Test and Reset. The test button will artificially pop the breakers so you can ensure they work, but critically, this requires having power. Once we had discovered the power wasn't working, we disconnected from the line in case whatever unknown problem was a fire risk. The real issue though was that to reset the GFCI, you need to push those breakers back in. They're designed to stop power within a tenth of a second, so they use powerful springs to open the connection quickly. That means you need to push a tiny button deep inside the outlet with incredible force, something we were incapable of doing last night! Thankfully in the morning we worked it out, and our power is back on without any expensive fix.

Maybe Avoid the "Nuke" Idiom

This week I saw the news that Caltech had completed a proof-of-concept mission demonstrating power transfer from space using microwaves. I was instantly transported to my childhood playing SimCity 2000, which offered a microwave solar power plant for more advanced cities. I hadn't realized such a plan was actually feasible, but it's been considered since the 90s. The main obstacle has been cost, since the project requires many solar panels, all delivered to space. With the price of solar panels going down though, it's come back into the realm of possibility. However, I was curious about the possible risks of such a system, since SimCity (a credible source if ever there was one) suggested the possibility of accidental fires set by the system.

Before getting into that though, let's discuss how these systems work. To get the maximum amount of power from a solar cell, we want it to be constantly illuminated, but for a panel on Earth this isn't possible, since it spends roughly half its time in night. To get around this, we could put the panel is space, where it can always face the Sun, but now we have a new problem: How do we get the power it produces back to Earth? The simplest solution is to send back electromagnetic waves, but why choose microwaves? To answer that, we need to look at the absorption spectrum of the Earth's atmosphere (click to enlarge):

Wikipedia

We're interested in the regions with low absorption, since we want our beam to go through the atmosphere to a receiver on the surface. Microwaves have a wavelength around 12 cm, which falls neatly in the dip on the right side of the plot.

Since we want to keep the beam fixed on a single receiving station, we want the satellite to be in geostationary orbit, which requires a distance of 36,000 km. This page gives the size of the receiving antenna as about 3 km in diameter, which corresponds to an angle of about 5 millidegrees from the spacecraft. That page also gives the total power transfer as about 1 GW. Given how tiny that angle is, it's easy to imagine the beam being knocked off center, so how much damage could it do?

With the numbers above, the spot would have a power density of 141 W/m^2. For thermal radiation, this is well below the level that can burn you. Of course, these are microwaves, commonly used for cooking, so how does it compare to what you have in your home? A typical microwave oven has an area of around 20" x 24", and puts out around 1000 W, which comes to 3200 W/m^2, almost 23 times what our beam is sending!

So you'd be able to take a nap on the receiving dish without getting cooked, but your WiFi uses the same 2.4 GHz frequency that this system does, so would you be able to read this blog? I found a paper discussing the power density from WiFi base stations as a function of distance:

Figure 9

Note the scale is in milliwatts per square meter, meaning this is several orders of magnitude below the beam's power. Even this weak microwave signal can knock out your wifi!

It seems my childhood fear (or fascination) of fiery beams from the heavens was unfounded, but if they do build one of these, it has the potential to knock out your internet nearby...