# How Tesla invented, I think, Tesla coils and wireless chargers.

I think I know how Tesla invented his high frequency devices, and thought I’d show you, while also explaining the operation of some devices that develop from in. Even if I’m wrong in historical terms, at least you should come to understand some of his devices, and something of the invention process. Either can be the start of a great science fair project.

The start of Tesla’s invention process, I think, was a visual similarity– I’m guessing he noticed that the physics symbol for a spring was the same as for an electrical, induction coil, as shown at left. A normal person would notice the similarity, and perhaps think about it for a few seconds, get no where, and think of something else. If he or she had a math background — necessary to do most any science — they might look at the relevant equations and notice that they’re different. The equation describing the force of a spring is F = -k x  (I’ll define these letters in the bottom paragraph). The equation describing the voltage in an induction coil is not very similar-looking at first glance, V = L di/dt.  But there is a key similarity that could appeal to some math aficionados: both equations are linear. A linear equation is one where, if you double one side you double the other. Thus, if you double F, you double x, and if you double V, you double dI/dt, and that’s a significant behavior; the equation z= atis not linear, see the difference?

Another linear equation is the key equation for the motion for a mass, Newton’s second law, F = ma = m d2x/dt2. This equation is quite complicated looking, since the latter term is a second-derivative, but it is linear, and a mass is the likely thing for a spring to act upon. Yet another linear equation can be used to relate current to the voltage across a capacitor: V= -1/C ∫idt. At first glance, this equation looks quite different from the others since it involves an integral. But Nicola Tesla did more than a first glance. Perhaps he knew that linear systems tend to show resonance — vibrations at a fixed frequency. Or perhaps that insight came later.

And Tesla saw something else, I imagine, something even less obvious, except in hindsight. If you take the derivative of the two electrical equations, you get dV/dt = L d2i/dt2, and dV/dt = -1/C i . These equations are the same as for the spring and mass, just replace F and x by dV/dt and i. That the derivative of the integral is the thing itself is something I demonstrate here. At this point it becomes clear that a capacitor-coil system will show the same sort of natural resonance effects as shown by a spring and mass system, or by a child’s swing, or by a bouncy bridge. Tesla would have known, like anyone who’s taken college-level physics, that a small input at the right, resonant frequency will excite such systems to great swings. For a mass and spring,

Basic Tesla coil. A switch set off by magnetization of the iron core insures resonant frequency operation.

resonant frequency = (1/2π) √k/m,

Children can make a swing go quite high, just by pumping at the right frequency. Similarly, it should be possible to excite a coil-capacitor system to higher and higher voltages if you can find a way to excite long enough at the right frequency. Tesla would have looked for a way to do this with a coil capacitor system, and after a while of trying and thinking, he seems to have found the circuit shown at right, with a spark gap to impress visitors and keep the voltages from getting to far out of hand. The resonant frequency for this system is 1/(2π√LC), an equation form that is similar to the above. The voltage swings should grow until limited by resistance in the wires, or by the radiation of power into space. The fact that significant power is radiated into space will be used as the basis for wireless phone chargers, but more on that later. For now, you might wish to note that power radiation is proportional to dV/dt.

A more -modern version of the above excited by AC current. In this version, you achieve resonance by adjusting the coil, capacitor and resistance to match the forcing frequency.

The device above provides an early, simple way to excite a coil -capacitor system. It’s designed for use with a battery or other DC power source. There’s an electromagnetic switch to provide resonance with any capacitor and coil pair. An alternative, more modern device is shown at left. It  achieves resonance too without the switch through the use of input AC power, but you have to match the AC frequency to the resonant frequency of the coil and capacitor. If wall current is used, 60 cps, the coil and capacitor must be chosen so that  1/(2π√LC) = 60 cps. Both versions are called Tesla coils and either can be set up to produce very large sparks (sparks make for a great science fair project — you need to put a spark gap across the capacitor, or better yet use the coil as the low-voltage part of a transformer.

Another use of this circuit is as a transmitter of power into space. The coil becomes the transmission antenna, and you have to set up a similar device as a receiver, see picture at right. The black thing at left of the picture is the capacitor. One has to make sure that the coil-capacitor pair is tuned to the same frequency as the transmitter. One also needs to add a rectifier, the rectifier chosen here is designated 1N4007. This, fairly standard-size rectifier allows you to sip DC power to the battery, without fear that the battery will discharge on every cycle. That’s all the science you need to charge an iPhone without having to plug it in. Designing one of these is a good science fair project, especially if you can improve on the charging distance. Why should you have to put your iPhone right on top of the transmitter battery. Why not allow continuous charging anywhere in your home. Tesla was working on long-distance power transmission till the end of his life. What modifications would that require?

Symbols used above: a = acceleration = d2x/dt2, C= capacitance of the capacitor, dV/dt = the rate of change of voltage with time, F = force, i = current, k = stiffness of the spring, L= inductance of the coil, m = mass of the weight, t= time, V= voltage, x = distance of the mass from its rest point.

Robert Buxbaum, October 2, 2017.