The Data are In!

Note: In this post I'll be referencing a couple papers, citations for which can be found at the post's end.

It's been a long time and a couple of science fairs since my last post, so I thought I'd take the time to update you on what I've been up to these past few months. With the end of the school year (and my high school career!) consuming a lot of my time, I haven't gotten to work on the project as much as I would have liked; in fact, I'm still not done experimenting or collecting data. I'll summarize the data I've collected so far and the methods I've used to collect it below.

A sample water arc explosion. Note the large fog jet and the bright arc inside the accelerator barrel.

Since I got my water explosion machine working reliably, I've been primarily concerned with quantifying the water explosions' energy inputs and outputs. The former has been relatively straightforward, as it only requires a couple of oscilloscope measurements (below). In each explosion I looked at, I measured the capacitor voltage before and after discharge, which I used to calculate the electrical energy delivered to the explosion chamber, as well as the peak discharge current, which is predicted to correlate with explosion strength (Graneau et al., 2000).

Sample capacitor voltage waveform. Each horizontal division represents 5 μs and each vertical division represents 5 kV. The voltage before discharge is about 15 kV.

Sample discharge current waveform. Each horizontal division represents 5 μs and each vertical division represents approximately 2.5 kA. The peak current is 4.5 kA.

I've made some minor tweaks to my charging circuit. I've also made this handy illustration to show them off (click for higher resolution):

The the EMF-sensitive shut-off circuit, which cuts power to the H-bridge inverter when it detects that the capacitors have discharged, now triggers more reliably and the switching transistors (in the H-bridge) are less prone to overheating. 

The latest incarnation of my water explosion chamber is pictured below:

The chamber comprises a cylindrical cup with an open top and two electrodes. One electrode projects through the bottom of the cup and the other forms a ring around the inside of the chamber. It's made from a giant block of wood because all the plastics I had available weren't strong enough to make a durable explosion chamber out of.

To determine a water explosion’s kinetic energy, I employed the same projectile-based method used by physicist Peter Graneau in his research on water explosions (Graneau et al., 2000). The idea is to place a porous projectile (mine was made of balsa wood) over the muzzle of the accelerator barrel so that a water explosion within the barrel launches the projectile into the air and some of the fog penetrates into the projectile during the process. The height to which the projectile ascends can be measured and used to estimate the initial kinetic energy (in the vertical direction) of the explosion.

Now for the maths: Let the balsa wood projectile have mass M and let the explosion launch it to a height h. Let m0 be the total mass of the expelled water, m1 be the mass of the water that bounces off the projectile, and m2 be the mass of the water absorbed by the projectile. The collision of m1 with M is elastic, and the collision of m2 with M is inelastic. Say m1 has initial average velocity μ01 and m2 has initial average velocity μ02. After the collision of m1 with M, say m1 has average velocity μf1 and M has velocity v1. After the collision of m2 with M, say the combined mass M+m2 has velocity v2.

M, m0, and m2 can be determined by before and after measurements of the projectile mass and water mass. h can be measured by taking a video recording of the explosion and analyzing the video frame-by-frame. The other quantities of interest, namely m1, μ01, and μ02, must be found indirectly.

Energy and momentum conservation in the collision of m1 with M require that:

Eqn. 1

and momentum conservation in the collision of m2 with M requires that: 

Eqn. 2

Examining the projectile, it follows from the conservation of mechanical energy that: 

Eqn. 3

where g is the acceleration due to gravity. Solving (3) for v2 yields:

Eqn. 4

By making some assumptions about the fog jet’s mass and velocity distributions, we can use (1), (2), and (4) to solve for m1, μ01, and μ02 and then calculate the fog explosion’s kinetic energy.

We can obtain a variety of approximations for the explosion’s kinetic energy using different models for the explosion’s velocity distribution. Each of the below velocity-mass functions could plausibly describe a water arc explosion given the behavior of the fog jets in the high-speed images presented in Hathaway et al. (1998).

Sample graphs of the velocity-mass function v(m) for a water explosion. The first has decay rate k = 5, while the second has k = 50

Eqn. 5

Using the methods described above, I was able to compute the input and output energies of several water explosions. A sampling of the data I collected is presented in Table 1 below (click to enlarge).

Table 1: The input and output energy data of 8 shots using distilled water at room temperature. In all shots, the charging capacitance C was 0.6 μF. The capacitor input energy was computed as E0 =  C/2*(V0^2 - Vr^2).

Key: V0: initial capacitor charging voltage; Vr: residual capacitor voltage after discharge; E0: electrical input energy; Ef: kinetic output energy; Ef/E0: the explosion’s efficiency, the ratio of output to input energy. M, m0, m2, and h are as defined in the Methods section. All kinetic energies are computed using a v(m) model with k = 300.

The results in Table 1 indicate that water arc explosions liberate stored potential energy (though these results are only as good as the assumptions made in obtaining them).  Looking at the last 3 columns of Table 1, it can be seen that the water explosion’s kinetic energy was even greater than the pulse discharge’s electrical energy in three of the eight shots displayed. These results are consistent with the findings reported in Graneau et al. (2000) and require that some sort of internal water energy is released during a water arc explosion.

If electrodynamic forces alone were responsible for the explosions, their kinetic energies should be only a fraction of the total input energy. For a typical shot with E0 = 60 J, a peak discharge current of 10 kA, and a pulse width of 10 μs,  the upward Lorentz pinch thrust should be about 0.1 N on average, corresponding to an explosion kinetic energy of 2.5 nJ = 2.5 x 10^(-9) J. This is about ten orders of magnitude off the observed values for Ef in Table 1.

Note that the accuracy of a kinetic energy measurement depends largely on the explosion’s actual velocity-mass distribution. If this actual distribution differs significantly from the model function in eqn. (5), the calculated value for Ef may significantly over- or underestimate the explosion’s actual kinetic energy. Furthermore, the parameter k in the velocity model (Eqn. 5) will have a significant impact on the calculated kinetic energies. In table 1 k is taken to be 300; a lower value of k will yield lower Ef. A more sophisticated measurement system (e.g. a phase Doppler particle analyzer) for determining a fog explosion’s velocity and mass distributions would be needed to calculate Ef with greater accuracy and eliminate the uncertainty surrounding k.

So, if water explosions do in fact release energy, where does this energy come from? I can't yet give you a definitive answer, but I have a hunch that the energy of a water explosion comes from the reorganization of the water’s hydrogen bonding network during the explosion process. Inelastic neutron scattering experiments have revealed the existence of two different kinds of hydrogen bonds in water, each bond with a different strength (Li and Ross, 1993). The “strong” and “weak” bonds have vibrational force constants of 32 meV and 24 meV, respectively, and exist in water in a strong-to-weak ratio of about 2:1 (Graneau, 1998). During an explosion event, forces associated with the high-current arc discharge tear the water into fog droplets. This consumes a certain amount of mechanical energy, which is stored as additional surface tension energy in the fog. However, the molecules in the fog droplets now have significantly fewer neighbors than they did in the bulk water (see my cheesy little diagram below) and can more easily reorient themselves into lower energy states. Thus, weak hydrogen bonds can reform into strong hydrogen bonds, and van der Waals attractions can collapse into weak hydrogen bonds. This bond reformation would, of course, release energy, and could conceivably supply the kinetic energy of a water explosion.



For a water explosion to release potential bond energy, as described above, the post-explosion fog must be in a lower-energy state (i.e. contain stronger intermolecular bonds) than the pre-explosion water. When the fog reforms into bulk water, it must therefore absorb energy from its surroundings to restore itself to its original higher-energy state. In other words, the fog must absorb heat as it agglomerates back into bulk water (this is rather unusual, as condensation is generally an exothermic process).

In light of this, I set out to determine the amount of heat (if any) absorbed by a quantity of post-explosion fog as it agglomerates back into bulk water. To do this, I strapped a balsa wood absorber over the muzzle of the accelerator barrel. The absorber would capture most of the water expelled by an explosion in the barrel. Before an explosion event, I would measure the temperatures of the water and balsa wood absorber , and after an explosion, I would measure the final temperature of the water—balsa-wood system by inserting a temperature probe into the absorber. Knowing these temperatures and the heat capacities of the water, balsa wood, and temperature probe, I could calculate and compare the initial and final thermal energies of the water—balsa-wood—temperature-probe system , revealing whether any heat was absorbed by the water after the explosion. In these calculations, I tried to account for the resistive heating that takes place in the water arc during an explosion; I calculated this heat using the current discharge's pulse width and underdamped ringing frequency.

 

Table 2 below displays the thermal energy measurements for several water explosions (click to enlarge).

Table 2: The initial and final thermal energy data for 5 shots using distilled water at room temperature.  The charging capacitance was 0.6 μF for all shots.

Key: m_w: mass of the water involved in the heat exchange; T_w: initial water temperature; m_b: mass of the balsa wood involved in the heat exchange; T_b: initial balsa wood temperature; T_f: final temperature of the water—balsa-wood—temperature-probe system (“the system”); Q_R: ohmic heat accrued by the water during the capacitor discharge; Q_i: initial thermal energy of the system; Q_f: final thermal energy of the system; ΔQ = Q_i – Q_f.

Looking at the last three columns, one can see that the post-explosion fog does indeed absorb energy. For each shot measured, the thermal energy of the water—balsa-wood—temperature-probe system decreased significantly over the course of the explosion event. The difference between the initial and final thermal energies was about 75 J on average, which is sufficiently large to account for the explosion kinetic energies reported in Table 1. Of course, since not all the exploded fog was captured in the balsa wood absorber in any of the shots summarized in Table 2, each reported value for ΔQ underestimates the total heat absorbed by the fog upon reforming into water, perhaps quite significantly. So, while Table 2 tells us that the fog does absorb heat as it reforms into bulk water, it can't really tell us how much heat is absorbed in total.

It is possible that evaporative cooling is partially responsible for the low values of Tf seen in Table 2. However, I imagine the evaporation of water from the balsa wood absorber was negligible while data was being collected since the absorber’s mass did not decrease significantly while its temperature was being measured.

Now, as to this project's future: I'm contemplating another (perhaps more accurate) method of quantifying the explosions' energies other than the projectile-based method I described earlier. In the late '40s, Sir Geoffrey Ingram Taylor derived an elegant expression for the energy released in an intense explosion using properties of the blast wave the explosion creates (Taylor, 1950). Taylor used his equations to estimate the energy released in the first atomic explosion in New Mexico based on frame-by-frame images of the "ball of fire" it created. I'm hoping to use Taylor's equations similarly to estimate the energy released in a water arc explosion based on the time it takes the sound of the explosion to reach various microphones placed around the chamber. This method, unlike the projectile-based approach, would account for all the energy (light, sound, heat, etc.) released by the water explosion. I'll elaborate more on the details in my next post after I've tried this new method out.


Works Cited:

Graneau, N. (1998). The anomalous strength of cold fog explosions caused by high current water arcs. IEEE, 40/1-40/3.

Graneau, P., Graneau, N., Hathaway, G., & Hull, R. (2000). Arc-liberated chemical energy exceeds electrical input energy. Journal of Plasma Physics, 63, 115-128.

Hathaway, G., Graneau, P., & Graneau, N. (1998). Solar-energy liberation from water by electric arcs. Journal of Plasma Physics, 60(4), 775-786.

Li, J. & Ross, D. K. (1993). Evidence for two kinds of hydrogen bond in ice. Nature. doi:10.1038/365327a0

Tag, P. M. (1980, June 23). On the diagnosis and conservation of energy during condensation. Journal of the Atmospheric Sciences, 37, 2347-2351. Retrieved February 11, 2013

Taylor, G. (1950). The formation of a blast wave by a very intense explosion: I. Theoretical discussion. Proceedings of the Royal Society of London, 159-174.

First Explosion Achieved!

I have, quite literally, made a major breakthrough in my research, the results of which are depicted below.

What you're seeing here are the remains of my homemade water accelerator barrel. The destruction of this small PVC cylinder represents the culmination of more than a year's worth of work. The bottom and side of the barrel were blown out when I fired a high-current pulse through the small amount of water it contained. If this isn't a veritable arc-induced water explosion, I don't know what is.

I was able to achieve the explosion by charging a 0.3 uF capacitor bank to near 15 kV and shorting the bank through about 3 mL of tap water. The explosion was accompanied by a loud pop and flash of light as the water ionized and several kiloamps of current rushed through it. In addition to fracturing the accelerator barrel, the explosion showered my workbench with little droplets of liquid water. Much of the water initially in the barrel ended up on the ceiling directly above it.

I had not expected the explosion to be nearly so powerful. In light of this, Peter Graneau's claim that water arc explosions are overunity reactions (i.e. that some sort of internal water energy is released during the explosion process) seems totally plausible, at least qualitatively.

Since I did not have my measurement equipment set up during the explosion event, I cannot be precisely quantitative about the capacitor input energy and the explosion output energy, but I can make some estimates. Naturally, one would think that the explosion's kinetic energy cannot exceed the capacitors' electrical energy. Let us see if this seems reasonable.

Assuming the capacitor bank was charged to 15 kV at the time of discharge and that the residual voltage on the capacitors was zero after discharge, the capacitor bank would deliver 33.75 J to the water accelerator. Since some energy was lost as light and sound in the arc and as heat due to the capacitors' effective resistance, and since energy was also expended to ionized the water, the capacitor bank would deliver slightly less than 33.75 J of energy to be put towards the explosion's kinetic energy.

Now consider that the explosion fractured my PVC accelerator barrel. Determining how much energy this requires is difficult; the tensile impact energy required to break PVC is generally between 500-750 ft-lb/in^2, or 108-163 J/cm^2 (as measured by an Izod-type strength test), but this varies hugely depending on the PVC's actual composition, its shape, and the direction of the applied force. For the sake of argument, let us assume that it requires 100 J/cm^2 to fracture my accelerator barrel (this is on the lowest end of the normal range I listed above). Since the barrel has a cross sectional area of about 1.75 cm, it would require 175 J to break it.

This is of course a very rough estimate and can only give a general idea as to how much energy was actually required to break the accelerator barrel. That said, our ballpark estimate of 175 J is more than five times as much energy as was initially stored in the capacitor bank! And this only accounts for the energy required to fracture the barrel; additional energy was required to break up the water into small droplets and spew those droplets all over my work space. Clearly, the kinetic energy of the water explosion cannot be supplied by the capacitor bank alone, so there must be some other source of energy contributing to the explosion.

Peter Graneau hypothesized that the extra energy was related to hydrogen bonding in water. In his paper "Arc-Liberated Chemical Energy Exceeds Electrical Input Energy" (Journal of Plasma Physics, 2000) he sates:

"...the most likely source of the explosion energy is that stored by hydrogen bonds between the water molecules. This bond energy is said to be equal to the latent heat of evaporation, and therefore could contribute up to 2200 J/g"

Graneau does not offer experimental evidence to support his theory, but I hope to procure some evidence either for or against the theory myself  in the near future (more on this in future posts).

Another theory (also propounded by Graneau) is that the explosion energy is related to longitudinal current forces (i.e. forces in the direction of current flow) predicted by Ampere's original force law but neglected by Maxwell-Lorentz-Einstein electrodynamics. The way in which my water accelerator barrel was destroyed may support this theory. As is particularly apparent in the second picture at the top of this post, the force of the water explosion seemed to be directed away from one electrode, viz. the negative one, and towards the other, viz. the positive one. Since the direction of actual (cf. conventional) current flow is from negative to positive, the explosion force seems to be in the direction of Ampere's longitudinal current force. I may experiment with different electrode configurations to explore this theory further.

I realize I've interspersed this post with fewer pictures than usual. To compensate, I'll give you a visual tour of my current charging/discharging apparatus.

High-level view of the entire system

Overhead view of my custom DC supply

My 0.3 uF capacitor bank. I can add an additional 0.3 uF if necessary.

Spark gap "switch" in a homemade Faraday cage. The capacitor bank
discharges when the spark gap breaks down.

My custom Rogowski coil, used to measure the discharge current.
Produces a voltage proportional to the time-derivative of the current through the primary "winding." 

Sample Rogowski coil output (dI/dt) during a discharge event.

Capacitor bank voltage during a discharge event. One vertical  division represents 20 kV.
The waveform is zeroed; I'm not sure what to make of the highly negative voltage the capacitors settle at.

Current and voltage waveforms overlaid in time. Each horizontal division
represents 5 uS.

My automatic shut off circuit. It cuts power to the HV DC supply when it detects an EMF
like the one produced by the spark gap firing mechanism. It keeps the capacitors from
charging to dangerous levels after a discharge event.

Stay tuned for more updates. They should be coming soon.

It's Alive! HV DC Supply is Fully Functional

After roughly half a year of design and troubleshooting, I finished fully testing my custom variable high voltage DC power supply yesterday. The supply is capable of producing up to 20 kV at 10 mA. I will be using it to charge my high-voltage capacitor bank and hopefully explode some water.

The finished and tested circuit

The circuit is essentially comprised of three subsystems: a DC mains supply, an H-bridge inverter, and a high-voltage multiplying rectifier. The first is visible in the top portion of the above photo. It's just your basic diode bridge rectifier, with a 60 Vp sinusoidal input and nearly 1000 uF of smoothing capacitance.

The H-bridge inverter can be seen in the middle of the photo. Its driving frequency can be varied between discreet values ranging from 40 to 47 kHz using the rotary switch visible towards the bottom of the shot. This small range of driving frequencies corresponds to a large range of output voltages (from about 8 to 20 kV), as my flyback transformer (not visible in the above photo) resonates at near 60 kHz. This phenomenon is illustrated below:

Above resonance

At resonance

Below resonance

The circuit's business end, the high voltage output stage, is depicted below:

Parts are (from left): flyback transformer, Crockoft Walton multiplier, protective diodes, output resistors

In order to prevent arcing between and within the high-voltage components, I had to submerge them all in mineral oil. While messy, this method was extremely effective at insulating the parts.

The small, innocuous-looking flyback transformer I used in my circuit is somewhat unconventional. Most HV hobbyists think of the large suction-cup sporting flyback transformers found in CRT's when they hear the term (see picture below). These transformers are frequently used to make spark generators and are easy to salvage from old TV's and computer monitors. They are generally much beefier than the downsized flyback I used in my supply, and one would expect to get more voltage from them.  I experimented with using one of these in my supply, but ultimately decided against it. All such flyback transformers I found were internally rectified (AC flybacks were outmoded in TV's thirty years ago) and therefore could not be used with my CW multiplying rectifier. So despite the traditional flyback's heft, I was able to get a greater voltage output from my miniaturized version by tacking a voltage multiplier onto its AC output.  

A classic flyback transformer, salvaged from an old computer monitor

Troubleshooting my circuit was hardly mundane; between prototypes, my circuit experienced some catastrophic failures. The aftermath of one is pictured below:

Casualties: four power transistors, one LCM555 timer, one breadboard

I was testing the circuit at the limits of its operating range, and I neglected to account for the heat my driving transistors would accrue. After operating for about five seconds, the transistors promptly burst into flames. Let it be known: the large heatsinks visible in the first picture are there for a reason!

Assuming my circuit doesn't spontaneously combust again any time in the near future, I will be using it to (carefully!) charge up some gargantuan high-voltage oil filled capacitors (see below). Once I'm confident  I can charge and discharge these monsters safely, I'll hook up my makeshift water accelerator barrel, and we'll see if I can replicate Peter Graneau's overunity water arc explosions.

Back to the Drawing Board: Making my own High Voltage Supply

Don't think I've been spending all that time since my last post idly!

That couldn't be farther from the truth. These past 6 months I've been laboring over a little circuit of mine, a custom high voltage power supply that should be able to source all the power I need to blow up some water. The power supply is built entirely from scratch and took a lot of troubleshooting to get working. And, it's not quite done yet, but its completion is in sight.

I will elaborate on the circuit later when the designs have finalized and thoroughly tested.

The system's components are outlined below (click for larger image):

Finally, some quick snapshots of the physical circuit:

Custom DC supply (from 120V mains):

110-90V tapped mains transformer

High-current 120 Vp rectifier, with about 500 uF of smoothing capacitance 

Pulse generator and H-bridge inverter:

The 555 timer is visible towards the far left of the breadboard.
The rear MOSFETS drive the H-bridge transistors in the foreground

Below is a little demo of the H-bridge's operating principle:

Flyback transformers:

A small, non-standard HV flyback, rated at 9 kV. Feeds into the CW multiplying rectifier pictured later.

A more traditional HV flyback, salvaged from an old CRT. Output is rectified internally.

CW voltage multiplying rectifiers:

Uses low ESL "doorknob" capacitors. DC output is nominally four times the AC input.

First Setup: No Success Yet

I have finally constructed and tested my planned version of the charging and discharging circuits. Unfortunately, I haven't gotten the circuit working properly; the circuit is functional, but can't induce a water explosion.

Below is a picture of my implementation of the charging circuit.

  
The leftmost breadboard holds the switching and control circuit. The silver box to its right is a  high-voltage neon sign inverter; it takes in 12 VDC and puts out 2 kV @ 60 kHz. The inverter powers a rectifying voltage multiplier (on the second breadboard). The circuit essentially consists of several Greinacher voltage doublers (described in the previous post) stacked on top of each other. The circuit is commonly known as a Cockroft-Walton (CW) voltage multiplier. It is named after physicists John Cockroft and Ernest Walton, who discovered the circuit in 1932 and used it to power their atom smasher. At no load, the output voltage is theoretically twice the peak input voltage times the number of Greinacher stages in the circuit. A simple two stage CW multiplier is shown below. If, for example, the circuit was powered by a 3 kVp AC source, the open-circuit output would theoretically be 12 kV DC.

The actual output is significantly less than the theoretical output, and decreases with increasing load current. The 5-stage CW multiplier used in my charging circuit should theoretically produce 30 kV at no load; the actual output is a little more than 10 kV.

The CW multiplier is connected through two 25 MOhm resistors to two special reed relays at the far right of the picture. The relays are each rated at 5 kV and are used to connect the charging circuit to the capacitor bank. 

The whole apparatus is mounted on top of a sealable plastic container lined with tin foil. Inside the container are the capacitor bank and discharge switching mechanism.

The plastic bin serves to insulate the capacitor bank from the circuit's operator. The tin foil lining is meant to act as a Farady cage. The capacitor bank's discharge is often accompanied by some arcing near the switching mechanism's electrodes; the tin foil Farady cage is there to neutralize the electromagnetic field generated by the arcs. This prevents the radio noise from interfering with oscilloscope measurements.

The obscure "switching mechanism" I referenced in the previous paragraph has consisted of two different devices over the course of this project. The first is a system of relays similar to the big red ones used in the charging circuit. These relays are switched manually when the voltage across the capacitor bank reaches the desired level. The second switching device is a much cruder spark-gap switch that breaks down at a certain voltage, determined (partly) by the spark gap's length (as well as by electrode shape, air moisture, and a slew of other factors). The former switching mechanism was eventually replaced by the latter because the relays didn't handle the capacitors' high-current discharges very well (to say the least: they may or may not still work).

I've also experimented with a couple different types of capacitors. The picture below shows a high-voltage "slapper" capacitor, rated at 15 kV with a maximum underdampened discharge current of >2 kA (yikes!). The capacitor is discharged using the spark gap switch described above.

When the discharge switch is closed, the capacitor bank is discharged through the water accelerator barrel. The accelerator is shown below, in actual and diagrammatic representation:

The accelerator's actual dimension turned out to be a lot smaller than those given in the diagram. Instead of 12 cm x 9 cm, the accelerator is approximately 5 cm x 3 cm. It's built from a couple of sections of 1/2" PVC pipe, a length of 6 AWG wire, and a 1 mm-thick aluminum sheet. It can hold up to 2 mL of water. 

During a discharge event, the voltage across the capacitor bank is measured using a specialty high-voltage voltmeter probe. The thing (pictured below) looks like it belongs on a Star Trek set, but it gets the job done. It has a 1000:1 attenuation ratio and an input impedance of 1 GOhm.

The discharge current is measured using a special current transformer, called a Rogowski coil, and an oscilloscope. The Rogowski coil produces an output voltage proportional to the rate of change of current through the wire it's wrapped around. To glean information about the current through the wire,  you can slap an integrator circuit on the end of your coil, or employ some other method of numerical integration.

During a discharge event, the coil voltage usually looks something like this:

This waveform is from the discharge of my 6 kV 0.3 uF capacitor bank. The spike seen towards the end of the waveform is an illustration of the effects arc-generated radio noise have on oscilloscope measurements. From the waveform, the derivative of the discharge current dI/dt can be modeled as follows:

where Io is a placeholder value for the initial current (in this case it is actually a quantity in volts taken from the oscilloscope readout), tau is the discharge time constant (approximately one fifth the pulse width, in seconds), omega = 2*pi*f is the angular ringing frequency (with f, the ringing frequency, in hertz), and t is time, in seconds. Integrating yields:

It's assumed here that the constant of integration is zero. If you want to double check this answer, I'll let you do the integration by parts yourself. Note that the proportionality constant of the Rogowski coil is negative, so it will flip the signs of the current waveform around. The above equation describes a waveform shaped similarly to the one seen in the oscilloscope screenshot. This derived equation for I(t) will be used to describe the current waveforms of discharges during this experiment.

In the case of the waveform shown above, Io = 24.6 V, tau = 56 uS/5 = 11.2 uS, f = 192 kHz, and omega = 2*pi*192 kHz = 1206 kHz. I(t) is minimized on the interval [0, inf) when t = 0, so the maximum discharge current will be the current at I(0). From our model for the Rogowski coil output, I(0) is:

Note that this quantity is in volt-seconds. Since the proportionality constant of our Rogowski coil is -1.1758 x 10^(-8) ohm-seconds, the actual discharge current is:

that is, 1.7 kA. Apparently, this isn't quite enough current to get any water to explode. To rectify this, I'm planning on revamping my charging circuit to increase the capacitor charging voltage (more power!). This should help increase the peak discharge current.

The results of my second attempt at the water explosion apparatus will be posted here shortly.    

Discharge Circuit Design

My latest work on the project has centered around the assembly and testing of my charging and discharging circuits. Below is a diagram of the circuit I ultimately implemented for charging and discharging my capacitor bank (click for larger resolution).

When the DPST mechanical switch S1 is thrown to the left, the leftmost DPDT relay S2 is actuated. This then turns the step-up converter on and actuates the SPST charging relay S3. The capacitor C towards the bottom right of the diagram is then charged to the converter's output voltage (approx 6 kV) through the 1.2M resistor. Once the capacitor is fully charged (if C = 0.3 uF, RC = .36 s and the capacitor C will be charged to within 1% of its charging voltage in 5RC = 1.8 s), S1 is thrown to the right, actuating the SPST discharging relay S4 and shorting the capacitor's positive plate to ground. The resultant pulse discharge can be used to initiate a water arc explosion.


The diodes in parallel with each relay are there to protect the switch S1 (and the contacts of relay S2) from the momentary voltage spike produced when the relays are switched off. Relays S3 and S4 must be specially made for switching HV signals. I used two 5kV reed relays connected in series for both S3 and S4. The reed relays are designed so that their contacts are contained in an evacuated chamber, thus allowing them to switch high voltage without sparks arcing between the contacts.


The box labeled "DC-AC-DC Step-Up Converter" will likely comprise the below circuit.

The unit after the step-up inverter is called a Greinacher voltage doubler; it takes an AC signal and outputs DC (with ripple) at twice the peak input voltage. The first capacitor is charged on the negative half of the AC  cycle to the AC peak voltage. On the positive half, the output is a superposition of the input AC waveform onto the capacitor's discharge current. The second capacitor reduces voltage ripple.

The circuit's open-circuit output is nominally 2Vp. Under load, the circuit's output voltage drops with load impedance. The circuit's exact voltage characteristics are given by

Where I(load) is the current through the load (A), f is the frequency of the AC input (Hz), and C is the capacitance of the doubling capacitor (F). Assuming a maximum load current of 6kV/1.2MOhm = 5 mA, an input frequency of 60 kHz, and an input capacitance of 270 pF, the circuit's minimum output voltage comes out as 5691.4 V. As the capacitor C charges, I(load) will decay to zero, and the output voltage will level off at 6 kV.