Analysis of the Cockcroft-Walton Circuit

Last time I noted that I had been unable to find a complete energy-based analysis of a Cockcroft-Walton voltage multiplier (or any other diode-capacitor charge pump for that matter) either on-line or elsewhere. So, when no-one else will do it, as is almost always the case nowadays, I have to do it myself.

Of the various kinds of computer analysis that I do, I have least experience with electric circuit analysis; but let's attempt the problem, anyway. Although purists might wish for more rigour, I think I've done enough, as set out below, to rule out the Cockcroft-Walton charge pump as a provider of excess energy. (And since it is ruled out, my analysis won't always go into every tedious last detail).

LTSpice

For electric circuit analysis I use LTSpice, the best freeware version I've found of the Spice software originally developed at the Electronics Research Laboratory of the University of California, Berkeley. LTSpice can be downloaded from http://www.linear.com/designtools/software/.

First attempt

For my first attempt, I decided to analyse a circuit that I had already built and tested nearly 20 years ago as a physical device. It was shown on the left in the image at the bottom of my post of 16 May 2015. It is a nine-stage Cockcroft-Walton (half-wave) voltage multiplier. It has a total of eighteen 0.47 μF metalized polyester film capacitors, and eighteen 400V 1A silicon diodes. Its label reads "HIGH VOLTAGE DC VOLTAGE MULTIPLIER. Input 238V AC, 50Hz. Output 5kV DC at R_L = 100MΩ. Polarities: HV Negative. LV Positive."

Fig 3. Circuit diagram, and output voltage vs time

Results

When modelled, this circuit gave the result shown in Figure 3. The steady state condition is reached in about 10 seconds. An output voltage of 4.856 to 4.950kV is obtained (4.903kV average), only slightly less than the experimental result. With an output load of R_L = 100MΩ, the steady state output current was only 49.03μA average, compared with a "spiky", 38% duty-cycle input current delivered from the energizing source, which reached a far higher peak value of 4.8mA. (Current graphs not shown above. And yes, strictly speaking the diode polarities should have been reversed in the model, but that would only change polarities, not magnitudes of the results).

Second attempt

I decided to make another model, of only three stages, but with "perfect" components this time, i.e. much larger 1 farad capacitors with zero equivalent series resistance, and diodes with zero volt drop when conducting. Besides the load resistor R2, now set at 1000Ω, another resistor R1 was added, of only 1μΩ, acting only as a current measurer in that part of the circuit. This model was energised from the same 238V (rms) AC source as before.

Fig 4. Circuit diagram, and output voltage vs time

Fig 5. Steady state input voltage (green) and current (red) vs time

Fig 6. Steady state output voltage and current vs time

Fig 7. Steady state input voltage (green), current (blue), and power (red) vs time

Results

When modelled, this circuit gave the full result shown in Figure 4, where the steady state condition is reached in about 2 seconds, with an output voltage of 2.01183kV.

Figures 5 and 6 show input and output voltages and currents for the last four cycles of steady-state operation. Output resistor R2, now 1000Ω, is "perfect" so there is zero phase shift in the traces, and (with LTSpice's auto-scaling) the I and V traces exactly superimpose. (Note that current is labelled on the right of the graph, and voltage on the left). Steady-state ripple is low so we get a sufficiently accurate result just from multiplying average current by average voltage i.e. 2.01183kV × 2.01183A = 4.0474kW of output power.

On the input side we must multiply instantaneous values of voltage and current to obtain a graph of power vs time. Analysing just the last cycle, over the brief periods of 0.000977s and 0.00102s during which current is flowing from the source, an average power of 102.624kW was obtained, by multiplying the (green) voltage and (blue) current graphs to get the red power graph shown in Figure 7. The duty cycle is (0.000977 + 0.00102)s out of 0.02s, so the average input power is 10.25kW, which is a lot higher than the output power.

No excess energy

The LTSpice simulation (including further investigation of the large current pulses returning as current I₂ in Figure 2 posted last time) shows that there is very little likelihood that the Cockcroft-Walton circuit could be developed into an "energy multiplier".

However, next time I'll look at a device apparently working on electrostatic principles that has been well-proven to deliver excess energy.

The Greinacher-Schaltung

Voltage multiplication

Last time, I mentioned that by bringing (shielded) charge to the inside of a spherical conducting dome, we would obtain "voltage multiplication," and that in the circumstances discussed, this would deliver excess electrical energy.

In existing orthodox electrical technology, there is already a circuit known as a voltage multiplier, which will be the subject of this and the next post.

The Greinacher circuit

In 1913 Swiss physicist Heinrich Greinacher (1880 — 1974) invented an "Ionometer" — basically an ionization chamber and an electrometer. The latter needed 200-300 V dc, but only a 110 V ac supply was available. So he also invented the voltage-multiplying circuit called the "Greinacher-Schaltung" (Greinacher circuit), the basis of all diode-capacitor charge pumps. In 1951 John Cockcroft and Ernest Walton were awarded the Nobel Prize in Physics, having used such a diode-capacitor charge pump in their successful transmutation of atomic nuclei. This device is commonly known today as the Cockcroft-Walton voltage multiplier.

Fig 1. Cockcroft-Walton full-wave voltage multiplier (foreground) housed in the Department of Materials Science and Metallurgy, Cambridge University, built by Emile Haefely & Co Ltd, Basel, Switzerland in the 1960's. ref http://issuu.com/cambridgealumnirelationsoffice/docs/cam_66_lo_res/18

Energy analysis

The input from the energizing source to a Cockcroft-Walton circuit is a charging current at low voltage. Some descriptions imply that its output is essentially the same current at an arbitrarily high voltage, thus giving an arbitrarily high power gain, in theory at least. For example, Wikipedia says "One way to look at the circuit is that it functions as a charge 'pump', pumping electric charge in one direction, up the stack of capacitors." 

Fig 2. Circuit diagram of a three-stage half-wave Cockcroft-Walton voltage multiplier

However, there are obviously two paths by which current can flow back to and through the energizing source, shown as I₁ and I₂ in Figure 2. Now, the question still remains: what are the relative magnitudes of these currents? If current I₂ is not a lot higher than I₁, could the charge pump still possibly provide excess energy? Could it be an "energy multiplier" as well as a voltage multiplier?

I have tried to find a complete energy-based analysis of a Cockcroft-Walton multiplier, on-line or elsewhere. There are many analyses, mostly only dealing with charge building up from zero value, i.e. the initial transient situation, which is only of minor interest. Although some analyses discuss steady-state operation, few will discuss currents, and none that I've found will make any comparison of input and output energies or powers.

I'll look further into the energy analysis of the Cockcroft-Walton circuit next time.

A Modified Van de Graaff Generator Part II

Transport of charge from outside the dome

In view of the difficulties with charge separation discussed last time, it seems that the only possibility for bringing charge to the inside surface of the charged dome without paying a full energy penalty is to generate it externally, and then shield it as far as possible while it is being transported.

There is no problem with getting rid of unwanted charge occurring from charge separation at an external location away from the dome, where the influence of the dome's voltage gradient is already low. There, all unwanted separated charge can easily be neutralised against Earth, for very little energy penalty.

The next requirement is to shield the wanted charge during its transport. Do the charge carriers really have to remain "exposed" against the entire voltage gradient between the dome and Earth?

Cross-sectional view of modified Pelletron design, with chains carrying hemispherical hollow metal shields.
Each shield has insulating links to its neighbours (not shown), similar to the pellet chain design.
The shields enclose, but never contact the charged pellets.

The above schematic drawing shows one way of solving this problem. It is a modified Pelletron design, with synchronized additional chains of hemispherical metal shields (green) that come together to enclose the charged pellets during their movement towards and away from the dome. The shields themselves are always uncharged, and so cannot experience a force from the dome's voltage gradient. (Any induced charge on the shields can be neutralised against Earth).

With charging/discharging by induction as before, and with the transported charges no longer paying an energy penalty from being exposed to the voltage gradient between the dome and Earth, it would seem entirely possible now to gain excess energy from this modification.

To recap: we could produce charge at relatively low voltage, and then ultimately access it at a very much higher voltage. We would obtain a "voltage multiplication", so to speak, while paying hardly any energy penalty for doing so. (Recall that the energy of a charge q at a voltage V is ½qV, so any increase in the voltage of the charge gives a corresponding increase in energy).

This is obvious?

The method discussed above is only a "first attempt" at a solution, with some problems still associated with it.

However, this general idea of transporting shielded charge and hence increasing voltage and electrical energy, without paying a full energy penalty, is so obvious that it surely must have occurred to others? It occurred to me many years ago as a physics student in the late 1960s. But I've not seen it described elsewhere except to some extent in the short article "Commentary and Experiment on Electric Fields and 'Free Energy'" by Philip Stone, Infinite Energy magazine Issue 53, Jan/Feb 2004, p56. That author also remarks on why something so obvious and of such importance should be so overlooked.

Increasing the low power levels

The major remaining practical problem is the very low power levels generally associated with electrostatic machines. There are other ways of transporting shielded charge, perhaps in higher quantities, which I'll discuss later. For now, I'll leave some last words to the foremost electrostatics expert of the last century, Professor Noël Felici of the University of Grenoble:—

"Production of electric flux does not require anything like a magnetic circuit or windings, but only electrified surfaces, with negligible power dissipation. The trouble arises due to the ease of ionization of the filling fluid, causing the field in the gap to collapse, and, therefore, suspending the operation of the generator.

...only a fluid of high permittivity (which must be a liquid) would allow wider gaps with smaller energy content. We know that such liquids, with sufficient resistivity, are not available at the present time. As long as the chemists have not found the clue to reduced ionization despite permittivity, any endeavour towards the immediate generation of direct current by electrostatic principles for power transmission purposes must be postponed." [Ref 1]

"So little effort has been spent in the whole world on these [electrostatic] problems, compared to the gigantic expenses devoted to many other developments, so great is still the path of empirical observation contributed by isolated individuals in the progress achieved so far, that we must consider the development of electrostatic machines as still following the paths of pre-atomic age. The fact that machines of very poor design, as seen from a purely scientific viewpoint, can be serious competitors for equipments embodying elaborate technology, is very comforting to the prospect of the immense wealth of possible progress still ahead of us." [Ref 2]

"The limitations of the electrostatic generator are related to the breakdown strength of the fluid medium surrounding the electricity carriers, not to the static character of their charge. If good insulating fluids were available, which allowed electrostatic forces comparable with the magnetic ones, nothing could prevent electrostatic generators from developing very high power outputs." [Ref 3]

Ref 1: "Developments in Regard to Electrostatic Generators for Direct Current" Professor N. J. Felici, DIRECT CURRENT, June 1953, p122.

Ref 2: "Recent Developments and Future Trends in Electrostatic Generation" Noël J. Felici, DIRECT CURRENT, December 1959, p192.

Ref 3: "Electrostatics and electrostatic engineering" N. J. Felici, 1967 Static Electrification Conference, p127.

Secrecy — in the past, and maybe today?

So, half a century later, are we really still waiting for the chemists to find suitable insulating fluids, which would solve the problem of low power levels, thus permitting high-power generation of free electrical energy generally as discussed above? Once again, many years ago, Prof. Felici made a comment on electrostatic technology that may still be generally relevant today (in Ref 1, my emphasis):— "Two important applications remain for the conducting segment type [of electrostatic generator]: infra-red tube supplies, and engine igniters. Since the first case is restricted by secrecy regulations, nothing may be published about it at the present time..."

A Modified Van de Graaff Generator Part I

Fig 1. The world's largest air-insulated Van de Graaff generator, built by Robert van de Graaff in 1931, now housed at the Boston Museum of Science. Image from http://en.wikipedia.org/wiki/Van_de_Graaff_generator.

Original Van de Graaff generator

The original Van de Graaff generator design is a good example of how electric charge continually brought onto the inside surface of an already charged spherical-shell conductor can be added to the shell for no additional energy penalty, hence increasing its voltage, no matter what voltage the shell has already reached. The charge is initially placed onto a motor-driven belt made of insulating material, at a location outside the shell. The belt transports the charge upwards into the shell, or "dome", where it is collected.

In a small to medium-sized laboratory Van de Graaff generator being charged by external excitation, charge would typically be sprayed onto the low-voltage end of the belt at say 10kV and 20μA. Ignoring losses, it would also be collected at the high-voltage end at 20μA, where the dome voltage can reach say 350kV. So if all energy losses could be eliminated, we would expend 0.00002 A × 10000 V = 0.2 W to get 0.00002 A × 350,000 V = 7 W, i.e. a 35 : 1 power gain.

A 7 watt output is of little practical use, and it's understandable in such a low-power device that not much attention would be paid to the possibility of generating excess energy. But if the current could be increased to say 200mA, still a very modest value in electromagnetic terms, the power output would be 70 kilowatts. That would certainly be a useful result, if it could be obtained without paying a full energy penalty to get it.

Even higher power gains would be possible for larger generators, able to operate with higher dome voltages.

Energy loss

For a Van de Graaff generator, the major energy loss associated with bringing charge onto the dome is obvious. Charge carriers on the belt have the same polarity as the charged dome, and so there is repulsion opposing the belt's movement towards the dome. The charge carriers have to be transported "exposed" against the entire voltage gradient between the dome and Earth.

Pelletron

Fig 2. Basic principle of the Pelletron. See http://www.pelletron.com/charging.htm for this image as an animated GIF.

This major energy loss also occurs in the "Pelletron" modification of the original Van de Graaff generator, which uses a chain of alternating insulating and conducting segments, instead of an insulated belt. (The conducting "pellets" can be conveniently charged/discharged by induction instead of by charge spraying or triboelectrification). In a Pelletron, not only are like charges repelling as the charged chain moves towards the dome, but unlike charges attract as it moves away, with a major energy loss in both cases.

Generation of charge within the dome

At first sight it would seem very easy to generate charge within the dome, rather than to generate it externally and then bring it in against the full voltage gradient. For example, simple metallic "comb" corona discharge generators could be connected directly to the inside of the charged dome (and would thus operate at its already high voltage). Or, a heated filament could be placed within the dome to generate electrons by thermionic emission, a method which can easily produce a current of 200mA or more.

Charge separation

However, we cannot produce any net charge by such methods, either within the dome, or elsewhere. Charge can only be separated, and the problem then arises of how to deal with the charges of unwanted polarity. A corona discharge will act on air molecules to create charges of the desired polarity; but also an equal quantity of charges of opposite polarity. These cannot be allowed to drift internally to the dome, where they would tend to neutralise the desired charge on it. Worse, if they are expelled outside of the dome, they will be attracted back to it, again tending to neutralise it. To get rid of them by neutralisation against Earth, they would have to be moved against the entire voltage gradient between the dome and Earth, thus destroying any possible net energy gain.

If a (negative) charge of electrons is expelled from a filament by thermionic emission, then the filament and whatever is energising it will develop an equal and opposite positive charge. This would soon cause a voltage breakdown (flashover) if the positive charge was not neutralised somehow, e.g. by connection to Earth. But electrons emitted from an earthed filament, no matter where it was positioned, would once again have to act against the entire voltage gradient between the dome and Earth, to reach the inside surface of the dome. This would again destroy any possible net energy gain.

So, charge separation within the dome certainly presents difficulties. Next time I'll look at the more promising method of bringing externally-generated shielded charge into the dome.

Comparing Electrical and Mechanical Quantities

Most students of physics will have seen the diagrams I've redrawn in Figure 1 below:—

Fig 1. Showing i) the exchange between kinetic energy of a mass and energy stored in a spring, in an oscillating mass-spring system; and ii) the exchange between energy stored in the magnetic field of an inductor and energy stored in the electric field of a capacitor, in an oscillating inductor-capacitor system.

Correspondences

By comparing the energy flow in an oscillating mass-spring system with that in an oscillating inductor-capacitor system, we see that there is a correspondence between certain electrical and mechanical quantities. For example:—

   charge corresponds to distance      (q corresponds to x)
   current corresponds to velocity      (i corresponds to v)
   capacitance corresponds to the      (C corresponds to 1/k)
      inverse of spring constant
   inductance corresponds to mass    (L corresponds to m)

Let's now look more closely at the correspondence between a stretched spring and a charged capacitor.

Fig 2. Energy added to a stretched spring, and to a charged capacitor

Energy triangles

In Figure 2A, a spring of spring constant k has been initially stretched over a distance x, by exerting an ever-increasing force F on it. The energy stored in the spring is given by the triangle (light blue) of base x and height kx  i.e. 

    U_S  =  ½ × x × kx  =  ½kx² 

Now, suppose we wish to stretch the spring a bit more, i.e. by Δx, and thus gain the extra energy represented by the darker blue area. The only possible way to do that is by increasing the exerted force F by ΔF, from an already high value to a new and higher one. Thus we have to pay a full energy penalty in terms of the force being exerted on the spring through the distance Δx, to gain our extra spring stored energy.

At first sight, the corresponding case with a capacitor seems to be exactly analogous. In Figure 2B, a capacitor of capacitance C has been initially charged with charge q, by subjecting it to an ever-increasing voltage V. The energy stored in the capacitor is given by the triangle (light red) of base q and height (1/C).q  i.e.

    U_E  =  ½ × q × (1/C).q  =  ½q²/C

Now, suppose we wish to add a bit more charge to the capacitor, i.e. Δq, and thus gain the extra energy represented by the darker red area. By analogy with the spring, we could do that by increasing the voltage V by ΔV, from an already high value to a new and higher one. Then we would once again pay a full energy penalty in terms of the voltage being applied across the capacitor to add the charge Δq, to gain our extra electrical energy.

Another way — but only for a capacitor

However, we could do something else, which as far as I know has no mechanical analogue. We could make our capacitor as a spherical conducting shell (or a major portion of one) which, by Gauss's Law, cannot retain any charge on its inner surface. Any net charge that is generated within the shell, or brought into it from outside, can then all be added to the shell at its inner surface for no added energy penalty, no matter what voltage the shell has already reached. Thus we might be able to obtain any desired increase in charge Δq without having to directly increase the applied voltage at all. That voltage increase, and the associated energy gain, would then occur incidentally, and automatically, as the charge was added.

Next time, I'll look further into the questions that I've now raised concerning a charged conducting shell:—

a) generating net charge within the shell

b) bringing an externally generated charge into the shell without paying the full expected energy penalty for doing so.