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.