Forgive them, Thevenin, for they know not how to reckon AC transient power.
"The worst error you can make is an unexamined assumption." ~Jed Rothwell, Lessons from Cold Fusion
About a year after CBS 60 Minutes aired their episode on Cold Fusion back in 2009, I followed up with Rob Duncan to explore Richard Garwin's thesis that McKubre was measuring the input electric power incorrectly.
It turns out that McKubre was reckoning only the DC power going into his cells, and assuming (for arcane technical reasons) there could not be any AC power going in, and therefore he didn't need to measure or include any AC power term in his energy budget model.
McKubre justified his fateful assumption thusly:
Under current control, the cell voltage frequently was observed to fluctuate significantly, particularly at high current densities where the presence of large deuterium (or hydrogen) and oxygen bubbles disrupted the electrolyte continuity. By providing the cell current from a source that is sensibly immune to noise and level fluctuations, the current operates on the cell voltage (or resistance) as a scalar. Hence, as long as the voltage noise or resistance fluctuations are random, no unmeasured RMS heating can result under constant current control, provided that the average voltage is measured accurately.
Together with several other people, I helped work out a model for the omitted transient AC power term in McKubre's experimental design. Our model showed that there was measurable and significant AC power, arising from the fluctuations in ohmic resistance as bubbles formed and sloughed off the surface of the palladium electrodes. Our model jibed with both the qualitative and quantitative evidence from McKubre's reports:
1) McKubre (and others) noted that the excess heat only appeared after the palladium lattice was fully loaded. And that's precisely when the Faradaic current no longer charges up the lattice, but begins producing gas bubbles on the surfaces of the electrodes.
2) The excess heat in McKubre's cells was only apparent, significant, and sizable when the Faradaic drive current was elevated to dramatically high levels, thereby increasing the rate at which bubbles were forming and sloughing off the electrodes.
3) The effect was enhanced if the surface of the electrodes was rough rather than polished smooth, so that larger bubbles could form and cling to the rough surface before sloughing off, thereby alternately occluding and exposing somewhat larger fractions of surface area for each bubble.
The time-varying resistance arising from the bubbles forming and sloughing off the surface of the electrodes — after the cell was fully loaded, enhanced by elevated Faradaic drive currents and further enhanced by a rough electrode surface — produced measurable and significant AC noise power into the energy budget model that went as the square of the magnitude of the fluctuations in the cell resistance.
Specifically, if the ohmic resistance is fluctuating R±r, then PAC ≈ α²PDC, where α = r/R.
To a first approximation, a 17% fluctuation in resistance would nominally produce a 3% increase in power, over and above the baseline DC power term. Garwin and Lewis had found that McKubre's cells were producing about 3% more heat than could be accounted for with his energy measurements, where McKubre was reckoning only the DC power going into his cells, and (incorrectly) assuming there was no transient AC power that needed to be measured or included in his energy budget model.
I suggest slapping an audio VU meter across McKubre's cell to measure the AC burst noise from the fluctuating resistance. Alternatively use one of McKubre's constant current power supplies to drive an old style desk telephone with a carbon button microphone. I predict the handset will still function: if you blow into the mouthpiece, you'll hear it in the earpiece, thereby proving the reality of an AC audio signal riding on top of the baseline DC current.
Transient AC Power and Wavefronts of Traveling Waves
Let's go back to McKubre's fateful assumption. McKubre writes:
Under current control, the cell voltage frequently was observed to fluctuate significantly, particularly at high current densities where the presence of large deuterium (or hydrogen) and oxygen bubbles disrupted the electrolyte continuity. By providing the cell current from a source that is sensibly immune to noise and level fluctuations, the current operates on the cell voltage (or resistance) as a scalar. Hence, as long as the voltage noise or resistance fluctuations are random, no unmeasured RMS heating can result under constant current control, provided that the average voltage is measured accurately.
Now let's parse that, one sentence at a time.
1) The cell voltage frequently was observed to fluctuate significantly, particularly at high current densities where the presence of large deuterium (or hydrogen) and oxygen bubbles disrupted the electrolyte continuity.
So we begin by observing that there is fluctuating resistance, and an associated fluctuation in cell voltage. So far so good.
2) By providing the cell current from a source that is sensibly immune to noise and level fluctuations, the current operates on the cell voltage (or resistance) as a scalar.
This is the key part of the unexamined assumption that needs to be carefully examined.
3) Hence, as long as the voltage noise or resistance fluctuations are random, no unmeasured RMS heating can result under constant current control, provided that the average voltage is measured accurately.
But wait! When the power supply is slewing (meaning the voltage is either rising or falling at the slew rate), the voltage pulse and the associated current pulse are in phase. In fact they amount to a transient wave front propagating from the power supply into the cell. There is real power in a transient pulse, which must be computed by the application of appropriate mathematical models for the transient AC power in the wavefront of a traveling wave. The appropriate mathematics for this can be found in the annals of telephony (among other places).
If the slew rate is fast (e.g. 1.25 A/μsec in constant current mode and 1 .0 V/μsec in constant voltage mode), then the Nyquist Sampling Rate to capture this brief interval when the voltage and current pulses are in phase has to be at an even higher frequency. Otherwise, the power in the AC transient will never be seen, never be measured, and never be reckoned in the energy budget model.
Note, also, that the transient AC power is independent of the actual slew rate. The same amount of transient AC power is injected whether the slew rate is fast or slow.
Fourier Analysis
Another way to model it is to use Fourier Analysis. Assume there is a sinusoidally varying load resistance going as R + r sin ωt. Then to obtain a true constant current, the active regulated power supply has to meet the rising and falling resistance. So, for example, if the power supply is trying to maintain a constant 1 A DC current (with no AC), the power supply has to produce a matching voltage given by 1 A × (R + r sin ωt) Ω. If the power supply can do this with no signal processing delay, and if there is no signal propagation delay in the medium between the power supply and the load, then this will indeed produce a perfect constant current and there will be no AC power.
But active power supplies have a non-zero signal processing time (given by the slew rate). Moreover, there is non-zero signal propagation delay in the circuit between the power supply and the load. Let this total round-trip delay be τ. Then the voltage produced by the power supply and delivered to the load will be 1 A × (R + r sin ω(t-τ)) Ω. The phase shift is given by φ = ωτ. The worst case is when φ = ωτ = π, in which case the AC power injected by the hapless power supply is PAC = [α²/sqrt(1-α²)] PDC, where α = r/R. The general formula, as a function of phase shift, φ = ωτ, for any harmonic, ω, in the Fourier Series is
PAC(ω) = ½[1 - cos(φ)] [α²/sqrt(1-α²)] PDC = ½[1 - cos(ωτ)] [α²/sqrt(1-α²)] PDC
where α = r/R and τ is the round trip propagation delay and signal processing delay at harmonic frequency, ω, in the Fourier Series for the time-varying resistance.
So when ω ≈ π/τ, there will be significant AC power that (to a simplified approximation for r ≪ R) goes as ½α²PDC, where α = r/R. If the fluctuating resistance arises from the formation of bubbles on the electrodes, then there will be very high-frequency components from the perturbation in load as bubbles form and slough off the surface of the electrodes. Note also that if the magnitude of the fluctuation, r, is very large (e.g. 80% of R), then the injected AC power can exceed the DC power.
Finally, note that the propagation delay isn't even an exact constant at any given frequency when the conducting medium is an electrolyte[1]. When the charge carriers are electrons, the propagation speed is about one-tenth the speed of light in a vacuum. But in an electrolyte solution with H⁺ or D⁺ ions (as well as other species of charge carriers), the portion of the signal carried by those ions of molecular weight, n, propagate more slowly, going approximately as C/(18360×n). The effect is to render τ to be an exponential distribution with the leading edge of a pulse traveling in about 0.1 μs and the trailing tail lagging by about 500 μs, depending on the mix of species of charge carriers in the electrolyte. It's worse in heavy water than light water because Deuterium ions have twice the atomic weight of Hydrogen ions, and so they travel at half the speed of protons.
[1] Horace Heffner, "10-meter Electrolytic Cell Experiment," April 1996.