Energy Anomaly  (13th May '08)


the next stage was to experiment with the circuit powering a resistive load (not just storing energy in another capacitor)

'textbook' treatments of the energy transformations in charging a capacitor suggest that the amount of energy converted as work in charging the capacitor will be the same value as the final amount of energy which gets stored in the capacitor - ie. to store 2 Joules in a capacitor will require an additional 2 Joules of external work

  eg.  see  http://farside.ph.utexas.edu/teaching/316/lectures/node47.html 
         (lecture notes by Associate Professor of Physics, Texas University)

in other words, if we measure the final voltage just stored on a previously discharged capacitor and calculate the equivalent energy stored and we double that value, then the result represents the total energy converted 

so i understand that with this type of capacitor-to-capacitor charge circuit, if there is not an appreciable reactive component in the charging path then there will be a minimum I^2*R loss of 50% of initial energy dissipated in any series resistive component and/or wiring

initial tests confirmed that this energy was indeed being lost in the discharge/charge process, so i modified the circuit to add a series resistor which could act as an intentional load both during the charge and discharge phase of an output capacitor

i also replaced the transistor switch arrangement with MOSFETs (1xFDN304P, 1xIRF540N) and i replaced the transformer with an inductor

since i was only attempting to investigate the impact of the charge anomaly on the energy behaviour of the circuit, and not construct an end-user Power Supply device, i didn't add the final MOSFET switch, Q3, to discharge the energy stored in the cap back thro' the load resistor which was just used to charge it - i made the connection manually instead 

 circuit modified with inductance and common load resistor

ok, now for some data:-

Input supply:  8.0V on 0.299F cap (2.392Coulombs; 9.568Joules)
discharged to:  7.0V (2.093C; 7.326J) in 45.8s
Energy supplied:  2242mWs(milliJoules)
(Power in: 2242/45.8 = 49mW av.)

Switched charge from Q1 output

Switched osc.
(cycle:  1.73ms;  charge/cycle duty:  0.12 / 1.73 = 0.07)
power:  0.73mA*7.5V = 5.5mW av.
Energy drawn by Switch osc.:  5.5*45.8 = 251mWs 

Output cap. charge and discharge, both via 10R load
(NB charging is switched, & takes longer than discharge thro' load))

Output cap:  charged from 0 to 2.67V on 0.342F (0.913C; 1219mWs)

Total start charge: 2.392C
Total end charge: (2.093+0.913) = 3.006C
(charge on 0.0005F switching cap can be ignored)

Energy stored into Output cap via load resistance:  1219mWs 
   = Energy discharged via load resistance

total energy thro' load resistance = 2 x 1219 = 2438mWs

unquantified losses:
dissipation by MOSFETS;
sound/vibration from inductor

Load Energy/Energy supplied = 2438 / 2242 = 1.09

Energy Quotient (EQ)
  = Useful energy used/Energy supplied = (251+2438) / 2242  = 1.2


INITIAL CONCLUSIONS
if the 'textbook' energy equations for capacitor charging are correct, then these results indicate that the Charge Anomaly has been accompanied by an Energy Anomaly - the switched capacitor circuit has converted more energy than was initially supplied to it !

'Energy Anomaly' revisited  (Jan '09)


my first energy tests relied entirely on measurements of the initial and final voltage on input and output capacitors

these results have prompted some discussion about the energy converted as work in charging the output capacitor, so i've repeated the tests using equipment which can monitor the charging voltage across Rload

 2-stage capacitor switching with load resistor

'textbook' explanations of energy-used and energy-stored when charging a capacitor give equations which indicate that the external work done in charging a capacitor has the same value as the energy which gets stored in the capacitor as a result (NB. it is  not  the 'same' energy - it is just the same 'value' of energy)

for example, if a capacitor has just been charged, from empty, with 2 Joules of energy then the conventional view states that an additional 2 Joules of energy must have been used as work to transfer the other 2 Joules into the capacitor - this means that a circuit has to convert 4 Joules of input energy to end up with 2 Joules of energy in the output capacitor

it follows from this statement that a simple capacitor-charging arrangement cannot be more than 50% efficient

(BTW this limit is not dependent on a particular series resistance in the charge path - larger resistances just reduce current and increase charge time, and smaller resistances do the opposite)


it seems to be common knowledge, however - in the power-supply design industry, at least - that it's possible to achieve efficiencies > 50% if a series inductor is included in the capacitor charge-path (theoretically up to 100% efficiency)

so are we now saying that a physical process which previously required us to do work to overcome the increasing internal polarisation producing the capacitor field has suddenly changed to NOT needing to overcome the increasing internal polarisation, just because an inductor has been added in the external current path?

in order to confirm what actual work done in charging the output capacitor, and also to cross-check the previous efficiency results, i've taken measurements from the load resistor placed in series with the output capacitor


ok, now for some results:- 

C1 = C3 = 4700uF nominal (measured values were used for the results)
C2 = 30uF nominal
(the capacitor values of the original tests were reduced for these tests to allow more accurate measurement of the charge-switching waveforms)

a 10 ohm load resistor, RL (Rload), was connected in series with C3 and used to monitor the current flow 

C1 was discharged from 8V to 7V by 2500 pulses of charge being switched across to C3
(switch-cycle period was approx 1.3  ms; C2 charge pulse-width was approx 40 us);
C3 charged from 0V to 3V

the charge anomaly was clearly seen:-

total initial charge (C1): 38.8 mCoulombs

final charge on C1: 33.9 mC (loss: approx 5mC)
final charge on C2: 14.3 mC (gain: 14.3mC)

total output charge: 48.2 mC (net gain: 9.4mC)


i was also able to replicate the situation where the final stored energy on C3 (21.5mJ) is greater than half the total input energy ( 36.7mJ)

the load waveform was measured at the mid-point of input energy conversion
 - ie. 50% had been discharged, 50% still to go

the average power on Rload at this point was 4.9mW


the duration of the test run was 3.25s so the total charging energy converted on Rload was approx 15.9mJ
(losses in inductor, L, were approx 0.004mJ and can be ignored here)


CONCLUSIONS

If the textbook teaching is correct and the final energy on C3 represents 50% of the total converted output energy, then that total would have been 43mJ, representing an efficiency greater than 117% (and there will also have been additional energy dissipated as losses)

however, i only measured a value of 15.9mJ for the charging energy - so if the 50% work relationship IS true then there was still approx 5.6mJ** of charging energy unaccounted for (ie 13% of the supposed output energy converted)



since the sum of the measured output energies, 37.4mJ, was so close to the input energy, 36.7mJ, it seems likely that the efficiency of the total system was just close to 100% (within experimental limits) 

however, this would mean then that the textbook teaching on the charging process is wrong, and the work done in charging a capacitor DOES NOT have to equal the final energy stored (even WITHOUT a series inductor between C2 & C3)

so - if there's NO 50:50 split in charging:stored output energy then it's likely there's NO excess energy here either - it's very efficient - but probably not over 100%