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


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 !