Questions Answered
Were the capacitor values measured
or assumed?
What level of test-instrument errors could have been introduced?
How was the Input Energy
calculated?
How was the Output
Energy calculated?
Why is the energy for the oscillator circuit included in the Energy Quotient factor?
Have you attempted to confirm the result by another method?
What effect does the equivalent series resistance (ESR) of the capacitor have on the results?
Doesn't
the ESR cause the on-load voltage of the capacitors to be lower than the
off-load voltage?
How did you account for ESR when calculating the stored charge?
How
did you measure pulsed waveforms accurately?
What's the difference between the operation of this switched-capacitor circuit and just connecting the charged input capacitor directly to the uncharged output capacitor?
Were the capacitor values measured
or assumed?
they were measured - i used the RC-constant discharge-time method:
i noted the time taken for a charge on the capacitor to
discharge to 37% through a measured resistive load and then divided that time by
the load resistance to give the capacitance (measurements were repeated for
confirmation)
the load resistor value (468R for the 0.25F; 10KR for the 4700uF) was sufficiently high to enable the internal
self-resistance of the capacitor to be ignored
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What level of test-instrument errors could have been introduced?
my voltmeter and 'scope readings were compared with a voltmeter calibrated to
National Standards - representative sample readings showed agreement of my test instruments to within approx 1% of the calibrated meter
later tests with a 2-channel PC
scope showed agreement within 2% of the calibrated meter
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How was
the Input Energy calculated?
the input capacitor was disconnected and pre-charged from a battery, via a
low-value resistor, to approx 8.1V and allowed to self-discharge to 8.0V; it was
then connected to the remainder of the circuit which was allowed to operate
until the voltage on the input capacitor had discharged to 7.0V (the switching
operation was adjusted to make the charge-transfer take approximately 45
seconds)
the energy input to the circuit was calculated as the difference between the
energy stored on that capacitance value at 8.0V minus the energy stored at 7.0V
the voltages on the input
capacitor were measured by voltmeter:
- the start-voltage was the
off-load voltage, taken before the circuit was connected;
- the end-voltage was an on-load
voltage, taken as the circuit was operating
later tests used a 2-channel PC scope which enabled the end-voltage on C1 to be
read at a point following the final charge pulse and therefore also off-load
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How was
the Output Energy calculated?
prior to a test, the output capacitor was discharged and then shorted until
immediately before the test started; the total energy stored on the output
capacitor was calculated using the end-voltage charge on that capacitance value
the output capacitor voltage was measured from the scope trace (in storage mode)
at the point just following when the circuit operation had been stopped; this
reading represents the off-load voltage of the capacitor
the final step of each test was to discharge the output capacitor through the
resistive load
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Why is the energy for the oscillator circuit included in the Energy Quotient factor?
i've defined the Energy Quotient to represent
'what you get' for 'what you pay'
'what you pay' is charge consumed from the capacitor (at cost, C)
'what you get for your money' is:
a) 'heat from the load' for time t (using energy, E1)
b) 'electrical activity in the oscillator' for time t (using energy, E2)
both a) & b) represent real work done
both a) & b) require energy to do that work
in the case of the switched-charge circuit:
the EQ = (E1 + E2) / C
(as mentioned in the results, there is also unaccounted additional energy being
used by losses in the MOSFETs and as noise)
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Have you attempted to confirm the result by another method?
let's say that, due to my failing eyesight and increasingly feeble mind, i measure my input or output capacitance wrongly
not only that, but i measure wrongly in such a way that the input capacitance holds more energy/volt than i think
- or the output holds less than i think
what effect would this have on the results?
well, if i was out by 10% on only one capacitance, then either:
- a larger-than-measured input capacitor would be providing more energy for a 1V drop than i thought
- or a smaller-than-measured output capacitor would need less energy than i thought to charge to 2.67V
the net effect would be the same: either of these measurement errors would tend to cause the
Energy Quotient (EQ) value to appear greater than it should be - the results would look
greater than 1 when in fact they were certainly under
of course, if i measured both capacitances wrongly, i would end up with a greater false-indication
of energy gain
so here is a second test which would confirm whether there was capacitance measurement error in the first test:
- swap the input and output capacitances and repeat the experiment
now, if there was a single capacitance measurement error, then either:
- a smaller-than-measured capacitor would be providing less energy for a 1V drop than i thought
- or a larger-than-measured capacitor would need more energy than i thought to charge to 2.67V
the net effect would be the same: either of these measurement errors would tend to cause the EQ value to appear lower than it should be - the results would definitely look
less than 1
if i measured both capacitances wrongly, i would end up with an even lower value of EQ
i've performed this second test and re-run the experiment with capacitors swapped and i'm measuring results around 3.1V on the 'new' output cap
the results of the second test [which also make EQ > 1] confirm that the capacitance measurements are good and also confirm the relative difference in values between C1 and C3
the 'Latest' section of this site contains the results of measuring the energy
converted on the resistive load to confirm whether the 'textbook' claim for
equal energy values stored and dissipated as work in achieving the stored energy
is true
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What effect does the equivalent series resistance (ESR) of the capacitor have on the results?
in the early days of supercaps the AC ESR values could be in the kilohm region but things have moved on since then - there is a greater range of characteristics for different application device groups - milli-ohm ESR values are easily achievable now
the caps i'm using exhibit sub 1R ESR under representative conditions
taking ESR into account:-
- when charging a capacitor in
this circuit, the RC time-constant will be affected slightly but not the final
off-load voltage - the timing can always be adjusted, if necessary, to enable
the capacitor to reach the required voltage level
- when discharging a capacitor in this circuit, again the RC time-constant will
be affected slightly (and the timing can be adjusted to compensate for this, if
necessary)
however, the terminal voltage will be lower than the cap plate voltage
according to its potential-division by the relative values of the ESR and the
load resistance - the effect of the ESR voltage drop for a given cap can be
reduced to a required level by adjustment of the load resistance - or the ESR
can be reduced by choosing a suitable capacitor
in terms of charge and energy measurement, an off-load reading of a charged
capacitor will be essentially unaffected by ESR in this circuit - an on-load
reading of a capacitor being discharged will tend to suggest that more energy
has been used than is the case (the extent depending on the relative values of
the ESR & load)
for the input capacitor , the effect makes the EQ factor appear less than it
really is because there is not as much energy being supplied to the circuit as
we think
for the output capacitor , the
effect makes the EQ factor appear more than it really is because not all of the
stored energy is being supplied to the load
the net result on the whole circuit is that these two effects tend to compensate
for each other and this can be optimised by suitable selection of components and
their values
the 4700uF capacitors have ESR values around 0.1 ohm
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When measuring the capacitance value,
shouldn't the ESR be added to the external load to get the correct C value?
it depends on the relative magnitudes of the actual ESR & load used
the load resistance in the measurement of capacitance, in this case, was 468 ohm
for the 0.25F stack (<1 ohm ESR) and 10K ohm for the 4700uF capacitors
(~0.1 ohm ESR)
so
practically, no - the ESR contribution is negligible
Doesn't
the ESR cause the on-load voltage of the capacitors to be lower than the
off-load voltage?
correct - this is completely separate from the self-recharge phenomenon, which occurs, over time, in the off-load state - ie. after the instantaneous on-load to off-load transition has occurred
by definition the ESR cannot influence the stored voltage once its serial connection to an external load has been opened
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How did you account for ESR when calculating the stored charge?
the calculation of stored charge (and energy) in the capacitors relies initially on measurement of their capacitance and subsequently on measurement of their terminal voltage
the capacitor voltage result measurements were made effectively off-load, using test devices presenting a load
which was several orders of magnitude greater than the ESR of the capacitors involved in the test, creating
negligible 'potential division' at the test nodes
the relevance of ESR in measuring the capacitance was addressed in the When measuring the capacitance
value... FAQ above
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How
did you measure pulsed waveforms accurately?
no pulsed waveforms were used for measurements in the original tests
when taking current-draw measurements of the supply to the oscillator circuit, potential noise was decoupled using 100uF and 0.1uF capacitors across the supply
all other measurements were taken as DC voltage readings on either the input or
output capacitors
in later tests, the digital data from the 2-channel PC scope was recorded to
file and then input to Excel to calculate instantaneous Power readings per
sample
the instantaneous power values
were then averaged and multiplied by the appropriate period to give energy
values
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What's the difference between the operation of this switched-capacitor circuit and just connecting the charged input capacitor directly to the uncharged output capacitor?
the measurement procedure of the input and output states can be the same but what happens in the two types of circuit between those two states is very different
the most 'visible' difference, of course, is the number of charge transfer 'cycles' which occur in each
method
i understand that the direct 'passive' circuit connection can exhibit a slow oscillation of charge between the input and output capacitors which eventually settles to a steady-state where the charge is divided between the two capacitors in proportion to their respective capacitance values - this corresponds to an equal terminal voltage on each (measured effectively off-load, to avoid ESR issues!)
a 'less-visible' but still significant difference between the two methods is that in the direct-connection case the energy can flow back & forth until equilibrium is reached - however, in the case of the
switched-capacitor circuit the charge is transferred in small 'packets' and the output cap gets disconnected from the input cap by the switch, on each cycle, either whilst still charging, or having just reached its maximum step charge
this regular disconnection prevents the oscillation of charge between the input and output capacitors - an asymmetry has been introduced into the circuit: charge can only flow in one direction - from input to output
the switching operation is like an 'integral' approximation of an analogue function - input charge still 'flows' to output, albeit in quanta, but in effect the charge pump has 'rectified' the ebb & flow of the single-cycle process
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