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Copyright © 2014 jsd

Capacitor-to-Capacitor Transfer of Energy and Gorge
John Denker

It is quite possible to transfer energy and charge (or rather gorge) from one capacitor to another with high efficiency. The energy-transfer efficiency can approach 100%, and the gorge-transfer efficiency can easily exceed 100%. These numbers far surpass the limits that are often assumed, “proved”, and/or “explained” in the physics education literature. We discuss a simplified version of a technique that is very widely used in the electronics industry.

*   Contents

1  Introduction

Let’s consider the so-called “Two Capacitor Problem”.

Suppose we have two capacitors, as shown in figure 1. Initially there is some energy stored in the left capacitor (shown in blue) but zero energy in the right capacitor (shown in red). The objective is to transfer “some” energy from one to the other.

capacitors
Figure 1: Two Capacitors

Without loss of generality, assume the right capacitor has N times the capacitance of the left capacitor:

C2 = N C1
             (1)

2  RC Scheme

Figure 2 shows one scheme for performing the transfer. When the switch is closed, a current flows.

capacitors-switch-r
Figure 2: Switched RC Circuit

Under this scheme, elementary circuit analysis tells us that the final voltage is:

V1(final) = V2(final)      
  = 
1
1+N
 V1(initial)
             (2)

The energy is proportional to the capacitance and the square of the voltage, so the final energies are:

E1(final) = 
1
(1+N)2
 E12(initial)         
E2(final) = 
N
(1+N)2
 E12(initial)         
E12(final) = 
1
1+N
 E12(initial)
             (3)

The subscript “12” refers to the whole system, i.e. both capacitors together.

We can also take a look at the gorge on the capacitors. (All-too-often people call this the “charge” on the capacitors, but that is a misnomer, as discussed in section 8.)

G1(final) = 
1
1+N
 G1(initial)      
G2(final) = 
N
1+N
 G1(initial)      
G12(final) = G12(initial)
             (4)

This works, but as we shall see, it is not the only possible scheme. Indeed it is nowhere close to being the optimal scheme.

The most remarkable thing about equation 3 is that the results are independent of R. So in some sense this a “universal” result.

One might even imagine that equation 3 remains valid even when R=0. However, this doesn’t make sense, because the dissipated energy is dissipated in the resistor. This is the infamous two-capacitor “paradox”.

The usual way of resolving this “paradox” is to argue that there is always some parasitic resistance (not shown in the circuit diagrams), so that in the limit where the explicit resistance R goes to zero, the parasitic resistance becomes dominant. If nothing else, there will always be some radiation resistance.

A tremendous amount of effort has gone into “proving” that this result is universal, and/or “explaining” why it is mandatory, and/or accounting in detail for the “lost” or “missing” energy. I would argue that all of this work is seriously misguided, because in fact the result is not universal and not mandatory.

3  LC Scheme : Transfer

In passing from figure 1 to figure 2, we added a switch and an explicit resistor. You could also add a parasitic resistor and/or an antenna to represent radiation resistance. However, there is no law of physics that says these are the only possibilities. The same physics that allows us to understand capacitors, resistors, switches, and radiation also allows us to understand inductors.

A particularly interesting possibility is to add an inductor and some switches, as shown in figure 3. Switch S3 is normally closed, and will remain closed until further notice.

capacitors-switch-l
Figure 3: Switched LC Circuit

Proper operation of this circut requires some deft timing, as we now discuss.

We start by closing the left switch (S1), connecting C1 to the capacitor. This creates an LC oscillator, a simple harmonic oscillator. The initial condition is shown by the highest point on the blue trace in figure 4. We call this the 12:00 point on the blue trace.

c-lc-xfer-phase-space
Figure 4: Phase Space of Switched LC Circuit – Transfer

The spreadsheet used to compute the waveforms and phase-space plots in this document is cited in reference 1.

After waiting a quarter of a cycle, the system has evolved to the 3:00 point. All of the energy has been transferred from C1 to the inductor. This is interesting, but not maximally convenient, because the current in the inductor is flowing in the “wrong direction” for our purposes. However, if we wait an additional half cycle, the system will evolve to the 9:00 point, where the current is flowing in the desired direction.

At this point we open the left switch (S1) and close the right switch (S2) connecting the inductor to C2. This creates a different LC oscillator, with a different period. The new period is longer than before by a factor of √N.

At this point in the diagram, the trace switches from blue to red. Now we wait an additional quarter period, i.e. a quarter of the new period. This takes us to the 12:00 point on the red trace. At this point – assuming ideal components – all of the energy has been transferred to C2.

V1(final) = 0             
V2(final) = 
1
N
 V1(initial)     
             (5)

E1(final) = 0        
E2(final) = E1(initial)         
             (6)

We should also look at the gorge:

G1(final) = 0             
G2(final) = 
N
 G1(initial)      
G12(final) = 
N
 G12(initial)
             (7)

In particular, comparing with equation 4, we find that this scheme produces more gorge on C2 by a factor of (N+1)/√N ... which is a factor of 2 already when N=1 and gets even bigger for large N.

Let’s be clear: In the circuit we are considering here, i.e. figure 3, we wind up with more gorge than we started with, for any N greater than 1 ... not only more final gorge than we would have gotten with figure 2, but more than we started with. There is no law of physics that requires gorge to be conserved.

You may have noticed that in figure 2 the final gorge was equal to the initial gorge, but this is not a consequence of any deep physical law; it is merely the consequence of some engineering choices that were made ... not even particularly clever engineering choices.

In addition to the phase-space plot in figure 4, we can get some additional insight by looking at the waveforms as a function of time, as in figure 5.

c-lc-xfer-waveform
Figure 5: Waveform of Switched LC Circuit – Transfer

The ordinate is the gorge on both capacitors combined. However, the capacitors take turns, so the blue part of the trace represents the gorge on C1, while the red part represents the gorge on C2.

The abscissa is time, measured in units of the period of the “blue” LC oscillator, namely

P1 = reference period       
  := 
L C1
             (8)

The period of the “red” LC oscillator (P2) is longer by a factor of √N, but this has no bearing on the units used for the abscissa in the plot. Note that to make the diagrams, the value N=3 was chosen.

Note that a full cycle of the red trace in figure 5 is identical to a full cycle of the blue trace, just scaled up by a factor of √N ... scaled up in both the gorge-direction and the time-direction. As a corollary, the slope is the same at corresponding points. In particular, at the point where the red trace splices onto the blue trace, there is no change in slope.

We can understand this in physics terms as follows: The time-derivative of the gorge is the current. At the splice-point, we are switching the current from one capacitor to the other, but it is the same current, so the slopes have to match.

As a further point of physics, we have chosen to do the splice at a point where there is zero voltage across the inductor, so in accordance with the following equations

φ = L I   (the flux) 
V = 
dt
   (one of the Maxwell equations)     
V = 
L 
dI
dt
  
  = 
L 
d2G
dt2
  
  = 0   (by choice)
             (9)

not only do the gorge-traces match as to slope, they also match as to second derivative. They both have zero second derivative, i.e. zero curvature. The chosen splice-point is an inflection point.

Here is a third way of understanding the scaling relationship. It follows from the Maxwell equation. There is some current flowing in the inductor. If the voltage is less, the current flows for a longer time.

4  LC Scheme : Equilibration

Let’s change gears slightly. Suppose that rather than transferring all the energy from C1 to C2, we only want to transfer enough so that the two capacitors come into equilibrium, i.e. so that they have the same voltage. We can do this using the same circuitry as in figure 3, just using different timing.

Everything is the same as in section 3 for the first three quarters of a cycle, up to the point where we throw the switches.

Recall that at the 9:00 point on the blue trace, both capacitors are at zero voltage. In this case, we leave switch S1 closed when we close switch S2. Thereafter the two capacitors remain locked together, with a common voltage.

The period of the combined oscillator (P12) is longer than the reference period by a factor of √1+N. To make the diagrams, the value N=3 was chosen, so P12 = 2 P1.

c-lc-join-phase-space
Figure 6: Phase Space of Switched LC Circuit – Equilibration

In figure 6, the magenta trace represents the combined system, blue and red together. When the magenta trace reaches 12:00, all the energy is in the capacitors, not the inductor. At this point we can open switch S3 and the capacitors will remain in equilibrium with each other. (The magenta trace in the diagram continues past this point, but if all you wanted was to establish equilibrium you would not allow the oscillations to continue.)

In figure 6 and also in figure 7, the behavior of C1 and C2 separately are shown beyond the splice point, using small blue and red circles. They are rather less interesting than the total system behavior.

The key results are:

V1(final) = V2(final)         
  = 
1
N+1
 V1(initial)     
             (10)

E1(final) = 
1
N+1
 E1(initial)         
E2(final) = 
N
N+1
 E1(initial)         
E12(final) = E12(initial)         
             (11)

G1(final) = 
1
N+1
 G1(initial)      
G2(final) = 
N
N+1
 G1(initial)      
G12(final) = 
N+1
 G12(initial)
             (12)

We can also look at the waveforms as a function of time.

c-lc-join-waveform
Figure 7: Waveform of Switched LC Circuit – Equilibration

Again there is a scaling law: The magenta curve is a scaled-up version of the blue curve, scaled in both the time-direction and the gorge-direction.

5  Implications and Ramifications

6  Easier-to-Operate Circuit

Note the contrast:

The operation of the circuit in figure 2 is simple. You can thrown the switch and then go get lunch. The capacitors will come into equilibrium and then stay in equilibrium while you are gone.   The operation of the circuit in figure 3 is much more complex. You have to open and close various switches at various times, and the timing is critical.

The circuit in figure 2 can easily be demonstrated in an introductory physics class.   The circuit in figure 3 is much more energy-efficient, but it is not convenient to demonstrate. It requires a great deal of supporting circuitry to get everything to work right.

We can split the difference using the circuit shown in figure 8. Compared to figure 3, this circuit is much easier to operate. It is not quite as energy-efficient, but still much more efficient than figure 2. In particular, for large N, more than 100% of the gorge on capacitor C1 gets transferred to capacitor C2.

capacitors-diode-l
Figure 8: LC + Diode Circuit

In this circuit, S1 is a double-throw center-off switch. The first step is to switch S1 to the left, so that capacitor C1 gets gorged up and V1 becomes equal to the applied voltage V0. We assume V0 is positive. We are not interested in the details of where V0 comes from; that is considered external to our circuit. Similarly R0 is considered external and not part of our efficiency calculation; it is just there for safety, to make sure the inrush current is not infinite when S1 is switched to the left. Our analysis begins after C1 has been gorged up; we want to see how efficiently energy and gorge can be transferred from C1 to C2.

The second step is to switch S1 to the right. This forms an LC circuit involving C1 and L. At this moment, C2 is not involved, since the diode is reverse biased and therefore non-conducting.

However, a short time later, the natural LC oscillation will bring the voltage V1 into negative territory. The diode is now conducting. If we arrange that the forward drop of the diode is negligible compared to the other voltages, the circuit is now equivalent to figure 3 with all three switches closed.

The voltage V2 will be negative, opposite to the sign of V1. This is slightly inelegant, but it is part of the price we pay for having a circuit with a low parts count and simple operating procedure.

After capacitor C2 is fully gorged, the diode will stop conducting. The LC circuit formed by C1 and the inductor will continue to oscillate with a small amplitude, less than one diode-drop.

At some point you will want to open switch S1. Ideally this should be done at a point in the cycle where the current I3 is positive, where the positive direction is “downward” as defined by the circuit diagram. However, this sort of fine timing is contrary to the spirit of ultra-simple operation. Therefore the recommended procedure is to simply wait until the LC1 oscillations have died out (due to the inevitable parasitic dissipation) before flipping the switch.

Another possibility is to use a Zener diode, where the Zener breakdown voltage is larger than V0. This protects the diode against the possibility that somebody might throw the switch in such a way that the inductor tries to force current through the diode the wrong way.

Consider the scenario where V0 = 24V and N = C2/C1 = 10. In this scenario, the circuit in figure 8 (compared to figure 3) delivers 3× more gorge and 3× more energy to the output capacitor C2. For larger values of V0 the advantage is even greater, because the diode-drop is less of an issue. Also larger values of N make the advantage even greater.

In any case, the fundamental physics point remains: It is possible to transfer energy and gorge from one capacitor to another with very high efficiency.

7  Pedagogical Remarks

Note that a lot of smart people have gotten this wrong over the years. A lot of people assumed – or even “proved” that the results in equation 3 were mandatory and universal. Let this be a warning: Just because you cannot think of a better way of doing things does not prove that no better way exists!

The advanced student may find it instructive to go over the various “proofs” and make a list of the false assumptions that went into them. It is a long list. Here are a few examples

  1. It is often asserted that there must be “some” parasitic resistance. It is then assumed that when the explicit resistance R approaches zero, the parasitic resistance begins to dominate the impedance. This is false, because it could also be that some parasitic inductance dominates instead.
  2. It is sometimes loudly asserted that things “naturally” happen in the most dissipative way possible. This is not generally true of the natural world, and it is certainly not true of living things or engineered things.
  3. There is pervasive confusion about the distinction between charge and gorge. Charge is strictly conserved, and the circuit in figure 2 happens to conserve gorge, but this does not mean that gorge is necessarily conserved. Really not.

And so on......

8  Appendix: Gorge versus Charge

We define gorge as follows: Consider the simple case where a capacitor has charge Q(upper) = +G on the upper plate, and a charge Q(lower) = −G on the lower plate. Then in this case we say G is the amount of gorge on the capacitor. By way of contrast, in this simple case the total charge on the capacitor is Q(upper) + Q(lower) = 0.

More generally, including non-simple cases, we define things as follows:

G := 
Q(upper) − Q(lower)
2
     (gorge)    
Q := Q(upper) + Q(lower)     (charge)    
             (13)

The concept applies equally well to batteries as well as capacitors. In real-world situation, the total charge Q might be 10 or 12 orders of magnitude smaller than the gorge G. In such a situation the total charge Q is sometimes called “stray” charge. As another way of saying more-or-less the same thing: In normal operation, to an excellent approximation, you have have a current flowing “into” a capacitor; instead you have a current flowing through a capacitor, i.e. in one terminal and out the other.

The distinction between charge and gorge is key to understanding the physics. In particular, charge is strictly conserved, but gorge is not. A real battery has some finite shelf life, and a real capacitor has some nonzero internal leakage. The battery just sitting there will disgorge itself. It would be a tremendous mistake to think that “charge” is being lost, because that would violate conservation of charge.

9  References

1.
John Denker,
Spreadsheet for Modeling Capacitor-to-Capacitor Transfer
./c-lc.xls

2.
Robert J. Sciamanda
“Mandated energy dissipation – e pluribus unum
Am. J. Phys. 64, 1291 (1996)
http://dx.doi.org/10.1119/1.18373

3.
Sami M. Al-Jaber and Subhi K. Salih
“Energy consideration in the two-capacitor problem” Eur. J. Phys. 21 341–345 (2000)
http://dx.doi.org/10.1088/0143-0807/21/4/307
http://www.tuks.nl/pdf/Reference_Material/Al-Jaber and Salih - Energy consideration in the two-capacitor problem.pdf

4.
Timothy B. Boykin, Dennis Hite, and Nagendra Singh
“The two-capacitor problem with radiation”
Am. J. Phys. 70, 415 (2002) http://dx.doi.org/10.1119/1.1435344

5.
A.M. Sommariva
“Solving the two capacitor paradox through a new asymptotic approach”
IEE Proceedings – Circuits Devices and Systems 150(3) 227–231(2003)
http://dx.doi.org/10.1049/ip-cds:20030348

6.
T. C. Choy
“Capacitors can radiate: Further results for the two-capacitor problem”
Am. J. Phys. 72, 662 (2004)
http://dx.doi.org/10.1119/1.1643371

7.
H.L. Neal
“Kirchhoff’s Rule and the Two Capacitor Paradox”
Bulletin of the APS : 74th Annual Meeting of the Southeastern Section (2007)
http://meetings.aps.org/Meeting/SES07/Session/HB.14

8.
A. P. James
“The mystery of lost energy in ideal capacitors”
arxiv.org (2009)
http://arxiv.org/abs/0910.5279

9.
Keeyung Lee
“The two capacitor problem revisited: simple harmonic oscillator model approach”
arxiv.org (15 Oct 2012)
http://arxiv.org/abs/1210.4155
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Copyright © 2014 jsd