Technical Report: How Lithimon Extends Flight Time
A few people have sometimes expressed skepticism about the claims we make for Lithimon's energysaving (and
hence flighttime lengthening) effects. Sure, the claims are pretty remarkable, but then, so is the product. This
report explains how and why Lithimon has its energysaving effect...
This article is intended to be read by those with at least a basic knowledge of physics and electrical theory.
Nonetheless we have tried to keep it as understandable as possible. Fig. 1 (no Lithimon): Fig. 2 (Lithimon fitted): Figures 1 and 2 are oscillograms taken from a Century Hummingbird helicopter powered by a threecell 1400mAh LiIon battery in actual flight conditions. The X scale is 0.5 ms/cm and the Y scale is 2 V/cm. Figure 1 was obtained with no Lithimon in circuit; Figure 2 was obtained with a Lithimon234 installed in accordance with its user manual and clearly demonstrates the voltagesmoothing effect discussed in the Lithimon description  the "ripple" on the waveform is reduced to just a few percent of its original value. The two figures are used as examples to elucidate the theoretical discussion which follows and help to demonstrate that the theory is sound. Similar tests on other Lipowered electric flying models have produced very similar results. A real battery is not a "perfect voltage source"; the latter exists only in theory! The theoretical perfect voltage source has zero impedance (resistance), and hence its voltage is invariant with changes in load. Real batteries are nothing like this, and have a finite internal resistance R. Thus, as a first approximation, a real battery can be modelled as a perfect voltage source in series with some resistance R (which is denoted the "internal resistance" of the battery). In practice, there are additional, nonlinear, elements involved too, due to electrochemical phenomena inside the battery such as ion transit effects. You can see the effects of the battery's internal resistance in Figure 1. Remember that an ESC (Electronic Speed Controller) "chops" the motor current between fullyon and fullyoff at a high rate (typically in the kilohertz range). The current consumed would appear as a fairly accurate squarewave switching between zero and some fullvalue current I. You can see that this is reflected to some extent in Figure 1: the battery's terminal voltage varies between the offload value of approximately 11.6 volts and an onload value of some 10.0 volts. The waveform is not as square as might be expected, the reason being the nonlinear effects mentioned previously. In particular it will be seen that the battery voltage takes a finite time both to "sag" and to "recover" as the ESC switches. With careful analysis of the electrochemistry of Lithium cells (and, indeed, most other battery types too, though LiIon and LiPo are especially susceptible), it can be seen that the effect of the nonlinearity is to make the battery's apparent internal resistance increase with increasing pulse currents, and hence this acts in the direction to exacerbate the voltage drop at higher pulse currents. The result of this nonohmic internal resistance, when combined with the chopping action of the ESC, gives rise to the Peak Pulse Effect already described. The remainder of this discussion neglects the nonlinear effects as they make the mathematics very complex. We have analysed the circuit only in terms of the basic, ohmic, internal resistance of the battery to demonstrate that even in the absence of any nonlinear effects, the Lithimon still achieves what is claimed for it, and as described above the reintroduction of the nonlinearity of a reallife system further increases the benefit of installing a Lithimon. Let our battery be represented by a perfect voltage V in series with a
resistance R (its "internal resistance"). Let the load (an electronic speed controller at about
halfstick) be represented by a pulsating current sink across the
battery, which takes a current I at a duty cycle of d=50% (i.e. fullyon for 50% of the time,
fullyoff for the remaining 50%).
A simple calculation involving Ohm's and Watt's Laws allows us to calculate the power dissipated in
the internal resistance R (which we designate P_{r}) as: Now let us imagine an arbitrarily large capacitance across the load.
(Of course, it is a slight simplification to assume that it is infinite, but at the
switching frequencies of typical ESCs and the capacitance provided by
Lithimon it is a fair assumption, as illustrated in Figure 2 which shows that the variation in battery voltage
is reduced to perhaps 5 percent of its unsmoothed value). This capacitance has the effect
of smoothing out the current such that the battery sees, instead of a
current I at d=50%, a current of 0.5I at a duty cycle of d'=1 (i.e. constant). Recalculating
the loss in the battery's internal resistance R for this case: From the above it is trivial to deduce that the energy loss in the battery's internal resistance  which in physical terms equates directly to the heat generated (i.e. energy wasted) in the battery  is halved under the conditions stated (50% duty cycle, equating to "halfstick"). It is now quite simple to generalise the calculations for other values of duty cycle (i.e. other settings than
halfstick). Equation 1 is already general, and equation 2 only requires that the 0.5 constant be replaced by d,
the original, unsmoothed duty cycle. Multiplying out, we find that with smoothing installed, Finally, by taking the ratio of P_{r} to P_{r}', we obtain a value of 1/d, showing that in the general case the reduction in wasted power is inversely proportional to the duty cycle, and hence the stick setting. Thus the reduction in power wasted ranges from very large at low power (e.g. 90% at 10% throttle), through 50% at halfstick, finally reaching no saving (but no loss either) at fullstick. Since full throttle is seldom, if ever, used in most applications, it can be seen that the energy saving, integrated over an entire flight, will be substantial, and this is borne out by inflight tests with several different helicopters of different sizes that all yield an approximate 2:1 (or better) improvement (reduction) in energy wastage. Since the energy saved remains stored in the battery, it is therefore available for continued flight, giving an increase in flight time. There is still another mechanism by which Lithimon lengthens flight time, and this is in relation to determining the endpoint of the battery's charge. If the operator were to judge the endofflight point by observing a loss of power, this would not only be errorprone (and misjudging it may damage the battery irreversibly!) but would also be prone to showing a premature loss of power in the absence of smoothing, because the power delivered to the propellor or rotors depends upon the "trough" voltage (the battery voltage while the ESC is actually drawing current). Likewise, if a simple (nonLithimon) voltage monitor is used, the troughs in the battery voltage may result in the battery appearing flat prematurely. With a Lithimon fitted, the endpoint is determined accurately based on the smoothed voltage waveform. A combination of the Litimon's sophisticated Digital Signal Processorbased monitoring algorithm and the effect of the smoothing capacitors helping to maintain the battery voltage above its minimum, allows every drop of usable charge to be squeezed out of the battery without risk of exceeding its safe minimum voltage and causing damage. The overall effect on flight time depends on many factors, but is always significant. The smoothing applied by Lithimon assures a significant reduction in wasted power (usually 50%) as explored in the equations above. Additional factors including endpoint determination are also improved by the addition of a Lithimon, such that in real flight tests we have regularly obtained increases exceeding 75% extra flying time from an otherwise identical setup. Not only that, but Lithimon's accurate endpoint determination prevents damage to batteries by overdischarge, and the reduction in waste heat generated in the battery can greatly extend the life expectancy of the battery (studies have suggested that each 10 degree Celsius drop in battery pack temperature can roughly double the number of operating hours that the battery will survive). Furthermore, in many cases a Lithimon will allow a smaller (in amphour terms) battery to be used than otherwise necessary, by boosting the effective maximum current that can be drawn, and this can be of great benefit in weightcritical applications. As an illustration of these aggregate benefits, in the setup which yielded Figures 1 & 2 it was found that the usable flight time more than doubled when the Lithimon was installed, and without the Lithimon it would have been necessary to install a larger, heavier battery in order to obtain a satisfactory flight duration  which in turn would have been unsatisfactory due to the added weight! Conclusion: This article has illustrated a number of mechanisms by which the Lithimon can increase flight time
and decrease energy wastage, and has quantified the latter both theoretically and experimentally; and furthermore the
overall effect of all the mechanisms described has been to increase flight time by between 75% and 130% in all the
test flights which were conducted using three differentlysized electric helicopters.

If you would like any further information about the Lithimon product family, please
contact our technical helpline:
tech@dawnmist.org.