1. Introduction (Top)

In a paper "Indication of anomalous heat energy production in a reactor device containing hydrogen loaded nickel powder" Levi et al describe a series of "black box" tests on the Rossi "Hotcat".

Unlike previous eCat models which heated water, producing steam, the "Hot Cat" was evaluated on its own, producing heat in the form of radiation and convection.

Three tests were performed:

November 2012. A test was started with a new "Hot Cat (HT)", but the eCat "ran away" -- reaching an external temperature of 850C, and glowing red hot. "The performance of this device was such that the reactor was destroyed, melting the internal steel cylinder and the surrounding ceramic layers. "
December 2012. A 96-hour run was made, at a temperature of 500C, giving a calculated COP of 5.6
March 2013. A 112-hour run was made, intentionally at a lower temperature, 300C, giving a calculated COP of 2.6

In all cases the output energy was calculated based on the radiation emitted by the device, and the calculated thermal convection. The AC input power was measured with an appropriate wide-band meter.

They did not report that they checked the DC component of the input power, but subsequent statements by one of the authors says that they did.

The investigators were not allowed to see the "magic powder" (they were allowed to see its container), or to look inside the controlbox, or to examine the waveforms on the wires between Rossi's controller and the eCat. None of these affect the calorimetric results. Apart from these restrictions the experimenters were allowed to bring any equipment they liked.

In addition to analyzing the energy performance, the authors commented on the unusual shape of the waveform of temperature with time, when a "square-wave" stimulus is applied,: the expected response is an exponential rise and fall. The actual curve looks more like a distorted sine wave.

They suggested that this "warrants further in-depth enquiry" : here it is.

2. Summary : my results (so far) in a (coco-) nutshell (Top)

I am using the Spice transient analysis program to perform thermal modelling. (There are direct relationships between electrical and thermal values -- for instance Temperature corresponds to Voltage).

I started with a simple lumped model, where the Capacitor C represents the thermal mass of the system, and the resistors represent the radiative and convective heat loss from it.

I applied a waveform to it :

The square wave produces the anticipated exponential rise and fall, but the triangular wave looks more like the actual response, so I made a more detailed lumped model (schematic not shown in this summary) and applied various waveforms to it.

The input from Rossi's controller is composed of two stages .. a STEP function to start it, and then a 35/65 PULSE to sustain it.

We can apply this to the "blank" or "dummy" ecat :

We can then apply various waveforms to the center of the hotcat, presuming that is where the anomalous heat is generated.

This is the result from applying a square wave, offset from the heater pulse:

This is a close view of the output for a Square wave, a Triangular wave and a Sawtooth:

The Triangular wave gives the best results at the output ... but when I moved to a Ladder model I preferred the Sawtooth.

The single lumped capacitor has been replaced by separate capacitors for the outer cylinder, the ceramic center, the inner cylinder, and the center. Between them are resistors representing the Thermal resistance (R-value) of the material. We are particularly interested in the temperature of the outer cylinder, and in the center which contains the "magic powder".

This is encouraging, because the output (LADDER) is a good match to the observed temperature, and the temperature at the CENTER bears at least a passing reemblance to the pulse which produced it, supporting the idea of a linear model. The ladder model is expected to be a little "softer" than the lumped model. The similarity of the LUMPED and LADDER models indicates that the Spice model is well-controlled and stable.

The square-wave pulsed input from the heater presents itself in the center as a 13 degree triangular pattern, offset slightly to the right of the pulse.:

(Hidden below that 13 degree ripple is a 100 degree "DC" component.).

I next considered the environment in which the reaction takes place. I inject the expected waveform for two cycles, and then "wait" for the reactor to respond:

Instead of seeing the expected "triangular" wave from the heater, we see that it remains almost flat. The reason for this is that the temperature from the PRECEEDING pulse is still falling, while heater pulse is ramping up. If we zoom in, we see:

WARNING : the curve looks as if it bottoms out to a minimum ... but this is a 0.2 degree ripple in a 300 degree signal in a simulation which is AT BEST 20% accurate.

This also affects the decision at the end of the cycle : the heater triangle peaks, and reverses direction. But at the center this will show up only as a small change of slope.

My guess at a rule : START a sawtooth when the temperature changes from FALLING to FLAT.

The March Hotcat HT2 is intentionally running at lower power (T=300C, COP=3) : let's look at the December equivalent (T=500C, COP=6):

This is a "clear" signal.

Now consider the center when it has to decide to turn OFF. We run the simulation up to the point where it has to decide where to continue rising, or to stop:

It is not at all clear from the rising waveform whether to stop, or to rise indefinitely.

My guess at a new rule : START a sawtooth when the downward temperature changes from to STEEP to SHALLOW. STOP .... when????

I can see how to do that with an arbitrary algorithm, but I don't see how to relate that to a physical process.

Next steps : Calibrate this model using actual values of specific heat and thermal resistance for the various components. This should improve the accuracy from 20% to 10%.

I expect that the exact shape of the "ripple" will be different, but that we will still see a levelling of the control signal, and not a rise.

Conclusion (so far) : The Sawtooth heat production looks like a good description of the heat production. The mechanism which produces this sawtooth is not obvious.

Missing/Confusing : The authors most likely have a lot of information about the applied power and temperatures when the hotcat was being started (by increasing the input in a series of steps), and in the cool-down stage -- both in the live and dummy tests. These could be used to calibrate the thermal model. In addition, they should publish pictures of the middle ceramic section and the way the heater wires are arranged (or confirm that it is similar to the Penon version). Also, confirm that the outer cylinder is steel in all tests.

3. Components of the eCat (Top)

This information is obtained from the above paper, plus an internal report by Penon of an internal test, using the same methodology, but with a different topology (the eCat was in the form of a tube, with a gap in the center, rather than the current cylinder).

The description of the July 2012 "Penon" ecat is:

The first component (outer cylinder) is made of AISI 316 paint-coated stainless steel: length 33 cm, Ø 8.559 cm, weight 1272.7 g, as measured before assembly. The cylinder was subsequently painted also internally with the same black coating, heat resistant up to 1200°.

The second component is the inner cylinder, made of the same stainless steel alloy: Ø 3.385 cm, weight 705 g, as measured before assembly. The cylinder was subsequently painted on both outer and inner surfaces with black coating, heat resistant up to 1200°.

The third component is a set of two 220 V resistors, total weight 2292.8 g; resistance values measured at the start of the test are 11.8 and 12.2 Ω. In parallel the recorded value was 6 Ω. Resistors were powered in parallel.

Lastly, there are four ceramic cable fittings for the resistors, each weighing 6.1 g, for a total of 24.4 g.

In the photo there is a clear gap at the middle of the length of the cylinder. Presumably the left and right halves were each made as two half-tubes.

For the November test we have:

Besides some minor geometrical differences, in the E-Cat HT used for the November test the
charge in the inner cylinder was not evenly distributed, but concentrated in two distinct locations
along the central axis. In addition, the primer resistor coils were run at about 1 kW, which might
be the cause of the ensuing device failure. For these reasons, a more prudent reactor design was
chosen for the test held in December and March, by distributing the charge evenly along its
container cylinder, and limiting the power input to the reactor to 360 W.

If one relates the length of the vertical line (32 pixels) to the diameter of the device (11 cm), one
may infer that each pixel in the image corresponds to a length of approximately 0.34 cm on the
device (with some approximation, due to the fact that the thermal image is a two-dimensional
projection of a cylindrical object). The thermal image shows a series of stripe-like, darker
horizontal lines, which are confirmed by the five temperature dips in the Y Profile. This means
that, in the device image, a darker line appears every 6.4 pixels approximately, corresponding to
2.2 cm on the device itself. As mentioned previously, the E-Cat HT needs resistor coils in order to
work; these are set horizontally, parallel to and equidistant from the cylinder axis, and extend
throughout the whole length of the device. By dividing the circumference of the base of the cylinder
by the number of coils, one may infer that the 16 resistor coils in this device were laid out at a
distance of 2.17 cm. one from the other. And, by comparing the distance between darker stripes
and the distance between coils, one may reach the conclusion that the lower temperatures picked up
by the thermal camera nicely match the areas overlying the resistor coils. In other words, the
temperature dips visible in the diagram are actually shadows of the resistor coils, projected on
the camera lens by a source of energy located further inside the device, and of higher intensity
as compared to the energy emitted by the coils themselves.

The description of the December 2012 "HT" ecat is:

The E-Cat HT-type device in this experiment was a cylinder having a silicon nitride ceramic outer shell, 33 cm in length, and 10 cm in diameter. A second cylinder made of a different ceramic material (corundum) was located within the shell, and housed three delta-connected spiral-wire resistor coils. Resistors were laid out horizontally, parallel to and equidistant from the cylinder axis, and were as long as the cylinder itself. They were fed by a TRIAC power regulator device which interrupted each phase periodically, in order to modulate power input with an industrial trade secret waveform. ...

The most important element of the E-Cat HT was lodged inside the structure. It consisted of an AISI 310 steel cylinder, 3 mm thick and 33 mm in diameter, housing the powder charges. Two AISI 316 steel cone-shaped caps were hot-hammered in the cylinder, sealing it hermetically. Cap adherence was obtained by exploiting the higher thermal expansion coefficient of AISI 316 with respect to AISI 310 steel. Finally, the outermost shell was coated by a special aeronautical-industry grade black paint capable of withstanding temperatures up to 1200° C.

For the March 2013 HT2" test they say:

The March test was performed with a subsequent prototype of the E-Cat HT, henceforth termed E-Cat HT2, differing from the one used in the December test both in structure and control system.

Externally, the device appears as a steel cylinder, 9 cm in diameter, and 33 cm in length, with a steel circular flange at one end 20 cm in diameter and 1 cm thick. The only purpose of the flange was to allow the cylinder to be inserted in one of various heat exchangers, which are currently under design. As in the case of the previous model, here too the powder charge is contained within a smaller AISI 310 steel cylinder (3 cm in diameter and 33 cm in length), housed within the E-Cat HT2 outer cylinder together with the resistor coils, and closed at each end by two AISI 316 steel caps.

For the "dummy" run they say:

The electrical power to the dummy was handled by the same control box, but without the ON/OFF
cycle of the resistor coils. Thus, the power applied to the dummy was continuous.
Power to the dummy’s resistor coils was stepped up gradually, waiting for the device to reach thermal
equilibrium at each step. In the final part of the test, the combined power to the dummy +
control box was around 910-920 W. Resistor coil power consumption was measured by placing the
instrument in single-phase directly on the coil input cables, and was found to be, on average, about
810 W. From this one derives that the power consumption of the control box was approximately =
110-120 W. At this power, the heat produced from the resistor coils alone determined an average
surface temperature (flange and breech excluded) of almost 300°C, very close to the average
one found in the same areas of the E-Cat HT2 during the live test.

...

This was calculated assuming an ambient temperature value of 14.8°C.

Since the Penon and the March systems both have "steel" outer cylinders, the description for December seems erronious : I will presume that the outer cylinder in all cases is STEEL, possibly COATED with a silicon nitride ceramic. (This probably has no impact on this thermal model, so it is omitted).

The individual resistor wires are spirals (from the photographs they appear to be twisted pairs), but they are higly umlikely to generate any significant electromagnetic field.

The eCat system consists of

An electronic controller (Blue and Yellow in Levi Fig 4 and 7)
A power-device, made up of 3-phase Triacs in December (Black and Yellow ) and an unspecicified 1-phase controller in March, which is not in any of the photographs
Power cables to the eCat. There are no obvious return cables providing feedback to the controller.
An outer steel cylinder, (possibly coated on its outside with silicon nitride ceramic), painted on its outside with black paint, (and on its inside with [...]?)
An inner AISI 310 steel cylinder, containing the reactive components, presumed to be nickel, an unidentified "catalyst" and (per Penon) a metal-hydride tablet providing hydrogen. [Note : The ends of the tube are welded by a technique historically and ironically known as "cold fusion". The ends are not considered in this model.]
A ceramic corundum cylindrical insert (shown only in Penon -- where it is actually made of two sections), which in the new hotcat probably fits tightly in the gap between the inner and outer cylinders.
A set of resistive heater wires in in troughs in the ceramic the length of the cylinder, and evenly spaced round the cyclinder. In the Penon version they are on the outside of the cylinder. Their position in the new versions isn't specified (although the meltdown November pictures shows a "cog" pattern on the outside) -- they are assumed to be near the outer edge.

July 2012 Hotcat (Penon)

This is a missed opportunity for the radiative calorimetry. You are looking into the orifice of a classical black body, where the inside is in radiative thermal equilbrium. They should have used the infrared camera from this angle.

With the resistor wires running the length of the eCat, close to the outside ceramic cylinder, and doubling back on themselves there is at most a small chance of their generating electric or magnetic fields which could communicate in any way within the outer cylinder, and a vanishingly small chance that they could penetrate to the inner steel cylinder.

We therefore have to conclude that ANY communication from the controller to the inner reactor chamber can ONLY be accomplished through heat transfer. (Some have suggested the possibility of creating thermal D-waves, but that is beyond the scope of this analysis).

November Hotcat (Levi) -- the reactor is now in the inner cylinder

This hotcat "ran away". The pattern of brighter and duller bands indicates that a similar ceramic insert is used, with the "cog" pattern holding the resistors, at the outside edge of the hotcat. NOT a shadow!

4. Temperature vs Time Plots of the eCat (Top)

Three plots of Temperature vs Time are available

Penon's plot clearly shows that power was supplied in a series of steps, resulting in an exponential RC rise of temperature for each step. Then the hotcat was turned off, with an exponential fall as it cooled.

The December Hotcat was already running when they started the test, and they do not record the cooldown phase. Since it was running at constant power, there is little relevant information in this plot.

For the March test the hotcat was heated up in a series of steps, and then switched to automatic control, where "full power" was applied with a 35/65 duty cycle with a period of about 450 seconds.

They also performed a test on a "blank" or "dummy" ecat, with an empty reactor. Again, it was heated in a series of steps, measurements were taken after it was stable, and then it was turned off. We only have the stable values, but they do provide important calibration information.

Unfortunately the Levi team did not include a thermal plot at these critical times, or for the cooldown phase. They do, however, include a representative plot of temperature for a few cycles of the resistor heater.

The question is : can we derive any useful information about the eCat from this little diagram?

(Note to self -- Insert H2 diagram)

5. Spice and Thermal Modeling

Spice is a program for electrical circuit simulation ... calculating the voltages and currents in circuits composed of resistors, capacitors and many kinds of transistors.

It can also be used to simulate heat, with the following equivalences:

Electrical	Thermal
Voltage	Temperature
Charge	Heat
Current	Heat Flow (Power, Watts)
Resistance	Thermal Resistance --- T/Watt
Capacitance	Specific Heat [Watt-Seconds/Kg] * Mass

Spice will be used to model the thermal behaviour of the HotCat.

I am using LT Spice , which includes a schematic editor and a waveform viewer.

(I will make my simulation files available for anyone who wants to try it out.)

Note : Robert Ellefson (on vortex) pointed me to an article on potential problems with Spice thermal anaylis, and their solution Stout & Billings. I believe I have taken all of these into account, and have a stable and reasonably accurate analysis.

Needed : the thermal resistance and specific heat of steel (easy to find), and the ceramic insert.

6. First Thoughts

This is a simple model, with a current source driving an RC circuit.

With a step input function (blue) we get the expected exponential rise and fall.

After some experimentation I discovered that a triangular wave produces something very similar to Levi's mystery waveform:

With the triangular waveform the initial response starts off coincident with the rising edge of the pulse, where it is concave (like a sin wave), rather than convex like an exponential.

And then, when it falls, the peak of the response is shifted to the right of the stimulus.

Of course the actual hotcat has a square-wave input, so we need to find some combination of input and response.

7. Lumped Models

This is used to explore the OUTPUT response.

At the bottom right are the lumped capacitor and resistor -- which includes both the radicative and convective terms.

At the top left are THREE current sources (For simulation reasons, in three stages). These ALL inject current onto the output node.

Note 1 : I am doing the simulation as if the ambient were 0 C -- freezing point -- represented by Spice's Ground node. This offset from the actual ambient of 17C (#check) is not significant for this analysis.
Note 2 : Although these results are "in the ballpark", they should be regarded as QUALITATIVE only.

7.1.Heater Input

First, consider the heat from the input resistors.

One input component describes the "starter" power, and another describes the repeating heater pulses.

Note : the input current was determined by increasing it until the final temperature reached 300C. (Not shown), and the resulting value was used in all subsequent runs.

7.2.Blank Hotcat

Let us apply these heater stimuli to a "Blank" hotcat:

In response to the step function of the STARTER signal, the output temperature (voltage) rises in a series of exponential steps to 220V (C)

When the pulse is applied, the output rises exponetially when the pulse is ON, and falls exponentially when it is OFF.

With the blank hotcat, this cannot sustain the starting temperature, so it gradually falls to about 35% (#check calculation) of the start-up value.

When the hotcat is turned off, the temperature just falls exponentially to room temperature.

Here is a detail of that plot:

The Time constant R*C is long compared to the pulse, so it looks almost like a linear rise and fall. (In electronics, this is often used to create a triangular wave from a square-wave input ... for instance, to drive the horizontal sweep of an old-fashioned CRT TV set).

7.3.eCat generates heat in a Square-Wave

The hotcat is presumed to generate excess heat in the form of a square wave, which is offset from the input stimulus, so that the peak is offset to the right.

The model is calibrated as follows:

The Input peak is adjusted so that the stable peak value matches the actual peak.
The Capacitance is adjusted so that the trough value matches the actual trough.

This is clearly not a good match to the actual waveform .. it looks more like a triangular wave than a sin wave.

7.4.eCat generates heat in a Triangular-Wave

The combination of the heater's square wave and the eCat's triangular wave looks like a very good match for the required output.

(I would have stopped there if I hadn't gone to a more detailed model).

7.5.eCat generates heat in a Sawtooth-Wave

This starts off well ... the fall is OK, but it is not displaced enough to the right.

Those are the main repeating waves which can be described with a Spice "PULSE" command.

The only other one which I tried (and didn't report) is trapezoidal rise-hold-fall, but this wasn't a good match.

Thinks .. I didn't try a "wedge" .. rise/hold/zero

7.6.Lumped Model Comparison

As indicated earlier, the Triangle is best, Sawtooth next-best and square-wave worst.

8. Ladder Models

Next up : a LADDER network of RC components.

That's the ladder circuit. Well, a ladder lying down. It is sometimes drawn in the vertical orientation, where the capacitors would be the horizontal steps.

At the top left are the three inputs : the STARTER and PULSE heaters, and the ECAT heat. But they go to different places.
At the bottom right are the two resistors, but the Capacitor now represents the specific heat of the OUTER cylinder.
Comments are in blue.
The black text at the bottom are the Spice commands describing the waveforms.

This is the ladder model, zoomed in. From right to left we have :

The load resistors
The Output Cylinder, represented by C_OUTER (its mass * specific heat)
We then make the transition into the Ceramic : since the resistor is close to the outside it has a small thermal resistance
This takes us to the MIDDLE. Here we have the Capacitor representing the specific heat of ALL the ceramic. This is where the STARTER and PULSED heat from the resistor is deposited.
Then we have to go left through the rest of the ceramic (with a larger resistance), to the inner cylinder and its capacitor.
Finally, we step into the void at the CENTER. We have to have a small capacitor and resistor even though it might in fact be empty space with little or no thermal mass. (Typo on the diagram : hydRide).

The capacitance of the lumped model has been spread between the various sections in an "eyeballed" manner. The resistances are hypothetical. At the next stage these will be replaced with values from the actual components.

Here we see that the eCat's anomalous heat is considered as originating in the center cavity.

This extra heat combines with all the other heat stored in, coming into, or leaving the hotcat, producing a resultant temperature on the surface of the inner cylinder.

This is transferred back into the center ... where it may affect the rate of heat production.

By applying the various possible waveforms I am emulating an unknown control mechanism.

We then play the various INPUT waveforms into both the LUMPED and the LADDER model.

The general shape is the same, so we will only look at the stable waveform. We will be looking at both the OUTPUT temperature, and at the CENTER temperature (which is what a control mechanism would know about).

Blank :

The lumped and ladder models are very close, the ladder being, as expected, a bit more rounded due to its passage through multiple RC's.

This waveform will show up as a 10 degree ripple at the center, even when the hotcat is generating heat at the 300 degree level at the output.

Square wave :

The output is now softer than the lumped version, looking much more like the observed waveform.

The center sees a sharp exponential rise and fall.

Triangle :

It turns out that I didn't calibrate the output correctly, so the lumped and ladder models disagree about the level.

The problem with this model is that the input has to turn off BEFORE the output reaches its peak. (Is this precognition?) The shape of the center temperature bears NO resemblance to the supposed triangular input.

Sawtooth:

This is encouraging, because the output (LADDER) is a better approximation to the required signal, and the waveform at the center has SOME resemblance to the assumed pattern (although it looks more like a triangle than the triangle did!)

So ... is there a control mechanism that fits into this environment?

9. The Control Model Environment

Lets's look at the Temperature environment at the center of the reactor. Here I initialized a hotcat, and show the result after TWO cycles of a Sawtooth waveform.

Then I OMIT my manually injected waveform, and consider what the pixie-dust would see.

So we don't have a lot of information about when to START. There is NOT a clear square-wave edge to trigger the reaction.

WARNING : the curve looks as if it bottoms out to a minimum ... but this is a 0.2 degree ripple in a 300 degree signal in a simulation which is AT BEST 20% accurate.

Just in case you are wondering if the pulsed heater signal got lost, let's drive the hotcat with CONSTANT heat, bringing the output temperature up to the required level:

So the heater pulse IS there -- with the expected ripple value of around 10 degrees. It's just swamped by the hotcat's output.

The March Hotcat HT2 is intentionally running at lower power (T=300C, COP=3) : let's look at the December equivalent (T=500C, COP=6):

This is a "clear" signal.

Depending on the COP, we can probably say that the center temperature either levels out, or falls less steeply.

Now consider the center when it has to decide to turn OFF. We run the simulation up to the point where it has to decide where to continue rising, or to stop:

It is not at all clear from the rising waveform whether to stop, or to rise indefinitely.

My guess at a new rule : START a sawtooth when the downward temperature changes from to STEEP to SHALLOW. STOP .... when????

I can see how to do that with an arbitrary algorithm, but I don't see how to relate that to a physical process.

Next steps : Calibrate this model using actual values of specific heat and thermal resistance for the various components. This should improve the accuracy from 20% to 10%.

I expect that the exact shape of the "ripple" will be different, but that we will still see a levelling of the control signal, and not a rise.

David Roberson (on vortex) is exploring the control mechanism, also through a Spice model. At some stage we will probably combine our two models and see what happens.

10. Thermalization

One more thing. I'm going to inject my personal opinion here.

One of the arguments against the anomalous heat generation is that IF it occurred ON a particle of pixie-dust, it would get too hot, melting, and presumably stopping the reaction.

My opinion is that the thermalization actually takes place elsewhere .. in the steel of the inner cylinder. The inner heats up, and heat is then transferred back to the center independently of its generation.

I have therefore moved the eCat's heat generation from CENTER to INNER.

(This also makes the modelling a little easier).

Looking at the temperature of the center, we can see that it still has the 0.2 degree ripple.

operties like thermal resistance and specific heat, and use the actual ambient rather than 0C.

But .....

11. Mesh Models

The ladder model I am using is strictly true only for heat propagation through a wall, extending sideways to infinty.

We should really model at least a "slice" of the cylindrical pie.

(There might be a mathematical formula to transform the results ... )

So .. the final level of modeling is a mesh model (2D should be enough), but I don't think it's going to be worth the effort. Beyond that, one needs to use a 3D finite-element model like Elmer.

12. Copyright

Permission is given to copy this document in its entirety, or to extract sections for review, provided my authorship is recognized.

The images from Levi and Penon were obtained by screen captures from their documents. I have stamped them as "my copyright" for the "DERIVATIVE" component I have added (Mainly labels). I am not claiming ANY copyright for the originals : I included the links to their documents in the images. All spice simulations and other data are exclusively mine. Permission is given to use my diagrams, and to ADD to them, provided you do not DELETE any of my information, and provided you leave MY copyright notice in the derivative work. (You may wish to add YOUR mark for any additions you make. This is similar to the ancient Chinese practice of putting Chops on copies.

When using these images on the WEB do not down-res the image (because all the fine-print will disappear) -- simply display the image with a smaller WIDTH, so that the user can open the full-res version. If you make a small scale image in a PRINT (or PDF) version, and the details disappear, then add the information to your caption.

(Technically, although I have given permission to use my versions of Levi/Penon images, you also need to make sure that the use of their underlying images constitute fair use.)

I wouldn't have bothered with this, but a well known Physics Organization site recently used one of my modified diagrams (which I prepared exclusively for Forbes), carefully cropped off my name, and didn't mention where they ~~got~~ stole it from.

/s/ Alan Fletcher