This invention relates to improving power transfer, and in particular to devices and methods for improving power transfer from ferro fluid used as part of heat engine or refrigeration cycle. The devices and methods disclosed herein build upon those laid out in earlier disclosures by the inventor R.O. Cornwall, including WO0064038, US6725668, JP2002542758, EP1171947, CN1376328, CN100388613, AU4307100 and US2008297290.

The present invention provides apparatuses and methods for converting energy, in accordance with the accompanying claims.

In order that the invention can be more readily understood, embodiments thereof will now be described, by way of example, in which:

Figure 1 shows an ideal Carnot cycle P-V and T-S diagrams;

Figure 2 shows P-V and T-S diagrams when the thermodynamic identity is altered by additional terms; Figure 3 shows two thermodynamic cycles laid out in the earlier patents of Cornwall;

Figure 4 illustrates the kinetic theory model;

Figure 5 illustrates Maxwell's Demon and a phase change and phase changing heat engine;

Figure 6 illustrates the Clausius and Kelvin-Planck statements of the 2 ^nd Law of thermodynamics;

Figure 7 is a schematic view of a regenerative means to cycle the magnetising fields;

Figure 8 is a graph of magnetisation vs. temperature and table of Curie points;

Figure 9 is a depiction of a ferro fluid particle;

Figure 10 shows the difference between the Langevin function and 0.9*tanh(x/3)

plotted to rn-H / kT = 6;

Figure 11 shows power loss in ferrofluids by Bode plots;

Figure 12 is a plant diagram;

Figure 13 is a T-S diagram for a cycle embodying the present invention;

Figure 14 illustrates a model for working substance and electrical load;

Figure 15 illustrates how a linear resistance leads to the return of the magnetising energy;

Figure 16 shows a model for working substance, series linear capacitance and electrical load or working substance, parallel linear capacitance and electrical load;

Figure 17 shows how a capacitor can lead to self-sustaining oscillations into an electrical load;

Figure 18 is a schematic view of a model for working substance and non-linear electrical load;

Figure 19 shows how the non-linear resistance can lead to energy in excess of the magnetising energy;

Figure 20 shows schematically the non-linear resistance used for space heating or a heat pump or as the hot reservoir to a Carnot cycle engine;

Figure 21 shows a possible configuration of the non-linear resistive element;

Figure 22 shows a model for working substance, series non-linear capacitor and/or series non-linear inductor and linear electrical load; and Figure 23 is a plant diagram for use of the two cycles as a conventional Carnot cycle.

1. Summary of the Conventional Art in Thermodynamics

Current thinking in thermodynamics requires that a heat engine operates between two reservoirs and that there is a limit to how much heat energy can be transferred to work. These concepts are derivable from the combination of the First and Second Laws in the general thermodynamic identity [1-3] for the control volume:

dU = SQ - SW

=> dU = T(dS + dS _gen ) - PdV + HdM + μ (η,Ρ, Τ,φ)άη

Shown is the contribution of dS _gen of entropy increase due to irreversibility. For illustration a variety of work terms have been included: pressure volume and magnetic work (field strength H and moment I). The chemical potential is a function of the amount, the pressure, the temperature and the potential energy of the substance.

We are concerned with cyclical devices meaning that the thermodynamic co-ordinates of the working substance are the same at the start and end of the cycle, so dU = 0. The conventional Carnot view is that this can only happen between two reservoirs giving the work as the net difference between the heat absorbed from the upper reservoir and rejected into the lower reservoir: W = Q _H - Q _L . This is accomplished by the 'work terms' P(T,...) or H(T,...) and the like. By a series of operations the working substance is transformed to effect equation 2, that is:

Where i is an index for the steps. An example is the idealised reversible (dS, 0) Carnot cycle (figure 1) which is completed by two reversible adiabatic and isothermal processes.

Step 1-2: Isothermal expansion. Q _H = W = JPdV = mRT _H lnCVz/V

Step 2-3: Adiabatic expansion. Q ₂ _3 = 0

Step 3-4: Isothermal compression. Q _L = -mRT _L ln(V4/V ₃ )

Step 4-1: Adiabatic compression. Q ₃ . ₄ =

For adiabatic processes temperature can be related to volume and the compressibility factor of the gas:

The efficiency after eliminatin

This result applies for all engines operating between two reservoirs because the result is independent of properties of the working substance and only relates to the reservoirs. Indeed, Rosenweig [4] covers a heat engine whose working substance is a magnetic fluid transiting a closed loop with variation in magnetic field. The working substance absorbs and rejects heat and experiences substantial changes in the pressure of the magnetic fluid. The pressure volume work of the magnetic fluid is once again Carnot limited.

Returning to the simple ideal gas case, in a sense we can see how the trick works by eqn. 2's summations and figure 1 (the area between the lines is the difference in heat taken in and rejected to the lower reservoir - the work): it doesn't sum to zero because 'adiabatics are steeper than isothermals' by the popular maxim. There we are constrained in our movement on the PV diagram by a family of curves: adiabatics, polytopies and isothermals. We have to alternate adiabatics with isothermals to get the path to map out an area on the PV diagram in a cyclical process otherwise we just come back the way we came and achieve nothing. Thus with this viewpoint we need two reservoirs. Also high grade heat energy is seen to be immutably turned to low grade heat.

2. The Thermodynamic basis of the disclosures of Cornwall We have summarised the conventional heat engine theory but the thermodynamic identity (eqn. 1) affords another method of building a cyclical heat engine. The trick of the heat engine is to sweep out an area between the trajectory of thermodynamic variables in a PV, HM or TS diagram, the working substance comes back to initial conditions (dU = 0) to complete the cycle. The implication of the Carnot result is that the working substance can only come back to initial co-ordinates after absorbing heat energy from the upper reservoir by rejecting some heat energy to the lower reservoir. The working substance is 'forced' around the trajectory by external influence of the two reservoirs (placing it in thermal contact). Merely cycling the thermodynamic co-ordinates of the working substance alone, in isolation (say along an adiabatic), would achieve nothing useful.

There is a way of getting the trajectory to 'jump off the adiabatic and render the thermodynamic identity inexact, by changing the phase of the working substance. We can make the working substance do work by the phase change and at the end (penultimate step to be more precise) of the cycle have lower internal energy than at the start - the working substance forms its own virtual cold reservoir. We then place the working substance in contact with just the one reservoir to bring it back to the start co-ordinates.

First let us 'unpack' the chemical potential's meaning: it is the thermodynamic potential per particle:

du = Tds - pdV + φ

Where s and p are the entropy and pressure per particle. The chemical potential has two parts [2] the 'internal' and 'external'. The internal potential is defined as the chemical potential if an external potential is not present. If we are to extract internal energy from the system, the expression for dU must be made inexact:

5u = Tds - pdV + φ + φ'

A change of μ can only correspond to a phase change, as this will introduce potential energy terms φ' such as latent heat (1 ^st order transition) or new magnetisation energy terms for instance (i.e. dipole work, 2 ^nd order transition). It is almost as if we have formed a different substance by the phase transition which then sweeps out a new path in TS space, figure 2 illustrates this.

At 1 : Working substance in contact with reservoir

Step 1-2: Work done adiabatically on substance

At 2: Working substance undergoes phase change forming its own

virtual cold reservoir

Step 2-3: Expands adiabatically but follows a different path and work leaves system

so internal energy decreases as does entropy.

At 3: Phase change back to original substance

Step 3-1 : Heat flow from reservoir to working substance along with phase change

Thus only one reservoir is needed.

To illustrate this more specifically and in more detail, figure 3 shows the two 2 ⁿ order magnetic cycles presented by Cornwall in earlier patents. We shall see later, in the electrodynamics and further thermodynamics sections, that the final step of each cycle has a work term MdM/dt added on to the thermodynamic identity what has the effect of the function φ'.

2.1. The Magneto-calorific Temporary Remanence Cycle

The first cycle in figure 3a is the magneto-calorific cycle. A magnetic material on the cusp of paramagnetic to ferromagnetic transition is cycled in a strong magnetic field:

1- 2: Field suddenly applied to sample and warms above ambient T _amb →T ₁ ii _g

2- 3: Sample cools to ambient in the field.

3- 4: Field switched off suddenly, sample cools below ambient T[ _ow

4- 1 : Sample below Curie point, has an independent magnetic moment (Appendix 1)

that is destroyed by the ingress of heat energy and does work MdM/dt.

4': Allows a variation where the induction is higher at step 4 than 3.

The magneto-calorific effect is an order-disorder 2 ^nd order phase transition [2] phenomenon whereby the magnetic field "freezes" out modes of energy partition in the system requiring it to be partitioned elsewhere, hence the rise in temperature. A similar effect can be experienced with a rubber-band in analogy to this cycle. Rapid stretching and then application to lips will show a temperature rise. On equilibrium with the body temperature, rapid contraction will cool the band below ambient as vibration modes frozen out by the stretching are re-established. Examples of materials displaying magneto-calorific effects are Gadolinium (Curie point approximately 290K) and Iron (700K). The rate of change of the flux is set by particle size as laid out in earlier disclosures.

2.2. The Delayed Relaxation Temporary Remanence Cycle

This is preferred cycle figure 3b for development of a device in that it doesn't require exotic,

materials like Gadolinium, large magnetic fields or operation about one temperature point.

1- 2: Field applied, sample magnetised.

2- 3: Field off suddenly leaving temporary magnetic moment.

3 - 1 : Temporary independent moment subj ect to buffeting from surrounding liquid

that does net electrical work MdM/dt.

Initially the magnetic moment and electrical work is large, the region surrounding the ferrofluid particle cools somewhat adiabatically before heat energy from the bulk fluid returns the particle to a temperature just below its starting point.

A further point of difference between these types of heat engine and conventional is that the latter heat engines run between two reservoirs and are Carnot cycled constrained; the change in magnetisation (or some other property) between two temperatures W = f |ΔΜ (7^— > T ₂ )} is the source of the power conversion. Whereas the magneto calorific and temporary remanence cycles affect a means of change in the magnetisation by the collapse of a temporarily remnant flux by Brownian motion such that

W = f |ΔΜ (i _j — > t ₂ )} ; power being generated, directly by Faraday's Law of Induction.

We shall return to more thermodynamic considerations in more detail after Kinetic Theory Analysis.

3. Kinetic Theory Analysis of New Cycles

There is no mystery to heat energy - it is random microscopic motion. In this section a simple kinetic model of a super-paramagnetic [4, 5] system, directly amenable to the two new cycles, shall be built modelling a 2-D array of coupled oscillators of nanoscopic dimension with Newtonian Dynamics. The model shows how macroscopic electrical work is obtained via the macroscopic flux collapse of microscopic domains randomising due to heat energy (figure 4).

First however some acceptable caveats shall be given. Newtonian Mechanics is justified because the energy and time scale of the process (the 'action') kTx » ht, where 'k' is Boltzmann's constant, T is temperature, τ is the time scale of the processes (greater than 10 ^"13 s the time of heat relaxation/phonon processes) and 'h' is Planck's constant. Physics is necessarily a science of approximation but enough features are introduced to simulate the 'mechanical' (electrostatic) and magnetic aspects of the system such that the system entropy and temperature can be defined.

A solid super-paramagnetic material (a 'ferroset', the magnetic analogue of an 'electret') is modelled where each moment interacts only with the external magnetic field B _ext . The basic equation describing the system is a system of coupled dipoles in a lattice, whose components couple to their nearest neighbours' dipole- dipole interaction forces,

W" <l .

The matrix of dipoles (i, j) of moment of inertia I, is represented by a state vector θ,, , θ„ , θ,, giving the angular acceleration, velocity and position respectively; motion is constrained in a flat plane with torques k _d i _p f( _" ) and μ x B. The effect of an external field B _ext is included in eqn. 6 too. In the second half of the temporary remanence cycle the collapse of the field of the dipoles as they become randomised induces a current in a surrounding coil which dumps its power into a resistance, R. The solenoidal field is given by B _∞ = μ ₀ ^η ί where n is the rums per unit length. Substitute i = '

where N is the number of turns, ψ is the flux from the dipole field collapse and ψ = μ ₀ ΜΑ where M is the magnetic moment per unit volume and A is the cross-sectional area:

Where

k _i is a dimensionless constant representing linkage

R is the resistance of the electrical load

e _j is a unit vector along axis j, only the flux down this axis causes the induction

By the Boltzmann relation [2, 3] S = k _B ln(w) where the entropy is related to the number of states, the magnetic and mechanical entropy of the array of dipoles in this classical setting is:

^m _a g ⁼ const x In (standard deviation O _i} )

S _mech = const x In (standard deviation Θ.. )

T = const x average (

On substituting eqn. 7 into e n. 6 and unpacking the last term this expression is developed:

That is, the average angular acceleration cancels the uncorrelated terms and tends to a simple expression which can be understood as a velocity damping term for each dipole which is a reflection of the mechanical work being turned to electrical work which is then dissipated in the resistance R (figure 4). Obviously this effect won't happen when there is zero resultant flux from the working substance as B _« . _xl (eqn. 6) must be greater than zero - it is hence reliant on the symmetry breaking of an order-disorder phase transition.

It is undeniable then that each paramagnet, although executing random Brownian motion, can be made to convert the micro-kinetic motions of heat energy to an external system. This proves the premise of this thesis that - low grade heat energy can be made into high grade energy again; because of this the next section shall discuss the "Maxwell Demon" problem in the light of these new thermodynamic cycles.

4. Literature Survey of the Maxwell Demon Problem

Reconstitution of heat energy to high grade energy is the subject of Maxwell's Demon[6-8] illustrated schematically in figure 5. This was thought impossible for the following reasons:

^■ A mechanical sorting element gets randomised too, until ineffective, Feynman[9] 'Ratchet and Pawl').

^■ Demon would be blind in black-body radiation to see anything, Kirchhofff 10].

^■ Or, to perform sorting would require measurement and expenditure of energy, storage of information, Szilard[7], Brillouinf lO], Landauer/Bennett[8].

A realisable Maxwell Demon is possible by the phase changing heat engines discussed herein because ultimately the sorting element is the phase change and hence the potential between the particles; this discriminant function is, as it was shown, the chemical potential between the phases and its action isn't ruled out by the considerations above. Figure 5 depicts a 1 ^a order system where those above the energy barrier and in the higher phase have undergone a direct sorting process. The author initially considered such a system with a water/calcium chloride (deliquescent/hygroscopic) solution and a reverse osmosis membrane: a tall column of the hygroscopic solution would favour water condensing and the head on the column would release water at the bottom through the reverse osmosis membrane. This was not practical. Another viewpoint of this is: At constant temperature (zero temperature difference) heat can be transferred to work with total efficiency (eqn.3) which would obviously be useless for a macroscopic heat engine as there would be no change in the system extensive variables (change in pressure, say), however microscopic work of partitioning particles across a phase boundary would occur. To achieve macroscopic work from this latent microscopic work we must form macroscopic regions of differing thermodynamic potential. An expression of microscopic properties at the macroscopic level is the phase of a substance and one phase can be made unstable, with respect to the other phase, to liberate this microscopic work.

Depicted in figure 5 too a second order system of the temporary remanence cycles (Magneto-Calorific and Delayed Relaxation) which were discussed earlier. Here an inverted sorting procedure is carried out: first (shown on the left-hand side of the figure) the magnetic field orders the magnetic part of the system which when the field is removed, it is randomised by the other modes of the system (shown on the right-hand side of the figure). It will be shown that the 'dipoie work', the electrical coupling forcing the system to perform useful electrical work can be made greater than the magnetising energy or losses of the process, later. The disruption of the order created in a finite region by the environment causes a lowering in the disorder of the environment.

4.1. Time's Arrow when dS = 0 or even dS < 0

So far in the preamble we have seen how the phase changing cycles have been justified by thermodynamics and Kinetic Theory. More thermodynamic justification will occur in later sections but it should be clear at this point that we are talking about a system that can operate and generate macroscopic work in a closed isolated system (dS = 0, figure 6). The two cycles discussed would form a cold sink in operation and even their losses (say hot electrical devices) could be coupled back to the input leading to the net conversion of heat energy into high grade energy (figure 6), the result is decreasing universal entropy.

The Second Law implies that there is a preferred direction of time but this is in conflict with the underlying equations of motion that are time reversible[ 1 1]. The Boltzmann H-Theorem[ 12] can be seen to be using circular logic - starting from the premise that the particles are statistically uncorrected (i.e. at maximum entropy) it derives a quantity resembling entropy that increased with time. This seemed satisfying. The true reason came from Chaotic Dynamics[ 13, 14-15] which was probably initiated by the new semi-quantitative methods of Poincare in his investigation of celestial mechanics. It had been known for sometime that there was no general solution of the three-body problem. Modern Dynamic Systems Theory has shown by toy models such as the Smale Horseshoe or Henon iterative maps, that distil the essence of complex dynamic systems, that points close in phase space will diverge exponentially (the so called Lyapunov exponents) leading to the churning of paths in a bounded chaotic attractor and the Ergodic Theorem. The system state vector in a short time accrues inaccuracy with exponential divergence and herein is the Arrow of Time: Loss of Information.

Violations of the 2 ^nd Law lead people to assume that type 2 perpetual motion devices are implied. This can never be the case but we argue 'heat reuse'. A machine utilising Maxwell Demon-like Thermodynamic cycles could exist in an isolated closed environment but it would eventually wear down and fail. It would have to have intelligence to repair itself (or self-propagate) but this couldn't be guaranteed for all time, as the materials from which it is made will get so diluted from wear and distribution that it couldn't refine them within the constraints of the power it could generate in the system. It also would have to have intelligence to cope with every scenario of malfunction or an incorruptible 'genetic code' to churn out perfect new copies of itself once the old one wasn't worth repairing. This is not perpetual but probabilistic.

Further sections now go into more detail and rigour than the Preamble.

5. Electrodynamics I

The characteristic feature of the two magnetic cycles presented is that they directly convert heat energy into electrical energy rather than generate pressure volume work, as in other magneto-calorific cycles[4]. The mode of doing work is by Faraday's Induction Law but let us use a unified starting point which derives that and the magnetisation energy needed to start the cycle:

Consider the total energy of a classical dipoie and then the first variation of this per unit time (the power): The mechanical and electrical energy in setting up dipole is taken into account [9] that is why the sign is positive, unlike the expression for the electro-static dipole. "Shaft-work" can be performed by the micro- mechanical rotations in the destruction of the ferrofluid remanence by Brownian motion. In fact observing the dipole equation, it matters not whether the magnetic moment is rotated wholesale or randomised

between the maximum and minimum energy configuration:

AE\ ^mm = ^r - B co O ^ ² or μ|° B cosO

The term "shaft-work" like a macroscopic electrical generator is appropriate and instructive. Indeed, the

Kinetic Theory analysis of an ensemble of microscopic dipoles showed that work from the flux collapse

looks like a "regular" macroscopic thermodynamic input.

However, unlike a generator, the magnetisation energy must be supplied on each cycle. Consider the

variation in energy without regard to angular variation (we have just proven variation of moment looks

exactly the same as rotation):

dE dE

dE =— du -\ dB

δμ dB

=Bd μ + μάΒ

The first term represents the energy change in the ferrofluid as it is magnetised. The field, B, is supplied by the solenoid. In the first step of the cycle the magnetising field is considered constant. Writing:

B = μ ₀ Η and μ = MV

eqn. 12

=> ^E _mag = μ ₀ ΗΜν

The power required to do this F times a second is of course:

P _mag.ws = W ₀ HMVF eqn. 13

Where η is the efficiency by which we can recoup this magnetising energy and re-use it each cycle.

On the second half of the cycle the flux change as the ferrofluid moment decays induces a current in the

power coil which resists the microscopic moments of the ferrofluid from decaying (Kinetic sections), in

short they are forced to perform electrical work. The variation in 'B' is then considered.

The following substitutions are made:

μ = MV

N

B = μ ₀ — i (ref. 6, from Maxwell Law/Ampere's Equation) where

L

N is the number of turns and L is the length of the solenoid

άλ/

i is the current, i = ¹ where λ is the flux linkage

R

=^> i =— N^- where ψ is the flux, ψ = μ _η ΜΑ, A is the cross-sectional area

R dt

Overall, putting n as the number of turns per unit length, the change in the energy of the working substance,

E _ws is.

Differentiating the above expression for E _ws gives the instantaneous power developed by the working

substance:

This result is also directly derivable from a Faraday Induction Law treatment. Note that in eqn. 14 only the time differential of the moment μ is taken and not the time differential of the product MdM/dt. Thus eqn. 10 is a complete statement of the behaviour of the working substance and one must note that although the maximum magnetisation/electrical energy (eqn. 10) is constant (indeed the time variation

ofrfi d-E Ίμ^ ₍ _ ₀ ) shaft- work can enter the system as expressed by the constant in eqn. 10 leading to

> δμ dt

, , · , r-p, , ·, , dE dE du \ dE dB dE άθ \

cumulative work. Thus eqn. 11 can be written as:— = — showing the energy from

dt θμ dt dB dt 3Θ dt

the power into the system as cumulative, the differential is still exact.

^' dM

The power generated by the working substance then is potentially squared in frequency, compared

dt j

to the power required to magnetise the working substance, eqn. 13. This analysis is a little naive and we

shall see in a later section that non-linearity is needed to extract net electrical power yielding:

Where κ is a positive constant. The goal then, is to have P _ws exceed Pmag.ws by good design. Let us now look at a good method of magnetising field cycling.

5.1. The Method of Cycling the Magnetising Fields

The section after this will explore the working substances most amenable to the two cycles presented.

However, it is a feature that the magnetic moment developed by the magnetising field is usually small:

M = χΗ eqn. 17

Where χ is the unit-less quantity called susceptibility and is usually positive and small. To develop an

appreciable magnetic moment in the working substance requires large magnetising fields. The power

required for this is:

Ρ _Η = ημ ₀ Η ² νΡ eqn. 18

1 2 f 2

Which is derivable from the field energy [9] assuming B and V are constant and replacing

2 ^J

the constants with the permeability constant. Taken together with eqn. 13, the total magnetising power is:

P _mag = W ₀ HVF (H + M) eqn. 19

Figure 7 shows a regenerative means to cycle the magnetising fields. To the left-hand side in figure 7, a low resistance LCR circuit 1 is commuted by a triac 2 at zero crossing. This is a good means to make η in

eqn. 19 small. A power coil 3 around the left hand magnetising coil [is then switched in after zero crossing in the first, magnetising circuit, collapsing the flux into an electrical load (for instance, in the form of a

resistor) in a separate load circuit. Some of the developed power can be used to re-charge the capacitor 4,

against the losses, for the next cycle.

Below in figure 7 are shown two oscilloscope traces: the upper trace is the field H and the lower is the

induced magnetisation M. It can be seen that the magnetisation typically follows the H field with a lag

(more on this in a later section) and when the H field is zero, the magnetisation decays providing an

independent flux (appendix 1). In the first trace 5, the H field cycles so slowly that there is hardly any lag

and hence remnant flux. In the second trace 6 a higher frequency of cycling ensures more lag and more

remnant flux, however a later section will show that this incurs more irreversible losses.

Electrical means have been shown to cycle the fields but mechanical methods can be used too provided that a window of zero field or discontinuous change in field is made. This can be achieved by techniques

employing pole shaping of the magnet.

6. Thermodynamics II

The work required to magnetise a substance[2-5, 9] is given by:

δΨ = -μ ₀ ΗάΒ eqn. 20 Substituting the magnetic field density [4, 5, 9] B = μ ₀ (H + M ) we obtain:

5W = -μ ₀ ΗάΗ - μ ₀ ΗάΜ eqn. 21

Around a cycle -μ ₀ HdH will certamly be zero for mechanical cycling and close to zero by electrical

methods (eqn. 18) so this term can be neglected for thermodynamic analysis leaving:

SW = -μ ₀ ΗάΜ eqn. 22

Let us write functions of the independent variables Η and T and then write terms for incremental work and heat, if M = M (H, T), c = (H, T) and g = (H, T) then:

c(H, T)dT + g(H, T)dH

Using the thermodynamic identity dU = SQ— 5W and relations between 1 ^st order and 2 ^nd order crossed

partial derivatives[4] these relations ultimately arise:

The entropy of the two types of working substance for the two types of cycles presented herein will be

discussed in the next section.

Another work term was identified in the electrodynamics section related to the rate of change of flux. This is not usually treated in thermodynamic texts or texts on magneto-calorific work conversion because the

matter is novel. The new work term to be considered was: oW o — M (see eqn. 14). Writing then,

dt

with the independent variabl

( dM δτ (Τ)

Thus a remnant flux heat engine is possible if and/or are functions of temperature. We

dT I dT

shall see that this is so in the next section.

7. Description of the Working Substance for the Two Cycles

The communality between the two cycles presented is decaying, independent, remnant flux at the end of

the cycle which leads directly to electrical work by Faraday's Induction Law. In the Magneto-calorific

cycle the decay rate is designed by the size of the particles in relation to the rate of heat flow from the

surrounds to the magnetic material. In the Temporary remanence cycle, the rate of decay is by a different

mechanism which, of course, can be designed too.

Key to both cycles and materials is the Brillouin function[2-5, 9]: _n , x 2S + \ , f 2S + 1 λ 1 ( χ

B ( x ) = COth COth eqn. 26

S S J 2S 2S J

Where,

m - H

X = eqn. 27 kT

Which is the ratio of the energy of a dipole in a field (eqn. 10 appropriately scaled since H rather than B is used here) to thermal energy, k being Boltzmann's constant. This can be written another way to show the

quantised nature of the magnetic moment:

gμ _B S ^■ H

X = eqn. 28 kT

Where g, the Lande spectrographic splitting constant and μβ, the Bohr Magneton are beyond the relevance of the discussion[2-5, 9] and refer to the relation between fundamental quantum spins (angular momentum) and the magnetic moment - they are just a proportionality constant. However the spin S, is relevant to what we consider our "fundamental" magnetic particle in our following discussion - be that single electrons or

clusters of atoms.

The Brillouin function is derived by statistical mechanics as the probability of finding the system in the

state of our concern. The partition/occupancy function[2,3] is defined as:

* =∑ ^< e ^kT eqn. 29

Where s, represents a numbered state and E _Si the energy of that state. The probability of finding the system

in a certain state (degeneracy is not discussed here) is:

P =— e ^kT eqn. 30 s Z

The expected value of a property is given by:

Of interest to the current discussion is the average magnetisation in the z direction:

eqn. 32

Where the energy of the state, E _Si has been shown as -m(n)H where the moment, m, itself is a function of

the amount of spin the particle has. Eventually this expression yields the Brillouin function.

The point of this discussion hasn't been to merely retread the argument but to show the underlying

relationship between three types of magnetism: Paramagnetism, Super-paramagnetism and Ferromagnetism

- the energy and the spin.

In the first instance for spin ½ and energy -m.H, the Brillouin function reduces to paramagnetic "spin gas" which many slightly magnetic materials (including solids) obey:

M (T) = M _s eqn. 33 Where M _s (0) is the saturation magnetisation at absolute zero and is related to the spin or magnetic moment density. For higher spins it is possible to truncate the Brillouin function[5]:

, \ (25 ^* + l) ² - l / , S + \ ( , \

B {x) = ± X + 0(x ³ ) = - X + 0(x ³ ) eqn. 34

It is very hard to saturate such materials and the relation can be linearised if H (eqn. 27) is not too high

giving the Curie Law[2-5, 9]:

M (T) = NS {S + l)( _{g ½} ) ² ^ = M _S {0) eqn. 35

Let us now change the energy function in eqn. 29 to include additional terms:

E _s = -∑ gμ _B S _i - H -∑'J ₉ S S _j -∑KV sin ² 0, eqn. 36 i ij i

The first term we have covered is of the form m.H and is the energy of a dipole in the field. The second

term is the so-called "Heisenberg Exchange Interaction" [2-5, 9] which is a very strong quantum

mechanical interaction of spins from the Pauli Exclusion Principle (via an intermediary electron - the

wavelength of participant electrons must overlap so this applies to solids) that tends to keep the spins in

alignment. The references analyse this further using the so-called "Weiss" or "Mean field approximation"

and what this amounts to is to a permanent field FT tending to align the field even without the external field

H. It is then possible to have a material with the spins near saturation even at normal temperatures. Thus

returning to the Brillouin function and letting S→∞ we arrive at the Langevin function:

L (X) = coth x -— eqn. 37

X

The third term in eqn. 36 is an anisotropy term which reflects an intrinsic tendency of the moments to align along the crystalline axis, or even the shape anisotropy where the magnetic fields of an elongated particle

self-reinforce along the long axis of the particle. K is the anisotropy constant and V is the volume of the

particle. In such a particle the magnetic moment swishes back and forth due to the influence of the thermal energy with a relaxation rate:

Where ¾ is called the Neel frequency and is the order of 10GHz. The particles of magnetic material then

potentially have a moment with temporal variation and depending on the time scale of interest, they are

called super-paramagnetic. The particle size is so-called "sub-domain" size.

However when the particle size becomes large (above "domain size") the spontaneous magnetisation in that particle persists for a long period of time and then the material is called ferromagnetic. On larger scales,

energy minimisation of the magnetic field will progressively overcome the exchange interaction breaking

the material up into domains separated by a domain wall where the moments gradually reverse from one

orientation in one domain to the opposite at the adjoining domain. It is these domain walls that give rise to the characteristic hysteresis in the B-H curve of ferromagnetic materials.

We have hence shown the inter -relatedness between paramagnetism, super-paramagnetism and

ferromagnetism as one of quantity of spin and the energy interaction term. Now the precise relation to the

working substances of the two cycles can be discussed and expressions for their entropy developed.

7.1. Materials for the Magneto-calorific cycle

Materials for this cycle rely directly on the variation of magnetisation with temperature. The materials can be paramagnetic (small induced moment, small susceptibility), super-paramagnetic (large induced moment, moderate susceptibility) or ferromagnetic (large induced moment and high susceptibility ie a permeability) and are operated near the Curie Point. Roseinweig[4] is able with a simple Curie Law to derive the

functions 'c' and 'g' (eqn. 23 and eqn. 24): eqn. 39

Where K is called the Pyromagnetic constant which is of the order of for Iron. Figure 8

shows a typical magnetisation vs. temperature curve and a table of Curie points for several materials.

Rosenweig then deduces:

C = c(r) = C ₀ eqn. 40 g = -μ ₀ ΚΤ eqn. 41

And the entropy in the transitory/paramagnetic regime is:

S = c ₀ In T - μ ₀ ΚΗ + const eqn. 42

The first term in the entropy is related to the heat capacity of the material and the second term shows how

the magnetic field causes magnetic ordering which lowers the entropy.

In the first magneto-calorific cycle, no temporal effects are studied on the magnetisation step. Thermal

relaxation effects are, in paramagnetic material of the order of the mean free path/the speed of sound or in

super-paramagnetic materials (eqn. 38), assumed to be much faster than the magnetisation field.

Conversely (and contradictory) in super-paramagnetic materials or ferromagnetic materials, the frequency

of relaxation eqn. 38 is assumed to be so long, because the particle size is relatively large, that the field in

the particle is constant and it rotates wholesale or domain walls move. If the materials is magnetically

"soft" there is little hysteresis and hence entropy generation in a magnetisation/de-magnetisation cycle[5,

9]·

However by design, since these cycles relate to temporal remanence effects, in the magneto-calorific cycle, we deliberately set the particle size of the working substance such that the rate of heat flow into the particle from the surrounds is set by a the heat flow diffusion equation.

Where T is the temperature, k is the thermal diffusivity and c is the heat capacity. According to eqn. 25

ί δΜ λ δτ (Τ)

both terms: and , the relaxation in the magnetisation due to the heat flow eqn. 43, allow

V dT J _T dT

a heat engine to be constructed. However for the magneto-calorific cycle this is over quite a narrow range

around the Curie temperature. This cycle also allows for the conventional pressure- volume work from the

transient decaying flux similar to that described in Rosenweig.

7.2. Materials for the Temporary Remanence cycle

The second cycle relies exclusively on a temporary remanence effect of super-paramagnetic materials. A

commonplace means of making a super-paramagnetic material is to suspend nanoscopic particles in a fluid with the help of surfactants to stabilise the colloidal suspension[4]. Figure 9 is a depiction of a ferrofluid

particle which consists of a core magnetic material of typically iron oxide or ferrite with attached surfactant chains. It is possible to dry down a ferrofluid to a solid so that higher induction and susceptibility can be

obtained.

The relaxation rate of the ferrofluids is controlled by two mechanisms: the Neel (eqn. 38) mechanism set by core size, V and anisotropy constant (shape or crystalline) and the Brownian mechanism (related to

wholesale rotation in the fluid) set by the viscosity η ₀ and core size, V:

KV

f

^{J 0} eqn. 44

3V _lo

^B kT A ferro fluid can have two such relaxation rates but slower rate will dominate considerations if it is within an order of magnitude of the other and the cycling frequency.

A feature of this cycle is that it is operated preferentially (though not exclusively) in the ferromagnetic

regime (figure 8) because it does not rely on

electrical work (eqn. 25). The temperature dependence of the Neel and Brownian relaxation rates makes it apparent that this is so and furthermore, the cycle can operate over a wide range of temperatures not just

near the Curie point:

dz _B {T) _ 3V _lt

eqn. 46 dT kT

Both of which are strong functions of temperature.

Strictly the ferro fluid core of some 10s of nanometres radius behaves as a particle of massive spin (over

10,000μ _Β ) and the Langevin equation (eqn. 37) applies to the variation of magnetisation with temperature.

However figure 10 shows that this is not too different than 0.9*tanh(x/3) (after Taylor expansion and

approximation) and is sufficient in accuracy and spirit to the Langevin function. Let us then write, for

brevity and simplicity, both the magnetisation vs. temperature and the impulse time response as:

The response to a step input magnetising field is:

M (t) = 0.9M _s (0)tanh eqn. 48

This also shows, not surprisingly, how super-paramagnetic materials, with their large intrinsic spin, are

somewhat easier to saturate than a simple spin ½ paramagnetic gas (eqn. 33).

The cycle benefits most by being in the ferromagnetic regime (figure 8) where the magnetisation is highest and there is little gained in analysing in the paramagnetic/transitory regime thus = 0 and straight

away the functions 'c' and 'g' (eqn. 23 and eqn. 24) can be found, giving the entropy in the ferromagnetic regime as:

S = c ₀ In T + const eqn. 49

That is, the reduction in entropy due to the magnetic ordering second term in eqn. 42 is negligible.

We shall now model irreversible processes in the ferrofluid. Figure 11 shows measured loss angle for a real ferro fluid supplied by Sustech Gmbh, the actual specifics of the ferrofluid are not important but the graphs show how well the first order pole approximation applies to real data[4, 5]. The susceptibility χ is the

measure of the response of a magnetic material to an external field (eqn. 17). On the graph X '(f) is the real or component in-phase with the magnetisation H of the susceptibility and X"(f) is the imaginary part.

Xo

X eqn. 50

\ + jan (T)

Where:

%o is the DC susceptibility, M = χ ₀ Η

τ(Τ) is the ferrofluid relaxation rate

ω the frequency in radians per second

When magnetising the ferrofluid, it behaves essentially as a lossy inductor. Inductance is defined[9] as the magnetic flux per unit current: λ μ ₀ ΜΑ

L =— = eqn. 51 i i

The flux is related to the magnetisation and cross-sectional area A of the coil, hence the inductance is

directly proportional to the susceptibility.

Let a current flow through the inductor be represented by the phasor method[9] as:

/ = I ₀ e ^jwt eqn. 52

The relationship between the complex voltage, current and impedance is:

V = IZ eqn. 53

Hence:

Z _ind = -j )L ( _a ) eqn. 54

Since V = .

Using the expression for the frequency response of the ferroiluid, given by eqn. 50, we proceed to find the real and imaginary parts of χ:

^{ReW =} _T7 ¾ ^{ImW =} l¾?¾ Tan _& - _→ ) e _q ,. ₅₅ a, _b ,c

Thus the imaginary part of the susceptibility is the resistive loss and the real part gives inductive behaviour.

The ratio of real to imaginary parts gives the power factor and this is linear in frequency, figure 11 shows a good approximation to this.

The irreversible generation of heat is then related to the magnetising energy (eqn. 12), thus:

5Q _irr = μ ₀ Η \πι{Μ)ν eqn. 56

Clearly at low frequency the magnetisation-demagnetisation of the ferrofluid tends to a thermodynamic ally reversible process.

8. The T-S diagram

Figure 12 shows one example of a plant diagram for a process embodying the invention. The apparatus 8 of this embodiment comprises a closed circuit 9 around which a ferrofluid 10 may flow. A pump 11 is

provided to pump the ferrofluid 10 around the circuit 9. A heat exchanger 12 is at a location on the circuit

9, and the heat exchanger 12 is in contact with a reservoir of environmental heat. Therefore, ferrofluid 10

flowing through the heat exchanger at a temperature below that of the environment will be heated up to the environmental temperature.

At another location on the circuit 9, preferably separated from the heat exchanger 12, is a power extraction area 13, which will be described in more detail below.

The process consists of two aspects: an adiabatic step (considered singular in this discussion) with rapid

magnetisation/de-magnetisation sub-steps in the power extraction area 13 and then a step of return of the

working substance (i.e. the ferrofluid 10) to the heat exchanger 12 to warm back to ambient.

Both eqn. 49 and eqn. 42 (when averaged over a sub-step so that the H field contribution averages to zero) say the same thing:

= c ₀ In T + const (specific entropy)

eqn. 57 8Q = mc ₀ dT We know too that there is work from the collapse of the remnant flux, thus for the adiabatic sub-steps using the thermodynamic identity, dU = T (dS + dS _lrr )— SW implies:

8Q + Q _irr = 8W

eqn. 58

^> SW -SQ _lrr = mc ₀ dT

Observing the work terms eqn. 25, eqn. 45, eqn. 46 and their negative signs dT must be negative too, the

working substance cools. If the work leaves the system and we consider the system boundaries to contain

just the working substance, then the internal energy has lowered. Furthermore, since the internal energy is the integral of the inexact net work, the internal energy is rendered an inexact integral dependent on the

path of the work integral:

5U = 5W _net eqn. 59

Our process and the plant diagram consists of two stages. Let us write the internal energy for step 1 as U'

and step 2 as U:

Step 1-2: Adiabatic cooling with work leaving system from temperature

Tamb to Tamb - dT path dependent by the consideration above

5U' = TdS -5W _net

= Tm [^ dT = mc ₀ dT -5W _net

Step 2- 1 : Isothermal warming at the heat exchanger from - dT back to

dU = TdS = mc ₀ dT

The relation at step 2 is precisely the variation in internal energy of the pure working substance with

temperature and with other variables held constant - that is, the heat capacity. The temperature entropy

relation is eqn. 49. However for step 1, the material moves from to - dT, as in step 2 but if:

And the step 1-2 looks like a "virtual heat capacity", which is smaller than Co and follows a different path in

T-S space. This is not surprising as electrical work is made to leave the system where it would go into the

vibrational heat storage modes of the working substance. In a sense, it is like "squeezing a sponge" where

by analogy, the heat energy is being "wrung out". On the step 2-1, the substance reverts back to the higher heat capacity and "mops up" the heat energy from the heat exchanger.

We now turn to the final electrodynamic section to investigate the work term further.

9. Electrodynamics II

A mathematical model can be constructed for the working substance and electrical output circuit. Let us

concentrate on the ferrofluid temporary remanence cycle as its features are applicable to the magneto- calorific cycle too.

With reference to figure 14, let us first consider the ferrofluid flux decaying into a linear resistor R, having a cross-sectional area A, a length of D and N turns:

The flux linkage is given by: λ = NAB = ΝΑμ ₀ (H + M) eqn. 62

The magnetic field is given by:

N_

eqn. 63 D

Where i is the current through the coil and D is the length

The ferro fluid or super-paramagnetic material in general obeys a 1 ^st order equation:

1

M =— (Μ - χΗ) eqn. 64

That is, the rate of change of the magnetisation is negatively proportional to the existing magnetisation

minus the driving contribution of the magnetic field, thus when Η is substituted, the following is obtained:

1 ( N

M =— \ M - x— i eqn. 65

The LR circuit, on analysis considering the voltages yields the following, another state space equation:

άλ

-m = o -λ _Μ - λ _Η - iR - 0

dt

N A di

- iR = 0

The two equations eqn. 65 and eqn. 66 are suitable for coding on a digital computer and figure 15 shows

the output from Matlab code (appendix 2) that solved this for the set of parameters listed in the code. It is

clear that the graph of energy initially obeys a simple Faraday's Law (eqn. 15) which then plateaus to

return just the field energy (the flat line at the top of the graph). The lower graphs of figure 15 show the

current vs. time and magnetisation vs. time and it can be seen how the decay of current becomes slower and slower at lower resistance, thus limiting the energy returned.

An analytical solution can be found by taking the Laplace Transforms of eqn. 65 and eqn. 66, then

substituting λ _Μ in eqn. 66 and solving for I(s), the current in the s-domain, we obtain:

The dominant pole near the origin sets the dynamics, and a binomial series expansion of the roots of the

denominator gives:

The dominant pole gives the response: τ +

R

The expression in the denominator of the first term is of the form: τ' = τ + L/R, the 2 ⁿ term being a purely electrical circuit effect (inductor -resistor circuit) which would dominate at high loading (R→0). If the

decay of the current is so slowed, another cycle cannot start until sufficient decay has occurred hence the

power output will be lower. It is easy to show that a purely resistive-inductive setup for the output power

couple will lead to only the magnetisation energy being returned and hence no power gain: The current is:

Where τ ' = -

If the time constant of the electrical circuit is much slower than the ferrofluid the energy dumped into the

load is:

Then only the magnetisation energy is returned: l _iT U -^E = -L

R 2 f { -¾ N J Y smce L= then E =— μ ₀ Μ ² ν

D 2 ^{0 0}

9.1. Inclusion of a series or parallel capacitance

Figure 16 shows the circuit schematic with the inclusion of a series capacitance. Considering the series

case, the circuit analysis becomes (with Q represent the charge on the capacitor):

N ² A■■ O

And in the Laplace domain,

DM,

LC

N

LCTS ³ +(LC{\ + X) + CRT)S ² + (CR+T)S + \

eqn.69

^LCs

I(S) ^: N

LCTS ³ +(LC{\ + X) + CRT)S ² +{CR+T)S + \

This is a 3 ^rd order system and standard solution of the cubic denominator[17] gives the poles for the system as follows:

If s ³ +a ₂ s ² + a ₀ = (s + + z ₂ )(s + z ₃ ) = 0

Then let q = ^a _l -^a ₂ ² and r = -^(α ₁ α ₂ -3α ₀ )--^-α ₂

The discriminant is given by: Δ = q ³ + r ² eqn.70

If Zi z ₂ and z ₃ are the roots of the cubic, then the roots can be classified by the discriminant:

A>0^>z ₁ eD, z ₂ eD and z ₃ = z ₂

A = 0 = z _l , z ₂ and z ₃ e□ and |z ₂ = z ₃ or z _l = z ₂ = z ₃

Δ < 0 = z _l , z ₂ and z ₃ e□ Let d, r + and d ₀ then the roots are:-

^Z 2 = -y - { ^d l + ^d 2 ) + { ^d l - ^d 2 ) eqn. 71

We can see that the roots of the system admit solutions such that Re(z) = 0 and Im(z)≠ 0 by correct choice of parameters (solution of eqn. 71) and the system will (after initial setup of magnetisation) self-sustain

oscillations, even into the electrical load. Figure 17 shows a graph of this and appendix 3 contains the

Matlab code. Non-linearities (such as a non-linear capacitor) will bring back the condition that Re(z) > 0 to the stable attractor at Re(z) = 0.

9.2. Inclusion of a non-linear resistance and non-linear co-material

Figure 18 shows the circuit schematic with the inclusion of a non-linear resistance which has a negative

dR (i)

dynamic conduction zone over at least some of its characteristic. The two state equations

di

become:

\ N

M =— \ M - x— i eqn. 72 di D

(μ ₀ ΝΑΜ + (ϊ) ) eqn. 73 dt μ ₀ Ν A

Substitution of eqn. 72 into eqn. 73 yields:

di

— = M 1 — iR (i )

dt Ντ τ AN μ ₀ ^; eqn. 74

= C _X M - C ₂ i - CjR {i)

With the positive constants Q C ₂ and C ₃ added for brevity.

In general, non-linear equations can be exceedingly difficult to solve and so we will use a brute force series solution technique where the current, magnetisation and resistance are written as power series:

M =∑M _t f eqn. 75

R =∑R eqn. 77

Appendix 4 contains a Matlab script for the tedious expansion and algebra to the 2 ⁿ order in all the series

eqn. 75 for eqn. 74. Thus we write a Taylor series expansion about a point:

M {t) = M ₀ + M _l t + M ₂ t ² + ... eqn. 78

' ^• (0 ^■ ■ I ₀ + I ₁ t + I ₂ t eqn. 79 And

R(t) = R ₀ +R _x i + R. i +... eqn.80

The result for i ² R(i} on series substitution into eqn.74 and comparing coefficients to 2 ^nd order is:

I ₀ + R ₀ ])

I ₀ ² (l _X R _x+ 2I ₀ I _X R ₂ )

eqn.81

-21 _A (RX + R _X I ₀ + R ₀ )[C ₂ I ₀ - C _X M _A + C ₃ / ₀ (RX + R _X I ₀ + R ₍

I ² (R ₂ I ² ₊ R _X I _{0 +} R ₀ )

Let us straight away make a simplification of starting from zero current, i.e. Io = 0 (modify the code in

appendix 4 to drop the I ₀ term):

'*(,·) =

_ΚΔ -ι ₊ ^Ά\ _+C ² M ² (R ₂ I ² ₊ I ₂ R _X )

2 2

C ₂ I _X C _X M _{X ,} C _X R,

-2C _X I _X M ₀ R _X

2 2

eqn.82

C ₂ I _X C _X M _{X t} C A

C _X I _X M ² R _X -2C _X M ₀ R ₍

2 2

C ² M ² R ₍

As seen in eqn.68 and the simulation of the Matlab code (appendix 2) and figure 15 are the terms due to Ro solely. Furthermore, as Ro tends to zero, they must correspond to the magnetisation energy terms when

integrated. Thus we must conclude that with the non-linearity, the energy output must be eqn.68 + some

unknown function:

Ε _οιιί =]ί ² 1ΐ(ί)άί = μ ₀ ΗΜν + /(Ί) eqn.83

So we note that these terms are, to the 2 ⁿ order, are:

-2C,I,M _n R ^{1 €χΜχ} eqn.84 _ di dR . . dM

The term in the second order term can be written: C, M _n

¹ dt di ° dt

It is noted that the current in the power coil must be increasing and the magnetisation decreasing, initially, dR

at least. Thus we can conclude that in eqn. 84 if that is , is negative then the mystery function in di

eqn. 83 is always positive. Furthermore, the new feature with the non-linearity is the term in the second order of effectively M ₀ x M _l our mystery function is this:

(^ )>o

This is precisely a function of the dipole work term eqn. 14 and therefore:

E out , > E mag

This analytical proof can be extended in a piecewise 2 ^nd order fashion for all time by use of eqn. 81 and a non-zero starting point for the current (I ₀ > 0), leading to the generalisation that energy production in excess of the magnetising energy input is possible, if the non-linear resistance has the following negative dynamic resistance characteristic:

dR (i)

< 0

di

The Matlab code is appendix 5 is a direct comparison to the same parameters for the linear case in appendix 2 but with a non-linear resistance of:

R = R (\ - i ² )

Where i is the normalised current (that is scaled to ⁰ ). This is an approximation to a diode's I-V characteristic or a thermistor. In the case of the two simulations, it can be seen in the energy vs 1/R trace of figure 19 that the energy gain is at least 2.5 times higher (figure 15) for these particular sets of parameters.

One manifestation of utility of the device is as a heat pump converting low-grade heat into high-grade heat. Figure 20 shows this application for space heating or running a conventional Carnot cycle engine at the output from the high temperature non-linear resistor.

Figure 21 has an implementation of the non-linear element with an active element (transconductance) device, a so-called Lambda diode. Other suitable elements are tunnel diodes and vacuum tubes. The active component can be biased to keep it in the negative resistance zone with the power coil output then inserted in series with the bias current or ac coupled with a bypass arrangement for the bias current.

9.3. Inclusion of a non-linear capacitance or non-linear inductance and a linear electrical load

The analysis of the previous section can be continued to investigate the affect of other non-linear elements in conjunction with a linear electrical load. Let us model the effect of a non-linear co-material, that is, a material with a non-linear B-H immersed or in juxtaposition with the working substance.

Μ = -- Μ - χ ^-ίμ _Γ (ή

T L ⁾

— = τπ ^~ ( μ ₀ ΝΑΜ + iR (i) ) The non-linear co-material is represented as \X _R (7) since the magnetic field it responds to is automatically a function of current. Substitution of eqn. 89 into eqn. 90 yields:

di D M X . D iR

-i

Ντ μ _Γ (i) τ ΑΝ ² μ ₀ μ _Γ (i)

eqn. 91 -

And a series expansion for μ (z ) along with eqn. 75 and eqn. 76, μ eqn. 92 ; ^ o di ¹

Lends itself to a series solution of eqn. 91. Appendix 4 has a second script dealing with this scenario to do

. . dM δμ _Γ

the algebra for the expansion. Our heuristic picks out terms which are a function of M for the

dt di

reason that: a) we know that excess power would come from the dipole work term eqn. 14 and b) there

must be variation in the permeability; all other terms just return the magnetisation energy. Such a term to

the 2 ^nd order is:

_ _! dt dt di

2

Clearly then co-materials with the following dynamic permeability constraint:

δμ

> 0 eqn. 94 dH

Over part of its characteristic meets the objective.

Let us now consider a non-linear capacitor in the power output circuit (figure 22) and a linear resistance.

The system equations are:

M =— M - x -Q eqn. 95

Where Q is the charge on the non-linear capacitor C(Q). Substituting eqn. 95 into eqn. 96 yields: d ² Q D _Ά X dQ D dQ D Q

-M R

Ντ τ dt ΑΝ ² μ ₀ dt ΑΝ ² μ ₀ C(Q)

eqn. 97

_KI M - K ₂ ^- - K^ R - K Q

dt dt C(Q)

All the constants ¾ to K ₄ are positive. A series expansion for C(Q) along with eqn. 75 and eqn. 76:

⁰⁰ Pt'C

C(Q) = Y— Q ^> eqn. 98 Lends itself to a series solution of eqn. 97 by the usual method of series substitution and comparison of coefficients. Appendix 4 has a third script dealing with this scenario to do the algebra for the expansion.

. , f.t/ dC

Our heuristic, once again, picks out terms which are a function of M :

it HQ

And the capacitance should have the following non-linearity: eqn. 100 dQ

Similar analysis for a series non-linear inductor (figure 22) leads to the result: dL (i)

^- > 0 eqn. 101 di

10. The improvement that this patent relates to and their embodiments

This method seeks to improve the previous disclosures of Cornwall by reference to the proceeding body of text in this description, in particular with reference to the embodiments described below.

In one embodiment, shown in figure 16, the load circuitry (e.g. as shown on the right hand side of figure 7) has in series or parallel with a linear or non-linear load R, whose I vs. V characteristic (if non-linear) has a

dR (i)

substantial portion where < 0 a linear or non-linear capacitor C, or an equivalent series or parallel

di

network of electrical components having a capacitance, whose C vs. V characteristic has a substantial

portion where > 0 . This arrangement self-sustains its oscillation delivering power to the load R

dv _c

via the power output circuitry without further need for the magnetisation circuit (e.g. as shown on the left- hand side of figure 7) that seeded the initial magnetisation.

In another embodiment, shown in figures 18 and 19, the load comprises a non-linear resistor R, or an

equivalent series or parallel network of electrical components having a resistance, whose I vs. V

dR (i)

characteristic has a substantial portion where < 0 . This arrangement requires field cycling

di

circuitry (e.g. as shown on the left hand side of figure 7).

In some examples of this embodiment, wherein the resistive load R acts as a radiant heater for a heat pump between the heat exchanger 13 and the said non-linear load R.

In some examples of this embodiment, the non-linear load R acts as the upper thermal reservoir to a

conventional heat engine, the thermal reservoir T _L is the lower reservoir. The output W is high quality

work.

Both of these examples are illustrated schematically in figure 20.

The non-linear load may be provided in parallel or series with a linear or non-linear capacitor (or equivalent network of components), and/or with a linear or non-linear inductor (or equivalent network of

components).

In another embodiment, the power output circuitry has a linear or non-linear electrical load in conjunction

δμ

with non-linear co-material, whose B vs. H characteristic has a substantial portion where > U ,

dH

immersed or in close proximity to the working substance and the magnetising cycle apparatus. In this embodiment the power output circuitry may still include at least one non-linear component, as discussed elsewhere, for instance a non-linear resistive load.

In a further embodiment, shown schematically in figures 22 and 23, the power output circuitry has a linear or non-linear electrical load in conjunction with a series non-linear capacitor, or an equivalent series or parallel network of electrical components whose C vs. V characteristic has a substantial portion where dC(v _c )

> 0 .

In another embodiment, also schematically shown in figures 22 and 23, the power output circuitry has a linear or non-linear electrical load in conjunction with a series non-linear inductor, or an equivalent series or parallel network of electrical components whose L vs. I characteristic has a substantial portion where

In a further embodiment, shown schematically in figure 23, the working substance arranged in regular separate spatial packets has its magnetisation remanence time altered such that the work produced is a function of the temperature difference between an upper (T _H ) and lower reservoir (T _L ) i.e.

Brownian or Neel process (eqn. 44) by placing the working substance 10 in contact with the first cold reservoir T _L , where it is also magnetised and then moving the working substance 9, by means of a pump 11, to a second location and hot reservoir T _H where the the flux collapse generates electricity into a power coil 14. The transit time from the first cold reservoir is set so that the flux doesn't decay appreciably in this time. As discussed elsewhere in this disclosure, the power coil 14 can be shorted into the load at timed instants when the working substance packets are within the second reservoir T _H and power coil 14, this reduces magnetic drag to the flow circuit.. Consultation of eqn. 44 and the standard Arrenhius factor[2,3] shows that in the case of Neel type fluid, this change can be of the order of a factor of 2 for every 10K change in temperature, for example.

In the above examples, certain components are stated to have non-linear characteristics. It should be understood that, as used in this specification, this non-linearity refers to a significant variation in the property in question, and normal components, supplied as providing a certain value of this parameter but having minor, unavoidable variations in the parameter across the component's operating range (particularly at the extremes of the operating range), do not fall within this definition.

In preferred embodiments, the parameter in question of the non-linear component varies by at least 10% over the operating range of the component. In other preferred embodiments, the parameter in question of the non-linear component varies by at least 20%o over the operating range of the component. In further preferred embodiments, the parameter in question of the non-linear component varies by at least 50%> over the operating range of the component. In yet further preferred embodiments, the parameter in question of the non-linear component varies by at least 100%o over the operating range of the component.

In this specification it is stated that the non-linear component is has a non-linear characteristic over a substantial portion of the characteristic. In preferred embodiments this substantial portion may comprise at least 10%o of the relevant operating range of the component. In further embodiments this substantial portion may comprise at least 20%o of the operating range of the component. In other embodiments this substantial portion may comprise at least 50%o of the operating range. In yet further embodiments this substantial portion may comprise at least 100%o of the operating range.

For any of the non-linear components mentioned in this document, the parameter in question may either rise or fall with respect to applied voltage, current or electric/magnetic field.

When used in this specification and claims, the terms "comprises" and "comprising" and variations thereof mean that the specified features, steps or integers are included. The terms are not to be interpreted to exclude the presence of other features, steps or components.

The features disclosed in the foregoing description, or the following claims, or the accompanying drawings, expressed in their specific forms or in terms of a means for performing the disclosed function, or a method or process for attaining the disclosed result, as appropriate, may, separately, or in any combination of such features, be utilised for realising the invention in diverse forms thereof. References

1. Abbott M. M., et al., Thermodynamics with Chemical Applications. Schaum's Outlines, McGraw- Hill. 1989.

2. Kittel C, K.H., Thermal Physics. W. H. Freeman and Company, San Francisco. Vol. 2nd ed.

1980.

3. Landau, L., A Course in Theoretical Physics: Statistical Physics. Butterworth Heinemann. Vol.

Vol. 5. 1996.

4. Rosensweig, R.E., Ferrohydrodynamics. Cambridge University Press. 1998.

5. Aharoni, A., Introduction to the Theory of Ferromagnetism. Oxford Science Publications 1996.

6. Maxwell, J.C., Theory of Heat. Longmans, Green and Co., London Vol. Chap. 12. 1871.

7. Szilard, L., On the Decrease of Entropy in a Thermodynamic System by the Intervention of

Intelligent Beings. Physik Z., 1929. 53: p. 840-856

8. Harvey L. S., A.F.R., Maxwell 's Demon: Entropy, Information and Computing. Adam Hilger, Bristol 1991.

9. Feynman, L., Sands, The Feynman Lectures on Physics. Addison- Wesley, Reading,

Massachusetts. Vol. Vol.1, Vol. 2 1989. 14.7, 15.1-6, 35.4.

10. Brillouin, L., Science and Information Theory. Academic Press Inc., New York 1956. Chaps. 13, 14.

11. Landau, L., A Course in Theoretical Physics: Mechanics. Butterworth Heinemann. Vol. Vol. 1.

1982.

12. Landau, L., A Course in Theoretical Physics: Kinetics. Butterworth-Heinemann. Vol. Vol. 10.

1981.

13. Coveney, P., Highlield, R, The Arrow of Time. Flamingo 1991.

14. Kapitaniak, T., Chaos for Engineers . Springer 1998.

15. Jordan D. W., S.P., Nonlinear Ordinary Differential Equations. Clarendon Press Oxford. Vol. 2nd ed. . 1991.

16. The Ergodic Theorem, in Encyclopaedia Britannica Inc.

17. M Abramowitz, I.A.S., Handbook of Mathematical Functions. Dover Publications 1965.

Appendix 1 - The Independent Flux Criterion

Let us explore this by means of a contrary proof: consider an inductor as some circuit element. The net energy for a cycle is given by:

dq> .

[vi - dt = - [— i ^■ dt

J ^J dt

Integrating the RHS by arts:

Since i(0) = i(T) and φ(0) =^φ(Τ^) ihe first two terms iance).' ^ ^ ^ ^

Let: i(t) = g((p(t)) eqn. A1.2 i.e. a dependent flux, the second integral of equation Al can be integrated by parts a second time by applying the chain rule:

Thus:

\ ^{φ(ΐ) ά8} ΐφ(!) ^ ^{φ(ΐ) =} [ ⁽ ' ^{)g ~} -I ^{" g ' 1 ' d(p(t) = G} ^ ⁽⁰⁾ ^ ^{" G =} °

The first term on the RHS cancels due to the flux being the same at the start and end of the cycle. The integrand on the RHS cancels for the same reason. The above result shows that a dependent flux, eqn. A1.2 cannot lead to net power.

The proof sheds more light on the necessary condition for an independent flux: the flux is constant for any current including zero current - it bares no relation to the modulations of the current. The proof also dispels any form of dependent relation, non-linear or even a delayed effect. If the eqn. A1.2 was:

i(t) = g ((p(t - n)) eqn. A1.3

This could be expanded as a Taylor series about g((l>(t)) but there would still be a relation, the flux would still be dependent. Thus it is a statement of the obvious (the First Law of Thermodynamics) that excess power production in an electrical circuit cannot happen by electrical means alone; flux changes must happen by some outside agency such as mechanical shaft-work to cause energy transduction.

Appendix 2 - Matlab code for the LR model

function PowerCoil 4()

% SiiiLiI t.es the basic LP see up of a simple pov~r output circuit end % shows that this can only ever be the mag n.etisat ion energy,

global handle wb

global MO

hi = findobj ( 'Name ' , '4. Energy vs i/R ^! );

if isempty(hl)

hi = figure ( ) ;

set (hi, ' U&me ¹ , '4. Energy vs 1/R' ) ;

xlabel ( ' Res istance ^: ) ;

ylabel ( ' Ene gy ¹ ) ;

end

h2 = findobj ( 'Nacae ' , 'Graph 4.1; Current:, vs tirae');

if isempty(h2)

h2 = figure ( ) ;

set (h2 , 'Name', 'Graph 4,1: Current vs titae' ) ;

end

handle wb = waitbar(0, ' ');

A = 0.01;

D = 10.5;

muO = 4.*pi.*le-7;

M0 = 1000;

N = 100;

tor = 20e-3;

X = 0.1;

R = logspace(-6, 4, 30);

Energy = zeros (1, numel (R) ) ;

for i = 1: numel (R)

s = sprintf ( ' PowerCoil 4 , ia busy, R ----- %0.2g ^! , R(i)); waitbar(0, handle_wb, s) ;

[t y] = Simulate (A, D, muO, N, R(i), tor, X) ;

% y ( 1 ) ----- cur enr.

I y { ^' 2 ) - flu.;; from M

Energy (i) = Compute_Energy (t, y(:,l), R(i));

plot_graphs (h2 , t, y, R, i) ;

end

figure (hi ) ;

elf;

loglog(l./R, Energy, 'b');

s = sprintf ( 'Energy vs Ϊ/Ρ: R ------ ¾Q.2g to %0,2g', R(l), R(end)); title ( s ) ;

xlabel ( ¹ 1 ,/R ' ) ;

ylabel ( 'Energy' ) ;

hold on;

loglog(l./R, mu0*X*M0 ^A 2/2. *ones (size (R) ) , ' · ' ) ;

close (handle_wb) ;

end.

function plot_graphs (h2 , t, y, R, i)

% y(i) ----- current

I y { ^' 2 ) - flu.;; from M

Rpick = [le-3, le-2, le-1, 1, le2 le4];

j = find(Rpick >= R(i)); if numel ( j ) > 0

figure (h2 ) ;

% Plot current

subplot (2, 6, j(l) );

plot(t, y(:,l) );

s = sprintfC? - ¾0.2g\.n ^! , R(i));

title ( s ) ;

xlabel ( ' time is) ' ) ;

ylabel ( 'Current (A) ^: ) ;

max i = max (y ( : , 1) ) ; % find ma.ximum current: to scale nicely indy = find(y(:,l) >= max i) ;

% find the next time when i < 0.05 ^,s ma.x i

indx = find(y(:,l) < 0.05*max_i) ;

if numel (indx) > 1

axis( [ 0 t(indx(2)) 0 1.05. *y (indy (end) , 1) ] ) ;

else

axis( [ 0 100 0 1.05. *y (indy (end) , 1) ] );

end

¾ :ίο'« magnetise ti on

subplot (2, 6, j (1) +6 ) ;

plot(t, y(:,2) ) ;

xlabel ( ' time (si ' ^' ) ;

ylabel ( 'Magnetisation. (h/M! ' ) ;

max i2 = max (y ( : , 2) ) ; % tied, maximum m g to scale nicely indy2 = find(y(:,2) >= max_i2);

% firid the next time «h~ri i < 0.05'ria i

indx2 = find(y(:,2) < 0.1*max i2);

if numel (indx2) > 1

axis( [ 0 t(indx2(2)) 0 1.05. *y (indy2 (end) , 2) ] ) ;

else

axis( [ 0 100 0 1.05. *y (indy2 (end) , 2) ] ) ;

e d

e d

end

function tmp = Compute Energy (t, current, R)

% Power I ' ^' 2. R

% Energy ^:;; :i. nt ( Power) 0 ΐ

% use interrhi function to interpolate between array value's

% Use quadlgko to integrate

tmp = quadgk(@isq, 0, t(end), ^: P.elTol ' , le-6, 'ftbsTol' , le-6) * R; function iisq = isq(tt)

iisq = interpl (t, current, tt, 'linear') .

e d

e d

function [t y] = Simulate (A, D, muO, N, R, tor, X

global M0

% y ( . ^" I ) ------ cu r rent

% ( 2 ) - flux from hi

yO = [0 M0];

options = odeset ( ^! Rel oi ' , le-7 , ' Ahsloi ' , le-7 )

Tend = 100; [t y] = odel5s (@flux, 0:le-3:Tend, yO, options, Tend, ...

A, D, muO, N, R, tor, X) ;

e d

function dy = flux(t, y, Tend, A, D, muO, N, R, tor, X)

% v(2) ^;: - fegnefi satio.n

global handle_wb

do_wait_bar ( ) ;

dy = [0; 0;];

dy(2) = -(l./tor) .* ( y(2) - N.*X.*y(l) ./D ) ;

dy(l) = - (D. /muO . /N. ^Λ 2. /A) .* ( muO . *N. *A. *dy (2) + y(l) .*R );

% nested u c ion

function do wait bar ( )

persistent Tlast

if (t <= eps)

Tlast = 0;

elseif (t > Tlast)

waitbar (t . /Tend, handle_wb) ;

Tlast = Tlast + 0.05.*Tend; % move i on hen 5% has ^; passed e d

end

end

Appendix 3 - Matlab code for the linear or non-linear model and self-sustaining action function PowerCoil 10 (k, Mi)

global handle wb

V = 5;

Tend = le-2;

Tstep = 0.2e-5;

fprintf (1, ^: \r; \ n.N3age ; PowerCoi 0 (k, Hi) ') ;

e d

handle wb = waitbar(0, ' ^' ');

D = (25*V) ^Λ (1/3) ;

A = V/D;

muO = 4. *pi . * le-7 ;

tor = le-3;

X = 0.1;

fprintf (1, ' valu¾ ^~ %0.2g' , k) ;

fprintf (1, ' \ ri \r· Ini ial Magnetisa fci or

fprintf (1, n;e - ¾0.2g', A*D)

fprintf (1, - %0.2g ^! , Tend);

MiT = Mi * muO;

MiG = MiT * le4;

fprintf (1, ' ia.1 magnetise

Mi, MiT, MiG) ;

fprintf (1, ' \ \r: agnetisai:ion E i

0.5*muO*Mi ^A 2*A*D) ;

R = logspace(-l, 6, 50);

C = k.*tor ./ R;

L = (k ^A 2 .* tor. ^A 2) ./4 ./(pi. ^A 2) ./ C;

N = sqrt( (L .* D) ./ (1+X) ./ muO ./ A) ;

N(N < 1) = 1; Energy = zeros (1, numel (R) ) ;

Rpick = [le-1, 5e-l, 1, 10, le2 5e5];

for i = 1: numel (R)

s = sprintf ( ' Pow rCoillO .IT; busy, P. - %0.2g ^: , R(i));

waitbar(0, handle_wb, s) ;

[t y] = Simulate (A, C(i), D, muO, N(i), R(i), tor, X, Tstep, Tend, Mi) ;

Energy (i) = Compute_Energy (t, y(:,2), R(i));

j = find(Rpick >= R(i));

if numel ( j ) > 0

figure (h2 ) ; ------ %

R(i) , L(i) , N(i) , C(i) ) ;

title ( s ) ;

xlabel ( ' time is) ' ) ;

ylabel ( 'Current (A) ^: ) ;

end

end

figure (hi ) ;

elf;

loglog(l./R, Energy, ' b ' ) ;

title ( ^: E ergy vs I/R: P le-4 to le.6');

xlabel ( ' 1 /R ' ) ;

ylabel ( ' Energy ' ) ;

close (handle_wb) ;

e d

functi tmp = quadgk(@isq, 0, t(end)) * R; function iisq = isq(tt)

iisq = interpl (t, current, tt, 'linear' ^' ) . ^A 2;

end

end

function [t y] = Simulate (A, C, D, muO, N, R, tor, X, Tstep, Tend, Mi)

yO = [0 0 Mi ] ; % MB Mi ir- A/M

options = odeset ( ^! Re O.J. ^! , le-4 , ' bs oI ', le-4 ) ;

[t y] = ode45(@flux, 0 : Tstep : Tend, yO, options, Tend,

A, C, D, muO, N, R, tor, X) ;

end

function dy = flux(t, y, Tend, A, C, D, muO, N, R, tor, X) global handle_wb

do wait bar ( ) ;

dy = [ 0 ; 0 ; 0 ; ] ;

dy(l) = y(2) ;

dy(3) = -(l./tor) .* ( y(3) - N. *X . *y (2) . /D ) ;

dy(2) = - (D. /muO . /N. ^Λ 2. /A) .* ( muO . *N . *A . *dy ( 3 ) + y(2).*R + d) O ; funct ion. do_wait_bar ( )

persistent Tlast

if (t <= eps)

Tlast = 0;

elseif (t > Tlast)

waitbar (t . /Tend, handle_wb) ;

Tlast = Tlast + 0.05.*Tend;

end

end

end

Appendix 4 - Matlab scripts for expansion of Equation 74 (etc.) as a series to 2 ^nd order syms CI C2 C3

syms AO Mi M2 Ϊ 3 M-j

syms 10 II 12 ¾I3 I ^; ; I¾

syms P0 R R2 ¾R3 R ?5

syms t

didt = diff(i, ^{! !} ) ;

f = didt - C1*M + C2*i + C3*i*R;

p = coeffs ( f , t) ;

pretty (p (1) )

disp ( ' t 0 ^! )

pretty (p (2) )

disp ( ' t ^! )

pretty (p (2) )

disp ( 'ί: ^Α 2 ^! )

11 = solve (p (1) , II) ;

pretty (II) ;

disp ( ^! 1 ' )

12 = solve (p (2) , 12) ;

pretty ( 12 ) ;

disp ( ^! 12 ^: )

disp ( ^! ' )

disp ( ^! ' ^' )

i = 10 + Il*t + I2*t ^A 2 ;

pretty (i) ;

disp('i ----- ;:o 2nd order')

disp ( ^: )

disp ( ' ^! )

I2R = i ^A 2*R;

p = coeffs (I2R, t) ;

I2Rtrunc = p(l) + p(2)*t + p(3)*t ^A 2;

pretty ( I2Rtrunc) ;

disp ( ¹ I2R to 2nd order.")

disp ( ^! ' ^' )

disp ( ' ' ) syms

syms

syms

syms

syms

didt = diff (i, '·:');

f = P*didt - C1*M + C2*i*P + C3*i*R; pn = coeffs ( f , t) ;

pretty (pn ( 1 ) )

disp ( ^! 0 coeiis ⁵ )

pretty (pn (2 ) )

disp ( ^! ;: co^ ;: ;:s ' )

pretty (pn (3) )

disp ( ' t ^'" 2 coei!is ' )

disp ( ^! ' ^' )

disp ( ' ' )

11 = solve (pn (1) , II) ;

pretty (II) ;

disp ( ' 11 ^ ' )

12 = solve (pn (2), 12) ;

pretty (12) ;

disp ( ^! 12 ------ ' )

disp ( ^! ' )

disp ( ^! ' )

i = 10 + Il*t + I2*t ^A 2;

pretty (i) ;

disp( ^! i to 2n.ci rder ¹ )

disp ( ' ' )

disp ( ' ^: )

I2R = i ^A 2*R;

pn = coeffs (I2R, t) ;

I2Rtrunc = pn ( 1 ) + pn(2)*t + pn(3)*t ^A 2; pretty (expand ( I2Rtrunc) ) ;

disp( ^! I2R to 2nd ord¾r ^: )

disp ( ^! ' )

disp ( ^! ' ) clear 3ΙΪ

syms Ki 2 3 K.1

syms MO Ml 2 M3 1M4 M5

syms QO 01 Q2 Q3 %Q4 Co

syms CO C C C3 ¾CM C5

syms t P

M = MO + Ml*t + M2*t ^A 2 + M3*t ^A 3 ;% ^■ ; ^■ K t" -i- 5*f ^" '5; Q = 0*Q0 + Ql*t + Q2*t-2 + Q3*t ^A 3 ; ΐ <- Q4*;:/ <3 + Q5 '' C ^"' 3;

C = CO + C1*Q + C2*Q ^A 2 + C3*Q ^A 3 ;¾ ^■ · ^■ C *Q" ^' 4 H- C3*Q"5; dQdt = diff (Q, ' v.' ) ;

dQ2dt2 = diff (dQdt, ^! t ^! );

f = dQ2dt2 - K1*M + K2*dQdt + K3*dQdt*R + K4*Q;

pn = coeffs ( f , t) ;

prett (pn ( 1 ) )

disp ( ^! 0 oei!is ^f )

pretty (pn (2 ) )

disp ( ¹ t coef is ' )

pretty (pn (2 ) )

disp ( ' t 2 coef fz ' )

pretty (pn (3) )

disp ( ' t 3 coeff¾ ^! )

disp ( ' ^'■ )

disp ( ^{: :} )

Q2 = solve (pn (1) , Q2) ;

pretty (Q2) ;

disp ( '0:2 - ' )

Q3 = solve (pn (2) , Q3) ;

pretty (Q3) ;

disp( ^! Q3 - ^: )

Q = 0*Q0 + Ql*t + Q2*t"2 + Q3*t"3;

pretty (Q) ;

disp('Q ^ to 3rd order')

disp ( ' ^! )

disp ( ^: )

i = diff (Q, ^{* :} ) ;

I2R = i ^A 2*R;

pn = coeffs (I2R, t) ;

I2Rtrunc = pn ( 1 ) + pn(2)*t + pn(3)*t"C2;

pretty (expand ( I2Rtrunc) ) ;

disp( ^! I2R to 2nd order')

disp ( ' ' ^' )

disp ( ' ' ^' ) Appendix 5 - Matlab code for the non-linear R model

function. PowerCoil 5()

global handle wb

global MO

hi = findobj ( ^! N-s;¾e ' , '5, Energy vs :i/R ^: );

if isempty(hl)

hi = figure ( ) ;

set (hi, ' atiis', '5. Energ vs 1/R ^! );

xlabel ( ^! Resistance ' ) ;

ylabel ( ' Energy ^: ) ;

end

h2 = findobj ( 'Rante' , ' Graph 5.1: Current vs tim ' ) ;

if isempty(h2)

h2 = figure ( ) ;

set(h2, ' N me ^! , 'Gr ph 5.1: Current vs time');

end

handle wb = waitbar(0, ' ');

A = 0.01;

D = 10.5;

muO = 4.*pi.*le-7;

M0 = 1000;

N = 100;

tor = 20e-3;

X = 0.1;

R = logspace(-6, 4, 30);

Energy = zeros (1, numel (R) ) ;

for i = 1: numel (R)

s = sprintf ( ' PowerCoil 5. IT; busy, R - %0.2g ^: , R(i)); waitbar(0, handle wb, s) ;

[t y] = Simulate (A, D, muO, N, R(i), tor, X) ;

Energy (i) = Compute_Energy (t, y(:,l), R(i));

plot_graphs (h2 , t, y, R, i) ;

end

figure (hi ) ;

elf ;

loglog(l./R, Energy, '?;.·');

s = sprint f ( ' Ene g y vs l./R: R %0.2g to 0,2q ^! , R(l), R(end)); title ( s ) ;

hold on;

loglog(l./R, mu0*X*M0 ^A 2/2. *ones (size (R) ) , ^: . ^' ) ;

close (handle_wb) ;

e d

function plot_graphs (h2 , t, y, R, i)

Rpick = [le-3, le-2, le-1, 1, le2 le4];

j = find(Rpick >= R(i));

if numel ( j ) > 0

figure (h2 ) ; subplot (2, 6, j (l) );

plot(t, y(:,l) );

s = sprintf('R - ¾0.2g s ^! , R(i));

title ( s ) ;

xlabel ( ' ti rae ( s } ' ) ;

ylabel ( 'Current (A) ' ) ;

max i = max (y ( : , 1) ) ; ¾ iind mar i κινι current to scale nicely indy = find(y(:,l) >= max i) ;

% find nerf ime when J. < ^' 0,05 mar ΐ

indx = find(y(:,l) < 0.05*max i) ;

if numel(indx) > 1

axis( [ 0 t(indx(2)) 0 1.05. *y (indy (end) , 1) ] ) ;

else

axis( [ 0 100 0 1.05. *y (indy (end) , 1) ] );

end

% ow n■aq neti satio

subplot (2, 6, j (1) +6 ) ;

plot(t, y(:,2) ) ;

xlabel ( ' tim ( s ) ' ) ;

ylabel ( ^< Magne <: :L sat er; (A/ ) ^: ) ;

max i2 = max (y ( : , 2) ) ; % find re.a.xii;;um ma.g to soa.Ie nicely indy2 = find(y(:,2) >= max i2);

t find the next time when i < 0.05*max i

indx2 = find(y(:,2) < 0.1 *max_i2 ) ;

if numel(indx2) > 1

axis( [ 0 t(indx2(2)) 0 1.05. *y (indy2 (end) , 2) ] ) ;

else

axis( [ 0 100 0 1.05. *y (indy2 (end) , 2) ] ) ;

end

end

e d

function tmp = Compute Energy (t, current, R)

t Power -~ T ^•s 2. R

% E ergy ------ i n.t ( Powe r ) 0..

% Use interp! funci: i on to interpolate between array values

% Use quadlgk;; to i r.teg ;:a he

tmp = quadgk(@isq, 0, t(end), ' Be lTol ' , le- 6 , ' AfosTol ' , le- 6 ) * R;

% nes d func ion

function iisq = isq(tt)

iisq = interpl (t, current, tt, ' linear ' ) . ^Λ 2;

end

end

function [t y] = Simulate (A, D, muO, N, R, tor, X)

global M0

t yd) cu r e t

% (2) - flux from

yO = [0 M0];

options = odeset ( ^! Re ?oi ^! , le-7 , ' AbsToi ' , le-7 ) ;

Tend = 100;

[t y] = odel5s (@flux, 0:le-3:Tend, yO, options, Tend, ...

A, D, muO, N, R, tor, X) ; ena

function dy = flux(t, y, Tend, A, D, muO, N, R, tor, X) globcil handle wb

global MO

do_wait_bar ( ) ;

ni = N. *y (1) . /D. /MO; ¾ normalised curren

dy = [0; 0; ] ;

dy(2) = -(l./tor) .* ( y(2) - N . *X . *y ( 1 ) ./D ) ;

dy(l) = -(D./mu0./N.-2./A) .* ( muO . *N . *A. *dy (2) + y(l) .*R.*(l - abs (ni) . ^Λ 1.001) ) ; % L:n¾ function do wait bar ( )

persistent Tlast

if (t <= eps)

Tlast = 0;

elseif (t > Tlast)

waitbar (t . /Tend, handle_wb) ;

Tlast = Tlast + 0.05.*Tend; % move it en e d

e d

e d