Electromagnetic heat engines and method for cooling a system having predictable bursts of heat dissipation

Magnetic heat engines directly converting heat to electricity, using emf induced by demagnetization. Generated power manifests as negative resistance, and almost any kind and shape of magnetic medium can be used. Electromagnetic engines are also tolerant to non-uniform heating, inherently non-contact and non-mechanical, easy to model and design, and operable at high frequencies. The engines are suitable for augmenting local heating, refrigeration without fluid refrigerants, efficient cooling of cryogenic components, synchronous cooling of digital circuits, completely solid-state power generation, and improvement of power plant efficiency and control.

FIELD OF INVENTION 
This invention generally relates to magnetic heat engines. More 
particularly, it is concerned with an electromagnetic heat engine for 
directly converting power between heat and electrical forms using 
magnetism. 
BACKGROUND OF INVENTION 
Most of the electrical power generated today comes from conversion of heat 
using steam turbo-generators. Magneto-hydrodynamic and magnetocaloric 
schemes, while reducing the solid moving parts, still involve an 
intermediate mechanical form as kinetic energy in a fluid medium, and 
furthermore, require the medium to be magnetic and conductive at the same 
time. The flow of fluids is particularly difficult to model and control, 
and in the above methodologies, the work transfers occur primarily through 
the physical boundary of the medium, which limits the throughput. The 
prior art lacks a heat engine in which the work transfer is not confined 
by the surface of the medium, and which converts heat to electricity 
directly without involving any intermediate mechanical form whatsoever. 
Hitherto only mechanical means have been used for the work transfers in 
magnetic engines, though they have been known for a century since Nikola 
Tesla's Thermo-Magnetic Motor U.S. Pat. No. 396,121, issued 15 Jan. 
1889!. The mechanism of inductive work transfer had not been conceived of 
for want of a negative resistance model of power generation. Inadequate 
thermodynamic insight is also to blame for the slow development of 
magnetic engines. 
Magnetism is made particularly difficult by the lack of a magnetic analog 
of the kinetic theory of gases. and by the multitude of units and 
conventions. Among the defects in the prior perception is the relative 
lack of interest in paramagnetism, the gaseous state of magnetization. 
Ferromagnetism basically means easier saturation, and holds even less 
energy than paramagnetism for a given magnetization intensity. Very high 
intensity fields are therefore needed in prior art magnetic engines to 
obtain useful power densities. 
The maximum magnetic energy densities are close to but less than the 
realizable energy densities in gases, because gases have no internal 
structure that can oppose or break under the stress. Higher operating 
speeds are required for magnetic engines to provide useful power 
densities, but the mechanical form of prior art magnetic engines severely 
limits their speeds. In cryogenic refrigeration, for instance, the speeds 
are down to a few cycles per second. 
Incidentally, thermodynamic ideas are lately being applied in the field of 
digital electronics, for reducing dissipation. Adiabatic switching and 
reversible computation require lowering of the operating speed, and are 
inapplicable to existing systems because different logic design principles 
are prescribed. Further, these schemes involve arguable extrapolations of 
thermodynamic ideas, particularly while a fundamental theoretical relation 
between abstract information and physical entropy is yet to be discovered. 
Much of the dissipation in modern CMOS technologies is due to motion of 
charges during logic state transitions. A more direct application of 
thermodynamics would appear appropriate to the problem, considering that 
the dissipation is periodic and driven by a clock signal, allowing a heat 
engine to be operated in synchronization to remove the heat rapidly. The 
dissipation in a given cycle is however relatively small and randomly 
located in the physical circuit. Further, it is the peak temperatures 
instantaneously reached within an individual logic gate or transistor 
structure that ultimately limits the performance and packing density, and 
therefore needs to be directly handled. Only magnetism can instantly 
couple the energy of such predictable bursts over the extent of a chip or 
system, and the coupling must be inductive rather than mechanical for 
practical reasons. None of the prior art heat engines qualify for this 
requirement. 
Accordingly, it is an object of the present invention to provide a 
thermodynamic means for directly converting power between heat and 
electrical forms using magnetism. 
Another object of the invention is to allow use of wider range of magnetic 
media and operating conditions in magnetic heat engines. 
Yet another object of the invention is to provide a means for operating 
magnetic heat engines at higher speeds, to obtain greater conversion power 
densities. 
A further object is to establish negative resistance as a useful model of 
power generation with the added feature that the power can be regulated 
and controlled by a low power input needed to setup the load current. 
A still further object of the invention is to provide a direct 
thermodynamic means for cooling synchronous digital systems. 
SUMMARY OF THE INVENTION 
In the present invention, these purposes, as well as others which will be 
apparent, are achieved generally by providing a heat engine which directly 
converts between heat and electrical power forms using magnetism. More 
particularly, the invention is concerned with an electromagnetic heat 
engine in which electrical power is absorbed or generated inductively by a 
magnetic medium. The invention also improves upon known magnetic heat 
engines by incorporating a wider range of magnetic media, and by allowing 
higher operating speeds than was hitherto possible. 
Advantage lies in potentially eliminating moving parts, directly converting 
to electricity at useful power densities, greater choice in selection of 
magnetic media and operating speeds, more flexible engine and system 
design, instant thermodynamic use of local hot spots, and the capability 
to control dissipation in digital systems by the instant conversion to 
electricity. The present invention thus consists of three parts: 
An electromagnetic heat engine, inductively converting power between heat 
in a magnetic medium and current in an electrical circuit. 
The magnetic Carnot cycle consists of magnetizing a magnetic medium at one 
temperature and demagnetizing the medium at a different temperature. If 
the susceptibility of the medium drops with rising temperature, as is 
commonly the case, heat gets absorbed during demagnetization and released 
during magnetization. Appropriate heat transfers are therefore required to 
keep the temperature steady during each of these isothermal operations. 
The temperature changes are effected by adiabatic magnetization or 
demagnetization operations. Work is done on the medium during 
magnetization, and the medium performs work as it demagnetizes, but the 
work transfers are unequal because the susceptibility changes with the 
temperature. Performing magnetization cycles in synchronization with 
temperature cycles thus results in net conversion between heat and 
coherent power. 
In the electromagnetic engine, the electric current does work in 
magnetizing the medium, and conversely, the medium does work on the 
electrical circuit by the induced emf during demagnetization. The engine 
thus converts between heat and electrical power. Since the induced emf is 
proportional to the current, the work done by the medium instantaneously 
appears as a negative resistance in the circuit. Correspondingly, 
magnetization induces a positive resistance in the circuit. The induced 
resistance also varies with the susceptibility, which depends on the 
instantaneous temperature, hence a net negative resistance is induced when 
operating as an engine. 
The negative resistance is not merely incremental, as obtained, for 
instance, in tunnel diodes, but is obtained even when the current is d.c. 
The negative resistance form of the induced power permits regulation and 
control through an auxiliary electrical power source needed to setup the 
current. 
Useful conversion power densities using high susceptibility media and high 
operating speeds. Greater freedom in system design. 
The work done per cycle by a heat engine is a direct fraction of the energy 
stored in the thermodynamic medium. The conversion power density is 
therefore directly proportional to the density of energy storage in the 
medium. Under practical operating conditions, gas engines are confined to 
a small range of power densities by the universal gas constant, and by the 
mechanical constraints on speed. 
The magnetic energy density depends on the susceptibility of the medium. 
Available susceptibilities span several orders of magnitude, ranging from 
the weak paramagnetism of fifty five elements to the strong ferromagnetism 
of iron. The corresponding energy densities range from a few microjoules 
per cm.sup.3 in many paramagnetic materials, to over 1 J/cm.sup.3 per 
Tesla in gadolinium and dysprosium in their ferromagnetic states. The 
comparative figure for gases is about 40 mJ/K-cm.sup.3 at ordinary 
pressures, and the range is limited to one order of magnitude. 
The range of susceptibilities, the various forms of magnetism available, 
and a wide range of operating speeds, mean a greater freedom in system 
design. This freedom was not realized in prior art magnetic engines 
because of restricted attention to ferromagnetism and mechanical 
operation. Though higher power densities are available particularly when 
using ferromagnetism, the theory and the practical aspects of 
paramagnetism remained unexplored. With the negative resistance approach 
and inductive work transfer, electromagnetic engines are better suited to 
exploiting the entire gamut of magnetic characteristics. 
Further, the operating frequency of electromagnetic heat engines is only 
limited by the speed of the temperature changes, making electromagnetic 
heat engines uniquely suitable for a particular application in modern 
technology, involving rapid local temperature changes at frequencies 
extending to several gigahertz. 
Synchronous cooling of digital systems. 
Dissipation in digital systems tends to be concentrated in bursts following 
clock edges that trigger the digital state transitions. An electromagnetic 
engine can therefore be constructed around a digital system, using a 
magnetic medium repeatedly heated by the bursts of dissipation. By 
synchronizing the engine current with the system clock, the current is 
made to build up inbetween bursts, when the medium is at a lower 
temperature, and is reduced during the bursts at the higher temperature 
caused by the dissipation. At least part of the dissipated heat thus gets 
converted to electricity. 
The per-clock switching dissipation of a logic gate is quite small, and in 
practice only a few gates change states at any given clock edge in many 
circuits. The overall dissipation becomes significant primarily because of 
the high operating frequencies. Consequently, the per-cycle conversion 
power density required for the cooling is quite small, and the performance 
is limited by the efficiency rather than by the conversion density. Even 
the efficiency can be considerably higher than expected from the overall 
operating temperatures, because the conversion depends on the 
instantaneous temperature reached in the immediate vicinity of the 
transiting gates. 
Other objects, features and advantages of the present invention will be 
apparent when the detailed description of the preferred embodiments are 
considered in conjunction with the drawings, which should be construed in 
an illustrative and not limiting sense as follows.

THEORY OF OPERATION 
Theoretical background 
The dynamical perception of magnetism has been so insufficient in the past 
that some researchers have indeed used the potential instead of the actual 
energy of magnetization see Chapter 18, Solid State Physics, A J Dekker, 
Prentice-Hall, 1957!, by associating the applied field B with the gas 
volume V, and the magnetization M, with the pressure p. This association 
is inappropriate because it ignores the fact that the thermal activity 
tends to gain orientational freedom for the elementary moments, so the 
magnetization, not the applied field, relates to a conceptual volume. The 
error is indicative of the theoretical difficulty in the prior art. 
A simple kinetic theory of magnetization is indispensable for understanding 
and designing magnetic engines. Magnetization is the angular confinement 
of atomic moments by an applied field, and is analogous to the spatial 
confinement of a gas by walls that withstand its pressure. 
Idealness, as in the term "ideal gas", means absence of significant 
internal interactions between the microscopic constituents. At low 
temperatures, interactions between atomic magnetic moments begin to 
dominate over the thermal activity, eventually causing the magnetization 
to condense around the Curie temperature. Ferromagnetism in general refers 
to the various condensed states of magnetization, while Curie law 
paramagnetism constitutes the ideal gas state. 
Basic heat engine theory 
In prior thermodynamic theory, the gaseous phase of matter describes the 
simplest linear relation between a pair of conjugate dynamical variables p 
and V: 
EQU pV=constant, (1) 
known as Boyle's law. The dynamical variables are conjugate in the sense 
that one represents a force and the other, a distance, such that while pdV 
defines a work differential, Vdp does not. 
Thermo-dynamics results when such a pair of conjugate dynamical variables 
is related through an intensive property called the temperature, for then, 
the dynamical work for a given motion dV can be varied. An analytical 
relation of conjugate dynamical properties with the temperature is called 
the equation of state, and suffices to define a cyclic thermodynamic 
process. 
The simplest equation of state is that of the ideal gas: 
EQU pV=rT, (2) 
where the constant r is the gas constant R=8.3144 J/K, times the number of 
moles of the gas. A very general equation of state would be the 
multinomial 
##EQU1## 
The higher order terms in this form relate to interactions between the 
molecules. 
The simplest cyclic process was first described by Carnot. The process 
theoretically confines the heat transfers to two given temperatures, and 
needs a pair of adiabatic operations to complete a cycle in the phase 
space. Adiabatic operations are described by the adiabatic equation, which 
takes the form 
EQU pV.sup..gamma. =constant (4) 
for the ideal gas, where .gamma. is a ratio of specific heats. No net work 
is done by the two adiabatic operations, which simply translate the medium 
between equal temperature intervals. Equation (2) shows that the 
isothermal energy transfers are proportional to the respective 
temperatures: 
EQU .delta.w.sub.i =rT.sub.i .delta.(1nV), i=h, l, (5) 
where the subscripts h and l stand for the high and low temperatures, 
respectively. The net work per cycle is the difference of the isothermal 
energy transfers: 
EQU .delta.w=r.delta.T.delta.(1nV). (6) 
The local conservation of energy requires an energy transfer 
.delta.q.sub.i, called heat, exactly compensating for each work increment 
.delta.w.sub.i : 
EQU .delta.w.sub.i +.delta.q.sub. =0, i=h,l, (7) 
because the internal energy does not change in isothermal operations on an 
ideal gas. For the same reason, all heat transfers are exactly equivalent 
to work increments of the same magnitude, so the theory and equations for 
Carnot cycles on an ideal gas are essentially dynamical, the only 
non-dynamical variable necessary being the temperature. The efficiency is 
therefore given by: 
##EQU2## 
Carnot's theory and the second law guarantee that the same efficiency will 
hold for all Carnot cycles, irrespective of the idealness of their media. 
With non-ideal media, the internal energy u is no longer constant during 
isothermal operations, for which the complete equation 
EQU du=-dq-pdV 
is required. The second law of thermodynamics requires that the 
corresponding entropy change ds=dq/T be an exact differential, hence the 
above equation leads to 
##EQU3## 
showing that in an isothermal operation, 
##EQU4## 
Magnetic engine theory 
Even simpler isotherms and adiabatics are obtained in the analogous theory 
developed below for magnetic thermodynamic cycles. 
We shall use Tesla as the common unit for both applied magnetic field B and 
magnetization H, so that the (volume) susceptibility .chi..ident.M/B 
becomes a dimensionless ratio related to the relative permeability 
.mu..sub..tau. as 
EQU .mu..sub..tau. =.chi.+1, 
and, further, the energy density of the fields can be compared directly. A 
strength of 1 T means 1/2.mu..sub.0 .apprxeq.397.89 J/m.sup.3, whether it 
be of B or M. One may convert from Tesla to amperes per metre by dividing 
by .mu..sub.0 =4.pi..times.10.sup.-7. 
Paramagnetism occurs when the alignment of the atomic moments is determined 
by thermal activity, giving for the magnetization: 
##EQU5## 
where N is the number density of the atomic moments, k.sub.B is the 
Boltzmann constant, B.sub.J is the Brillouin function, and .mu. is the 
component of individual atomic magnetic moment along or against the 
applied field. The component .mu. is related to the total atomic magnetic 
moment as: 
##EQU6## 
where p is the effective atomic magnetic moment in units of the Bohr 
magneton .mu..sub.B. In weak fields, the magnetization is quite linear, 
adequately described by 
EQU M=N.mu..sup.2 B/k.sub.B T. (11) 
B and M are conjugate dynamical variables because they involve the work 
dw=-BdM/.mu..sub.0, where the negative sign reflects the fact that the 
work BdM/.mu..sub.0 gets done on the medium in magnetization. They are 
evidently related via the temperature by equations (10) and (11), which 
therefore constitute the paramagnetic equations of state. 
FIG. 1 shows that it is appropriate to associate B, the applied field, with 
the confining pressure (as in a cylinder with a piston), and 
m.ident.M.sup.-1 with the confinement volume, ie. the angular freedom 
.theta. in the figure, since the atomic moments are unconfined (m=.infin.) 
In the figure, the applied field B causes an aligning torque 
.tau.=.mu..sub.B .times.B, on the atomic moment .mu..sub.B, so as to 
compress the latter's angular degree of freedom (.theta.). Thermal 
interactions however tend to disrupt the alignment, tending to expand the 
angular distribution of the atomic moments. The effect is an expanding 
statistical torque -.tau. proportional to the temperature T. The 
equilibrium magnetization is a statistical balance between the aligning 
torque and this expanding "angular pressure". 
Classical Langevin theory represents the aligning torque by its potential 
energy, and the "angular pressure", by the Boltzmann distribution. Both 
torques are replaced by transition probabilities in the quantum picture, 
with substantially the same physical results. 
The association of B with pressure indeed conforms to dynamical 
considerations, for work is done on the medium by magnetization under 
constant applied field, dw.varies.-BdM, but no work is done on the medium 
by increasing the applied field if the magnetization does not change, 
which happens when the medium is saturated, ie. MdB does not represent 
work. This compares with the theory of gases, in which pdV represents work 
but Vdp does not. 
It helps to think of m as being similar to V, but the analogy is not exact 
because the incremental work is proportional to -BdM, not Bdm. The 
thermodynamics of magnetization follows from these dynamic variables. 
Equation (11) may be rewritten as 
EQU 1/.chi.=B/M=k.sub.B T/N.mu..sup.2 =kT, (12) 
where k=k.sub.B /N.mu..sup.2. This is known as the Curie law, usually 
written as 
##EQU7## 
where C.ident.k.sup.-1 is called the Curie constant. 
The adiabatic equation corresponding to equation (12) is derived next. The 
adiabatic condition requires that any work done on the medium should 
exactly equal the rise in its thermal energy content: 
##EQU8## 
giving 
EQU -kdm/.mu..sub.0 cm.sup.3 =(dm/m+dB/B) 
or 
EQU B/M=Ae.sup.kM.spsp.2.sup./2.mu..sbsp.0.sup.c =Ae.sup..kappa.M.spsp.2, 
where c is the applicable heat capacity, .kappa.=.kappa./2.mu..sub.0 c, and 
A is a constant of integration depending on the thermal energy density at 
any particular state: 
EQU A=kTe.sup.-.kappa.M.spsp.2. 
The magnetic adiabatic equation is therefore: 
EQU 1/.chi.=B/M=kT.sub.0 e.sup..kappa.(M.spsp.2.sup.-M.sbsp.0.spsp.2.sup.),(13) 
where T.sub.0 and M.sub.0 describe a reference point on the adiabatic. 
In the B-M space, the Curie law isotherms form a family of straight lines 
through the origin, while the adiabatics are a family of parabolas that 
clearly intersect the isotherms, allowing one to define Carnot cycles. 
Note that the applicable heat capacity c is most likely to be c.sub.p, 
because the medium would most likely be subject to a constant 
(atmospheric) pressure. 
When B and M are both measured in Teslas, the applied field holds the 
energy density: 
EQU E=B.sup.2 /2.mu..sub.0, (14) 
while that in the magnetization is: 
EQU U=BM/2.mu..sub.0 =.chi.E=E/kT. (15) 
U differs from the potential energy only in the sign. 
The total energy is: 
##EQU9## 
It is also useful to define a pseudo-energy F.ident.M.sup.2 /2.mu..sub.0. 
The isothermal energy transfers integrate to: 
EQU .delta.w.sub.i =.delta.U.sub.i,i=h,l, (17) 
corresponding to equation (5) for a gas. 
FIGS. 2a-c describe the magnetic Carnot cycle abcda working with non-zero 
magnetic fields. 
FIG. 2a is the classical temperature-entropy (T-S) diagram of a heat engine 
working between the temperatures T.sub.h and T.sub.l (T.sub.h &gt;T.sub.l). 
FIG. 2b is the corresponding B-m diagram, which is similar to the p-V 
diagram of a gas Carnot cycle. The isotherms ab and cd are hyperbolae 
exactly as in the gas engine theory. The adiabatics bc and da differ from 
the gas theory, but are somewhat similar in form, since they intersect the 
isotherms. FIG. 2c is the equivalent B-M plot. 
The cycle consists of isothermal demagnetization ab at the higher 
temperature T.sub.h, adiabatic demagnetization bc lowering the temperature 
to T.sub.l, isothermal magnetization cd at this lower temperature, and 
finally adiabatic magnetization da raising the temperature back to 
T.sub.h. 
The magnetic Carnot cycle can be operated in the reverse (adcba) to 
function as a refrigerator or a heat pump. 
Equation (17) is insufficient to determine the cycle, since the isothermal 
energy transfers .delta.U.sub.i are as yet unrelated. The common 
temperatures relate the ends of the adiabatic steps bc and da as: 
EQU .chi..sub.a =.chi..sub.b and .chi..sub.c =.chi..sub.d. 
By the adiabatic equation (13), these connectivity conditions are 
equivalent to: 
EQU M.sub.a.sup.2 -M.sub.d.sup.2 =M.sub.b.sup.2 -M.sub.c.sup.2. 
Transposing terms leads to 
EQU .delta.F.sub.l =.delta.F.sub.h .ident..delta.F. (18) 
This is simply equivalent to the condition 
EQU .delta.M.sub.l =.delta.M.sub.h .ident..delta.M, (19) 
ie. the high and low temperature magnetization changes must be equal. The 
corresponding connectivity condition in gas engine theory is the equality 
of the volume ratios for the isothermal operations. The pseudo-energy F is 
clearly related to the paramagnetic entropy. 
The last equation is equivalent to 
EQU .chi..sub.l .delta.U.sub.l 32 .chi..sub.h .delta.U.sub.h, (20) 
giving 
EQU .delta.w.sub.i =-kT.sub.i .delta.F, i=h,l. (21) 
From this, it is easy to show that the magnetic Carnot cycle has the same 
efficiency, equation (8), as the gas Carnot cycle. The work per cycle is: 
EQU .delta.w=k.delta.T.delta.F=.eta..sub.c .delta.U.sub.h. (22) 
It is helpful to note that 
##EQU10## 
Equation (21) means that the work transfer of larger magnitude occurs at 
the higher temperature, just as in a gas engine, even though the work 
transfers are reversed in direction. This does not affect the ability of 
magnetic engines to do work, however, as will be seen in the detailed 
description further below. 
Equation (22) says that, for a given temperature difference, the power 
density of a magnetic engine using a paramagnetic medium is proportional 
to the susceptibility of the medium at the higher temperature. Comparison 
with equation (6) shows that the pseudo-energy .delta.F plays the same 
role as the compression factor .delta.(1nV), but can vary by several 
orders of magnitude depending on the material. 
Extension to ferromagnetism 
As in the thermodynamics of gases, the Carnot efficiency also holds for 
non-ideal paramagnetism, which involves a non-zero interaction between the 
atomic moments. 
The most important non-ideal paramagnetism is that associated with 
ferromagnetism, and is described by: 
##EQU11## 
giving 
EQU B=k(T-.theta.)M, 
where .theta. is called the Curie temperature. This is commonly written as 
the Curie-Weiss law: 
##EQU12## 
A variant of this law due to Neel is associated with antiferromagnetism, 
and has the form: 
##EQU13## 
where .theta..sub.N is the Neel temperature. 
All forms of condensed magnetism reduce to paramagnetism at temperatures 
above the respective Curie temperatures. Provided the specific heat c 
remains largely constant, the magnetic adiabatic equation (13) and the 
connectivity conditions (18)-(20) continue to hold. The adiabatics are the 
family of parabolas in the B-M space as in Curie law paramagnetism, and 
the isotherms are again straight lines, though no longer passing through 
the origin. 
The isothermal involvement of the internal energy u follows from the 
association of pressure p with -B/.mu..sub.0 and volume V with M in 
equation (9), giving 
##EQU14## 
hence 
EQU .delta.q.sub.i 32 kT.sub.i .delta.F, i=h,l. (26) 
The work transfers are directly integrated from equation (24) as 
EQU .delta.w.sub.i =.delta..intg.-BdM/.mu..sub.0 =-k(T.sub.i 
-.theta.).delta.F,(27) 
and the internal energy changes are 
EQU .delta.u.sub.i =-.delta.q.sub.i =.delta.w.sub.i =-k.theta..delta.F.(28) 
Clearly, the same work is obtained per cycle as with the Curie law 
material, and at the same efficiency, but using smaller work transfers. 
The Neel law is equivalent to the Curie-Weiss law with a negative .theta., 
so antiferromagnetism merely requires larger work transfers. In electrical 
terms, a ferromagnetic medium appears to be less reactive, and an 
antiferromagnetic medium, more reactive, than a paramagnetic medium 
performing the same work per cycle. 
The Curie constants for the paramagnetic elements listed in Table 1 further 
below are generally less than unity, except for dysprosium (18.75K) and 
gadolinium (41K). Cohesive interaction between the atomic moments lead to 
much higher effective C, of the order of 10.sup.3 in superparamagnetic 
media, for which the Curie law takes the form 
##EQU15## 
where d is the average diameter of the ferromagnetic particles, M.sub.d is 
the domain magnetization, and .phi. is the volume fraction of the solid 
present chapter 2.7, Ferrohydrodynamics, R Rosensweig, Cambridge 
University Press, 1985!. The increased C is due to the larger constituent 
magnetic moments, because the liner the particles, the lower is the 
(initial) susceptibility obtained. 
Rosensweig also derives the efficiency of a magnetic engine cycle operating 
entirely under saturation just below .theta. where the intrinsic domain 
magnetization drops linearly to zero with rising T. chapter 6, 
Ferrohydrodynamics!. As in the prior art, the treatment given by 
Rosensweig is specific to saturation magnetization, approximated over the 
operating range by 
EQU M.sub.sat =K(.theta.-T). (30) 
The engine cycle is necessarily non-Carnot, so a regenerative mechanism is 
proposed to restore the efficiency. The engine is also restricted to 
mechanical work transfers, since a superconducting magnet is required for 
maintaining a saturating field. 
A more general theory using the same approximation is developed below for 
the electromagnetic engines. Since the work must extracted through the 
magnetizing current, the applied field cannot be maintained at saturation, 
but must vary with the electrical load. Following standard engineering 
practice, the ferromagnetic behavior is approximated by an initial 
susceptibility, assumed to be proportional to the domain magnetization, to 
obtain the linear ferromagnetic equation of state: 
##EQU16## 
where C is the equivalent "Curie constant" for the modified Curie-Weiss 
law 
##EQU17## 
FIG. 2f shows the variation of magnetization with temperature under 
non-saturating applied fields. For consistency, the applied field is still 
designated as B, though, particularly when dealing solely with 
ferromagnetism, one usually uses the letter H. 
The corresponding linear ferromagnetic adiabatic equation follows from: 
##EQU18## 
which would reduces to the constant A, and thus to the isothermal form, if 
the second order term in the square root were to vanish. The adiabatics 
clearly intersect the isotherms, and Carnot cycles can once again be 
defined. 
In the B-M space, the linear ferromagnetic isotherms are again straight 
lines through the origin, while the adiabatics are ellipses centered on 
the origin, intersecting the B axis (T=.theta.) at B=.sqroot.kc.mu..sub.0 
, and the M axis at M=(.theta.-T).sqroot.c.mu..sub.0 /k, as shown in FIG. 
2g. 
Equation (32) leads to the connectivity condition 
##EQU19## 
or equivalently, 
##EQU20## 
and the ratio should be completely determined by the temperature range for 
the Carnot cycle. The connectivity condition is clearly more like that for 
gas engines, unlike the condition for paramagnetism, equation (18). 
The work transfers integrate to 
##EQU21## 
but the .delta.Fs differ. The isothermal heat transfers are as follows: 
##EQU22## 
giving for the internal energy changes 
##EQU23## 
Since cyclic operations cannot change the internal energy, the net work per 
cycle .delta.w=.delta.w.sub.h -.delta.w.sub.l, and the net heat transfer 
per cycle .delta.q=.delta.q.sub.h -.delta.q.sub.l, must be equal and 
opposite. Appropriate algebraic manipulation of equations (34) through 
(36) results in the condition 
##EQU24## 
This can be further reduced, using the modified Curie law (31), to 
##EQU25## 
If T.sub.h be quite close to .theta., .delta.T would be almost the same as 
.theta.-T.sub.l, so ideally the entire .delta.U.sub.l becomes available. 
Since ferromagnetic media are easily magnetized to saturation, the 
saturation energy density can be conveniently assumed to be the maximum 
work per cycle. 
The theory for more complex magnetic equations of state, including 
non-linear behavior below or across the Curie temperature, can be derived 
analogously. For example, in some materials, such as ferrites, the 
susceptibility distinctly rises with the temperature a little below the 
Curie point, so the heat transfers become reversed in direction when 
operating over such temperature ranges. The theory however remains 
substantially the same, and an electromagnetic engine can be operated 
between any two temperatures that yield different susceptibilities. 
Susceptibility and power density 
It remains to explore the power densities actually realizable from the 
various forms of magnetism. 
Linear paramagnetic susceptibilities are limited, from equation (11), by 
the numerical density of atoms, N, and the square of the effective atomic 
moment .mu.. Though neither N nor .mu. can vary by more than an order of 
magnitude, the variation of .mu. leads to large variations in magnetic 
behavior. 
Table 1 contains a short list of the (volume) susceptibilities .chi., with 
the corresponding energy densities U, of several paramagnetic elements, 
computed from handbook cgs molar values using the relation 
EQU .chi.=4.pi..times.10.sup.-6 .chi..sub.m .rho./M, 
where M is the molar weight (10.sup.-3 kg), .rho. is the density (10.sup.-3 
kg/m.sup.3), and 4.pi..times.10.sup.-6 converts the cgs mass 
susceptibility (in cm.sup.3 /mole) to SI units (m.sup.3 /mole). 
TABLE 1 
______________________________________ 
Susceptibilities 
.chi.m .chi. U 
(10.sup.-6) .rho. M (10.sup.-6) 
.mu.J/T-cm.sup.3 
______________________________________ 
Al 17 2.7 27 21 8 
Cr 180 7.2 52 313 125 
Dy 98k 8.5 163 64k 26k 
Gd 185k 7.9 157 117k 46k 
Mg 13 1.7 24 12 5 
Mn 529 7.4 55 900 358 
Pd 567 12.0 106 804 320 
Pt 202 21.5 195 279 111 
Ti 153 4.5 48 179 71 
______________________________________ 
(k .ident. 10.sup.3, gadolinium data at 350K) 
The table lists the total susceptibilities, including the inherent 
diamagnetism and the temperature-independent electron paramagnetic 
contributions in metals. Only the temperature-dependent paramagnetic 
component is useful to thermodynamics, which means that the effective 
values are likely to be somewhat different. The numbers however serve to 
illustrate the magnitudes involved. 
Vacuum represents a temperature-independent capacity of 
E.ident.1/2.mu..sub.0 .apprxeq.400 kJ/T-m.sup.3 or 400 mJ/T-cm.sup.3. Air 
incidentally has a paramagnetic susceptibility, due to its oxygen content, 
of about 1.4.times.10.sup.-7, so air adds 56 nJ/T-cm.sup.3 to the 
capacity. The mechanical power of an "air-cored" magnetic engine would 
therefore be quite undetectable. For example, the air flow in an aircraft 
jet engine is of the order of 100 m.sup.3 /sec, representing only 5 W/T of 
magnetic storage capacity. 
Gases have energy densities of the order of 831.44/22.414.times.10.sup.3 
=37 mJ/K per cm.sup.3 at ordinary pressure and temperature. The rare earth 
elements have per Tesla magnetic energy densities of the same order: 26 mJ 
in dysprosium and 46 mJ in gadolinium. 
Practical operating temperature ranges are in hundreds of degrees, so the 
limit on gas engine power density is actually of the order of 3.7 
J/cm.sup.3 or 185 kW/litre at 50 Hz operating frequency. This number must 
be further scaled up by the log of the compression ratio, according to 
equation (6). Throughputs of the order of 150 kW/litre are actually 
obtainable. 
Practical applied field variations can usually be no more than 0.1 T (about 
80 kA/m), and further the operating temperature difference does not factor 
into the throughput with the energy density, according to equation (22). 
This means a maximum power density of only about 130 W/litre using 
dysprosium or 230 W/litre using gadolinium at around 60.degree. C. 
Magnetic engines using linear paramagnetism would clearly have to operate 
at least 1000 times faster to yield power densities comparable to gas 
engines. This is not a serious problem for electromagnetic engines, whose 
operating speed is limited only by the circuit time constants and the 
relaxation rate of the magnetism, which is very high in the case of linear 
paramagnetism. 
Saturation energy densities at a hypothetical 1 T applied field are listed 
in Table 2 below. The applied field actually required for saturation 
varies with the material. For example, in soft iron at ordinary 
temperatures, the initial susceptibility is of the order of 10,000, 
suggesting onset of saturation at a mere 0.2 mT (160 A/m). Higher applied 
fields are actually required because the susceptibility diminishes as 
saturation is approached. 
The easier saturation in the ferromagnetic regime, means that the 
saturation energy densities do become pg,27 
TABLE 2 
______________________________________ 
Saturation energy densities 
M.sub.sat (T) 
U.sub.sat (J/T-cm.sup.3) 
______________________________________ 
Dy 3.67 1.46 
Gd 2.59 1.03 
Ni 0.64 0.25 
Mg 0.56 0.22 
Fe 2.18 0.87 
Co 1.82 0.72 
______________________________________ 
available at the same O(0.1 T) applied field variations, when operating at 
temperatures below the respective Curie temperatures. Thus, soft iron can 
yield 0.087 kJ/litre, or over 4 kW/litre at 50 Hz. To compare, dysprosium 
can convert 0.146 kJ/litre or over 7 kW/litre at 50 Hz, so the difference 
is not much. A 0.1 T applied field corresponds to about 100 turns/cm at 8 
A load current. The soft iron engine would therefore be contributing 500 
V, at an effective negative resistance of 62.5 .OMEGA.. At 10 turns/cm and 
the same load current, the engine power would reduce to 400 W, the 
voltage, to 50 V, and the effective negative resistance would be only 6.25 
.OMEGA.. 
If the applied field ("ampere-turns") can be raised, possibly by using 
superconducting wires, to 1 T say (approximately 800 kA/m), the soft iron 
engine would produce about 40 kW/litre. For comparison, paramagnetic 
gadolinium would still be limited to under 2.3 kW/litre at this applied 
field intensity. 
Ferromagnetism is clearly advantageous in terms of power density, which 
cannot be raised further merely by using superparamagnetic media, which 
have the same saturation as the bulk ferromagnetic materials from which 
they are derived. Higher power densities can still be obtained by 
increasing the operating speed. A 0.1 T applied field variation 
corresponds to about 40 W/litre-Hz per Tesla of saturation magnetization. 
Ferrofluid relaxation speeds are better than 100 ns, or 10 MHz, at under 10 
nm particle size, so a 1 MHz operation is feasible from standpoint of the 
medium itself, which sets an upper bound of about 40 MW/litre per Tesla of 
saturation. The speed would be generally limited by the heat transfers, 
though the temperature of the medium is ideally changed by the adiabatic 
operations. In applications like synchronous cooling, where high speed 
heat input is not a problem, it is indeed possible to drive 
electromagnetic engines at microwave frequencies using paramagnetic media. 
In general, reciprocating electromagnetic engines are largely limited in 
power density by the realizable speed of thermal diffusion. This 
limitation is somewhat alleviated in turbine design, in which the heat 
transfer is effectively speeded up by the motion of the medium through the 
active portions of the engine. This comparison is equally valid for gas 
engines. 
Negative resistance generation theory 
A negative resistance is a useful form of power generation, though it needs 
an auxiliary source to setup the load current. Consider an electrical 
circuit containing just the auxiliary voltage source V.sub.b and a load 
resistance R.sub.L. The current in the circuit would be i.sub.b =V.sub.b 
/R.sub.L, and the power flow is P.sub.b =V.sub.b.sup.2 /R.sub.L. These 
base values are modified when a resistive device R.sub.t is introduced 
into the circuit as shown in FIG. 3. 
In the figure, R.sub.L represents the total electrical load, V.sub.b is the 
voltage due to the auxiliary source, i is the load current, and R.sub.t is 
the equivalent resistance of the device. When the device is not operating, 
R.sub.t =0, so the load current is i=V.sub.b /R.sub.L. When the device 
supplies electrical power to the circuit, R.sub.t becomes negative, and 
the load current becomes V.sub.b (R.sub.t +R.sub.L)&gt;V.sub.b /R.sub.L. 
For a given voltage V.sub.b, the device resistance changes the current to 
##EQU26## 
where 
##EQU27## 
The power drawn from the auxiliary source also changes by the same factor 
.beta.: 
EQU P.sub.i=.beta.i.sub.b V.sub.b =.beta.P.sub.b. (42) 
However, the load power changes much more: 
EQU P.sub.o =i.sup.2 R.sub.L =.beta..sup.2 P.sub.b =.beta.P.sub.i.(43) 
The factor .beta. is thus the power gain caused by the resistive device. 
Correspondingly, .alpha. may be called the cohesion factor, for it 
represents power brought in, rather than being dissipated, by the device. 
If .alpha.&lt;0, the device is dissipative and drains power, while a positive 
.alpha. means the device is adding power to the circuit. 
It is easy to show that the power due to the negative resistance is 
EQU P.sub.t .ident.P.sub.o -P.sub.i =(.beta.-1)P.sub.i -.alpha.P.sub.o.(44) 
Also, the voltage V.sub.L seen by the load is 
EQU V.sub.L =iR.sub.L =.beta.V.sub.b, (45) 
showing that the voltage delivered to the load is also magnified by the 
gain .beta.. For the load voltage to remain constant, V.sub.b must be 
reduced as the .beta. increases. 
FIG. 4 shows how the gain and the cohesion can vary with the device 
R.sub.t. As .alpha. tends to unity, the load power is almost totally 
supplied by the device. A vanishing device resistance R.sub.t =0 is 
represented by .beta.=0, .beta.=1. 
Electrical loads are dissipative, including motors, heat pumps and 
refrigerators, and a dissipative device is characterized by the region 
.alpha.&lt;0, R.sub.t &gt;0. 
Negative resistance devices must operate in the region 0&lt;.alpha.&lt;1, 
.beta.&gt;1, since they must mean additional power to the load circuit. 
Equations (40) and (42) show that more power is also drawn from the 
auxiliary source by the factor .beta.. 
For example, consider a load resistance R.sub.L =10 .OMEGA., and an 
auxiliary source of 10 V. The base value current i.sub.b is then 1 A, and 
the base value load power is 10 W. Introducing a device of -7.5 .OMEGA. 
shifts the circuit to .alpha.=+0.75, .beta.=4. The current rises to 4 A, 
so the auxiliary source now sources 40 W. The load however gets 160 W, so 
the circuit gains 120 W from the negative resistance device. The load sees 
40 V, of which only 10 V is from the auxiliary source and 30 V is due to 
the negative resistance. 
Consider what happens if the auxiliary source is a current source instead, 
driving exactly 10 A through the same load R.sub.L. When R.sub.t is 
inserted into the circuit, the current source holds the current down to 10 
A, but only needs to put in V.sub.b /.beta.=2.5 V to sustain the current. 
The load power P.sub.o does not change, but only 2.5 W now comes from the 
auxiliary source, and the remaining 7.5 W comes from the negative 
resistance. The load still sees 10 V, but only 2.5 V now comes from the 
auxiliary source, and 7.5 V is from the negative resistance. 
The infinite gain at .alpha.=1 is equivalent to taking zero power from the 
auxiliary source, and all the load power should come from the negative 
resistance device, as indicated by equation (44). Over unity .alpha. would 
mean that the device supplies power to the auxiliary source as well. 
Clearly, neither of these modes of operation can occur with 
electromagnetic engines. 
The delivered power comes from the auxiliary source and the negative 
resistance device, so both inputs are means of controlling the delivered 
voltage: 
##EQU28## 
and power: 
##EQU29## 
The sensitivity of the overall system to fluctuations in the load 
resistance is easily derived from the above. 
Electromagnetic engine theory 
The dynamical variables B and M directly interact with electric currents, 
allowing a direct coupling of the electrical and thermal energies. The 
direct coupling distinguishes electromagnetic engines from the prior art 
magnetic engines using electromagnets, including cryogenic magnetic 
refrigerators, in which the work per cycle is purely mechanical. 
An electric current does work in magnetizing a medium. Conversely, 
demagnetization induces electrical energy into the magnetizing circuit, in 
the form of a back-emf. It has been noted before that in adiabatic 
demagnetization, the atomic moments in the medium do work on the 
withdrawing field as they demagnetize Feynman, in Lectures, Vol. II, 
section 35-5!. The demagnetization is thus limited by the speed of thermal 
relaxation, which is however sufficiently fast for engine applications. 
Actually, even the magnetization is limited by the thermal relaxation, 
which makes the thermal dependence symmetric. Other magnetic issues, such 
as the energy of the applied field, the diamagnetism of all materials, and 
hysterisis effects of various components and neighboring materials, are 
not directly concerned with the thermodynamics of the electromagnetic 
engines. 
FIGS. 2d and 2e show the magnetic Carnot cycle more suited to the 
electromagnetic engines. Since the magnetization is due to an electric 
current, which can reduce to zero cyclically, especially when using a.c., 
the Carnot cycle of FIGS. 2b and 2c must be modified to include the zero 
magnetization. The corner points b and c are moved to m=.infin. in FIG. 
2d, and to the origin, M=0, in FIG. 2e. Note that the graphs ignore the 
signs of the variables, otherwise a symmetric third quadrant would be 
needed to show both phases in a.c. operation. 
The instantaneous emf due to magnetization or demagnetization caused by the 
current i in a coil, is given by 
EQU .xi.=-N.phi..sub.M =-L.sub.M di/dt, (48) 
where N is the number of turns in the coil, .phi..sub.M is the rate of 
change of the magnetization part of the total flux, and L.sub.M is the 
magnetization part of the inductance, defined next. A magnetic circuit of 
effective length l and effective cross sectional area A has the reluctance 
and inductance 
##EQU30## 
respectively. The magnetization contributes 
##EQU31## 
while the applied field relates to 
##EQU32## 
respectively, giving 
EQU R.sup.-1 =R.sub.M.sup.-1 +R.sub.0.sup.-1 and L.sub.M =.chi.L.sub.0.(52) 
This allows us to handle the magnetization energy L.sub.M i.sup.2 /2 
independently from that of the applied field, L.sub.O i.sup.2 /2. 
The instantaneous power flow to and from the medium is i.xi., and can be 
characterized by an instantaneous resistance R=.xi./i. Since the 
temperature changes between the magnetization and the demagnetization 
operations, the induced emfs, and the corresponding induced resistances, 
are unequal in magnitude. The mean resistance R.sub.i over a complete 
thermodynamic cycle is therefore non-zero, relating to net conversion of 
power. 
The magnetization and demagnetization energy transfers are 
EQU .delta.w.sub.l =L.sub.l i.sub.l.sup.2 /2 and .delta.w.sub.h =L.sub.h 
i.sub.h.sup.2 /2 
respectively, where i.sub.l is the current magnitude reached during the 
magnetization, i.sub.h is the magnitude reached in demagnetization, and 
L.sub.l, L.sub.h are the inductances L.sub.M due to the magnetization at 
the temperatures T.sub.l and T.sub.h respectively. Equation (20) means 
that 
##EQU33## 
when using linear paramagnetism, which means that i.sub.h is the larger 
value. FIG. 2e shows that this is also the likely peak current, so with 
the help of equation (22), the work per cycle can be written as 
EQU .delta.w=.eta..sub.c .delta.U.sub.h =.eta..sub.c L.sub.h i.sub.h.sup.2 
/2=.function..eta..sub.c L.sub.h i.sub.rms.sup.2, (54) 
where .function. is a form factor that depends on the current waveform. 
Small deviations from the Carnot cycle and other losses of efficiency can 
be bundled into the factor .function., so the equation is of a general 
form. 
The mean resistance R.sub.t would be 
EQU R.sub.t i.sub.rms.sup.2 =v.delta.w (55) 
where v is the operating frequency of the engine, ie. the number of 
thermodynamic cycles per second, giving 
##EQU34## 
By the ordinary sign convention, a positive resistance represents a drain 
of electrical power, hence the net induced resistance of an 
electromagnetic engine operating as a refrigerator or heat pump, is 
positive, appearing as a load to the magnetizing circuit. When operating 
as an engine, the induced resistance is negative, since it adds power to 
the circuit. 
DETAILED DESCRIPTION OF PREFERRED EMBODIMENT 
FIGS. 5a-c describe mechanical heat engines using magnetism, while FIGS. 
6a-g describe the corresponding electromagnetic forms that can convert 
heat directly to electricity. The reference numerals used in these 
drawings are listed in Table 3 below. 
Description of magnetic engines 
A simple reciprocating engine is shown in FIG. 5a, in which a magnet (20) 
is constrained to move cyclically over a limited distance from a magnetic 
medium (10) by a crankshaft attachment (50). The engine is analogous to a 
conventional gas engine, with the magnet (20) serving as the piston. The 
medium becomes magnetized when the magnet approaches it, and demagnetizes 
as the magnet recedes. The medium is thermally isolated for the adiabatic 
steps, and heated and cooled in the isothermal steps, by means not shown 
in the figure, in synchronization with the motion of the magnet (20). 
As the magnet (20) approaches the medium (10), the medium gets magnetized 
and tends to warm up. The medium is simultaneously cooled to maintain the 
temperature, and its entropy decreases. This corresponds to the isothermal 
magnetization cd in FIGS. 2b and 2c. In the adiabatic portion da, the 
medium is thermally isolated while being magnetized further, so its 
temperature rises from T.sub.l to T.sub.h. The isothermal demagnetization 
ab is then performed by adequately heating the medium as the magnet is 
withdrawn. In 
TABLE 3 
______________________________________ 
Reference numerals 
# Denotes Figures 
______________________________________ 
10 magnetic medium 5a,6a,b 
12 wheel with paramagnetic rim 
5b 
14 moving magnetic medium 
5c,6f,g 
20 permanent magnet "piston" 
5a 
22 stationary magnet 5b 
24 annular magnet in turbine 
5c 
26 magnet to propel medium 
6f 
30 magnetizing coil 6a,b 
32 magnetizing coil in turbine 
6f 
34 magnetizing coil in cascade 
6g 
40 heat source (candle) 
5b 
42 heat exchanger (hot) 
5c,6f 
44 heay exchanger (cold) 
5c,6f 
46 a heating element 6g 
50 crankshaft 5a 
______________________________________ 
the segment bc, the medium is not heated while the magnet continues to 
recede, so the demagnetization drops the temperature back to T.sub.l. 
In the traditional gas engine, the high temperature heat intake does work 
against the mechanical load, while the flywheel action of the crankshaft 
mechanism must do work on the medium during the low temperature 
compression. The net work of the gas engine is thus done in the outward 
movement of the piston. 
Flywheel action is likewise involved in the magnetic engine, though the 
direction of the mechanical forces is reversed from a gas engine. The low 
temperature inward motion converts some of the field energy in the space 
between the magnet and the medium into kinetic energy of the flywheel. The 
flywheel supplies the work in the high temperature outward motion. Only 
part of the flywheel energy returns to the field, because of the lower 
susceptibility at the higher temperature. The remainder is available to 
operate a mechanical load. Heat is thus converted first into magnetic 
field energy and then to mechanical power in the magnetic engine. 
Particularly at higher power densities, the flywheel action alone may be 
insufficient to transfer the field energy to the load. High field energy 
density means strong attraction between the magnet and the medium. 
A mechanical spring appropriately connected to the magnet, or compressed 
gas between the magnet and the medium, may be used to reduce the net 
attraction and ease the outward motion. Indeed, the gas may be used 
thermodynamically as well, for additional conversion of heat within the 
same system. 
It also helps conceptually to think of the field energy as equivalent to 
the confined gas in a gas engine. The outward motion of the piston allows 
this field energy "gas" to expand and absorb heat, while the inward motion 
compresses the "gas" and releases heat at the lower temperature. 
FIG. 5b shows a non-reciprocating engine, a simple "magnetic turbine", in 
the sense that the working medium moves continuously, with the various 
thermodynamic steps occurring simultaneously in different parts of the 
system. A wheel (12) with a paramagnetic rim constitutes the stator, and a 
magnet (22) applies a magnetic field over part of the rim as shown. A heat 
source (40), a candle say, warms the rim close to the magnet. Since the 
magnetic susceptibility decreases with temperature, the warmed portion of 
the rim is less attracted to the magnet (22) than the cooler region on the 
other side of the magnet. The differential susceptibility results in a 
torque, which causes the wheel to spin in the direction shown, from the 
magnet to the heat source. This thermodynamic conversion of heat to 
kinetic energy of the medium is used in magnetocaloric power generation. 
Since the rim is not specially cooled to maintain the lower temperature as 
it approaches the magnet (22), this simple turbine deviates significantly 
from the Carnot cycle and has a much lower efficiency. 
A variant of this design uses a wheel with a ferromagnetic rim. The heating 
must be sufficient to reach the Curie temperature (770.degree. C. for 
iron, 20.degree. C. for gadolinium) above which the rim becomes 
paramagnetic, thus providing a large change in susceptibility. 
A more general magnetic turbine using magnetism is shown in FIG. 5c, in 
which an annular magnet (24) produces a concentrated axial field. A 
magnetic medium (14) moves axially through this field at a speed v as 
shown, propelled by magnetocaloric effect. A heat exchanger (42) maintains 
the medium at the higher temperature T.sub.h for some distance as it moves 
out of the magnet, to cause approximately isothermal demagnetization. 
Beyond the heat exchanger, the medium continues to demagnetize 
adiabatically. A second heat exchanger (44) maintains the medium at the 
lower temperature T.sub.l for some distance before it reaches the magnet, 
causing almost isothermal magnetization. 
Between the exchanger (44) and the magnet (24), the medium continues to 
magnetize adiabatically. A non-Carnot cycle is always less efficient, so 
operation as close as possible to the Carnot cycle is always desirable, 
which means that the temperature changes must be completed entirely in the 
adiabatic portions, and the passage through the heat exchangers must be 
isothermal. The two heat exchangers are appropriately shaped and 
positioned relative to the magnet (24) for this purpose, since the heat 
transfers must be proportional to the respective rates of magnetization or 
demagnetization. 
The magnetic turbine of FIG. 5c is analogous in structure to a gas turbine, 
in which air (the medium) is compressed (magnetized) as it enters and 
expands (demagnetizes), providing thrust (magnetocaloric effect), at the 
exhaust at a higher temperature. 
Description of electromagnetic engines 
FIGS. 6a-b show that electromagnetic engines involve two power inputs, a 
heat supply q and an auxiliary electrical source V.sub.b, which can both 
be varied to regulate and control the load power. 
In FIG. 6a is shown a simple reciprocating electromagnetic engine 
consisting of the working magnetic medium (10), which is magnetized by a 
coil (30). The coil is connected by a switch alternately to an auxiliary 
source of voltage V.sub.b and to a load resistance R.sub.L. When connected 
to the auxiliary source, an electric current builds up in the coil and 
magnetizes the medium. When the coil is switched to the load, the medium 
demagnetizes by dissipation in the load. As in the engine of FIG. 5a, the 
medium must also be heated, cooled or isolated by an external means to 
effect the thermodynamic operations. 
It is useful to note that the Carnot cycle does not require the temperature 
of the medium to be changed by thermal means, because the temperature 
changes are ideally performed by the adiabatic operations. Heat transfers 
in the Carnot cycle are ideally isothermal. It therefore suffices to 
provide a thermal switching means to place the medium in thermal contact 
alternately with high and low temperature "reservoirs", and to isolate the 
medium during the adiabatic operations. This observation is useful in 
designing the thermal components. 
The switching embodiment of FIG. 6a suffers from large transients and high 
losses. The a.c.-excited system of FIG. 6b is smoother and eliminates 
switching, but requires an a.c. auxiliary source. The medium gets 
magnetized while the current is increasing in magnitude, and is 
demagnetized as the current falls in magnitude. Alternate thermodynamic 
cycles thus involve magnetizations in opposite directions, and the 
thermodynamic frequency is twice that of the alternating current. 
The timing diagrams for power generation and refrigeration, FIGS. 6c and 6d 
respectively, show that the difference in the operation is in the phase of 
the temperature cycling relative to the current. 
For generating electrical power, the temperature must be high whenever the 
current is falling in magnitude, as in FIG. 6c. The reverse, with the 
temperature high while the current increases, (FIG. 6d), makes the cycle a 
refrigerator. The temperature waveform is idealized as a square wave, with 
the horizontal regions corresponding to isothermal operations. The 
adiabatic operations are confined to the vertical changes in the 
temperature. 
The equivalent resistance of the engine (FIG. 6c) is positive during the 
magnetizations (R.sub.m) and negative during the demagnetizations 
(R.sub.d), since electrical energy is lost to the medium during the former 
and gained from it in the latter. Without the temperature changes, the 
average magnetizing resistance R.sub.m &gt;0 would equal the average 
demagnetizing value R.sub.d &lt;0, and the average resistance R.sub.t would 
be zero. With the temperature cycling, the average demagnetizing 
resistance dominates to give a net negative resistance R.sub.t. Note that 
the total load resistance R.sub.L includes the ohmic resistance of the 
coil (30). 
Changing the phase of the temperature cycle relative to the current, by a 
half cycle as in FIG. 6d, makes the net induced resistance R.sub.t 
positive, and the engine then draws electrical power, since it operates as 
a refrigerator. 
FIG. 6e shows the timing diagram of FIG. 6c in greater detail for 
implementing a Carnot engine cycle. The medium is adiabatically raised to 
the maximum temperature T.sub.h by the current rising to i.sub.h. 
Thereafter, the dropping current causes demagnetization, and heat must be 
transferred into the medium to maintain its temperature at T.sub.h, till 
the current drops to i given by 
##EQU35## 
where c is the applicable specific heat of the medium. The medium is then 
isolated, and allowed to cool adiabatically till the current drops to 
zero, when the temperature will have reduced to T.sub.l. The medium is 
maintained at T.sub.l by drawing out the heat generated as the medium is 
magnetized by the rising current. When the current reaches i.sub.l given 
by equation (53), the medium is isolated again for adiabatically warming 
up to T.sub.h for the next cycle. The induced resistance of the engine 
correspondingly varies with the temperature as shown. 
The heat transfer subsystem could be controlled electronically in response 
to the instantaneous current. It may be useful instead to take the 
instantaneous load voltage as the heat transfer control parameter. This 
provides tolerance for changing loads and avoids runaway voltage or 
current as follows. 
Let the load current be usually maintained within a small relative phase of 
the voltage. If the current goes a little out of phase, its magnetic field 
becomes correspondingly out of phase with the temperature cycle. From the 
preceding theory, R.sub.t would drop in magnitude, and the engine would 
deliver less power to the reactive load. In the event that a load puts the 
current more than quarter cycle out of phase, ie. it sends power back, the 
engine would operate as a refrigerator and take out the excess power by 
pumping heat. This is particularly useful for regenerative braking when 
using the electrical output of the electromagnetic engine to drive a 
motor. 
The electromagnetic turbine in FIG. 6f is substantially similar to the 
mechanical version in 5c, with a coil (32) replacing the magnet (24). The 
net resistance R.sub.t is a negative value when generating electrical 
power. A second magnet (26) may be optionally included for keeping the 
medium in motion using the magnetocaloric effect. 
FIG. 6g illustrates how multiple stages of an electromagnetic turbine can 
be cascaded. Each stage consists of a magnetizing coil (34) followed by a 
heating element (46). The magnetic medium (14) passes through the stages 
in succession. As the medium emerges from the field of a magnetizing coil, 
it passes over the adjacent heating element, and induces power into the 
coil as it demagnetizes. The medium is assumed to cool between the stages. 
The coils could be connected in series or in parallel, or to different 
load circuits. 
Equation (48) shows that the instantaneous equivalent resistance R depends 
on the instantaneous susceptibility as well as on the instantaneous rate 
of change of the current. One ordinarily has much less control over the 
shape of the temperature changes, which tend to be smoother than the 
electrical parameters. Especially when not using a linear paramagnetic 
medium, the instantaneous resistance is not likely to be a smooth function 
of time. Some degree of control may be obtained through the rate of change 
of current, but in general, a reciprocating electromagnetic engine would 
generate significant harmonics and the output power needs filtering. 
Although equally operable with a.c., the turbine form is essentially a d.c. 
device, in which the positive and negative resistance phases are 
contributed simultaneously by different portions of the continuous medium, 
so the instantaneous resistance is constant at R.sub.t. The instantaneous 
resistance is due to the net emf contributions of the elementary magnetic 
moments magnetizing and demagnetizing in the different portions of the 
medium. 
Synchronous cooling 
Synchronous dissipation in digital circuits, ie. dissipation in bursts 
triggered by a clock, can be alleviated by instantly converting some of 
the heat to electrical power using an electromagnetic heat engine. 
Consider an engine as in FIG. 6b applied for cooling a simple digital 
circuit to which the magnetic medium is thermally coupled. The engine is 
constructed with a coil around the medium, and the auxiliary source 
voltage is derived from the clock, so that the bursts of dissipation occur 
during the demagnetization phase of the current waveform. 
With the quiescent dissipation alone, the digital circuit soon reaches 
thermal equilibrium, so the B-M relation would be linear and would follow 
the line ox in FIG. 6h as the engine current varies in the coil. State 
transition of one or more gates, causing dissipation, raises the 
temperature locally. The demagnetization phase is arranged to occur 
throughout the transitions, so that the local thermodynamic state follows 
the typical path abcda in the figure. 
The segment ab corresponds to the sharp rise in the local temperature as 
the gates begin to transit. This onset of dissipation drops the 
magnetization rapidly without much change in the engine current. The local 
temperature remains high for the duration of the gate currents, while the 
engine current drops, producing the isothermal demagnetization along bc. 
When the gate currents cease, the local temperature drops to the mean chip 
temperature raising the magnetization along cd. The engine current is 
raised again along da to prepare for the next burst of transitions. 
The diagram is obviously simplistic and represents the thermodynamic state 
only in the vicinity of the transiting gates. Only a relatively small 
fraction of the gates in a digital integrated circuit chip actually change 
state at any given clock. Conditions near non-transiting gates merely 
follow the line ox in the diagram. The cycle abcda applies to the 
neighborhood of the transiting gates, and the area of the cycle represents 
the heat converted away as electrical energy. The total heat converted in 
a given cycle is the integral of the local contribution over the 
transiting gates: 
##EQU36## 
where the outer integral is over the physical volume of the digital 
circuit, and the local integral is proportional to the area of the path 
abcda. 
The inductive mechanism permits the use of a single coil of one or more 
turns around the digital circuit to construct the cooling engine, at least 
when the clock frequency is low. One does not need to place separate coils 
around the individual logic gates. Since only a few gates transit each 
cycle, the per-cycle dissipation is quite small, and the power conversion 
density of the engine need not be high. Hence neither very high 
susceptibility media nor very high applied fields and engine currents to 
produce them, are particularly necessary. 
For example, a CMOS processor chip dissipating 30 W at 100 MHz is only 
dissipating 300 nJ per cycle. If two-thirds of this power is synchronous 
dissipation, ie. in response to the clock and only about 1 thousand gates 
are transiting each cycle on an average, this represents 200 pJ per gate. 
Over the die volume, including substrate, of about 0.1 cm.sup.3, the 
dissipation amounts to only 2 .mu.J/cm.sup.3, well within the capability 
of paramagnetic elements. 
The inductive mechanism means that the conversion efficiency depends on how 
much and how quickly the magnetic medium is locally heated by the gate 
transition currents. High thermal conductivity of the circuit and close 
thermal contact with the medium are desirable. If the dissipation takes 
more time to reach the medium than the clock period, the dissipation is no 
longer distinctly periodic and the engine would not be able to cool the 
circuit 
At lower gate densities and in simple circuits, a layer of magnetic 
material in the substrate or deposited uniformly over the chip would 
probably suffice for synchronous cooling. At higher densities and 
operating speeds, atoms with high magnetic moments, such as those of iron 
or gadolinium, may be ion-implanted in the substrate or an overlay in 
greater numbers in the expected regions of dissipation. Conduction 
electrons cannot themselves function as magnetic thermodynamic media, but 
magnetic atoms in the lattice can, and if present, would provide the 
strongest instant coupling of ohmic dissipation. 
At the higher gate densities, on-chip electromagnetic delays become 
significant, and different regions of the circuit would be operating out 
of phase, so a simple coil around the circuit may not be sufficient for 
synchronous cooling. It would be also unsuitable at higher operating 
frequencies because of its time constant. In either case, a transmission 
line approach becomes useful for locally synchronizing the applied 
magnetic field with the bursts. The field is then applied by a 
"magnetizing clock wave" travelling slightly ahead of the logic clock. 
Embodiments in detail 
FIGS. 7a-b and 8a-b describe a reciprocating magnetic engine and an 
electromagnetic turbine, respectively, in greater detail. The reference 
numerals used in these drawings are listed in Table 4 below. 
TABLE 4 
______________________________________ 
Embodiment numerals 
# Denotes Figures 
______________________________________ 
100 left working element 
7a 
102 right working element 
7a 
104 working material 7b 
110 ferromagnetic block 
7a 
112 center pole in 110 
7a 
120 permanent magnet piston 
7a 
140 pipes within 104 7b 
150 crankshaft 7a 
152 wheel 7a 
160 spark mechanism 7b 
162 injected gases 7a,b 
164 exhaust gases 7a,b 
210 ring of magnetic material 
8a,b 
220 coil 8a 
242 heat exchanger (hot) 
8a 
244 heat exchanger (cold) 
8a 
260 hot liquid 8b 
262 pan collecting 260 
8b 
280 furnace 8b 
______________________________________ 
A reciprocating magnetic engine 
FIG. 7a shows a reciprocating magnetic engine using two engine elements 
(100, 102) made mostly of a magnetic working material, and a moving 
permanent magnet (120). The magnet (120) functions as the "piston", being 
constrained to move linearly between the engine elements (100, 102). A 
ferromagnetic block (110) confines the magnetic flux. The block has a 
small center pole (112). The flux tends to concentrate between the engine 
elements and the center pole. When not operating, both the engine elements 
are cold, and the magnet will rest in stable equilibrium on one or the 
other side of the center pole (112). The magnet (120) is connected to a 
crankshaft (150) to transmit power to a wheel (152). 
The engine elements (100, 102) can be heated and cooled independently as 
described below. The engine operation begins with heating the element 
closer to the magnet (120), which increases its reluctance, so the magnet 
is now attracted to the other side of the center pole (112). The heating 
is then switched to the other side, while the first element cools, to move 
the magnet back to the first element. The operation is analogous to a 
typical steam engine, in which the piston is pushed in either direction by 
steam. 
The engine elements (100, 102) may be designed for internal or external 
combustion. An internal combustion engine element might consist of a block 
of the working magnetic medium (140), as shown in FIG. 7b. For heating the 
element, a gaseous air-fuel mixture (162) is injected into pipes (140) 
embedded in the medium, and ignited by an electric spark mechanism (160) 
at a strategic position within the element so that the heat spreads 
rapidly through the embedded pipes in a short time compared to the 
mechanical motion of the magnet. For example, at 50 Hz (.ident.3000 rpm) 
operation, the combustion and the heating should be complete in 10 ms. For 
cooling, the burnt gases are flushed out (164) by injecting cool air and 
then a fresh charge of air-fuel mixture. The internal combustion element 
functions like a petrol (Otto) engine. 
An engine element with embedded pipes can be used with external combustion. 
The element is heated by injecting a hot fluid, and cooled by injecting a 
cold fluid. Note that most of the heat is removed as work done during 
demagnetization. 
Calculations for reciprocating engine 
Let the working elements (100, 102) be made of soft iron. At 50 Hz, 0.1 T 
applied field variation, the saturation power density is over 4 kW/litre. 
With a backoff-cure-efficiency factor of 0.4, the engine can yield up to 
1.5 kW/litre. If the working elements are each of 1 cm.sup.3, the maximum 
power is about 3 W. 
Clearly, a magnetic engine is not very advantageous over a gas engine at 
the same frequency with respect to power density. 
But similar figures are applicable to the completely stationary 
electromagnetic engine of FIG. 6b. Using an internal combustion element 
constructed as in FIG. 7b and containing 1 litre of soft iron to serve as 
its magnetic medium (10), the electromagnetic engine potentially generates 
up to 1.5 kW at mere 50 Hz from heat without using moving parts or a 
thermodynamic fluid. 
Since it can conceivably be operated at 100 or 200 Hz instead, this 
electromagnetic engine can potentially yield 3 or 6 kW respectively. 
Incidentally, a 100 Hz or 120 Hz is preferable for operating at the common 
a.c. supply frequency of 50 or 60 Hz respectively. 
An electromagnetic turbine 
The electromagnetic turbine of FIG. 8a uses a horizontally mounted ring 
(210) of a solid magnetic working medium spinning through a coil (220), 
which is connected to an electrical load circuit and an auxiliary electric 
source to setup the load current. 
A heat exchanger (242) heats the ring immediately as it emerges from the 
coil, and a second heat exchanger (244) cools it as it enters the coil. 
The hot exchanger may be built as shown in FIG. 8b, in which a liquid 
(260) carrying heat from a furnace (280) is simply poured on the ring and 
collected below by a pan (262). The cold heat exchanger would use a liquid 
coolant in like manner. In high power application, the heat exchangers may 
use water for cooling, and a liquid metal for heating, with a nuclear 
reactor replacing the furnace. 
Equations (50) and (56) allow the required number of turns N to be computed 
from the magnetic circuit as 
##EQU37## 
It should however be noted that the ring (210) is not really the magnetic 
circuit for the coil (220), because the engine operation depends on each 
portion of the ring becoming magnetized as it approaches the coil, and 
becoming demagnetized as it recedes. Consequently, the magnetic circuit 
must be completed, possibly through stationary ferromagnetic segments, 
outside of the ring medium, such that the flux can enter or leave the ring 
in the region of the heat exchangers, where the magnetization is intended 
to change the most for ideally following the Carnot cycle. 
Since the applied field depends on the current and the number of turns per 
unit length of the magnetic circuit, one requires 80 kA-turns/m for 0.1 T 
applied field, irrespective of the length of the magnetic circuit. Thus, a 
corresponding number of turns may be wound on the ferromagnetic segments, 
so that the coil (220) only requires turns proportional to the effective 
portion of the magnetic circuit passing through the ring (210). 
Calculations for electromagnetic turbine 
In any case, if the ring be of soft iron, the magnetic circuit would 
probably be at least as long as the ring circumference. If the ring is of 
4 cm radius and 1 cm.sup.2 cross-section, the circumference is about 25 
cm, requiting 2500 turns at 8 A. The total volume of the medium being 
about 25 cm.sup.3, the engine can generate up to 100 W at 50 Hz, producing 
12.5 V emf at a negative resistance of 1.5625 .OMEGA.. For the same power, 
it can be operated alternatively at 80 A using 250 turns to produce 1.25 V 
at -15.625 m.OMEGA.. 
A small amount of ferrofluid can be used to provide a low reluctance bridge 
between the moving ring and the stationary ferromagnetic segments to 
reduce the overall magnetic circuit and the total number of turns. 
ADVANTAGES 
From the foregoing, it will be appreciated that the present invention 
provides an electromagnetic heat engine having the following advantages. 
The electromagnetic engine overcomes the limitation on speed due to 
mechanical operation in prior art magnetic engines, thus providing higher 
power densities. 
The electromagnetic engine is generally useful over prior art heat engines 
because it eliminates moving parts, including, in the a.c. reciprocating 
embodiments, that of the medium. 
For example, small solid-state electromagnetic heat pumps operating at high 
frequencies can augment local heating applications including heating irons 
and electric ovens. The desired higher temperature is first established in 
an object of low heat capacity by other means, such as a resistance, and 
the electromagnetic heat pump then supplies enough heat to maintain the 
temperature. Such applications would generally require fast thermal 
switches to thermally connect the magnetic medium alternately to the two 
temperatures. 
At sufficiently high operating frequencies, electromagnetic heat engines 
can become useful for refrigeration at ordinary temperatures, thus 
providing an alternative to the fluid refrigerants in use today. 
Though magnetic refrigerators have been employed in cryogenic applications, 
their non-contact nature has not been particularly useful because of the 
mechanical motion required. Electromagnetic engines allow the non-contact 
nature to be better exploited, since the medium need not move, nor have 
particular shapes for optimal operation. Indeed, as explained for 
synchronous cooling, the magnetic medium can be closely coupled with or 
even constitute part of the object to be cooled. The electromagnetic 
engine is particularly suited for cooling digital Josephson circuits. 
The elimination of motion and shape requirements means the electromagnetic 
engines are realizable in unusual shapes and sizes, and further, reaps a 
thermodynamic benefit generally unavailable to prior art magnetic and 
non-magnetic heat engines, in that local hot spots are directly coupled 
and utilized for higher efficiency in the electromagnetic engine, as 
particularly described for synchronous cooling. 
At the high power end, electromagnetic engines are likely to be simpler to 
design, construct and control than the prior art gas engines and steam 
turbines, apart from their unique no-movement operability. For example, 
spacecraft power generators cannot involve mechanical means for converting 
the heat from an on-board reactor, and frequently depend on thermocouples 
giving about 4% efficiency. Electromagnetic engines address one end of 
this conversion problem, viz. eliminating intermediate mechanical form, 
while potentially providing useful power densities. The foregoing 
calculations show that if a high power thermal switch be available for 
operating at 400 Hz, a completely solid-state electromagnetic engine can 
typically provide about 25 kW/litre of soft iron medium. 
With the exception of the cryogenic magnetic refrigerators, prior art heat 
engines have invariably involved complicated fluid dynamics, necessitating 
expensive and time consuming experiments. Turbulence and 
magneto-fluid-dynamics add to the analytical and computational complexity 
of the prior art engines. Even in cryogenic magnetic refrigerators, both 
the paramagnetic medium and the field of the superconducting magnets must 
be specially shaped for approaching the Carnot cycle. 
Electromagnetic engines are also much easier to model and design, using 
existing thermal diffusion and electromagnetic field analysis tools, and 
do not require fluid media even in the turbine forms, considerably 
reducing the modelling complexity. The resulting simplicity also means 
much better design and control are possible with electromagnetic engines. 
The electromagnetic engine also permits a wider choice of thermodynamic 
media, since the medium need not be electrically conducting or even fluid. 
With special properties, including fluidity and conductivity, being no 
longer required, more magnetic materials can be investigated and developed 
for thermodynamic and power applications. 
Superconducting magnets have hitherto been used as very strong permanent 
magnets, and almost never in series with electrical loads. Superconductive 
coils can now be utilized in a more active role for carrying very high 
load currents to generate high applied fields in electromagnetic engines 
for power generation. 
Although the invention has been described with reference to preferred 
embodiments, it will be appreciated by one of ordinary skill in the arts 
of thermodynamics, magnetics and electricity that numerous modifications 
are possible in the light of the above disclosure. 
For example, electromagnetic heat engines are not limited to Carnot cycles. 
Non-Carnot cycles with or without regeneration are equally applicable to 
electromagnetic engines. Also, an electromagnetic engine can be operated 
between any pair of temperatures that yield a difference of 
susceptibility. Further, electromagnetic engines may be used in 
conjunction with alternative means of heating or cooling. For instance, 
cryogenic magnetic refrigerators are operated only after the temperature 
has been lowered to 10 K or less by other means. 
Likewise, synchronous cooling is not limited to CMOS integrated circuits, 
but is equally applicable to any system potentially generating periodic or 
predictable dissipation, including digital optical circuits. Even 
superconducting circuits, particularly those using high T.sub.c materials, 
are known to produce some dissipation, and are possible candidates for 
synchronous cooling. 
All such variations and modifications are intended to be within the scope 
and spirit of the invention as defined in the claims appended hereto.