Wireless Electrical Energy Transfer Technology

Introduction:

Electrical energy transmission has always depended on an easily available medium to transport the energy.

Wires developed with increasing conductivity and ability to carry large amounts of current have gone on hand in hand with the growth of energy demands for our civilization

The first DC transmission lines were useful for short range applications, however they could not be used for long range transmission as the effect of resistance would build up to the point at which to negate losses, huge amounts of energy would be required from an endless stream of DC generators along the cable's trajectory.

On September 4th 1882, The American Inventor Thomas Edison used his company, General Electric, to open the first electric distribution system in the heart of lower Manhattan financial district, New York, after many delays and cost overruns.

Edison knew that this newly created product was going to be expensive and need to reach many financially powerful customers in order to survive, Hence lighting up the "Pearl Street" financial district was a good place to start.

This was the start of many projects that would However Edison’s choice of Direct Current (DC) made his product into a limited range and he could not transmit the power very far without losing tremendous amount of energy. So basically he would need a power plant every kilometer to provide consistent power to the public.

As a consequence of it Edison’s distribution system and being a major investor in DC power, had a web of electric wires overhead, it has sometimes said that they blocked the sunlight at some places.

      Thomas Edison's DC distribution system network in the financial district of Manhattan, New York.


Meanwhile, Thomas Edison’s competitor George Westinghouse, founder of Westinghouse Electric, the main rival of General Electric company, invested in the development of Alternating Current (AC) as Westinghouse saw the future of the electrical industry hinge on long distance transmission.

In this "war of currents", Nikola Tesla, a Serbian born inventor perhaps the most important contributor to the development of electrical energy in human history as he developed the foundations of Alternating Current and most of the technology associated with it, from electrical transformers, AC generators to the first radio towers and countless more technological wonders that created the Second Industrial Revolution, ushering in the electrical age.

From car motors, remote controls, robotics, radio, X-Rays, lasers and wireless communications, Nicola Tesla invented the 20th century and his visions continues to generate new ideas about  technology today.

Alternating Current, AC, where the electric field varies in a frequency of time, increased the transmission capability of wires by an analysis and optimization of the skin effect, which can use electrical transformers to transfer the alternating electrical field into a magnetic field and transform it back into an electric field on the skin of the transmission wire which removes the resistance experienced by the current which creates loss.

Alternating Current placed through a loop of wire, by Ampère's Law, will create a time varying magnetic field, the flow, or flux, of which can be directed around another coil of wire, which by Faraday's law of electromagnetic induction, will transform this alternating magnetic field (B-field) back into an electric current.

The nature of the transformation of voltage, i.e. the ratio of the difference between the input and output voltage, depends on the ratio between the number of coil turns between the primary and the secondary coil.

If the Primary coil has less turns than the Secondary coil, then the amount of magnetic flux transformed into an electric field density, and hence voltage, will be less than the voltage input to the primary coil. So the ratio will be < 1 and the voltage will have been down-converted.

This is how high voltage AC is down-converted for safe, domestic uses in homes from the higher voltages used to transmit the power across lines from power plants and sub-stations.

Although the magnetic flux is directed along the iron core of the transformer, magnetic flux ill be lost through heating of the iron core. Cores are therefore typically insulated, in oil for example.

This is the basic principle of how an electrical transformer works and is used to convert DC to AC and vice-versa.

Moreover, For all conductors, as the frequency of the time varying magnetic field is increased the depth at which the current is induced in the conductor will decrease. This is the skin effect, and is named due to the current being attenuated at the center of the conductor and having to exist in the outer “skin”. Because the current is confined to the outer skin, the resistance the current experiences in the whole wire can be greatly reduced.


The skin effect occurs in conductors where the time varying magnetic field is parallel to the conductor surface which can be designed by placing a conducting rod within a solenoid so that the magnetic field lines are parallel to the axis of the rod or by having a polarised frequency of radiation which will have the magnetic field components parallel to the axis of the rod as is the case in pickup antennas where a certain frequency of polarised radio or microwaves induce a current on a pickup coil in an antenna at a certain frequency depending on the skin depth of the metal the antenna is made of. 

The magnetic field lines initially penetrate the metal uniformly which induces a uniform current throughout the conductor. However, separate eddy currents within the surface of the conductor will form, as in the case of the conducting rods used in the experiment. These currents circulate around the time varying magnetic field in a direction that flows against the current in the center of the conducting rod which cancels any current going through the center of the rod. This is shown in the picture below: 

Here the primary current, I, is generated from the homogenous magnetic field B which is generated by the primary solenoid. the primary current creates a magnetic displacement field, H, around its direction which induces smaller eddy currents, Iw, which flow in a direction against the direction of the current near the centre of the conducting rod and flow with the current at the surface of the conducting rod. This negates current flowing through the centre of the rod and isolates the current in a skin at the surface of the rod. 

As the frequency of the A.C current which generates the time-varying magnetic field increases the eddy currents will penetrate through a much greater depth in the conductor, in this diagram the red circles will have much larger radii. This will restrict more and more current to the surface of the conductor for higher frequencies.


This means that the current induced by the time varying magnetic fields can only exist at the surface of the metal conductor, or the “skin” as the eddy currents do not negate the current here.

Good conductors will also have lower skin depths by the shielding of induced currents caused by the eddy currents being more prominent, good conductors will also yield a more constant induced AC current. Also the eddy currents themselves generate their own magnetic field which opposes the change of current within the metal so this current does not vary over time, it is a direct current where as the current which generated the time varying magnetic field would have to have been an alternating current.

So, in conclusion, unlike DC, where the current is distrubuted all over the diameter of the wire (and hence experiences all of the resistance of the wire) AC is concentrated near the surface of the wire, with virtually no current travelling in the bulk.

An important conclusion is that with an increase in the diameter of the wire, the skin depth increases. Hence to reduce losses by resistance we can make AC wires stranded in braids.

This is the optimum design and is used in virtually all AC cables, from household power lines to high tension pylon cables.

                        Cross-Section of a High Tension Power Cable Used in Overhead Pylons

As our dependence on electricity continued and accelerated over the 20th century new wire materials were developed to optimize transmission power of electricity. Copper and Aluminium had ideal properties for AC transmission and were readily available and were light enough to be set up in transmission towers quickly. Now, new materials such as Graphene, are being developed which can hold greater volumes of current in ever more lightweight wires .

High Temperature Superconductor (HTS) wires are also becoming mediums for transmission of electrical power, as superconductors can carry amounts of current per unit area several orders of magnitude beyond conventional wires.

However, these wires are expensive to mass produce, require advanced insulation techniques which often involve vacuum systems and, as of 2014, require a constant supply of liquid nitrogen to cool the materiel down to superconducting temperatures. There are also problems with flexibility,as most of the HTS materials are ceramics and are brittle. Hence there are still a long way to go before this technology can be used without existing power transmission lines and these will be most likely incorporated in hybrid systems with existing power grids.

Room temperature and beyond superconductors, if developed by continuing research, promises transmission of huge amounts of electrical current with no losses whatsoever, which would not only minimize the amount of cable required by the generator motors themselves would be far smaller in size. 

Despite all of our developments, in summary we have always focused on how we can transmit energy via particles in matter, i.e. electrons in a wire. There is however another way, we can instead transmit energy in the form of fields.

Wireless Energy Transfer

The most ubiquitous way we can do this is through near-field electrodynamic induction, which transmits the electrical energy through free space as an induced magnetic field over a fraction of the wavelength of AC current used to generate the magnetic field (the B-field) . 

Electrical transformers and solenoids use this effect, however in these cases a material is used to guide the magnetic field as it is a near field effect.

Other technologies, such as EM transmitter and receiver devices, TV/Radio antennas can transmit electrical energy radiatively over long ranges as the transmitters and receivers are tailored for receiving in the far field.

Radio telescope arrays, in order to have good angular resolution, are large because the wavelengths they receive are large and will hence the efficiency for power collection is a function of the spacing of the telescopes, w.r.t. the wavelengths being observed, rather than the efficiencies of the telescopes themselves.

So, in summary, using fields we can transmit energy in 2 ways:

Radiative transfer - i.e. electromagnetic radiation which offers long range but is inefficient because it radiates in all directions (1/r squared)

Electrodynamic induction - is short range (1/r cubed) but is efficient due to stationary fields in a fixed direction around the coils.

In the case of radiative transfer, for EM transmitters we can collimate the beam in a single direction, as a laser or maser, but we have to fix these very precisely. Projects involving solar-powered geostationary orbiting platforms, in sync with Global Positioning System technologies could make space-based energy generation a real possibility if a space program had the will to accomplish such a mission as the gains, from even small space-based solar power plants, would be substantial.

Such projects are ambitious, and require a way to make orbiting outposts self-sustaining and also providing a sustainable space flight system that can send humans to such outposts should there be any problems. Such projects probably would not be attempted until a new reusable space-vehicle, to replace the retired space shuttle, is developed which may not be until mid-century under conservative estimates.

Moreover, large scale rectenna infrastructure needs to exist on the earth's surface to collect the microwave energy from space in a secure location away from other antennas in towns and cities. Hence careful and cautious planning needs to be implemented by a large hierarchy of people before any such project would get the go ahead.

Nevertheless such projects are worth considering, particularly for space exploration. As the reliance on the sun for power increases, more and more ways of relaying power collected from the sun will have to be found.

Future plans for powering aircraft and spacecraft remotely using laser technology are also future areas in technology which could yield promising results over the next century, particularly in the field of laser and microwave energy delivery systems to aircraft engines or spacecraft propulsion systems powered by an earth-based laser/microwave source which would reduce the amount of fuel weight required to be carried by the craft to practically nothing (apart from the gases needed to produce propulsion plasma in space itself).

Probably the best known of these projects is the "Lightcraft" prototype laser propulsion system designed at the Rensselaer Polytechnic Institute and launched at the White Sands Missile Range with a CO2 laser which is normally used for testing Intercontinental Ballistic Missile (ICBM) components for vulnerability to laser countermeasures from strategic missile defense systems which could be used as space-based weapons for destroying missiles.

                                     The "Lightcraft" laser-powered, plasma propulsion engine

Using a 50 cm wide beam from a 10 kw CO2 Infrared laser focused onto a parabolic mirror at the base of the vehicle, a ring shaped plasma is generated from the gases in the air. Air at the focal point is heated to between 10,000 and 30,000 Kelvin at tens of atmospheres by kilojoule laser pulses lasting 30 microseconds and repeated 10 times per second. After each pulse the plasma expands and cools very rapidly producing thrust and a brief flash of visible light. Air flowing through the vehicle replenishes the supply of gas near the focal point. The aluminum craft is 15 cm in diameter, has a mass of 50 grams and is gyroscopically stabilized.

It is the highest average power, pulsed carbon dioxide laser presently operating in the U.S. The team is planning to use a more powerful 150 kw CO2 laser in preparation for flights to the edge of space (100 km altitude). Laser propulsion costs several orders of magnitude less than chemical-fueled rockets. Near term applications include launching microsatellites.


The current laser technology used in the Lightcraft engine can in principle be used to detonate the air with shock-waves generated by the plasma ahead of a spacecraft, or high altitude aircraft, to induce an airspike ahead of the vehicle reducing the drag due to the atmosphere considerably. A plasma would be generated ahead of the vehicle by means of a focused laser or microwave beam from a source onboard the craft. Pulsing the output produces a series of detonation waves. The shock wave thus generated can then control the airflow around the vehicle reduce drag and hence increase fuel efficiency and decrease the amount of fuel needed. Such a system would also help negate turbulence at high speeds, allowing aircraft and spacecraft to be pushed at higher speeds and also assisting with atmospheric re-entry for craft in orbit. Such hybrid plasma and chemical engine systems could easily find their way into spaceplane technologies before complete plasma propulsion becomes possible.

Technological challenges are similar to those described in literature concerning the use of plasma to reduce the radar cross section, RCS, in stealth aircraft and to shield against electromagnetic pulse using plasma to deflect radio and microwave radiation around certain sections of an aircraft. Such challenges are nontrivial, with research on how dielectric plasma shells affect radio waves going back as far as Sputnik the first artificial satellite launched by the Soviet Union on October 4, 1957.

While trying to track Sputnik it was noticed that its electromagnetic scattering properties were different from what was expected for a conductive sphere. This was due to the satellite traveling inside of a plasma shell. While Sputnik was flying at high velocity through the ionosphere it was surrounded by a naturally-occuring plasma shell and because of it there were two separate radar reflections: the first from the surface of the satellite itself and the second from the plasma shell. If one of the reflections is greater the other one will not contribute much to the overall effect. When the two reflections have the same order of magnitude and are out of phase relative to each other cancellation occurs and the RCS becomes null. The craft thus becomes invisible to radar due to this effective "plasma forcefield". 

Attempts to replicate this by active plasma technology are ongoing and therefore similar to the attempts to make plasma propulsion by means of directing energy, by a laser or microwave beam, in front of or around a craft to eliminate air drag.


Using current technology, some systems onboard aircraft and possibly spacecraft, could also use lasers and/or microwaves to recharge at least some systems remotely in a situation where a landing is being avoided, for example to increase the time of flight of the aircraft or to reduce time spent on refueling if the craft needs to carry out an operation quickly.

NASA's laser-powered plane. Using a laser beam centered on its panel of photovoltaic cells, a lightweight model plane makes the first flight of an aircraft powered by a laser beam inside a building at NASA Marshall Space Flight Center.

Such developments could become a possibility in the future, particularly with the increasing efficiency of solar cells which could allow some aircraft to reduce dependence on fossil fuels, which not only contribute to global warming but whose weight decreases the running efficiency of the craft itself.



Alternatively, wireless power transfer by Electromagnetic induction  is a function of the frequency and the intensity of the conductor ís the current and voltage that produces the magnetic field, B.
The higher the frequency, the greater the induction effect. Energy transfers from the conductor that produces the magnetic flux density fields (The Transmitter) to any conductor on which the magnetic flux flows through (The Receiver).

The coupling between two conductors is increased by winding them into coils and placing them close together on a common axis, so the magnetic field of one coil passes through the other coil.

Coupling must be tight to achieve high efficiency. If the distance from the primary to the secondary increases some of the magnetic field will miss the secondary, lowering the coupling.


For this near-field effect we should also ask the question:
Can we modify the electromagnetic induction effects, which give us a large degree of freedom, to transport electricity in an efficient way without wires?

It turns out we can enhance wireless power transmission with the application of resonance effects. This technique employs transmitter and receiver inductors tuned to a mutual frequency. This enables power to be transmitted over a distance of up 1/4 or 1/3 times the size of the primary coil. Transmitting and receiving coils are usually single layer solenoids or flat spirals with series capacitors, which, in combination, allow the receiving element to be tuned to the transmitter frequency.

Using The rapidly oscillating electrostatic field it is therefore possible to have mobile energy transfer on a considerably large scale using a magnetic field gradient collected by the capture coil, thereby transferring the power.

The amount of energy transferred by the magnetic field lines is associated with the amount of torsion in the magnetic field lines themselves. Hence for optimum efficiency the majority of the field lines have to be transferred from the transmitter coil such that the field lines bind the two coils together, i.e. from the center of the transmitter to the center of the receiver.

In outline the energy transmission circuit will connected to the induction coil
The energy receiver will be connected to the ground.
The air acts as a dielectric between the transmitter and receiver, hence this is essentially a capacitive coupling.

The lamp can be moved anywhere and will be illuminated without being connected.

There are 2 designs for this circuit, the AC-AC Design:

The AC-DC Design

The fact that the electric field gradient arises through the dielectric in the capacitor, the air, as a result of an alternating electrostatic field, there should be an reactance in the electric field gradient between the transmitter and receiver which acts to oppose the changes in the voltage across the capacitor.

The reactance, added to the intrinsic resistance of the capacitor  + inductor circuit generates the impedance of the electric field gradient which is the dampening of the electric field across the capacitor. 

Impedance decreases as the frequency of the AC current increases.

in a transformer, the impedance is eliminated if ratio of the number of coils at the output to the input coil turns is equal to the square root of the resistance at the output coil to the resistance of the input coil:

Turns Ratio = [OutTurns/InTurns]

Turns Ratio = √[(OutResistance)/(InResistance)]

This is known as Impedance matching.

The power that can be taken from a homogeneous magnetic field B is dependent on the induced voltage used in the receiver coil. Considering it as a loop, for sinusoidal signal shape it results as:

Reciever = 2πf.B.A
where f = frequency and A = loop area.

With the same magnetic flux density, a higher power can be transferred at higher frequencies. This means that the product of maximum magnetic flux density times the frequency is relevant for the power transmission.

For the electrostatic induction transfer, the size and number of turns in the input coil are also physical parameters which naturally affect impedance since the energy transfer is between capacitor plates, impedance matching can occur if the frequency of the AC current placed into the capacitor transmitter matches the frequency of the radiative transfer across the capacitor plates: which is the resonant frequency.

This allows electrostatic, oscillating fields to effectively tunnel from the transmitter capacitor to the receiver capacitor if the frequency input to the coil matches the resonance frequency of the receiver with zero impedance.

From the beginning of inductive power transmission, pioneered by Nicola Tesla, resonant circuits were used to enhance the inductive power transmission. Tesla himself, from the beginning, used resonances in his first experiments about inductive power transmission more than hundred years ago.

For systems with a low coupling factor, a resonant receiver can improve the power transfer. Resonant power transmission is a special, but widely used method of inductive power transmission and is limited by the same constraints of magnetic fields emissions and efficiency. The phenomenon of resonant coupling, in which two objects are tuned to the same resonant frequency, exchanges energy strongly between the two objects but interact only weakly with other objects.

To understand the effect, it can be compared to mechanical resonances.
In mechanical terms, resonance is simply another word for vibration. When you tap a wine glass, it makes a sound at a particular pitch, like a chime. It is 'resonating' at that pitch which has a particular frequency, for example 400 beats per second. You can make a wine glass vibrate by tapping it, but you can also make it vibrate by making a sound that is very close to its resonance frequency. This is called 'sympathetic resonance' because the glass is vibrating in 'sympathy' with the original sound.

You can see this effect in the following science experiment if you take two identical wine glasses, tap one and hold it close to the other. The other one will begin vibrating a little, even though you didn't tap it:



Considering the wire is tuned to a certain tone as mechanical resonator. Even at a far away distance and low sound level, a generation of acoustic waves in the air can excite the string to vibration, if the tone pitch is matched.

A more extreme example of this involves a set of wine glasses, each filled to a different level so that it vibrates at a different sound frequency. If your acoustic generator, which can even be a singer, hits a pitch that matches the frequency of one glass, the glass can absorb so much acoustic energy that it will shatter; while the other glasses can remain unaffected.

Electronically, the resonator in the receiver consists of the receiver inductance and a capacitor. Also the transmitter can have a resonator. The transmitter and receiver coils can be considered as weakly coupled transformer. For this, an equivalent circuit diagram consisting of magnetizing and stray inductance can be derived.  

In the diagram above, also the resistances of the windings are shown. The diagram shows clearly, that the resonant capacitors cancel out the stray inductance in the receiver and the magnetizing inductance in the transmitter. Now, the only remaining limit for the power transmission is the winding resistances of the coils, which impedance is one or two orders of magnitude lower than that of the inductances. Therefore, for a given generator source, much more power can be received.

This resonance dependence was discovered in 2006 by researchers at MIT wanting to make wireless energy transmission a reality for small rooms

in 2007, the MIT team published a paper detailing a successful demonstration of their prototype. They used resonating coils to power a light bulb over a distance of about two meters

If we use a larger transmitter coil, with the same number of coil turns on the outside of the ring, and a smaller receiver coil, with the same number of coil turns on the inside of the ring, the amount of field lines that pass through the transmitter and receiver would increase hence increasing the range of the transmission capability to a certain point.

This would only work up to a certain point as the magnetic field density has a sharp drop off rate. Hence for large distance energy transport radiative transport, via microwave or laser radiation is the only solution.

Dimensions for these calculations have been scaled to the larger diameter coil 'D', which can be either the transmitter or receiver.
The values are shown as a function of the axial distance of the coils (z/D). The variable is the diameter of the smaller coil D2.
The figure shows that
The efficiency drops dramatically at larger distance (z/D > 1) or at a large size difference of the coil (D2/D < 0.3)
A high efficiency (>90%) can be achieved at close distance (z/D < 0.1) and for coils of similar size (D2/D = 0.5..1)

This shows that inductive power transmission over a large distance, e.g. into a space, is very inefficient. Today, we cannot afford to waste energy for general power applications by using such a system.

On the other hand, the figure shows that inductive power transmission is competitive with wired solutions under close proximity settings. Wireless proximity power transmission combines comfort and ease of use with today’s requirements for energy saving with domestic technologies.

The system, being contactless, also provides benefits for technologies where mobility is an inherent property. We can already think of mobile phones and computers finding use for this but even electrical vehicles can be charged by a contact-free system, as demonstrated in this video featuring 2 wireless drones, each with a different antenna, which can land and be "refueled" at a wireless depot without any wire connections. This increases the capability of automation, as nobody needs to manually connect or disconnect the wires.

Hence the design of even short range wireless energy transfer may be ideal for portable devices that depend on a central wireless energy transmitter in a room or office for recharging electronics.

The use of such a system would also be of enormous benefit for installation of lighting systems such that holes don't need to be drilled in ceilings or walls to connect a light and switch to the electricity in a building, the energy can just pass through instead.



                            



This also opens up the possibility for an easier installation of solar panel systems, where you can plug them in on the roof and with no wiring having to go through the walls of the buildings.
Moreover, if there is a fault with the wiring, the wall does not have to be teared down for repairs.

This would increase the flexibility of lighting systems without causing major disruption to existing infrastructure.

Nevertheless, a wireless power system is limited by the power losses which appear in the system. This is wasted energy and the losses generate heat, which sets an upper limit of power which can be transferred. Therefore, an optimization strategy aims in minimizing the losses.

The losses can be expressed as loss factor

which is the sum of all losses related to the transferred power. A deeper analysis results in a minimum loss factor, which can be achieved by a given wireless power system, if generator and load are proper matched:

Which can be simplified to

     (For a detailed solution of this loss optimization please check the notes section)

The equation is shown graphically in Figure 3. The equation is only dependent on the two basic parameters of the wireless power system: The coupling factor k, between the receiver and transmitter coil and the system quality factor Q. The system quality factor is the geometrical average of the transmitter’s and receiver’s quality factors.
The equation suggests the product of the system quality factor Q and the coupling factor k as a general figure of merit (FOM). This means, that the system quality factor and the coupling factor determine the performance in an equivalent way. Poor coupling can be linearly compensated by a better quality factor and vice versa.

The ratio of the inductance L to the resistance R of a coil remains constant for different winding arrangements in the same volume and shape. It makes sense to define this value as a figure of merit to distinguish different coil structures. The quality factor Q is defined by this ratio.
The voltage, which is induced by the same current in an inductor scales with the frequency f and thus the apparent power in the device. The general definition of the quality factor is based on the ratio of apparent power to the power losses in a device. From this definition, the quality factor of a coil results to:

with ω = 2πf:

The quality factor Q can have a value between 0 and infinity, although it is difficult to obtain values far above 1000 for coils. For mass production you may expect values around 100. A quality factor below 10 is not very useful. These values have to be considered as the typical order of magnitude.

For a fixed operating frequency, the quality factor Q is mainly dependent on the shape and size of the coil as well as the materials used. Quality factors are generally provided for standard coil techniques (e.g. wire-wound coils, PCB coils).

Depending on the distance between the transmit and receive coils, only a fraction of the magnetic flux generated by the transmitter coil penetrates the receiver coil and contributes to the power transmission. The more flux reaches the receiver, the better the coils are coupled. The grade of coupling is expressed by the coupling factor k.

The coupling factor is a value between 0 and 1.1 expresses perfect coupling, i.e. all flux generated penetrates the receiver coil. 0 expresses a system, where transmitter and receiver coils are independent of each other.
The coupling factor is determined by the distance between the inductors and their relative size. It is further determined by the shape of the coils and the angle between them. If coils are axially aligned, a displacement causes a decrease of k. 

The definition of the coupling factor, k, is given by:

It results from the general equation system for coupled inductors:

where U₁ and U₂ are the voltages applied to the coils, I₁ and I₂ are the currents in the coils, L₁and L₂ are the self inductances, L₁₂ is the coupling inductance and ω = 2πf is the circular frequency.
The coupling factor can be measured at an existing system as relative open loop voltage u:

If the two coils have the same inductance value, the measured open loop voltage u equals k.

A good tool to analyze Resonant Coupling is 'reflected impedance'. Fig.1(a) shows the coupled circuit model with capacitor, Cs, added in series with secondary winding to form resonant tank. Rp, Lp, Rs and Ls are the resistance, and inductance of the primary and secondary winding, respectively. M is the mutual inductance between the primary and the secondary. RL represents the equivalent resistance of the load. Fig.1(b) is the equivalent primary circuit model with reflected impedance.

The reflected impedance, Zr, can be expressed with following equations:

in which ReZr is the real-part of the reflected impedance. It needs to be maximized for highest primary efficiency.
By analyzing the above equations, it can be found that when the secondary works under resonant condition

the reflected resistance, ReZr, exhibits maximum property and is equal to

if

Moreover, if

 

then there isn't a maximum point in finite frequency range.

It can also be found that said maximum ReZr can be further increased with the increase of frequency, the increase of mutual inductance, or the reduction of load resistance and secondary winding resistance. But it must be noted that the substantial decrease of load resistance may influence the secondary efficiency, because the secondary efficiency equals

                                                         

Indeed, other resonant topologies (like parallel resonance, or the combination of series and parallel) can also be employed at secondary side. They can be analyzed and optimized with similar approach described above.

Future Enhancements in Wireless Transfer: Electromagnetic Band Gap Metamaterials

The term "Metamaterial" comes from the greek work "meta" which means "beyond", i.e. a material with properties beyond normal materials. What does that mean? The simplest scientific definition of a metamaterial is that it is a macroscopic composite having a man-made, three-dimensional periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to electromagnetic excitation.
In other words each cellular component of the metamaterial must absorb and re-emit both the electric and magnetic components of electromagnetic radiation under excitation for the material to be a metamaterial. 

However, in a metamaterial, the sum of all of the cellular components do not obey the “rule of mixtures”, as seen in "normal" composite materials. The rule of mixtures is a method used to estimate a composite material’s properties assuming that these properties are a simple volume-weighted average of the properties of a single component dispersed in a matrix and or phase. In other words, the characteristics of the entire metamaterial is not simply the multiplication of the properties of a single cellular component. 

A common cellular component of a metamaterial for radio and microwave frequencies are split ring resonators which behave completely differently as individual structures than they do in a composite metamaterial structure hence metamaterials violate the "rule of mixtures".

Several features of these metamaterials definitions are worth noting. An engineered material can be a combination of different types of materials and/or structures used to obtain desired material properties. Examples of this are the use of periodic or aperiodic grid surfaces and/or the use of different applications of materials in a suitable combination. Synthetic materials can include ways of introducing vias, voids or cavities into a conventional dielectric with the voids/cavities potentially filled to include dielectric and/or magnetic materials with properties different from the surrounding bulk medium. Layering of such composite materials can then be adopted to yield a bulk material with the desired material properties. 

The study of metamaterials was initiated with a scholarly article in 1968 by Russian Physicist Victor Vesalgo who hypothesized that negative refraction, wherein a light ray is bent in a negative direction as compared to the conventional bending of a ray in the positive direction for a positive index medium, can occur if both the electric permittivity and the magnetic permeability of a material are negative. 

            Physicist Victor Vesalgo who hypothesized the existence of metamaterials in 1968


his prediction was confirmed 33 years later when David Smith et al., created a composite material with negative refractive index, and Sir John Pendry showed that the planar lens proposed by Veselago can provide greatly improved resolution who was studying electromagnetic transmission properties through a hypothetical medium with a refractive index that was assumed to be negative. 

After Smith's and Pendry's accomplishments with metamaterials, Veselago realized that the most important contribution of his original paper is not that a composite material can be designed to produce a negative refraction, but that a composite material can be designed to produce any value for permittivity and permeability. At least a part of his research goals was then to critically reconsider all formulas of classical electrodynamics that involve permittivity, permeability or refractive index.

A flood gate of extensive research was then opened up the possibility of using a variety of periodic surfaces as a way to provide unique material properties. It should be noted that, although these properties were demonstrated with periodic structures, periodicity is not required to create a metamaterial. 

While headlining applications — such as cloaking and invisibility — gain much public fanfare, there are many practical but often overlooked challenges in applying metamaterials to real-world applications. Due to the resonant nature of most metamaterial solutions, the approaches for achieving broadband effective properties are challenging. The principle of loss, or dissipation of the electromagnetic energy through its interaction with the material, presents a significant barrier for applications requiring transparency or high-efficiency transmission. 

Additionally, the creation of manufacturable bulk materials, beyond a few stacked surfaces, can be a significant challenge for materials and manufacturing engineers. The difficulty of this process is amplified as metamaterial features are pushed down into the nano-scale required for optical frequencies. Ultimately, the success of metamaterials requires an emphasis on the ability to model, design and manufacture them for system applications. Veselago himself perceived that the next big breakthrough with metamaterials will be the fabrication of transparent low-absorption metamaterials, which could open up the possibility of broadband negative refraction in the infrared to visible spectrum range.

Metamaterials can provide a means to enhance the performance and size of wireless components — for example, by making antennas multi-functional, and reducing the size and cost of front-end filtering. As the antennas become smaller however, the quality of classical metamaterials decreases as the amount of isolation decreases, as the resonators are closer together hence creating cross talk between them which causes sections of the metamaterial to damp out the signal.

An Electromagnetic Band Gap (EBG) metamaterial structure can be used to increase the isolation between antennas close to each other. The decoupling effect is not only a function of frequency but also polarization and coupling-plane configuration. In short, an EBG metamaterial affects photons in the same way semiconductor materials affect electrons as it, in effect, allows one to gate electromagnetic waves propagating through the metamaterial. This research is derived by finding means to create circuits using electromagnetic waves instead of electric currents for use in sensor and integrated circuit technology.

When designing an EBG structure, one needs to make sure to not apply an incorrect frequency and polarization, since this would increase the coupling between devices. This design challenge highlights the need for detailed simulations of such structures.

EBG's can be based on passive metamaterial split ring resonators with diodes that exhibit a variable capacitance, i.e. the varicap effect, which when inserted into the split ring resonator to give a variable frequency response. 

                                           Varicap Split Ring Resonantor Metamaterial Circuit


Experimentation with the varicap effect need not remain in the realms of advanced electronics or solid state physics labs. All semiconductor junction devices exhibit the effect, some to a surprising degree. Although many common devices exhibit the effect, they are not designed for that purpose so the effect can vary widely between one batch of a certain device and another.

The electronic component below is a Varicap diode used in an EBG, conductive carbon split ring resonator metamaterial.

Varicap diodes typically used in EBG Metamaterials are on the order of 16 picoFarads.

In practice, the Philips BA 102 varicap and a common rectifier diode, the 1N5408, exhibit similar changes in junction capacitance. So there can be many inclusions of cheap electronics in metamaterials, allowing them to be mass produced easily and cheaply. 

However, for quality testing of split ring resonators in prototype metamaterials, high quality varicap diodes are recommended for initial testing, after a success it is then reasonable to see if cheaper diode types can be used for manufacturing the metamaterial type en mass.

Using tuned EBG metamaterials, the varicap split ring resonators act as multiple cooperative flux generators which can be used to increase the range and flexibility of wireless energy transfer technologies by focusing the magnetic induction, dispersed over a large area from a large transmitter into to a narrow receiver as the metamaterial focuses the induction like a lens.

Metamaterial acting as a lens, focusing a broad transmission of magneto-inductive waves. Such a lens increases the distance of wireless power transfer, just as an optical lens in a telescope increases the intensity of light at a single focus 


By focusing magneto-inductive waves and related metamaterial propagation modes, EBG metamaterials provide a potential means for wireless power transfer to near field coupled receiving terminals. 

Using EBG Metamaterial Antennas in a wireless energy reciever circuit allows for a significant increase in the range of wireless transmission of energy. The following schematic illustrates such a receiver 

along with a video demonstration of its creation and operation

These Metamaterials were made using a conductive epoxy printer ink I made uing graphite powder mixed with an epoxy glue which was mixed in a 3:5 mixture with a standard black ink for an inkjet printer which, when the printer was set to print the metamaterial at the highest resolution possible (HD photo quality) allowed the metamateiral circuit to be printed and dried directly from the computer. 

Comparative testing of the conductive carbon ink metamateirals with standard copper pcb metamaterials produced using photo-lithography was done to asses the quality. The metamaterial ciruits are not as good as pure copper pcbs as they degrade faster nevertheless they are easier and cheaper to mass produce, are more flexible and are biodegradable.

Power transfer in magneto-inductive wave, MIW, devices is limited by two major factors, attenuation and electrical breakdown, with attenuation being reduced if coupling is high (and cell resistance is low) whilst breakdown is largely limited by the components used to fabricate the MIW cells.

Hence by making the cells out of a material with a high resistance to attenuation and electrical breakdown, i.e a material that can carry a high electrical load, then we can make serious progress. 

Graphene therefore offers huge promise in this field. therefore the manufacture of graphene-based metamaterials for use in magneto inductive transfer can greatly improve the range of wireless resonant induction transfer of electricity. Research into making a lightscribe graphene metamaterial structure from nanostructured graphene oxide is underway to try and make this a possibility.

In this sense, metamaterials of all kinds may hold the key to increasing the range of electromagnetic induction transfer, to the point where we could imagine at least small appliances (tv screens, laptops, phones, lights, ect) in the range of several meters, being powered wirelessly from a single, local source, i.e. a small power storage unit for solar, wind or thermal energy.

Phase shifting is another function that could easily achieved by tuned metamaterials, providing further design possibilities for wireless equipment. If the metamaterial structure is made active, then even more gains could be made, such as creating an antenna and filtering front end that could operate across multiple frequency bands allowing for broadband wireless energy transfer that can be tuned to specific bands for specific energy requirements. 

As progress continues, the development of wireless energy technologies will unites a lot of different fields of engineering and physics with the overall result being that this technology will become more widespread as traditional, centralized power distribution systems are upgraded and hybridized with decentralized power distribution systems over the coming decades.

Notes:

Optimization Solution for Increasing Link Efficiency

Enhancements in this technology which are currently available lie in 2 major domains: 

(1) a resonance tuning of effective load resistance between transfer and pickup coils

(2) ferromagnetic core shielding to enhance the magnetic coupling factor

Both of which pertain to increasing the link between the transmitter and receiver. 

The effectiveness of the electrodynamic induction is called link efficiency, η_link. It is a measure of the primary and secondary coils ability, to transfer energy from one coil to the other:

Where:

k = Magnetic coupling factor

= Unloaded quality factor of the primary circuit

= Unloaded quality factor of the secondary circuit

= Q_E = Effective Q

Where:

R_E = Effective load resistance which models the rectifier including the output filter capacitor and the actual load resistance R_L. The relationship between R_E and R_L is

If series resonance is used on the secondary side

ω = Primary angular velocity (2πf)
ω ₂ =Secondary angular velocity (2πf₂)
L ₁ = Primary inductance in Henries
r ₁ = Primary dc resistance in Ohms
L ₂ = Secondary inductance in Henries
r ₂ =Secondary dc resistance in Ohms

The loaded quality factor is:

For maximum efficiency:
Tune the link to the secondary resonant frequency
Set the effective load resistance to:

To maximize kQ (figure of merit):

Q is called the system quality factor. It should be noted that a low magnetic coupling can be compensated by an increased system quality factor.

Therefore, optimum efficiency =

The optimum link efficiency as a function of the figure of merit for the coils, which is η _opt vs. kQ.

One way to manage a high k value is to use a ferromagnetic shield (core) to enhance the magnetic coupling factor and reduces the stray magnetic field. This is why electrical transformers use an Iron core as illustrated in the introduction section of this article.

Another way to manage a high efficiency inductive solution is maximize the Q or Quality factor for the inductive coils.

Generally, for any type of coil, the magnetic coupling factor, k, can be improved by reducing the vertical distance between the coils and aligning the coils vertically, as lateral misalignment and angular misalignment will degrade efficiency.

Particle Physics Software and Financial Analysis Mechanics

Many people ask what are the benefits of massive particle physics experiments such as the Large Hadron Collider at CERN. Some go back to the basics and say that fundamental research inevitably generates spin offs in technology. After all, hospital equipment such as PET scans and MRI machines would have never found their way into hospitals without modern detector and superconducting magnet technology which are used in the massive particle detectors.

However it is important to note that the technology used in modern accelerators, such as superconducting magnets and silicon detectors were not invented by particle physicists, rather they were developed by materials scientists and engineers.

Particle Physics has benefited from Materials Science much the same way as Medical Physics has benefited from Particle Physics, simply in a different direction. However Particle Physics research, often labelled "Blue Sky" not only tests proof of principle of many important technologies but also does create applications in the here and now, most important of which is new software technology.

So when talking about the benefits of Particle Physics, and more importantly why people should fund it, what are the benefits?
The real benefits of Particle Physics research has been the vast amount of software that has been developed for these projects. Grid computing and Data Mining systems have been integral in the whole process from the beginning as has the ability to develop faster computer programs the analyse vast amounts of variables.

Physicists and Engineers have had to examine how they can program computers to perform these tasks as quickly as possible and it has led to them developing new programming languages to do so using existing techniques. One such example is CERN's ROOT programming language which uses the more familiar C++ language as a template.

CERN based ROOT off of C++ because even though it is a bit harder than most other languages it is very fast and powerful. Computer programming languages are chosen in the same way you would chose to buy a car, either for safety from crashing or for speed. If Fortran and Visual Basic are stable, yet slow programs, equivalent to a family car say, then a C++ program would be like a Ferrari. It is fast but crashes easily.

C++ is the computer language for mathematical models where you need speed. For models with closed form solutions you are naturally doing fine in almost any language, but when it comes to large scale Monte Carlo C++ is really a plus.

Therefore it is not surprising that CERN uses C++ in ROOT, as Monte Carlo simulations are used all the time in theoretical particle physics models. Before, CERN had used Fortran to run its simulations in its Geant-3 particle phsycis platform. CERN adopted C++ for Geant-4, which was the first such program to use object-oriented programming. CERN ROOT truly is an amazing piece of physics software with far greater potential than people give it credit for. ROOT works on virtually all operating systems however for optimum performance I have found that Red Hat Linux's Fedora is the OS of choice for CERN ROOT, at least in my experience.

Windows is useful for doing work too, for quick jobs and modifications, however I find it crashes far too easily and is not good if you are planning on working with ROOT for hours on end. Moreover, since the advent of the infamous Windows 8, most would consider Windows to be a generally bad operating system for doing any form of high level work on.


Apart from particle physics, Monte Carlo Simulations are also an important tool in modern finance, as they can be used in models to make predictions, namely in integrating certain formulas but incorporating random fluctuations as well as boundaries.

An elementary example of using the Monte Carlo method is to find the value of pi

Where humans can solve integrals symbolically, which is more mathematically complete, computer programs can only solve integrals numerically. To understand how a computer program does this, we need to think back to the basic definition of an integral from fundamental calculus:

The following example generates a solution, using standard integration, of  the area under the curve drawn from the polynomial f(x)

An integral can also be solved by approximation using a series of boxes drawn under the curve (see figure below); as the number of boxes increases, the approximation gets better and better. This is how a computer program can calculate integrals - it calculates the area of a large number of these boxes and sums them up.

Image credit: http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html

In a similar fashion, a value for the mathematical constant π is represented or “modeled” by definite integrals, such as

we can simplify this by the approximation:

This forms our new representation or "model" for as

As a simplifying approximation we can use a Monte Carlo method where we choose points randomly inside of the square. The points should be uniformly distributed, that is, each location inside the square should occur with the same probability. Then the simplifying approximation which allows us to compute π is

A simple algorithm to implement this Monte Carlo calculation is:

• Choose the radius of the circle R = 1, so the enclosing square has side 2R = 2 and area (2R)^2 = 4

• Generate a random point inside the square.

–> Use a random number generator to generate a uniform deviate, that is, a real number r in the range 0 < r < 1. The x coordinate of the random point is x = 2r−1.
–> Repeat to find the y coordinate.
–> If x^2 + y^2 < 1 then the point is inside the circle of radius 1, so increment the number of inside points.
–> Increment the total number of points.

• Repeat until the desired number of points have been generated.

• Compute

Implimented in CERN ROOT, we get the following distribution:

All well and good but what does this have to do with anything beyond abstract math? It turns out that the notion of a fixed constant whose value is computed based on random external counting is very similar to the concept of a priced stock option, a fixed value but under the influence of random fluctuations caused by the externalities that every system of transaction inherently has.


SPX Corporation is a Fortune 500 multi-industry manufacturing firm. SPX's business segments serve developing and emerging end markets, such as global infrastructure, process equipment, and diagnostic tool industries.

How do we know this company is lucrative? Simply calling it a "Fortune 500" Company should not be the only index we use in measuring a company's worth. If we really wanted to invest in a company we should see how its worth changes over time.





















From this we can learn a lot about what happened, particularly how the company was affected by the 2008 banking crisis; namely it suffered a minor collapse and returned to stability but this did not last and suffered an even larger crash over the course of the following months. It is noteworthy however how quickly it recovered and how it began to flourish again.

However we should remember that the banking bailouts promised a large trickling down of money into companies like these, so was this a result of actual worth or was the company benefiting some way from a bailout? President Obama's stimulus package also went into major Fortune 500 corporations, hence the overall speed of recovery should not be a de-facto litmus test of the company's competitiveness as state intervention in any corporation by any means, by bailout or stimulus, is a violation of capitalist market principles by which the system is supposed to be, at least in theory, self-correcting, i.e. the companies which are in fact doing better are allowed to survive whereas the companies doing worst must die out. Therefore measuring the price over time is not a clear indicator of a companies value in a true capitalist market system.

With smaller businesses we can however usually check the profits that a business makes in a year and use this as a accurate measure of competitiveness. The profit of such a business letting us measure its trading power and thus a competitive market system, independent from state intervention, is often more apparent in smaller business. However with bigger businesses, which often are intervened by a state we should use a marker which lets us monitor the trading power more closely, on a daily basis, hourly and even on the minute basis.

One way to do this is to see how much the company's stock was traded over time.

Volume–price trend (VPT) (sometimes price–volume trend) is a technical analysis indicator intended to relate price and volume in the stock market. VPT is based on a running cumulative volume that adds or subtracts a multiple of the percentage change in share price trend and current volume, depending upon their upward or downward movements

VPT total, i.e. the zero point, is arbitrary. Only the shape of the resulting indicator is used, not the actual level of the total.

The VPT is a similar to On Balance Volume (OBV) in that it is a cumulative momentum style indicator which which ties together volume with price action. However the key difference is that the amount of volume added to the total is dependent upon the relationship of Today's close to Yesterdays close
VPT is interpreted in similar ways to OBV. Generally, the idea is that volume is higher on days with a price move in the dominant direction, for example in a strong uptrend more volume on up days than down days. So, when prices are going up, VPT should be going up too, and when prices make a new rally high, VPT should too. If VPT fails to go past its previous rally high then this is a negative divergence, suggesting a weak move

The VPT can be considered to be more accurate than the OBV index in that the amount of volume apportioned each day is directly proportional to the underlying price action. So then, Large Moves account for large moves in the Index and small moves will account for small moves in the Index. This way the PVT can be seen to almost mirror the underlying market action, however as shown above, divergence can occur, and it is this divergence that is an indicator of possible future price action.

VPT is used to measure the "enthusiasm" of the market. In other words, it is an index that shows how much a stock was traded.

Using ROOT we can make animations showing the Price vs Volume movement of the SPX Corp over 3 years. A 50-day moving average line is also drawn over the scatter plot. (note: if animation stops and you want to see it again, open it in a new window)

We can see from this that the motion is truly random and looks very similar to the concept of Brownian Motion from physics. Using Monte Carlo Simulations physicists have been able to create Brownian Motion Simulations on computers to predict the possible random paths of particles.

Stock prices are often modeled as the sum of the deterministic drift, or growth, rate
and a random number with a mean of 0 and a variance that is proportional to dt
This is known as Geometric Brownian Motion, and is commonly model to define stock price paths.  It is defined by the following stochastic differential equation.

Where

St is the stock price at time t, dt is the time step, μ is the drift, σ is the volatility,  Wt is a Weiner process, and ε is a normal distribution with a mean of zero and standard deviation of one .

Hence  dSt is the sum of a general trend, and a term that represents uncertainty.

We can convert this equation into finite difference form to perform a computer simulation which gives

Bear in mind that ε is a normal distribution with a mean of zero and standard deviation of one.


We can use ROOT to perform such a geometric Brownian Motion Simulation

This ROOT script reads in data file which contains 32 days of closing prices.
The script then takes all 32 days, produces a log-normal histogram which is fit with a Gaussian to get the volitility (σ) and drift (μ) assuming Geometric Brownian Motion (GBM).  Once these two parameters are obtained from the data, a simple Monte Carlo model is run to produce 5%, 50%, and 95% CL limits on future price action. On top of the CL contours, 10 world lines of possible future price histories are also drawn.

The future price histories can be "hacked" in a new code to examine the world line histories to compute the probability for a given line to cross a specified price threshold. The next plot here shows 1000 world lines.

Red lines are those which never exceed $350 closing price and green are those which do exceed the $350 price threshold at least one time. For 1000 world lines, the result is that 161 trials had prices exceeding $350 at least one time, thus implying a 16% chance for the closing price to exceed a $350 price threshold.



A computer simulation is not a proof positive way to predict how the trading power of a business will increase or decrease over the course of a year, but considering that it labels the possibility of growth and decay, it is at least a more reasonable way of prediction than the apparent instinctual way people invest in business and do trade, sometimes depending on betting schemes that they don't fully understand and with which carry externalities such as systemic risk .

The management of risk—especially systemic risk—in the financial world was evidently deeply flawed in the 2008 finiancial crisis. An important part of the problem was that core financial institutions had used a shadowy secondary banking system to hide much of their exposure. Citigroup, Merrill Lynch, hsbc, Barclays Capital and Deutsche Bank had taken on a lot of debt and lent other people’s money against desperately poor collateral. Prior to the US bank deregulation and UK privatizations of the 1990s, the exotic forms of risky investment in corporations that banks all across the Anglo-American system were doing would have been barred by the Glass–Steagall Act of 1933 from dabbling in retail finance. Banks, Credit lenders and building societies would have been less exotic and venture capitalist, more boring in the eyes of neoliberalization, but would have nevertheless remained stable and solid institutions.

Hence, paying attention to risks involved in all financial tools is of utmost importance. At the end of the day, all our beautiful graphs, fancy theorems and newest computer models are no more than decorative pieces if the economy is going completely out of whack, as it did in the early 2000's with essentially 8 Trillion dollars in the US alone existing out of thin air supporting the construction and housing boom. Considering the possible hits and misses helps reminds ourselves that no form of trade is ever too big to fail, an important lesson to avoid future crisis.

It is ironic to think that just before the great banking deregulation between the mid-1990's and early 2000's, in 1990, the grand old man of modern economics, Harry Markowitz, was finally awarded the Noble prize:

                                                         Professor Harry Markowitz


Markowitz' work provided new tools for weighing the risks and rewards of different investments and for valuing corporate stocks and bonds.

In plain English, he developed the tools to balance greed and fear, we want the maximum return
with the minimum amount of risk. Our stock portfolio should be at the "Efficient Frontier", a concept in modern portfolio theory introduced by Markowitz himself and others.

A combination of assets, i.e. a portfolio, is referred to as "efficient" if it has the best possible expected level of return for its level of risk (usually proxied by the standard deviation of the portfolio's return).

Here, every possible combination of risky assets, without including any holdings of the risk-free asset, can be plotted in risk-expected return space, and the collection of all such possible portfolios defines a region in this space. The upward-sloped (positively-sloped) part of the left boundary of this region, a hyperbola, is then called the "efficient frontier". The efficient frontier is then the portion of the opportunity set that offers the highest expected return for a given level of risk, and lies at the top of the opportunity set (the feasible set).

To quantify better the risk we are willing to take, we define a utility function U(x) . It describes as a function of our total assets x, our "satisfaction" . A common choice is 1-exp(-k*x) (the reason for the exponent will be clear later) .

The parameter k is the risk-aversion factor . For small values of k the satisfaction is small for small values of x; by increasing x the satisfaction can still be increased significantly . For large values of k, U(x) increases rapidly to 1, there is no increase in satisfaction for additional dollars earned .

In summary:
small k ==> risk-loving investor
large k ==> risk-averse investor

Suppose we have for nrStocks the historical daily returns r = closing_price(n) - closing_price(n-1) .
Define a vector x of length of nrStocks, which contains the fraction of our money invested in each stock . We can calculate the average daily return z of our portfolio and its variance using the portfolio covariance Covar :

z = r^T x   and var = x^T Covar x

Assuming that the daily returns have a Normal distribution, N(x), so will z with mean r^T x and variance x^T Covar x

The expected value of the utility function is :

E(u(x)) = Int (1-exp(-k*x) N(x) dx = 1-exp(-k (r^T x - 0.5 k x^T Covar x) )

Its value is maximized by maximizing  r^T x -0.5 k x^T Covar x under the condition sum (x_i) = 1, meaning we want all our money invested and x_i >= 0 , we can not "short" a stock


How can we do this? We need to use a technique called quadratic programming

Let's first review what we exactly mean by "quadratic programming" :

We want to minimize the following objective function :

c^T x + ( 1/2 ) x^T Q x    wrt. the vector x

where c is a vector and Q a symmetric positive definite matrix

You might wonder what is so special about this objective which is quadratic in the unknowns.

Well, we have in addition the following boundary conditions on x:

A x =  b
clo <=  C x <= cup
xlo <=    x <= xup  ,

where A and C are arbitray matrices and the rest are vectors

Not all these constraints have to be defined . Our example will only use xlo, A and b Still, this could be handled by a general non-linear minimizer like Minuit by introducing so-called "slack" variables . However, quadp is tailored to objective functions not more complex than being quadratic . This allows usage of solving techniques which are even stable for problems involving for instance 500 variables, 100 inequality conditions and 50 equality conditions .

what the quadratic programming package in our computer program will do is

minimize    c^T x + ( 1/2 ) x^T Q x    
subject to                A x  = b
                  clo <=  C x <= cup
                  xlo <=    x <= xup

what we want :

  maximize    c^T x - k ( 1/2 ) x^T Q x
  subject to        sum_x x_i = 1
                   0 <= x_i

We have nrStocks weights to determine, 1 equality- and 0 inequality- equations (the simple square boundary condition (xlo <= x <= xup) does not count)




For 10 stocks we got the historical daily data for Sep-2000 to Jun-2004:


GE   : General Electric Co
SUNW : Sun Microsystems Inc
QCOM : Qualcomm Inc
BRCM : Broadcom Corp
TYC  : Tyco International Ltd
IBM  : International Business Machines Corp
AMAT : Applied Materials Inc
C    : Citigroup Inc
PFE  : Pfizer Inc
HD   : Home Depot Inc

We calculate the optimal portfolio for 2.0 and 10.0 .

Food for thought :

- We assumed that the stock returns have a Normal distribution . Check this assumption by histogramming the stock returns !

- We used for the expected return in the objective function, the flat average over a time period . Investment firms will put significant resources in improving the return predicton .

- If you want to trade significant number of shares, several other considerations have to be taken into account :

+  If you are going to buy, you will drive the price up (so-called "slippage") . This can be taken into account by adding terms to the objective (Google for "slippage optimization")

+  FTC regulations might have to be added to the inequality constraints

- Investment firms do not want to be exposed to the "market" as defined by a broad index like the S&P and "hedge" this exposure away . A perfect hedge this can be added as an equality constrain, otherwise add an inequality constrain .




This was just a brief taste of some of the overlying fields of study that exist between fundamental experimental and theoretical physics research and the world of finance and trade, demonstrating that the two fields which are both very different and abstract can be nevertheless unified to some degree.

I understand that finance is never a universally popular subject among scientists, as most science is critically underfunded worldwide, however I believe that by cosying up to finance in the same way that science has cosied up to industry it will give us more and more chances to "sing for our supper" and get more funding for fundamental science in the future. Moreover it may also help us to do trade with business that are shown, statistically, to be the best people to trade with and therefore will bring the cost of big projects down, helping the process of science to get done with the least amount of cost.




(If you want a copy of any of the CERN ROOT codes used to develop the images and graphs above, please contact me by leaving a comment below)