Power Line Induction Energy Harvesting Powering Small Sensor Nodes

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I , EXAMENSARBETE DESIGN OCH PRODUKTFRAMTAGNING 300 HP AVANCERAD NIVÅ

STOCKHOLM SVERIGE 2016,

Power Line Induction Energy Harvesting Powering Small Sensor Nodes

OSKAR THORIN

KTH KUNGLIGA TEKNISKA HÖGSKOLAN

SKOLAN FÖR INDUSTRIELL TEKNIK OCH MANAGEMENT

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ii

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Master of Science Thesis 2016:06 MDA492 Power Line Induction Energy Harvesting Powering Small

Sensor Nodes Oskar Thorin

Approved: Examiner: Supervisor:

2015-01-10 Martin Grimheden Hans Johansson

Commissioner: Contact Person:

Watty AB Hjalmar Nilsonne

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Abstract

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This thesis is examining the possibility of powering small wireless sensor nodes using induction and magnetic fields, generated by alternating current in conductors. As more devices becomes online, powering them all using batteries is not a good enough solution.

The application in this thesis is a current sensor mounted near the electric distribution box where there are typically no power outlets, instead powered by an inductor.

Two theoretical models are created. One basic, with calculations in 2d and one more complex in 3d. Experiments show that the 3d one is significantly more accurate and a good way of estimating an inductor’s possible power output. Along with the theoretical models, an electric one is also created showing the performance to expect while connected to a load.

The combination allows for estimation and optimization of the inductor output during the design process.

A hardware prototype is then constructed and evaluated according to the requirements of a real world application, as a way of minimizing the number of battery changes of a small wireless sensor node. The results show that it would not be reasonable to use an inductor for charging batteries this way as it fails to reach the requirements in several areas. At reasonable sizes and weights it will not be able to produce enough power for the sensor’s needs. With an inductor big enough, the required output could be big enough. Although batteries of equal size would last for a very long time.

These tests are performed with the Swedish power grid as the source conductor. The higher currents and frequencies of the American power grid shows a big increase in output power, as both parameters are squared in the equations. In Sweden the output power is only about 16 percent of what would have been possible in America. But even then the inductor would have to be placed in close proximity of the source conductor, as its output power is quickly decreasing with distance.

The conclusion is that this way of powering electronics has very limited use. There are few situations where the load are close enough to conductors with strong alternating current and the size of the inductor required to power a minimal load makes it unpractical.

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Examensarbete 2016:06 MDA492

Power Line Induction Energy Harvesting Powering Small Sensor Nodes

Oskar Thorin

Godkänt: Examinator: Handledare:

2015-01-15 Martin Grimheden Hans Johansson

Uppdragsgivare: Kontaktperson:

Watty AB Hjalmar Nilsonne

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Sammanfattning

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Detta examensarbete undersöker möjligheten att driva små trådlösa sensornoder genom induktion och magnetfält genererade av växelström i ledare. När fler och fler enheter kopplas upp duger inte batterier för att driva allting. Appliceringen för detta examensarbete är en stömsensor monterad nära ett elskåp, där det typiskt saknas eluttag, istället strömförsörjd med via en spole.

Två stycken teoretiska modeller skapas. En enkel med beräkningar i 2d samt en mer avancerad i 3d. Experiment visar att 3d-modellen är betydligt mer precis och ett bra sätt att estimera en spoles strömuttag. Dessutom skapas en elektrisk modell som visar prestandan man kan förvänta sig med en last ansluten. Kombinationen möjliggör en estimering och optimering av spolen under designprocessen.

Sedan skapas en hårdvaruprototyp vilken utvärderas enligt kraven för en verklig tillämp- ning, med målet att minimera antalet batterbyten för en liten sensornod. Resultaten visar att det inte är rimligt att använda en spole för att ladda batterier i det här fallet eftersom flera av kraven inte uppfylls. Inom rimliga restriktioner för storlek och vikt skulle inte tillräckligt med energi levereras för sensorns behov. är spolen tillräckligt stor skulle skulle dess utmatning räcka till. Dock skulle batterier i motsvarande storlek räcka väldigt länge.

Undersökningen är gjord med det svenska elnätet som källa. De högre strömmarna och frekvenserna i det amerikanska elnätet visar en betydlig förhöjning av levererad energi eftersom de båda parametrarna är i kvadreras i ekvationerna. I Sverige kan endast 16 procent tas ut jämfört med vad som vore möjligt i USA. Men även där måste spolen placeras i direkt närhet till källan, eftersom den levererade energin snabbt avtar med avståndet.

Slutsatsen blir att detta sätt att strömförsörja elektronik har mycket begränsat använd- ningsområde. Det är få tillfällen där lasten är nära nog ledare med kraftig växelström och storleken på spolen som krävs för att driva en minimal last gör det opraktiskt.

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Introduction

1

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2 CHAPTER 1. INTRODUCTION

1.1 Background

Internet of Things is rapidly growing, putting more and more devices online. Several of these devices are small sensor nodes placed near what they measure and report data to the cloud. They typically does not require much power, but whatever they use have to come from somewhere. A solution gaining in popularity is Energy Harvesting, meaning that the sensor node itself gathers energy from its surroundings. How depends on the application type and what is available.

The gain is that the end user no longer has to worry about changing batteries. With a small power consumption and large batteries, a simple sensor could live for years at a time on batteries. But eventually they have to be changed and with the uncertainty of when, combined with a large number of devices, there is a need for a better solution.

This thesis will focus on harvesting energy through induction, which is the typical way of transfer energy wirelessly. Today many consumer devices uses the charging specification Qi [18] consisting of two resonating coils in close proximity to transfer energy magnetically.

The application in this thesis is a bit different, where the device supposedly being charged is a small current sensor node, located next to existing AC wiring.Harvesting from such a generic source has a huge disadvantage, as the transmitting medium is not matched to the receiving inductor. Gathering only the already available fields of the wiring will produce far from what would be possible otherwise.

1.2 Watty AB

As our planets resources are diminishing, supplying the modern world with enough power is one of the biggest challenges ahead. Huge savings can be made by optimizing the systems we already use. Many of which were constructed when power was considered practically unlimited. Today the situation is different with risk of not having enough energy and also the energy usage impact on the environment.

Watty is a company that are developing energy analysis tools for consumers and companies, so that they can use energy more efficiently. Today little is done with the large amount of energy data that is available according to them. By using data measured more intelligently, there are lots of savings made possible.

Disaggregation Algorithms

An emerging field in this area is disaggregation algorithms that can extract much more information from data than traditional services. A successful algorithm is able to correctly decompose total energy usage into its constituent components.

In modern homes there are energy meters that digitally keep track of energy consumption. These meters are equipped with a LED that is indicating the current power usage by blinking a specific number of times per used kWh. This provides an easy way of connecting external hardware for gathering data to existing equipment without the need of authorized electricians. By collecting the number of blinks it is possible to constantly log the energy flow through the energy meter and send that data for analysis.

Energy Sensor

The sensor used to measure and transmit the data needs to be very power efficient. Often there are no wall sockets close enough to the energy meter and the sensor unit needs to operate on battery power. As a consumer device it is important to keep maintenance low

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1.3. PURPOSE 3

Figure 1.1: Total energy consumption pattern split between devices

and make it as easy as possible for the end user. If the sensor needs too much attention, the idea of a background analytics tool fails. It is critical for the product success that the sensor stays online for as long as possible. Additionally, setup must be possible without calling in an electrician. Internal connections to the distributor box will not be possible.

The sensor unit consists of the LED sensor, a clock, storage, communication device and control logic. Each blink input is represented by a timestamp and stored in a buffer while waiting to be transmitted for analysis. The current prototype uses the cellular network for communication with Watty’s servers because of its simplicity to set up. In comparison to the other components however, wireless communication require much energy.

There are several methods of extending the battery life of the sensor. The obvious one is to make all components use as little power as possible. This also includes controlling logic and transmissions to only operate when necessary. The next step is to make the sensor charge itself. This thesis will focus on the possibilities for a sensor suitable for Watty’s needs to harvest energy from its surroundings. The solution is then evaluated based on potential to be implemented in Watty’s technology platform. Mainly by considering the method’s impact on energy gain, costs, design requirements and user experience.

1.3 Purpose

Using ambient energy to power electronics is currently a very interesting topic. Much research has been done on the subject of energy harvesting and commercial products are already available. One area where the research is lacking is energy harvesting using induction. This is probably because of the fairly limited use, since the device must be in close proximity of high current conductors. However, there are situations where induction might be favorable, such as with the Watty sensor.

Induction is being used in energy transfer contexts, though mainly as a method of wireless charging. In that use case, both a transmitter and a receiver is working together to transfer energy with as small losses as possible. This technique is therefore different from energy harvesting, where a lone receiver is harvesting ambient energy. The setup resembles the traditional transformer with only one revolution on the source side and an open core, not surrounding the source conductor.

This thesis aims to answer the question whether powering electronics with induction harvesting is reasonable or not. The starting point is the Watty sensor, however the rea-

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4 CHAPTER 1. INTRODUCTION

soning around the subject will be more general. The research questions to be answered are:

1. Is there a way of estimating the power output of an open core inductor used for energy harvesting?

2. Is open core induction energy harvesting for electronics feasible?

1.4 Methology

The theory of induction, obtained in Chapter 2 and 3, with an alternating current conductor as source is used in Chapter 4 and 5 to build two models of the energy transfer. One simple, with approximations on a 2d plane, and one more advanced, counting in the wires physical size in 3d. Combined with models for the electrical behavior of the circuit connected to a load, formulas for the maximum amount of power that can be transferred are constructed.

These formulas are then validated against constructed inductors of various sizes, to make sure they are accurate and usable in future designs. They are then altered in Chapter 7 to enable faster computation for large ranges of parameters, required for the analysis in Chapter 8.

Several parameters of the inductor construction affects the output power. The analysis discusses them individually and demonstrate how they affect the end result. Based on the conclusions drawn, an inductor is designed to the requirements of the Watty sensor in Chapter 9.

To demonstrate the findings, a full prototype capable of harvesting and storing energy is build in Chapter 10. The performance of the prototype in combination with other findings are then discussed in Chapter 11, where future work also is suggested.

1.5 Limitations

There are several limitations to the created models and designs as they use idealized sets of conditions which will not be found in real life. Instead, they serve to showcase the theoretical limit for harvesting power in this way. Also, where the sensor is to be placed causes limitations on which designs that are possible.

Methology Limitations

• Only one configuration is modeled and evaluated.

• The second research question is dependent of the result of the first one.

Model Limitations

• Unknown interference with multiple conductors placed at random.

• Current flowing in opposite directions canceling out the generated fields unless the conductors are separated enough.

• Inductor modeled as a perfect rectangle in perfect alignment to the magnetic field.

• Resistance and capacitance of the load might change over time causing phase shifts.

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1.5. LIMITATIONS 5

Implementation Limitations

• Too few locations to be representative for what is possible.

• Manual assembly of components causes low precision.

• Not able to test with other power grids, such as the American, where output should be higher.

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Chapter 2

Frame of Reference

6

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2.1. ENERGY HARVESTING 7

2.1 Energy Harvesting

Energy harvesting is winning traction as a concept, as more and more devices are getting connected to the Internet. The Internet of Things (IoT) is the Internet for physical objects, exposing and exchanging data. A common type of connected device is a sensor of some sort. Placed where it needs to be to gather data and then transmit it up to the cloud for analysis.

Even though battery technology is advancing, there still is a lot to wish for. Eventually, the battery will run out and with an increasing number of devices, changing batteries will become a hassle. Therefore the devices would ideally be able to power themselves. No need for physical interaction at all. This is called energy harvesting and there are numerous sources from which the power can be harvested.

These are still very weak power sources, which means that the devices in addition still needs to be very power efficient. But optimizing it to function on minimal power which can be automatically gathered is a huge advantage and worth investigating.

Watty’s current sensor could easily be described as an IoT-device, as it is pushing data from the location up to the cloud for analysis and augmentation. It does not do much itself and should ideally be installed and forgotten, while the benefit of the data is being used elsewhere. The measuring itself does not draw much energy, but the data transfer do. On batteries, it will last a finite amount of time. If it were to gather energy itself, it would be much more convenient.

Solar Power

Probably the most traditional type of energy harvesting - Solar Power has been around for long time. The sun is a powerful source of energy and a lot of research is put on increasing the solar panels efficiency. For devices located outdoors, with generous amounts of sunlight available, solar power provides a simple and robust powering solution, as no moving parts are required and sunlight generally aren’t restricted to only a small space.There could be issues with size as the panels power output is relative to its surface area and since the sun is not shining around the clock, appropriate measures must be taken for the divice to store energy if it needs to function in the dark [14].

Vibration

If the sensor is moving, simple vibrations could be used to generate power. Often this technique uses a piezoelectric element which generates an electric charge when applied to mechanical stress. The same technique is used in certain types of microphones to capture sound. A piezoelectric element typically generates power in the mW range when the material is in mechanical resonance. Sensors mounted on vibrating industrial machinery could benefit from using the piezoelectric element as energy source [4]. Another interesting example is remote control wall switches, gathering enough power to not require batteries, only by exploiting the energy used to push the button. Companies such as Leviton and Philips are using this technique for it’s wireless wallswitches [3] [2]. The Philips Hue Tap Remote is based on an off the shelf component from EnOcean [1], with a mechanical energy bow which when pressed converts the movement to electricity, using a magnet and coil.

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8 CHAPTER 2. FRAME OF REFERENCE

Heat

Different temperature levels, means difference in energy that can be captured. Thermo- electric generators provide a simple way of converting temperature differences directly into electrical energy without any moving parts. Keep one side of the place cold and the other warm to get a voltage across the poles. Although the efficiency of the conversion is very low, the output is still enough to be useful in small applications [12]. Since it works both ways:

cold to hot and hot to cold; the thermoelectric generator can benefit from temperature changes caused by the sun during the day.

Electromagnetism

Alternating current flowing in a conductor causes magnetic fields to appear. The faster the frequency and the bigger the current, the stronger fields. Unless harvested, these magnetic fields won’t produce any losses, but with appropriate equipment they can can be used to transfer power from the conductor to a receiver system. It is the same principle as a regular transformer, only a lot less ideal. The transformer encapsulates the fields it is using within a high permeability core and successfully transfer power without much losses. Capturing the magnetic field from a single conductor will only provide fractions of the power used in the conductor.

The sensor to be used by Watty needs to be placed near high power electrical installa- tions. That is where the measuring equipment is and where the sensor gets its data. This results in an opportunity very specific to this use case. Since the power possible to harvest from magnetic fields decreases quadratically with distance; It is only realistic to use this technique in close proximity of the conductor. That significantly reduces the number of possible applications where this method of powering the device is even remotely plausible.

2.2 Related Work

A lot of the challenges of energy harvesting are common among the different approaches.

They all are about gathering energy and using it as efficiently as possible. That means there are numerous studies made on different areas of the concept. A few of them, related to this project, are presented below.

Power Management

One very specific feature is to capture the energy and convert it for storage. Here there have been lots of work done in optimizing the process. There are papers describing novel methods to reduce losses, such as [15], where the inconvenient voltage drop of the diode is improved. Big IC manufacturers are already producing great numbers of ready made solutions, tailored at specific input sources. One small package containing everything needed to capture, convert and store the power efficiently. One of these will be used later on.

Communication

Also there are strategies of how to best handle communications, which is often the most power hungry part of the device. Simplified, it means to use the radios as little as possible.

For a single unit communicating with a standby receiver it can decide for itself when to transmit to achieve the optimal information/battery drain ratio by not sending data that

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2.2. RELATED WORK 9

would not make a difference at the other end or compress data and send as bigger chunks if the receiver is not dependent on real time updates. Several concepts are discussed in [11].

For several devices cooperating, transmission in between is a bit trickier. Sending data demands that the other device is listening and active listening costs power. One solution is to make all the devices wake up simultaneously to check for news. But that requires careful synchronization of the device clocks. Should one of them be incorrect or drift, the device is potentially lost. One way of reducing listening time is presented in [6] there the authors present a method of minimizing waiting time by synchronizing the different nodes using a routing tree.

Wireless Power

The idea of Wireless power is well over a century old [7]. Today its even quite common, considering all the devices having the capability of charging without connecting a cable.

The common practice is to use both sender and receiver rectennas. Transferring power wirelessly becomes much simpler when the two nodes are constructed for each other. The resonance phenomena improves the power transfer capabilities drastically as described in [8]. Using it to harvest from existing power lines will not work though. The sensor Watty needs should be able to operate in very unspecified environments, with electrical cabinets different from each other.

Rotation Harvesting

Another use of inductive harvesting is in rotation. A standard electric motor used as a generator is the obvious example how rotational energy can be transferred into electrical energy. Another way of harvesting the rotational energy is discussed in [13] . Here, magnets are placed on the rotating object transferring energy as they pass the harvesting coils.

Power Grid Induction

Harvesting energy radiating from the power grid is made possible due to the alternating current. The current produces a varying magnetic field, which when passing through a loop of wire induces a current in it. This induced current will be alternating in the same frequency as the source and the voltage will be proportional to the number of loops. It is the same principle as for an ordinary transformer.

Vikram Gupta et al [17] uses this phenomena to experimentally examine how much power you can get from off the shelf inductors placed close to AC wiring. Their experiments included both a laboratory setup with the source conductor separated from its return path and in inductors placed in office landscape power conduits. The result was a promising 1 mW to 2 mW, which could potentially power some sort of device.

For future work, they suggests constructing inductors specifically made for harvesting and see if the performance could be improved. Also, they plan to examine the rest of the electronics required to enable the usage of the power.

Another similar approach is one taken by Evans Sordiashie [16]. He is constructing inductors specifically made for energy harvesting from power lines. It is done with an inductor where its core is allowed to surround the source conductor, creating a closed permeable path. This design allows for much higher performance, as the magnetic fields are captured in the core, directed through the wire loops of the inductor and does not have to transition between the core and the surrounding air.

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By using this technique both simulation and experimental results showed a resulting voltage of over 3 V and a short-circuit current of 34 mA for the specific inductor created.

Significantly higher than for inductors placed at the side and very promising for powering electronics.

In this thesis, the approach is to design and construct open core inductors specifically for energy harvesting, where the inductor is matched both to the source conductor and the load it is powering. The behavior of the inductor as well as the remaining electronics is considered to create a system with as high power output as possible.

2.3 Power Transfer

Depending on the power supply, there are different approaches when transferring power to the load. The common goal though, is that the total amount of energy reaching the load is as high as possible.

Infinite Power

For loads plugged into the electrical grid, power is close to limitless. Here, no special care needs to be taken unless the load uses as much energy as it significantly affects the costs for the user. Loads placed in remote locations suffering from power outages or when the grid is only on periodically, the load could be equipped with a big rechargeable battery.

Battery Operation

Running only on batteries, the most important goal is to transfer as much energy from the battery to the load as possible. This means the efficiency of the transfer matters most. Components causing unnecessary losses on the way should be carefully evaluated. A challenge for battery power transfer is that the voltage of the battery drops as it is getting discharged. Simple circuits will stop working when the voltage drops too much, even though there are still energy in the battery. There are solutions for this where the voltage is held steady at a specified level in exchange for higher currents and lower efficiency. In this case, lower efficiency is preferable to not working at all.

Energy Harvesting

For harvested power, efficiency does not matter at all. It is not of importance how much energy is used from the source at a given moment, since it is not a finite resource. What matters is the amount of energy reaching the load. A low efficiency energy transfer using large amounts of the source energy but results in more energy for the load is far better than an efficient one, if it results in the load getting less power.

Energy harvesting is often combined with a battery solution to provide power when only harvesting is not sufficient and store excess energy when there is plenty. Designing the entire circuit requires care when optimizing for harvest input, power transfer, storage, loads and duty cycles.

2.4 Impedance Matching

According to Jacobi’s law, the maximum power transfer theorem, power transfer is at its maximum when when the load impedance equals the complex conjugate of the source impedance. For purely resistive sources and loads, there is no reactance and maximum

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2.4. IMPEDANCE MATCHING 11

power transfer occurs when the load resistance equals the source resistance. With the same resistance in both load and source, equally much power is being dissipated in them.

Therefore efficiency is only 50 percent, since the load uses only half of the total power.

Us

Rs

Rl

Il

Ul

Figure 2.1: Generic, purely resistive voltage supply

Using the standard representation of a voltage source, which can be seen in Figure 2.1 , the following equations can be constructed.

η = Pl

Ptot

= Rl

Rs+ Rl

(2.1)

U = RlI = E − RsI (2.2)

I = E

Rs+ Rl

(2.3)

P = U I = RlI²= Rl

E

Rs+ Rl

2

(2.4) Where Ptot denotes the total power usage of the circuit.

Using the maximum power transfer theorem, the highest power available to the load will be

Pmax= Rs

E

R_s+ R_s

²

= E²

4R_s (2.5)

This can be verified by finding the extreme values of the power formula, which are the same as the roots of its derivative.

0 = dP dRs

=E²(R_s− Rl) (Rs+ Rl)³

⇒ Rl= Rs, Rl6= 0

That being said, high efficiency is still important for the load itself. To make the most out of the incoming power, the load should be as efficient as possible and minimize potential losses. Maximized power transfer from the source output to the load input, enables the load to save excess energy in a battery or similar, while high efficiency from the load input to load actions makes sure the battery lasts as long as possible.

Harvesting energy from magnetic fields

Harvesting energy from magnetic fields is done using an inductor. The internal resistance of an inductor comes from the wire in the coil winding and is proportional to the length.

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0 _R ¹

s=R_l

0%

50%

100%

Rl

R_s

P_l Pmax

η

Figure 2.2: Power transfer vs. efficiency

As the wire normally does not change over time, the resistance stays constant. An inductor connected directly to a load matches the general voltage supply in Figure 2.1 well, when the inductance of the coil is cancelled out with a matching capacitance. The purely resistive model can then be used.

According to the model, the voltage is linearly dependent on the current. Plotting the voltage as a function of the current, creates a straight line with −Rs as gradient. This is called the I-U curve and can be seen in Figure 2.3.

E Rk

E

I

U

U src I src

Figure 2.3: I-U curve of a power supply

Maximum power transfer corresponds to the point on the line that creates the largest

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2.4. IMPEDANCE MATCHING 13

area, which is at

U =E

2 (2.6)

I = E 2Rs

(2.7)

U = RlI ⇒ Rl= Rs (2.8)

P = U I = E² 4Rs

(2.9) (2.10) Since the source resistance stays constant, it is important that the load and source from the beginning are matched to each other. As will be shown later, the resistance of the inductor can be arbitrary selected without changes in performance or size, which means that the load probably will determine what resistance and therefore length of the coil winding to use. When possible to select a low load resistance, this is beneficial to the total power usage, since a low source resistance leads to a low dissipation in the inductor.

To be able to harvest energy from a magnetic field, it needs to be changing. Otherwise, power will not be induced in the inductor. The field to be used for the harvesting is created by current flowing in a hight power conductor. If the fields need to constantly change, the current needs to be alternating, or diverge to infinity. Conveniently, this is what is mostly used for power circuitry used in homes and where the sensor node will be placed.

However, batteries and low voltage electronics are based on direct current and thus incompatible with what the inductor harvests. Converting between the two causes losses which is unfortunate at already small power levels. Another problem is that for this specific placement, the wires will be hidden away inside the cabinet. It would be highly beneficial to surround the conductor with a high permeability core to encapsulate the magnetic flux, as is done with the common AC transformer. Having a small air gap in such a construction lowers the performance considerably, and not surrounding the conductor at all will be even worse. There will still be some power induced, and using a high permeability core for the inductor will improve the result somewhat, but since the flux have to travel trough air most of the time, power transfer will be poor.

Solar Power

For comparison, harvesting solar power uses the same principle of matching the load resistance to the source. However, the problem is more complex since the I-U curve is not linear, but depends on several properties of the solar panel. An important characteristic of the photovoltaic cell is that the resistance to match, called its characteristic resistance, changes with the level of illumination. For maximum power transfer the load has to alter its own resistance accordingly for them to match. This makes it hard to directly connect the load to it, since the load usually has a static resistance. To solve this problem, solar panel setups use a regulator in between the panel and the load. This regulator alters its own input resistance to match the solar panel while keeping the output resistance static to match the load.

Based on plots of I-U curves for a solar panel, for most voltages, it acts like a constant current source. However, maximum power transfer, creating the largest area, occurs after that, in the non linear region. This is called the maximum power point, MPP. In Figure 2.4 I-U curves are shown for different levels of illumination, but the curves are also affected by temperature, age and in case of a multi cell panel, if it is partially blocked.

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Figure 2.4: Solar panel I-U curves [19]

The regulator is monitoring changes, trying to find the MPP and adapt accordingly.

There are several ways of doing it, like there are several ways of numerically finding the extreme values of a graph.

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Chapter 3

Induced Power

15

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16 CHAPTER 3. INDUCED POWER

3.1 Model

The idea is to capture energy from changing magnetic fields, generated by the current flowing in a conductor. The conductor would be the wiring inside the electrical cabinet, where the main fuses are located. This has two advantages. All the power is passing here and the line and neutral wire are separated when passing the fuses. With current flowing in different directions in the two wires, their magnetic fields close to cancels each other out when they are located next to each other.

For calculation, a much simplified model will be used. All electrical cabinets will be different and even if the wiring inside each is known, it is still very hard to calculate all interfering magnetic fields. The most basic representation is an infinitely long, straight conductor, surrounded by air. It will create a magnetic field with a strength relative to the current it is carrying and decreasing exponentially with distance. A specific field strength could be visualized with a tube with the conductor in the center. To harvest energy from this field it needs to be changing, which it is in AC systems.

Figure 3.1: Magnetic field at a specific distance, generated by current flowing in the conductor

Introducing other closed loop conductors in this changing field will cause current to be induced, called eddy current, according to Faraday’s law of induction. The direction of the induced current will be such that it counteract the magnetic field, which is inducing it. As explained in Lenz’s law, this is why the induced electromotive force, EMF, in Faraday’s law is defined with the negative sign.

Combining several loops of conductors sums up their EMF, resulting in more power.

Several loops of wire connected to each other forms an inductor, which is what will be used throughout the report. The inductor can be in any shape as long as the area the wires surround, fully or partially aligns with a plane cutting through the main conductor. As the magnetic field does not change along the direction of the main conductor, calculations will be greatly simplified if the inductor shape stays constant orthogonally to the wire. This report will consistently use a rectangle as it is both good for calculations and prototyping.

Connected to the inductor will be a load. Any load that accepts the low alternating power being induced could be used, but the idea is to connect a AC/DC converter, energy storage and a micro controller capable of gather and transmit information. As discussed earlier, it is important for the load to be matched to the inductor. The inductor naturally has inductance and to be able to use the purely resistive version of the maximum power transfer theorem, the inductance have to be cancelled out using a capacitance. Normally, a load causes some capacitance on its own, which have to be taken into consideration when matching the inductance.

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3.1. MODEL 17

The model to be used consists of an infinite conductor, an inductor aligned to the plan orthogonal to the direction of the magnetic field, connected to a load represented by a capacitor and resistance. A graphic representation of the setup can be seen in image 3.2.

Figure 3.2: Harvesting setup

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Chapter 4

Theoretical Inductor Model

18

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4.1. PHYSICAL MODEL 19

4.1 Physical Model

An alternating current i flows through a conductor. At a distance r there is an inductor, made of wire with a diameter of d and resistivity of ρ, wound around a solid core of width w and height h with a relative permeability of µ_r.

i_s

w

h

r

N, L, d, ρ, µr

C

Rl

il

Figure 4.1: Setup of induction calculations

The strength of the magnetic field B generated by the current in the conductor is depending on the distance according to

B = µi_s

2πr (4.1)

where µ is the permeability of the surrounding material. This means that B will not be constant inside the inductor. The magnetic field will be stronger at the side closer to the conductor than the one further out.

The permeability of air is µ0 and other materials’ permeability are often expressed as µ = µ₀× µr where µr represents the material’s relative permeability. A material with a higher permeability than air placed in the magnetic field will concentrate the magnetic field through it and make it possible to harvest more energy. However, since the magnetic field is not homogenous - around the conductor it is almost discontinuous - and the material needs to encapsulate the setup, all calculations are done using the permeability of air.

Performance will still be improved if another material is used as a core for the inductor.

How much the gain will be is harder to predict. Different materials behave in different ways. Their permeability may be varying with frequency, limited at specific intervals or vary according to alignment. A core surrounding the conductor would capture the magnetic field generated by it and increase the possible power output even if surrounded by air. This will unfortunately not be possible due to the placement restrictions of the sensor.

The alternating current through the conductor generates an alternating magnetic field.

By placing an inductor in an alternating field, there will be an electromotive force (EMF,

), according to Faraday’s law, that could provide power to a separate circuit.

 = −NdΦB

dt (4.2)

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20 CHAPTER 4. THEORETICAL INDUCTOR MODEL

ΦBis the magnetic flux through an area and N the number of turns the wire goes around the area. In this case, the area of the inductor. Magnetic flux through an area of a varying magnetic field is defined as

ΦB = Z

B · dA (4.3)

where A is the area.

Both a changing magnetic field and a changing area induces power in the coil. In this case the area is constant but the magnetic field is alternating. By using an inductor with a high permeability core the change in the magnetic field increases in amplitude and thus more power can be harvested.

Every area is facing the same direction as the magnetic field. This means that the dot product is no longer necessary. The magnetic field is only varying with the orthogonal distance to the conductor and thus the dA can together with equation 4.1 be rewritten to only depend on the distance.

The resulting equation is

Φ = Z r+w

r

µis

2πxhdx = µish 2π

Z r+w r

dx

x = µish 2π

h ln

1 + w r

i

(4.4) where x is the distance from the conductor to the infinitesimally small area dA = hdx.

To find the induced electromotoric force in the inductor, the change of magnetic flux over time must be defined. The only parameter that changes over time is the current is

which can be separated in the derivation.

 = −N ΦB

 di_s dt

= − ln 1 +w

r

N µh 2π

 di_s dt

(4.5) Assuming pure sinusoidal AC

is= ˆIssin(ωt + α) = Is

√

2 sin(ωt + α) = Is

√

2 sin(2πf t + α) (4.6) dis

dt = 4Isπf

√2 cos(2πf t + α) (4.7)

the equation for induced EMF becomes

 = −N I_sµhf√

2 cos(2πf t + α) ln 1 + w

r

(4.8) Conditions at t = 0 is irrelevant and therefore the equation can be shortened:

 = N Isµhf√

2 sin(2πf t) ln 1 +w

r

(4.9) The resulting EMF has the same frequency f as Isand a RMS value of

E = N Isµhf ln 1 +w

r

(4.10)

4.2 Electrical Model

The circuit is modeled as an EMF-source in series with an inductance and resistance, connected to the load described as a resistance and capacitance in series.

In the circuit, both the inductance and capacitance are able to create a phase shift between the voltage and current, resulting in more complex calculations and power losses.

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4.2. ELECTRICAL MODEL 21

E L

R_w

C

R_l il

U

Figure 4.2: Induction calculation model

The inductance value depends on how the inductor is constructed, while the capacitance depends on the components chosen as load. Either way, if both are matched to each other according to

ω₀= 1

√LC (4.11)

their effects can be neglected and the circuit can be modeled as only resistive.

E Rw

Rl

I_l U

The purpose of the circuit is to transfer as much power as possible to the load, now only represented by a resistance R_l. The EMF-source is not ideal, since the wire used in the inductor is creating a resistance, represented by R_w. This makes the voltage U and current i_l, generated by the inductor, depend on the load resistance R_l.

The resistance in the inductor depends on the length, diameter and material of the wire being used according to

Rw= ρ L Aw

= ρ2N (w + h) Aw

(4.12) where ρ is the resistivity of the material, L is the length, A is the cross section area and d is the diameter of the wire.

Without this resistance voltage, current and therefore power would go towards infinity with, for example, the number of wire turns. More turns, generating more EMF, would also generate more current through a constant load. The resistance in the wire results in both voltage and current converging.

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22 CHAPTER 4. THEORETICAL INDUCTOR MODEL

Rl

U

Rl

Il

Rw

P_max

Rl

Pl

Figure 4.4: Voltage, current and power as a functions of the load resistance

The electromotoric force will create a current, flowing through both Rw and Rl. Il= E

Rw+ Rl

(4.13)

U = RlIl (4.14)

P = U Il= Rl

E

Rw+ Rl

2

(4.15)

P = R_l N I_sµhf ln(1 +^w_r) ρ^{2N (w+h)}_A

w + Rl

!2

= R_l

ln

1 + w r

N Isµ0µrhf Aw

2ρN (w+h) + A_wR_l

²

(4.16)

Maximum power transfer occurs when the source and destination have the same impedance, as stated in section 2.4. Since the phase shift of the impedance can be cancelled out by selecting a matching capacitor for the circuit, only the resistance matters when it comes to the inductor construction. To harvest as much energy as possible the resistance of the inductor must match the load resistance. The formula for maximum power is constructed by substituting Rlwith Rwand is valid when the resistances for the inductor and load are matched.

Pmax= Rw

E

Rw+ Rw

2

= AwLE²

4ρ = AwL(N Isµhf )²

4ρ ln

1 +w r

²

(4.17)

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Chapter 5

Extended Inductor Model

23

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24 CHAPTER 5. EXTENDED INDUCTOR MODEL

5.1 Physical Model

When using inductors created with high numbers of thicker wire, the actual space it occupies starts to make a difference. Both the area enclosed and the distance to the source conductor will be significantly different between the largest loop and the smallest loop. In this section a new model dealing with this problem will be introduced.

Three dimensional inductor

With this new model the calculations will be applied to an inductor taking up space in all three dimensions. The previous model was based on dimensionless loops of wire encapsu- lating an area parallel to the X-Y-plane which made calculations simple without the need for vector multiplications. Now the inductor will grow because of wire dimensions in that X-Y-plane, but also along the Z-axis. This means that the inductor will have depth as well, which results in the magnetic flow not being orthogonal to the encapsulated area of certain loops and thus requires vector multiplication.

x y

z

Previous formulas

The same basic laws from the previous chapter still applies. Equation 4.2 with N = 1 gives the EMF generated by a single loop of wire surrounding a specific area. Previously, in the case of a dimensionless coil, all the loops of wire would be in the same place, surrounding the same area. Thus, to get the combined EMF of the loops connected together, all that have to be done is to multiply with the number of turns.

This will be exactly the same as calculating the sum of all loops

 = −N ΦB

 dis

dt

= −

N

X

n=1

Φⁿ_B diⁿ_s dt

(5.1)

With every loop taking up space, Φ will be a function of n, where n determines the area which the loop encapsulates plus the position along the depth of the inductor.

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5.1. PHYSICAL MODEL 25

Magnetic flux in R³

The only addition for calculating the magnetic flux for a single turn will its Z position. The loop will be in the same place for X and Y .

x y

Figure 5.2: X-Y plane of the 3d inductor

x

z

Figure 5.3: X-Z plane with source conductor, coil and magnetic field

Using the previous definition for r, together with the new coordinate z the effective distance to the point where B measured will be r_z= sqrt(r²+ z²). The usable part of B will be its projection on the Z axis.

Power Line Induction Energy Harvesting Powering Small Sensor Nodes