Wireless Clock

8/8/2019 Wireless Clock

http://slidepdf.com/reader/full/wireless-clock 1/85



ii



Abstract

An analytical method is presented to model a system for simultaneous transfer of power anddata. This system consists of two patch antennas, to receive the power and data, a splitterto extract as much power as possible from the received signals, whilst still being able toreliably receive the data, and a rectifier to convert the received RF-power in DC-power.

These subsystems are modeled, analyzed, designed, manufactured and measured individu-ally. The antenna model is used to design a dual-polarized patch antenna with a bandwidthof 5.9% around 2.45GHz. Another, single-polarized, antenna with a complex impedancematched to the rectifier has been designed. The presented model is able to predict theinput impedance and radiation pattern within a few percent. This is accurate enough forour initial design.

A nonlinear diode model is presented which is used to design a voltage-doubler rectifier.This model is able to determine the diode impedance for the different harmonics within

five percent difference between measurements and modelling results.

A Wilkinson splitter has been designed and compared to an analytical model based onmicrostrip transmission line theory. This has resulted in reflection and isolation coefficientsthat are accurate within a few percent.

The antenna and rectifier are combined to form a rectenna with a rectenna efficiency of 52% for 0dBm received power. The unloaded DC-voltage is larger than 1.0 V at 0 dBmreceived power. The matching circuit between antenna and rectifier has been eliminatedresulting in a rectenna with smaller losses and which is smaller in dimensions than the

conventional rectenna with external matching circuit.

All the subsystems have been combined, which makes it possible to wirelessly receive powerand (AM) data. This system has a rectenna efficiency of 25%. The combination of theindividual models predicts the DC output voltage within 8% percent.



ii



Contents

1 Introduction 5

2 Wilkinson Splitter 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Microstrip line theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Microstrip Patch Antenna 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Cavity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Radiation of a Single Side . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Total Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Input impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 Lossless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.2 Including Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.3 Input impedance model . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Effective dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Rectifier 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Schottky diode model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1



4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Total System 59

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Single Rectenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.1 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 A rectenna-powered wall clock . . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Prototype design of the total system . . . . . . . . . . . . . . . . . . . . . 66

5.4.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Conclusions and Recommendations 77

2



Acknowledgements

This is the final report of my Master project, performed at TNO Science and Industry, in

Eindhoven, which took me twelve months to complete.

This graduation project concludes my studies of Electrical Engineering at the EindhovenUniversity of Technology (TU/e). I have chosen to specialize in the area of electromagnet-ics, and to perform my graduation project from the Electromagnetics chair. I would liketo thank prof. dr. A.G. Tijhuis of TU/e and ir. J.C. Sirks of TNO Science and Industryfor creating the opportunity to do this project.

During my research I received support from numerous people. Especially I would like tothank my supervisor ir. H.J. Visser of TNO for his daily support throughout this project,in particular with the analytical challenges and for his confidence in my capabilities. Mygratitude goes out to dr. ir. M.C. van Beurden of TU/e who, as supervisor, has put a lotof time and effort in this project, making sure every detail was covered.

Further, I would like to thank ing. G.J. Doodeman of TNO, for sharing his enormouspractical knowledge regarding design issues, and A. Reniers of TNO for his help with thedifferent measurements.

Finally I would like to thank M. van Lierop, my girlfriend, for all the times she convincedme that I was able to cope with all the obstacles one stumbles upon during a Master project.

3



4



Chapter 1

Introduction

Nowadays more and more electronic applications become wireless. This adds an enormousdegree of freedom to the experience of the application for the user. Transferring requireddata wirelessly is very common nowadays, but electronics need power as well. The powersource is usually included in the device in the form of a battery. This brings along thedisadvantage that the power source becomes exhausted after using it for a while and willhave to be replaced or recharged. It is desirable to transfer power to the device wirelesslyas well, thus bypassing the need for a battery, or to recharge the battery.

In this project it is attempted to supply power to the application using a rectenna. Theword rectenna is a combination of the words antenna and rectifier. Electromagnetic wavesare used to transfer power to the application. Besides the power we would like to transmitdata as well. It is tried to use the same antenna to transmit power as well as data, tominimize the occupied space.Another issue with batteries is that they sometimes demand precious space in an appli-cation. Some antennas can be designed in such a way that they can be integrated in theapplication more easily. It is tried to make the rectenna as small as possible, to make iteasier to include a rectenna in a bigger system design.

Sometimes replacing or recharging the battery of an application is hard to do, for examplefor active sensors at places that are hard to reach. When these sensors do not need muchpower to function, the power can be supplied wireless and with this system the measureddata can be read wirelessly as well. In this way it is no longer necessary to reach for thesensor once it is placed and no wires have to be connected to it. An example is a sensormeasuring the air pressure in a tire or other sensors in enclosed volumes.

Research in the area of wireless power transfer has been going on for quite a while, see [1].The idea of combining the data and power is well-known for the normal in-house electricitynetwork. For wireless applications this is a fairly new research topic. Transferring data

5



and power at the same time can be done by two separate systems, but what we wouldlike to do is to use one system for both purposes. In this way an optimum combination of

power and data transfer can be achieved.

In this report two frequencies will be considered. In a previous research [2] the licensefree 2.4 GHz band was used. Here, we elaborate on this research. Further, 868 MHz willbe considered, because some ultra-low-power applications are available in this band. Thedisadvantage of using 868 MHz is that the antenna might become quite large because thewavelength at this frequency is roughly 35 cm.

A rectenna system consists of three major parts. First of all an antenna is used to collect thepower and the data. After that there is a part that extracts power from the received signalsand recovers the sent data. Finally a so-called rectifier is used to convert the RF powerinto usable DC power. More and more circuits need only very little power to operate. Itwould be very useful to be able to activate integrated circuits with this system. Integratedcircuits need a certain minimum voltage to operate, some already work at as little as 1.0Volt. In this project it is attempted to supply an output voltage of at least 1.0 Volt. So wewould like to maximize the output voltage for a given input power. Because only limitedpower is allowed to be transmitted in the frequency bands of interest it is assumed that thereceived power by the rectenna is 1mW. This power level might seem arbitrary, however itlies within the range of expected power levels of our applications.

To extract as much power as possible from the received signals while still being able to

collect the data, a so-called Wilkinson Splitter (or combiner) is used. This splitter is usedin a somewhat unusual manner. The typical characteristic of a Wilkinson splitter is that itpossesses a resistive part to dissipate power that would otherwise have been reflected at theports. This results in a reflection-free lossy splitter. What we try to do is to use a rectifieras resistive part and turn the power into direct current (DC ) before it is dissipated. Inthis way the dissipated power in the splitter that is normally lost has now become useful.The splitter is actually used here as a combiner; meaning that two receive antennas willbe connected to the splitters and the combined output of the splitter is the data output.The power is extracted from within the splitter. This part is analyzed in Chapter 2.The antenna that will be used is a microstrip patch antenna, which is described in Chap-

ter 3. The main reasons for this choice are its small dimensions, the ease of manufacturingand the low costs. It is chosen to use a voltage doubler as rectifier, as described in Chap-ter 4. In this way the output DC-voltage is maximized for a given input power.Each of these subsystems will be modeled using analytical techniques. For each subsystema prototype design will be designed and measured. Finally, in Chapter 5, these subsystemswill be combined to design the total system. The models will be combined and comparedwith measurements on the total system.

In this project we have developed mathematical models for the individual systems based onphysical properties. This gives us insight in the factors that determine the behavior of the

6



system. We can rely on computer simulation results, which are in general more reliable.However, these simulations consume much more time and do not give the desired insight.

The results of our models are verified by these computer simulators or by measurements.

We accept certain accuracy margins for our models. We will use the mathematical modelsto design a system optimized at a certain frequency and we accept a frequency offset of roughly 3 % in a realized design. Further we use the models to predict reflection coeffi-cients for the individual systems, which define the amount of power that is accepted. Forthe antenna design we aim to reach a reflection coefficient of less than -10 dB. For theWilkinson splitter we aim to reach a reflection coefficient of less than -20 dB. Both thesevalues are common in industrial RF-design. We aim to be able to determine the outputDC-voltage and DC-power of the total system within 10 % accuracy.

7



8



Chapter 2

Wilkinson Splitter

As mentioned before we would like to design a rectenna system for simultaneous power and data transmission. To realize such a system a part is needed which extracts power from thereceived signals, while still being able to collect the data that is included in these signals.A so-called Wilkinson Splitter as described in [ 3 ] is chosen for this purpose. A Wilkinson splitter is used for this purpose because it makes a system more broadband and power can be extracted from it, to be turned into DC-power, while still being able to receive data. Thispower extraction acts as the required resistive part in the splitter.

2.1 Introduction

A Wilkinson Splitter may be realized as a Printed-Circuit-Board (PCB)-based splitter,formed by microstrip lines. In Fig. 2.1 the schematic layout of a Wilkinson Splitter isshown. The resistive part, R, in this kind of splitter is used to prevent any mismatchesin the system by dissipating power. We would like to extract this power and turn it intoDC power. We are actually using this splitter as combiner, using two ports of the splitter

as input and the combined port as output. However, due to reciprocity the splitter isanalyzed as splitter instead of combiner, because this is the most common way.

2.2 Analysis

Microstrip lines can be treated as simple transmission lines as shown in [4]. Power divisionby transmission lines is achieved by using transmission lines with different characteristic

9



Figure 2.1: Basic layout of a Wilkinson splitter

impedances. A Wilkinson splitter is used to divide power in a predefined way. The poweris provided at port 1 and the output ports are port 2 and port 3. The splitter is designedin such a way that the power out of port 3 is K 2 times that out of port 2. If Z in2 and Z in3are the input impedances right after the tee (the point where actual splitting takes place)and P 2 and P 3 are the output powers then it can be derived from transmission-line theorythat

P 3P 2

= K 2 =Z in2Z in3

. (2.1)

Furthermore, the splitter should be designed in such a way that an impedance match isseen at port 1. This means that the input impedance at the tee should be equal to Z 0, thecharacteristic impedance of port 1. This impedance is the parallel impedance of Z in2 andZ in3 . So

Z in2 · Z in3Z in2 + Z in3

= Z 0,

combining this with Equation (2.1) shows us that

K 2 · Z in31 + K 2

= Z 0. (2.2)

A resistive element is placed between the ends of arm 2 and arm 3. When the total systemis matched and the lengths of arms 2 and 3 are a quarter of a wavelength the voltages atthe ends of arm 2 and 3 are equal. At the desired frequency no current flows through thisresistive element and no power is dissipated here, because there is no voltage differenceover this resistor. When a mismatch occurs due to a frequency shift or a mismatch in

10



the impedance levels, power is dissipated in this resistive element instead of reflection of power at the different ports. Further, two extra arms are included after arms two and three

to transform the output impedance levels back to Z 0 (or any other desired characteristicimpedance). Quarter-wavelength transmission line impedance transformers are used forthis purpose, the theory of which described in [5].To satisfy all the conditions stated above the lengths of arms 2, 3, 4, and 5 should be aquarter of a wavelength. Arms 2 and 3 should be this length to achieve the equal voltageof the resistor at the desired frequency. The length of arms 4 and 5 should be a quarter of a wave length to perform the impedance transformation. The wavelength, λ, is defined asthe wavelength in the substrate. This wavelength is determined using λ = c/(f

√εe) and

the procedure described in [6]. The characteristic impedances of the different arms shouldbe

Z 02 = Z 0

K (1 + K 2), Z 03 = Z 0

1 + K 2

K 3,

and

Z 04 = Z 0√

K, Z 05 =Z 0√

K . (2.3)

The analysis of the impedances of arms 4 and 5 and the impedance of the resistive part isdescribed thoroughly in [3]. There it is shown that this choice of characteristic impedancesresults in a identical voltage level at both sides of the resistor. This results in no powerdissipation at the desired frequency. The resistive part of the splitter should be

R = Z 01 + K 2

K . (2.4)

This value yields infinite isolation and a perfect match at the center frequency.

2.3 Microstrip line theory

As shown in the preceding section, the splitter that we would like to use consists of diffe-rent microstrip lines with each its own characteristic impedance. These microstrip lines areformed by a PCB. A PCB consists of different layers. First there is a conducting groundplane, usually copper. In our model this layer is assumed to be perfectly conducting. Ontop of the ground plane a dielectric medium is present with a certain thickness, h. Thissubstrate has a dielectric constant, εr, which determines the electromagnetic behavior of the substrate. On top of that another conducting plane is positioned. By giving the toplayer certain shapes transmission lines can be formed. The characteristic impedance of thismicrostrip line is determined by its width, the thickness of the substrate and the permit-tivity of the substrate. Given the height, permittivity of the substrate, and K 2, a model

11



for the characteristic impedance of a microstrip line is used to determine the width of eacharm of the splitter.

To determine the characteristic impedance of a microstrip line the effective dielectric con-stant εe has to be determined. A microstrip line has a dielectric between the strip and theground plane but it has air above the strip. The field is traveling in the dielectric as wellas in the air. The effective dielectric constant is an approximation of the average dielectricconstant of the two materials. So the effective dielectric constant will always be boundedby εr and 1. Using this frequency-dependent effective dielectric constant the wavelength inthe medium can be determined. The effective dielectric constant is determined accordingto [7] and the characteristic impedance is determined by the model described in [8]. Thesemodels are numerical models based on empirical relations.From the effective permittivity the wavelength in the substrate can be determined. If c isthe speed of light and f represents the working frequency then

λ =c

f √

εe. (2.5)

The models that are used to determine the dimensions of the different pieces of microstripline first calculate static values for the characteristic impedance and the wavelength andfrom these values the frequency dependent values are determined. The validity of thesemodels is checked using the full-wave simulator Ansoft Ensemble. This simulator deter-mines the behavior of a RF structure using a method-of-moments analysis. The modelused agrees very well with the simulation results, as can be seen in Fig. 2.2(b). For ourfrequency range of interest the impedance is predicted within 3 %.

2.4 Design

To be able to determine the accuracy of the model presented in the previous sections, severalWilkinson splitters have been produced. To determine the dimensions of the differentpieces of the splitter, one has to know some parameters of the material to be used. Wehave chosen to use a FR4 PCB for the splitter. This material has a height of 1.6 mm, a

top and ground layer made of copper which has a conductivity σ = 5.8 · 107

Ω−1

m−1

. Thesubstrate has a dielectric constant between 3.5 and 4.5. The dielectric constant is not thesame for each slab because the distribution of the different materials in the substrate canbe different between different batches. This dielectric constant is assumed constant over asingle slab. So first of all the dielectric constant has been determined by measuring a pieceof transmission line etched from this material.The structure used to determine the dielectric constant consists of a piece of transmissionline of roughly 50 Ohm (determined using the average value of εr for this material), followedby a piece of transmission line with a certain width, different from the 50 Ohm piece. Theline is terminated with a 50 Ohm load. The initial piece of the 50 Ohm transmission line

12



(a) Layout (b) Results

Figure 2.2: Estimation of εr by comparison op input impedances

is used to minimize unwanted reflections at the connector due to a change in impedance.In this way the beginning of the line is matched to the connector. For this microstrip-

line measurement the reflection coefficient is measured at the beginning of the line by anetwork analyzer. From this reflection coefficient the input impedance is determined. Thetransmission line is measured over a broad frequency range, from 10 MHz to 4 GHz, so awide range of reflection coefficients is available. The model for determining Z 0 and εe isevaluated for a range of εr and for each value it is evaluated how well the measured andcalculated input impedances fit. In this way the value of εr for the material is chosen tobe the one that coincides best with the measured values. In Figure 2.2(a) the piece of testmicrostrip line is shown.

When designing such a Wilkinson splitter one encounters some physical challenges. Theresistor that is placed between the two arms must be connected directly to the microstriplines. Otherwise another piece of transmission line has to be included. So after the tee thetwo transmission lines should bend back towards to each other. We have chosen not to useelliptical (or any other techniques that does not use right angle bends) arms, although thebend backwards can be constructed more easily with them. However, the length and widthof these lines can not be determined unambiguously. Rectangular microstrip transmissionlines are used in our case, as shown in Figure 2.3. In case the splitter is designed to resonateat 2.4 GHz the wavelength is roughly c/2.4 · 109 12.5cm or less (because of the effectivedielectric constant). So when an arm of the splitter should be λ/4 long, the physical lengthof an arm is about 3cm. This length is not enough to make the complete bend backwards.

13



(a) 868 MHz initial design (b) 868 MHz improved design

Figure 2.3: Design of a Wilkinson splitter, K 2 = 2

So the lengths of arms 2 and 3 are chosen to be 5λ/4.

The design of a 2.4 GHz splitter is shown in Figure 2.3(a). A drawback of this design

is that the splitter becomes more narrowband, compared to the design with λ/4 arms,because we are actually using the second operating frequency. If f des denotes the desiredfrequency, then the first operating frequency is f des/5, this is the frequency where arms 2and 3 are exactly λ/4.

Three splitters have been designed and manufactured using this approach. A 1:1, a 1:2and a 1:3 splitter have been manufactured.

By using the εr found with the previous measurements the dimensions for the splittercan be determined. The design of this splitter requires the transmission lines to bend a

few times. These right-angle bends introduce some capacitive and inductive effects. Todecrease the influence of these effects, especially the capacitive effect, on the performanceof the splitter a part of the bend is cut off. This cut-off bend is called a miter and isdescribed in [9]. Furthermore, one should make sure that the supply lines at the ports arelong enough for the higher order modes, arising from steps in width, to vanish. One of thedesigned splitters is shown in Fig. 2.4.

14



Figure 2.4: A 1:3 Wilkinson Splitter

2.5 Results

First of all a piece of transmission line is measured to be able to determine εr. The resultsof the measurement and the estimation of εr are shown in Fig. 2.2(b). We can see that themodel agrees very well with the measurements, although for frequencies above 2.5 GHzthe model differs more and more from the measurements. For our frequencies of interestthe error is smaller than 2 %. This is mostly due to the fact that this transmission lineis created cutting copper tape in the desired shape. This is a really quick but not veryaccurate way of creating these kind of structures. Because we are mainly interested infrequencies below 2.5 GHz this method for determining εr is satisfying. The estimation of the dielectric constant resulted in a value for εr of 4.3.

With this dielectric constant the dimensions of the different splitters have been determinedand several splitters with different power ratios have been created. Three splitters havebeen created to be used at 2.4 GHz and two splitters have been created for 868 MHz(K 2 = 1 and K 2 = 2). The splitters designed for 2.4 GHz have been measured over afrequency range from 2.0 GHz to 2.8 GHz. Some results of these measurements for K 2 = 2are shown in Fig. 2.5. The reflection coefficient (S 11) and the isolation (S 23, not shownhere) agree reasonably well with the model, but it can be seen that the S 33 parameterdoes not show the expected behavior. We see that the levels do agree but the shape doesnot. However, the reflection coefficients are small enough to be accepted to be used furtherfor our purposes as explained in Chapter 1. It turns out that this error can be explained

15



(a) S11 reflection (b) S33

Figure 2.5: Results of the 2.4GHz K 2 = 2 splitter

by the higher order modes introduced at the bend near the resistor (verified by simulationresults). At this bend a change in line width and direction is formed.

To prevent the propagation of higher-order modes in the splitter the initial design is alteredat the bend near the resistor, additional lines are added to sufficiently suppress the higher

order modes caused by the bends. An example of this new design in given in Fig. 2.3(b).This is a design of a splitter to be used at 868 MHz with K 2 = 2. Furthermore, becausethis splitter is designed to be used at 868 MHz, the wavelength increases and it is notnecessary to make arms 2 and 3 five quarters of a wavelength, so this splitter will be morebroadband, as explained before. This splitter design has been used to produce a K 2 = 1and a K 2 = 2 splitter. These new splitters have also been measured. In Fig. 2.6 some of the results for the K 2 = 2 splitter are shown. Again, the results of the 2.4 GHz splitter areshown, however the results for the 868 MHz splitter are similar. Now it can be seen thatall measured scattering parameters show the expected behavior and are within the designmargins of Chapter 1.

16



(a) Reflection coefficient S11 (b) Isolation S23

(c) Reflection coefficient S22 (d) Reflection coefficient S33

(e) Power division

Figure 2.6: Results of the new 2.4 GHz K 2 = 2 splitter

17



2.6 Conclusion

In this chapter the Wilkinson Splitter theory has been discussed and it has been shown howthese splitters have been designed. These splitters are to be used as a part of a system tosimultaneously receive power and data. After an initial design an improved design has beenpresented to prevent higher-order modes from propagating. It has been shown that themodel satisfies our design demands. Further it has been shown that an impedance matchis seen at the desired frequency at ports 2 and 3, where the antennas will be connected, sothere will be no power reflected towards the antennas.

18



3.1 Introduction

A patch antenna, in its basic form, consists a of conducting ground plane, an intermediatesubstrate and a conducting top layer. This top layer is called the patch and is usuallyphotoetched from a copper clad PCB. Some examples of patch antennas are given inFigure 3.1. In the model that is presented in this section it is assumed that both the toplayer and the ground plane are perfectly conducting. The size of the patch is denoted bya and b as shown in Figure 3.2.

A patch antenna can be fed in several ways. In Figure 3.1 some examples are shown; aprobe feed; an edge feed, and an inset feed. All these kind of feeds can be used with themodel presented in this chapter, however the model can only handle the inset feed undercertain conditions. These conditions are given in Section 3.2. The method used assumes aprobe or a piece of transmission line to excite the patch at a certain location, this locationis denoted by (x0, y0). When an edge-fed microstrip patch antenna is used, either x0 or y0

is zero. The feed has certain dimensions denoted by dx and dy. In the case of a probe-fedantenna these dimensions are chosen in such a way that dx = dy and dx × dy is the crosssection of the probe. When the patch antenna is fed with a microstrip transmission lineeither dx or dy (depending on a horizontally or vertically aligned microstrip line) is set tothe width of the microstrip and the other one is set to zero. There are other methods forfeeding a microstrip patch antenna (such as slot coupled feeding) which are beyond thescope of this report, and these methods require major changes in the model.

Important parameters of an antenna are its input impedance and its radiation pattern.Both will be treated in this chapter. First a model will be presented to calculate theseparameters and next calculated parameters will be compared with measured values.

The modeling presented in this chapter follows the same approach as the one presentedin [2], that dealt with other aspects of the same subject. However, the model presentedhere is more extensive, since the model shown here can handle microstrip feed lines as well,something that was not included in [2], where only probe fed microstrip patch antennaswere treated.

Further, the model shown here has been improved to determine the radiation and gainpattern much more reliable, comparing Section 3.6 with Section 2.5 of [2]. The reason forthis improvement is not clear.

20



Figure 3.2: Layout of a microstrip fed patch antenna

3.2 Cavity model

A patch antenna can be seen as a radiating cavity. This way of analyzing a patch antennawas introduced by Richards et al in [10] and was improved by Carver and Mink in [11]. Toanalyze a patch antenna the Maxwell’s equations are used, i.e.,

× E = − jωµH ,

× H = J + jωεE , (3.1)

assuming an e jωt-time dependence. In these equations E denotes the electric field strength,H the magnetic field strength, ω the radial frequency, J the electric current source and µand ε represent the permeability and the permittivity of the medium. The height of theantenna is small compared to the wavelength. This results in electromagnetic fields in theantenna that are (almost) independent of z. The resulting fields will be transverse magnetic(TM) with respect to the z direction, since an electric field is applied in z direction. Since

the electric field is alligned purely vertically the Helmholtz equation for the z direction isgiven by

∂E z∂x2

+∂E z∂y2

+ k2E z = jωµJ z, (3.2)

where k = ω√

µε = 2πλ

. In this expression λ represents the wavelength in the dielectric,between the patch and the ground plane.

The cavity is assumed to have a perfectly electrically conducting patch and ground layer.The feed is placed in such a way that when the patch is excited, the electric field lines will

21



be vertically directed. When the electrical field lines are vertically directed the side wallsmay be assumed to be perfectly magnetically conducting. These assumptions result in a

complete set of boundary conditions for the cavity. First of all the electric field, E , canonly have a z component at the top and bottom layer. Secondly the magnetic field, H ,can only have a component notmal to the side walls of the cavity. Thus,

n × H = 0 at n = ux ,

n × H = 0 at n = uy ,

n × E = 0 at n = uz . (3.3)

These boundary conditions and the Helmholtz equation (3.2) can be used together todetermine the fields inside the cavity.

First, the homogeneous wave equation

∂E z∂x2

+∂E z∂y2

+ k2E z = 0 (3.4)

is evaluated. This equation describes the electric field inside the cavity when sources areabsent. The solution of this equation under the boundary conditions of equation (3.3) isof the form

E mnz = Amnψmn(x, y). (3.5)

In this equation E mn

zrepresents the z component of the electric field of mode mn. The ψ

mnfunctions are the eigenfunctions of the left-hand side of Equation (3.4). The electric fieldis split up in different modes. Each mode represents a given number of half patch lengthsin the x and y direction of the cavity. This way only discrete solutions (m,n) exist. Anexample of a m = 2 and n = 1 mode, called (2,1)-mode is given in Figure 3.3(a). Applyingthe first two boundary conditions of Equation (3.3), using separation of variables, resultsin

ψmn(x, y) = cos(kmx)cos(kny), (3.6)

where

km = mπa

,

kn =nπ

b. (3.7)

These last two expressions for km and kn represent the (transversal) wavenumber compo-nents of the m, n mode in the cavity in the x or the y direction. When Equations (3.5)and (3.4) are combined with the expression for the wavenumber k the resonance conditioncan be derived:

k2 = k2m + k2

n = k2mn. (3.8)

22



(a) TM21 mode (b) Magnetic surface currents

Figure 3.3: An example of a cavity with E fields and equivalent magnetic surface currents

Inside the cavity multiple modes can coexist. The total electric field is then simply thesum of the electric fields of the individual modes.

So far an expression has been derived for the homogeneous Helmholtz equation inside thecavity with boundary conditions. When a source is present in the cavity the correspond-ing Helmholtz equation is Equation (3.2). The solutions of the homogeneous Helmholtzequation can be substituted in this equation. Together with the summation of the different

modes this results in m

n

(∂E mn

z

∂x2+

∂E mnz

∂y2+ k2E mn

z ) = jωµJ z (3.9)

When Equations (3.4) and (3.8) are combined this results in

∂E mnz

∂x2+

∂E mnz

∂y2= −k2

mnE mnz . (3.10)

Substituting this relation in Equation (3.9) gives the expression

m

n

(k2E mnz − k2

mnE mnz ) = jωµJ z (3.11)

This expression can be used to determine the amplitude coefficients, Amn, of Equation (3.5).To do so we, integrate this expression over the total volume of the cavity and multiply itby ψmn, independent of the ψmn of Equation (3.5). This results in

V

m

n

(k2mn − k2)AmnψmnψmndV = − jωµ

V

J zψmndV. (3.12)

23



The left-hand side of this expression involves an integral of two independent ψmn functions.This part of the equation is treated first. Because the modes are orthogonal the summation

can be left out and the integral can be evaluated for each single combination of m, mand n, n. The part of Equation (3.12) where ψmnψmn is integrated can be written as

V

ψmnψmndV =

ax=0

by=0

hz=0

ψmnψmndxdydz

=

0 for m = m or n = nabhχ2mn

for m = m and n = n. (3.13)

In this equation

χmn =

1 for m = 0 and n = 0√2 for m = 0 or n = 02 for m = 0 and n = 0.

(3.14)

Substituting these expressions in Equation (3.12) yields

(k2mn − k2)Amn

abh

χ2mn

= − jωµ

V

J zψmndV. (3.15)

This equation must be satisfied for each single mode, so the amplitude coefficients are givenby

Amn = j χ2mn

abhωµ

k2 − k2mn

V

J zψmndV. (3.16)

Now that the left-hand side of Equation (3.12) has been treated, we are left with the right-hand side of the equation. This side involves an integral over the current density insidethe cavity. This current density is caused by the feed of the patch antenna. As mentionedbefore there are different ways of feeding the patch antenna. The ways treated here are aprobe feed, an inset microstrip feed and an edge microstrip feed.The model is only capable of accurately analyzing a patch antenna with an inset feed whenthe radiating modes of the antenna do not result in a varying E z field in the direction

perpendicular to the feed line. This way the feed line does not disturb the electric field inthe cavity, compared to a probe feed. Further, some capacitive coupling will arise with aninset feed, between the feed line crossing the patch and the patch itself, which make theresults of the model less reliable.Feeding a patch antenna with a microstrip line comes down to impressing a voltage at acertain place at the patch. This place is defined as (x0, y0) and has dimensions, dx × dy,with one of these to set to zero. This impressed voltage is transformed to a current densityJ e, using Huygens theorem, J e = n × H . The magnetic field in the cavity, under themicrostrip line is directed perpendicular to the feed line in the horizontal plane, so theequivalent electric current is directed in the z direction. This process is described in [4].

24



When a probe is used to feed the antenna an electric current can be defined directly asthe physical current flowing through the probe. Again, (x0, y0) is the position of the probe

and the dimensions are dx × dy. When a microstrip line feed is used either dx or dy is setto one. These dimensions are set as described in Section 3.1. The excitation current isdefined as

J z =

I 0

dxdyfor |x − x0| dx

2∧ |y − y0| dy

2

0 elsewhere.(3.17)

The integral of the right-hand side of Equation (3.15) can now be determined as

− jωµ

V

J zψmndV =

− jωµI 0

sinmπdx2a

mπdx

2a

sin

nπdy2b

nπdy

2b

= − jωµhI 0sinc

mπdx

2a

sinc

nπdy

2b

= − jωµhI 0Gmn. (3.18)

Now the electric field inside the cavity can be determined when Equations (3.5), (3.16)and (3.18) are combined. When using η =

µε

, the intrinsic impedance, the electric fieldinside the cavity of the microstrip antenna can be written as

E z =

mn

Amnψmn(x, y), (3.19)

with

Amn = jI 0ηχ2

mn

ab

k

k2 − k2mn

ψmn(x0, y0)Gmn. (3.20)

3.3 Radiation Pattern

So far, we have determined the electric field inside the cavity. As mentioned before wewould like to determine the radiation pattern of the microstrip antenna. The electric fieldin the cavity is used to find this radiation pattern. To calculate the radiation pattern, thecavity is modeled by four independent radiating apertures. The radiation pattern can bedetermined from the equivalent magnetic current density on the side walls of the cavity.These equivalent magnetic current densities are defined as M = E × n, as described in[12]. An example of these magnetic surface currents is shown in Fig. 3.3(b).

25



3.3.1 Radiation of a Single Side

To determine the radiation pattern of a patch antenna, first the radiation pattern of onesingle side is calculated. Next, these radiation patterns are combined to form the totalradiation pattern. As mentioned before the equivalent magnetic surface current is used todetermine this radiation pattern. One single side is modeled as an aperture above a per-fectly conducting ground plane, stretching to infinity. The consequences of the assumptionof an infinite ground plane, as described in [13], are mainly the absence of edge diffractions,which do not harm the functionality of our antenna . This allows us to define the electricvector potential at any given location, as described in [14], as

F =ε04π

S

M se− jk0R

RdS

. (3.21)

In this equation ε0 represents the permittivity of free space and k0 is the free-space wavenumber. M s is the aforementioned surface current density. The integrand is evaluatedover the total area of a radiating aperture and R represents the distance between theobservation point and a point on the side wall.

We are mainly interested in the far-field radiation pattern of the antenna. This allows usto perform some simplifications in Equation (3.21). We can use r, the distance from theorigin to the observation point, instead of R, in the denominator. The R in the phase termof the exponent can be approximated by R ≈ r − r cos(ξ), as described in [14]. Where r

represents the distance from the origin to the source point on the aperture and ξ is theangle between the lines from the origin to the observation point and from the origin to thesource point on the aperture.

To transform the vertically directed electric field lines into the magnetic surface currentthe relation M s = 2E × n is used. The fact that the amplitude of the magnetic surfacecurrent correspond with twice the electric field strength is due to mirroring in the groundplane. When we take for example the x = 0 side of the cavity, without losing generality,the electric field in this aperture can be written as

E aperture = Amn cosnπy

b

az. (3.22)

Here, Amn is the amplitude coefficient as given in Equation (3.19), which depends on themode number and the frequency. Combining Equations (3.21) and (3.22) the electric vectorpotential is found to be

F =ε0ay

2π

hz=0

by=0

Amn cosnπy

b

e− jk0[|r|−|r| cos(ξ)]

rdydz. (3.23)

To evaluate this integral in Cartesian coordinates the relation

|r| cos(ξ) = r · ar,

= y sin(ϑ) sin(ϕ) + z cos(ϑ) (3.24)

26



can be used, with ar the unit vector pointing from the origin to the observation point.Now, Equation (3.23) can be written as

F = jε0ay

2πrAmne− jk0r

hz=0

by=0

cosnπy

b

e− jk0[y

sin(ϑ)sin(ϕ)+z cos(ϑ)]dydz,

= jε0bhay

πrAmne− jk0r

Y [(−1)ne− j2Y − 1]

4Y 2 − n2π2sinc(Z )e− jZ , (3.25)

with

Y =b

2k0 sin(ϑ) sin(ϕ),

Z =h

2k0 cos(ϑ). (3.26)

Now that we have determined the electric vector potential the relation

E rad =1

ε0× F (3.27)

can be used to determine the radiated electric far field. The radial component of E radcan be neglected compared to the ϑ and ϕ components, due to the radiation properties of electromagnetic waves.Using ay = sin(ϑ)sin(ϕ)ar + cos(ϕ)aϕ + cos(ϑ) sin(ϕ)aϑ the radiated electric field in thefar-field region can be expressed as

E rad = − jk0

ε0F y(cos(ϑ) sin(ϕ)aϕ + cos(ϕ)aϑ). (3.28)

3.3.2 Total Radiation Pattern

We have calculated the radiated field of a single side of the microstrip patch antenna andnow we can combine the radiated fields of the different side walls to determine the totalradiated field. To do so we employ array theory, as the total radiated field consists of contributions from two arrays. Each array consists of two opposing apertures of the cavity.Two opposing apertures radiate the same field except for a phase shift. This phase shiftoccurs due to the spacing of the radiating slots and another phase shift of π radians mightoccur depending on the mode mn. As described in [5] the array factor (AF ) of two identicalradiating sources, with phases α1 and α2 and d as the vector between two similar pointson the different sources is

AF = 2 cos

k0d · r

2+

α1 − α2

2

e j

α1−α22 . (3.29)

27



The total radiated field can now be determined as

E rad = E aradAF a + E bradAF b. (3.30)

In this expression Equation (3.28) is used to determine the individual radiated fields of theapertures, expressed as

E arad =k0bh

πre− jk0rAmn

Y [(−1)ne− j2Y − 1]

4Y 2 − n2π2sinc(Z )e− jZ (cos(ϑ) sin(ϕ)aϕ + cos(ϕ)aϑ),

E brad =k0ah

πre− jk0rAmn

X [(−1)me− j2X − 1]

4X 2 − m2π2sinc(Z )e− jZ (cos(ϑ)cos(ϕ)aϕ − sin(ϕ)aϑ).

(3.31)

Here

AF a = 2 cos(X − αm)e jαm

,AF b = 2 cos(Y − αn)e jαn, (3.32)

X =a

2k0 sin(ϑ) cos(ϕ),

Y =b

2k0 sin(ϑ) sin(ϕ),

Z =h

2k0 cos(ϑ), (3.33)

and

αi = π

2for i even,

0 for i odd. (3.34)

With these expressions the total radiated field of a microstrip patch antenna can be de-termined for a single mode m, n. The radiated field of multiple modes can be found byadding the fields of individual modes.

To determine the total radiated power of a patch antenna on an infinite ground plane theradiated field must be integrated over a hemisphere, so

P rad =

π/2ϑ=0

2πϕ=0

1

2η0

|E rad|2r2 sin(ϑ)dϕdϑ, (3.35)

where η0 =

µ0/ε0 represents the intrinsic impedance of vacuum.

3.4 Input impedance

Another important property of an antenna is its input impedance. If it is possible todetermine this impedance accurately the feed impedance can be matched to the antenna’sinput impedance and maximum power transfer can be accomplished.

28



3.4.1 Lossless

As described in [15] the input impedance is defined as

Z in = − 1

|I 0|2 V

E · J ∗dV. (3.36)

In this expression I 0 represents the incident current at the feed of the antenna. If we useEquation (3.19) in the expression and perform the integral over the cavity, the resultinginput impedance will be purely imaginary because no losses have been taken into account.For each mode a reactance, X mn, can be determined and the total input impedance is thesum of these individual impedances.

Z in = − jm

n

ηχ2mn

abk

k2 − k2mn

ψ2mn(x0, y0)Gmn

= − jm

n

X mn. (3.37)

These reactances, X mn, can be seen as an inductance and a capacitance in parallel. Foreach mode such an equivalent circuit can be modeled. The values of the inductance andcapacitance then will be

Lmn =√

µεηχ2

mn

abk2mn

ψ2

mn(x0, y0)Gmn

C mn =√

µεab

ηχ2mn

1

ψ2mn(x0, y0)Gmn

. (3.38)

Whenever the wavenumber in the dielectric equals a modal wavenumber, so at a modalresonance, this input impedance reaches infinity, which will not happen in practice. Thereason for this non-realistic behaviour is found in the neglection of the losses thus far,including the radiation losses, meaning that the fields have been assumed to be restrictedto the cavity.

3.4.2 Including Losses

In the preceding section no losses were taken into account. Some losses have to be includedin the model through, for example radiation loss is an important non-negligible propertyof an antenna. Further, there are other losses, such as dielectric losses of the substrateand conduction losses of the patch and the ground plane. To include the losses in theaforementioned equivalent circuit of parallel inductance and capacitance for a particularmode a resistance is added as shown in Figure 3.4. The input impedance of this circuit is

29



Figure 3.4: Impedance circuit model for a single mode

Z in =jωRL

R − ω2RLC + jωL

=jω/C

ω2 − ω2r (1 + j/Q)

. (3.39)

In this expression ωr represents the resonance frequency ωr = 1√LC

and Q equals the quality

factor Q = RωL

.

Since we have no explicit expression for the losses in the system we use the quality factorto determine the losses. The quality factor can be determined from field concepts, meaning

Q =magnitude of reactive power

magnitude of dissipative power. (3.40)

We can determine the quality factor for each individual loss factor and combine theseindividual quality factors to a total quality factor which will be explained later. First of all we can determine the quality factor of the radiation losses as

Qr = 2ωW E

P rad. (3.41)

In this expression P rad is the total radiated power as determined in Equation (3.35). Thestored electrical energy, W E , is determined as

W E =

V

ε|E |2dV,

= |Amn|2 abhε

χ2mn

. (3.42)

30



To determine the quality factor for the conduction losses in the patch and the ground layerwe use the expression that can be found in [16], i.e.,

Qc = h

ωµ0σ

2. (3.43)

Here, σ represents the conductance of the ground plane and patch layer. The last type of losses we include in the model are the dielectric losses given by

Qd =1

tan(δ). (3.44)

In this expression tan(δ) represents the loss tangent of the dielectric. Further, tan(δ) = σdωε

,with σd the conductivity of the dielectric.

Now the total quality factor can be determined using the relation

1

Q=

1

Qr

+1

Qc

+1

Qd

, (3.45)

as described in [15].

Now we can determine the input impedance for each single mode including losses, usingEquation (3.39). To determine the total input impedance the individual impedances cansimply be added up.

3.4.3 Input impedance model

In the way described in the preceding section an accurate estimation for the input impedancecan be found. However, calculating the input impedance in this way requires quite somecomputation time, especially solving Equation (3.42) is computationally expensive. Thiscomputation time can be decreased by making some assumptions about the different modes,because the microstrip patch-antenna’s input impedance is mainly (but not completely)

determined by the dominant or resonant mode(s).

First of all the TM00 mode will have no inductance and a capacitance of C 00 = abεh

,according to Equation (3.38). So the impedance of the TM00 is modeled as a parallel RC circuit with impedance

Z 00 =j/ωC

1 − j/Q. (3.46)

Here, Q is calculated from Equation (3.45). The TM00 mode will hardly radiate any energyand therefore Qr can be set to zero.

31



Figure 3.5: Impedance model for a microstrip patch antenna

Next, the radiating modes of course do have radiation losses and also have inductances. So,for these modes the complete RLC circuit has to be determined. The so-called higher-order

modes, which are non-radiating modes, can all be added up into one single RL parallelcircuit. As described in [10] the inductance of this impedance can be determined as

X L =

µ/ε tan(ωh√

µε) (3.47)

and the resistance of the higher order modes can be determined using Equation (3.45) withQr = 0, since these modes do not radiate. The higher-order modes impedance can then becalculated as

Z ho =jωL

1 + j/Q. (3.48)

Now we can derive the total input impedance of the microstrip patch antenna with sim-plifications concerning non-radiating modes as shown in Figure 3.5. The losses are used todetermined an effective k to be used in Equation (3.20) so this value can not reach infinity.

3.5 Effective dimensions

The model for the input impedance and the radiation pattern of a microstrip patch an-tenna as described in the preceding sections has been implemented and analyzed. Theresults from this model have been compared to full wave simulations performed with An-soft Ensemble. It turns out that the model does not agree with the full wave simulations,the result forms are similar but we observe a considerable offset. The resonance frequencyas well as the radiation pattern of the presented model do not match with the full wavesimulations. Commercially available full-wave simulators have shown to be able to predictthe electromagnetic behaviour of antennas with an excellent degree of accuracy. It hasbeen checked that the simulations converge. So the discrepancy between the model andthe full-wave simulation is caused by imperfections in the cavity model.

32



Figure 3.6: Electric fringe fields of the cavity

This discrepancy is caused by the so-called fringe fields of the antenna, [ 17]. These fringefields are responsible for the radiation of the patch antenna. It has been suggested beforethat the electric field lines are perfectly vertically directed in the cavity of the patch.Besides these fields some fringe fields outside the cavity will exist as well. These fieldsbend outwards of the cavity as shown in Figure 3.6, where a side view of the cavity isgiven. To account for these fringe fields an extension of the dimensions of the cavity isintroduced. It is assumed again that all the electric fields inside the (extended) cavity arevertically directed using these extensions. These same fringe fields occur at an open-endmicrostrip line and the effect of these fringe field is described in [10]. The extension length,δ, for this case is given by

δ = 0.412h(εeff + 0.3)(w/h + 0.262)

(εeff − 0.258)(b/h + 0.813),

(3.49)

with

εeff =εr + 1

2+

εr − 1

2(1 + 10h/b)−1/2 , (3.50)

where w is the width of the microstrip line. When the effective length in the length orwidth direction of the patch is to be determined w is replaced with the length or the widthof the patch. The extra length δ is added twice in each direction because the patch hastwo open ends in each direction.

33



The dimensions of the patch as analyzed in the preceding model are now as shown inFigure 3.7. These effective dimensions are still an approximation to account for the extra

electric field lines in the cavity and the extensions suggested here are not always correct.In Section 3.6 it is explained how the approximation can be improved.

Figure 3.7: Effective dimensions of the patch

3.6 Results

The model that has been presented in the preceding sections has been used to designmicrostrip patch antennas. The starting point of the first design is to create an antennathat is as broadband as possible to be able to receive power over a frequency range as wideas possible. Further, it is intended to design a patch antenna that is fed by a microstripline, because of the ease of manufacturing (it is even possible to use copper tape on a

34



grounded RF substrate). Further, it is chosen to use an edge feed, because an inset feedintroduces some capacitance between the feed line and the patch itself which can not be

included in the model unambiguously. When an edge feed is used the number of degrees of freedom to choose the feed location is reduced, but it is still possible to design an antennawith practically any required complex input impedance needed.

First, an antenna was designed to operate at a frequency of 2.45 GHz with a feed impedanceof 50 Ω. This feed impedance was chosen, because in this way the antenna can easily bemeasured with existing equipment and can be used in conventional systems. To makethe antenna broadband, a double resonance was introduced. Around 2.45 GHz resonanceoccurs in both perpendicular directions of the patch. These resonances have a slight fre-quency offset making the bandwidth of the antenna roughly twice as large compared to asingle-resonance microstrip patch antenna. A disadvantage of this approach is the fact thatit is difficult to obtain a 50 Ω input impedance at the edge, when a half wave fits in boththe horizontal and the vertical direction. Therefore, we have made not a half-wavelengthbut a full wave fit in the a direction of the patch and now the 50 Ω input impedance isfound at the side of the patch.The antenna is etched from 1.6 mm thick, grounded FR4 material. This material has aloss tangent, tan (δ), of 0.016 and the relative permittivity is determined to be εr = 4.28,by the method described in Chapter 2. The conductivity of the copper on the patch andground layer is σ = 5.8 × 107Ω−1m−1.

The determination of the dimensions and feed location of the patch antenna followed the

following approach. The model presented in the preceding sections is used to design apatch antenna that satisfies the performance demands.Next, this antenna is analyzed with a full-wave simulator. In general the predicted behaviorby our model and the and the full-wave simulation differ. This difference is mainly causedby the inaccuracy of the length extensions of the cavity due to the fringe fields. The lengthsextensions are now altered in such a way that the model and full-wave simulation agree.Now our model can predict the behavior of the patch antenna much more accurate and animproved design using the cavity model is made that satisfies our performance demandsonce again. Again the behavior of the antenna is verified with a full-wave simulator. Whenthe differences in the predicted behaviors is still too large the process of adjusting the length

extensions can be repeated. However, in all our cases this was not necessary. In this waya rapid and reliable design of a patch antenna can be reached, since only two full-wavesimulations are needed. The cavity model turns out to require at least a factor of ten timesless CPU time, compared to full-wave simulations.At 2.45 GHz this design approach resulted in an antenna with dimensions a = 58.3 mm,b = 29.2 mm, x0 = 20 mm and dx = 3.1 mm, as in Figure 3.2. This antenna wasmanufactured, as shown in Figure 3.8, and measured.

The results of this antenna are shown in the Figures 3.9-3.11. No reliable measurementsconcerning radiated fields have been performed, due to problems with the measurement

35



Figure 3.8: The 2.45 GHz broadband patch antenna

equipment, probably caused by radiating cables in the anechoic chamber. However, asmentioned in Section 3.5, we can rely on full-wave simulations performed in Ansoft En-semble. We can see in Figure 3.9, that the input impedance is predicted within a fewpercent, well enough to use the proposed cavity model as a design tool for the antenna of our rectenna system taking the margins given in Chapter 1 into account. The reflectioncoefficient, assuming a 50 Ohm feed, in Figure 3.10(a) shows us that the model is capableof determining this coefficient with enough accuracy for our purposes. The 10 dB band-

width of the reflection coefficient of this patch antenna is 5 .9%, around 2.44 GHz. We cansee that the center frequency of resonance is very close to the 2.45 GHz, for which it wasdesigned. Compared to other results from the literature [18] and [19] this bandwidth islarge. Comparable bandwidths were only achieved by stacked patch antennas or changingthe shape of the patch. Stacked patch antennas result in an increase in size and costs.When the patch shape is changed from the rectangular forms presented in this chapter wecan only rely on full-wave simulations because the different models presented in literaturecan not cope with exotic shapes, or we have to invest in the development of new models,which is not desirable.It can be seen that the reflection coefficient has two minima around 2 .45 GHz, these corre-

spond to two different resonating modes, the (1, 0) and the (0, 2) mode. These two modescorrespond to two different polarization states, resulting in elliptic polarization. The formof the ellipse changes with frequency within the resonance band. The polarization changes(approximately) from linear to elliptical to linear in the other direction. This makes thesystem particularly suitable for frequency-redundant modulated systems.We can see in Figure 3.10(b) that the radiation pattern determined by the model and thefull-wave simulator show a good resemblance for angles up to ±75 degrees. For angleslarger than this value we see a discrepancy because we assume the absence of dielectricoutside the cavity in our model while the simulator assumes an infinite dielectric layer, butthese angles are outside our main interest and so this discrepancy is not a major concern.

36



The gain patterns as shown in Figure 3.11 also show us that the model is capable of deter-mining the magnitude of the radiated field within 1 or 2 dB. In the figures the gain in the

φ and θ direction is denoted by Gφ and Gθ. When considering Figure 3.11(d) it should benoted that the scale shown in this figure is only very small, from -29 to -27 dB, and theseemingly different behaviour of the model and the full wave simulation is actually of noconcern.

37



(a) Real part

(b) Imaginary part

Figure 3.9: Input impedance of the rectangular patch antenna

38



(a) Reflection coefficient

(b) Normalized radiation pattern

Figure 3.10: Characteristics of the rectangular patch antenna

39



(a) (b)

(c) (d)

Figure 3.11: Gain patterns for angle ϑ of a rectangular patch antenna(a) Gϕ in the ϕ = 00 surface (b) Gϑ in the ϕ = 00 surface(c) Gϕ in the ϕ = 900 surface (d) Gϑ in the ϕ = 900 surface

40



(a) Real part (b) Imaginary part

Figure 3.12: Input impedance of patch antenna matched to rectifier

Next, an antenna is designed to be used mainly for rectenna purposes. Therefore we aimedat matching the input impedance of the antenna directly to the input impedance of avoltage doubler. More details about this setup is given in Chapter 4, where it will beshown that this complex input impedance is typically around 40 − j45 Ohm. To matchthis impedance to the antenna, the input impedance of the antenna has to be the complex

conjugate of the diode’s output impedance, as described in [14]. Hence, to match thiscombination of diode and antenna, the antenna should have an input impedance of 40 + j45 Ohm. Again, a microstrip edge-fed antenna design is chosen, but now a single-moderesonance is chosen, resulting in a nearly square patch. This resulted in a patch antennawith dimensions a = 27.7mm, b = 30.8mm, x0 = 0.4mm and dx = 0.45mm, using the samesubstrate as used for the previous antenna. The microstrip feed line width of 0.45mm waschosen because this is the width of the pins of the intended diode.In Figure 3.12 it can be seen that, using the presented methods, an antenna with therequired complex input impedance can be obtained. More generally, an antenna with anydesired input impedance (within practical limits) can be quickly and accurately designed.

3.7 Conclusion

In this chapter a model has been presented to analyze a microstrip patch antenna. Thismodel is based on a radiating cavity. First, the electric field inside this cavity was deter-mined. From the electric field at the edges of the cavity we have determined the radiatedfield. From the electric field inside the cavity together with different losses (radiation,

41



substrate and copper losses) the input impedance is determined.The results for two different patch antennas have been presented. The input impedance

and the reflection coefficient have been measured and the model is able to determine theseproperties with a high degree of accuracy. The input impedance was predicted within afew percent. Further, the full-wave simulations of the radiated field and the gain patternof these antennas were compared to the model. Here, the model was also able to calculatethese patterns accurately. The radiation pattern was predicted within 1 or 2 dB.The first antenna that was presented has two radiating modes around 2.45 GHz makingit a broadband patch antenna. With the second antenna that was presented it is shownthat it is possible to design a microstrip patch antenna with any desirable complex inputimpedance, within practical limits.The presented model is able to determine the antenna properties with a degree of accuracythat enables us to use it to design a microstrip patch antenna for our needs.

42



Chapter 4

Rectifier

In the previous chapter it was described how to receive RF signals. In this chapter it is described how the RF-power in these signals can be converted into usable direct current (DC)-power, using a so-called rectifier. This conversion is necessary because an application can normally not be powered with an alternating current (AC) source, especially not at theRF frequencies that we use.

4.1 Introduction

Rectifier circuits are well known in electronics. In these circuits a diode is used to rectify avarying voltage waveform. A diode only conducts in one polarity and will therefore createa DC voltage. When we suppress the components, at frequencies other than the desiredone, using for example a capacitor, we are left with a pure DC voltage. After this, the DCpower is normally dissipated in a load, represented by a load resistance. This circuit, inits basic form, looks like the one shown in Figure 4.1.

Figure 4.1: Diode rectifier

43



Figure 4.2: Diode V-I relation

In Figure 4.2 the practical and ideal V-I characteristic of a diode is shown. Here, it can beseen that a diode is a nonlinear device. This makes the complex calculus for linear passiveelectronic circuits unsuitable and a non-linear analysis has to be employed.

In this chapter a method will be presented that can cope with the non-linearity of thediode in the rectifier circuit. The method will be used to determine the input impedanceof the rectifier circuit and to determine the output voltage for a given input power and loadimpedance. The input impedance of the rectifier circuit is important because a maximum

power transfer through the rectifier requires an impedance that is matched to the rest of the circuit.

The method to analyze the nonlinear behaviour of the diode that will be presented in thischapter will be used to design a so-called voltage doubler rectifier. This voltage doubler isshown in Figure 4.3, compared to Figure 4.1 a second diode is added. A voltage doubleruses this second diode to (approximately) double the output voltage of a rectifier, assuminga large load impedance. The voltage doubler puts the diodes in parallel with respect to theinput RF signal, which lowers the input impedance and this generally reduces the difficultyof matching the rectifier to the other circuitry. However, the diodes appear in series with

respect to the output load, which (approximately) doubles the output voltage. If the twodiodes are contained in a single package, the cost associated with the second diode is verysmall, making the doubler an interesting circuit for wireless powering. More informationconcerning voltage doublers is given in [20].Some rectifiers use a bias current to use a certain working point for the rectifier. However,because we assume no other power source is present other than the RF input power, wecannot bias the rectifier circuit. The diode that will be used is a Schottky diode. Thisdiode has been chosen because it has a low forward bias voltage, meaning that it starts toconduct at a low voltage, compared to other types of diodes. This makes it more suitablefor low-power rectifier circuits.

44



4.2 Schottky diode model

As mentioned before, a (Schottky) diode is a nonlinear device. This nonlinearity arisesfrom the composition of the materials in a diode. A Schottky diode is a semiconductor de-vice, where a piece of metal is placed on top of a n-type or p-type semiconductor substrate.Here, we use an n-type semiconductor. One of the contacts of the diode is connected tothe semiconductor, the other to the metal. When a semiconductor and a conductor arebrought into contact a so-called contact potential is formed. For a Schottky diode, thiscontact potential is called the barrier potential.Let us assume that the semiconductor contact is connected to ground. When a negativevoltage is applied to the metal contact the free charge carriers in the semiconductor areexpelled from the metal-semiconductor transition and no current can flow from the semi-conductor material to the metal. This situation is called reversed bias. When a positivevoltage is applied to the metal contact free charge carriers are attracted from the semicon-ductor to the metal surface and now a current can flow. This situation is called forwardbias.The area where no free charge carriers are present is called the depletion layer. A typicalproperty of a Schottky diode is that it has a thin depletion layer giving it a very shortswitching time between reverse and forward bias. This property makes this diode evenmore suitable for our application.

By using this structure of a Schottky diode a circuit model of the diode can be formed.

The substrate is not a very good conductor (compared to metal) and therefore a substrateresistance, Rs, is introduced to account for the conduction losses in the substrate. Thedepletion layer forms an isolation between two conducting plates and can be modeled asa so-called junction capacitance, C j. As the hight of the depletion layer changes with thebarrier voltage, the value of C j also changes with this voltage. In [21] the expression forthis junction capacitance can be found as

C j(vd) =C j0

[1 − vd/φ]1/2. (4.1)

Figure 4.3: Voltage doubler diode rectifier

45



In this relation vd represents the voltage over the nonlinear part of the diode as shown inFigure 4.4. The zero-bias differential barrier capacitance is given by C j0 and the barrier

potential of the diode is denoted by φ. Both these values can usually be found in thedatasheet of the corresponding diode. The nonlinear conductance of the diode is modeledas a nonlinear resistance with V-I relation

id = I s(eαvd − 1). (4.2)

In this relation id is the current through the nonlinear impedance d, as in Figure 4.4.Further, I s represents the saturation current and can be found in the datasheet of thediode. This saturation current represents the current that can still flow through the diodewhen it is reverse biased. This current is caused by thermal emission in the depletion layer.The factor α is given by

α =q

nkT , (4.3)

where q is the electron charge, k Boltzmann’s constant and T the absolute temperature inKelvin. The ideality factor of a diode is denoted by n and can be found in the datasheet.For most diodes the value of n is close to one. Now we can use a circuit model for thediode as shown in Figure 4.4. This model is widely used in literature, as can be seen forexample in [22, 23].

The circuit model that has been described so far represents a single Schottky diode. A

normal ‘off the shelf’ diode is put in a package, so it can be used in an external circuit.The effects of this packaging can also be included in the model turning the circuit modelof the diode into a package model. This packaging results in parasitic capacitance, C p, andinductance, L p, which are also given in the datasheet of the corresponding diode. Thisdiode package model will be used in the analysis of the rectifier and is shown in Figure 4.5.

Figure 4.4: Diode circuit model

46



Figure 4.5: Model of a diode in a package

4.3 Nonlinear analysis

4.3.1 Methods

There are many methods to deal with the nonlinear behaviour of the Schottky diode. Acommon method is to linearize the system around a working point. However, in our setupthe diode is used in a so-called large-signal region. This means that the system can not belinearized unambiguously, because the system shows the nonlinear behaviour within theworking region.

A truly nonlinear method is the so-called harmonic balance method. In this method a signalwith a certain discrete frequency is used as input and it is assumed that the nonlinearityin the system causes the output signal to be composed of different frequency components.These frequency components are all harmonics of the input frequency (including a DC-term and the fundamental frequency). To solve the system it is attempted to matchall the harmonics in the system in such a way that the nonlinear constraints, as givenin Equations (4.1) and (4.2), are satisfied. This method tries to find a balance for allharmonics in the system, so it uses a frequency-domain approach. This method is describedin [2].

There is another truly nonlinear method that employs the time-domain formulation. In thismethod, first the (nonlinear) differential equation governing the system is determined. Nexta starting condition is chosen. This starting condition can be chosen relatively arbitrarily,as long as the system will evolve to a steady state. Now a time-domain input signal isimpressed on the system. An integral, using a small time step over the differential equationis then used to determine the system state at the next time instance, i.e.

yn+1 = yn + ∆t · f (tn, yn), (4.4)

where yn is the system state at t = tn and the function f (tn, yn) represents the derivativeof y at t = tn. The timestep is called ∆t. When this time interval is chosen small enough

47



(a) Circuit (b) Model

Figure 4.6: Analysis setup

this is an accurate way of determining the behaviour of the system. Often, this methoduses a Runge-Kutta integration method and this is described more thoroughly in [24].

We have chosen to use the latter method to analyze the rectifier circuit, because it ismore flexible and it gives the impression to be computationally more efficient. It is moredifficult to include the nonlinearity of the junction capacitance in the harmonic balance

method. Furthermore harmonic balance has already been used in a similar study [ 2], so acomparison can be made concerning accuracy and computational costs. This comparisonwas requested from within the project for choosing future analysis methods. In Section 4.5the two different methods are compared.

4.3.2 Runge-Kutta method

To determine the parameters of the diode, the setup shown in Figure 4.6(a) was used. We

assume a voltage source with a single frequency ω as input, i.e.,

V g = |V g| cos(ωt). (4.5)

The generator impedance is denoted by Rg. The package model of the nonlinear diode, asdescribed in the preceding section, is used to analyze this circuit as shown in Figure 4.6(b).

48



The electrical behaviour of this circuit can be described with the following expressions.

V g

= I gR

g+ L

p

dI g

dt+ V

C p

V Cp = V d + V Rs

V Rs = RsI Rs= Rs(I C j + I d)

I Cj = C jdV ddt

I d = I s(eαV d − 1)

(4.6)

I g = I d + I C j + I C p

I d = I s(eαV d − 1)

I Cj = C jdV ddt

I Cp = C pdV Cpdt

.

(4.7)

These expressions can be written as two coupled differential equations where the diodevoltage, V d, and the generator current, I g, are the unknowns. These differential equationsinclude a second-order derivative of the generator current. A third differential equationcan be added to form a coupled set of three first-order differential equations. The set of coupled first-order differential equations that will be used is

V g = L pI g + V d + RgI g + RsC jV d + RsI s(eαV d − 1)

I g = I s(eαV d − 1) + C jV d + C pV g − C pRgI g − C pL pX

X = I g.

(4.8)

The last differential equation is added (resulting in the elimination of second order deriva-tives) because most techniques to numerically integrate differential equations can onlyhandle first-order differential equations. In these equations the time derivative of a vari-able is indicated by a dot above the variable, to make the equations more readable.

4.3.3 Implementation

The equations that have been derived in the preceding section, to analyze the diode havebeen implemented in a numerical routine. As mentioned before a method is used wherethe next state of the system is determined from the present (known) state of the systemand the differential equation of the system. At time instance t = t0 the system parametersV d, I g and X are known. The system state at t = t0 + ∆t is determined by a Runge-Kuttamethod.

49



To do so the expressions in Equation (4.8) are rewritten in such a form that each equationexpresses an unknown derivative in terms of the known parameters. The derivative of the

impressed generator voltage, V g, in Equation (4.8) is known. It can simply be determinedfrom Equation (4.5). Now we have expressions for V d(t, V d, I g, X ), I g(X ) and X (t, V d, I g, X, V d),i.e,

V d =1

RsC j

V g − RgI g − V d − XL p − RsI s(eαV d − 1)

(4.9)

I g = X (4.10)

X =1

C pL p

I s(eαV d − 1) + C jV d + C pV g − C pRgX − I g

(4.11)

V g and˙

V g are time dependent, as can be seen in Equation (4.5), and therefore the totalsystem is time dependent.

The Runge-Kutta method multiplies each expression with a time period ∆t and uses thesevalues to determine the state variables at time instance t + ∆t, as depicted in Equation(4.4).Our routine uses a so-called Runge-Kutta4 routine, see [24]. This means that to determinethe state variables at the next time instance, derivative information at the current timeinstance, at the end of the interval and twice at the midpoint is used. The midpoint isevaluated twice, assuming two different system states. Next, these four derivatives arecombined in a predefined way to minimize the error term at the end of the interval. Thisapproach cancels the first four orders of error terms, making this method a fourth-ordermethod. The step-size ∆t is adjusted with an adaptive step-size algorithm as described in[24]. This step-size algorithm compares the result using a certain step size with the resultobtained by using half the step size twice. If the difference between these two results isbelow a certain minimum accuracy limit the step size is accepted.

Suppose we know the system state at time t = tn. We call the system variables at thistime point V dn, I gn and X n. We would like to determine the system variables at momentt = tn+1 = tn + ∆t. The derivatives that are used to find these system variables aredetermined as follows. First the derivatives are determined at the current point in time.

k1 = V d(tn, V dn , I gn, X n)

l1 = I g(X n) (4.12)

m1 = X (tn, V dn , I gn , X n, V d(tn, V dn , I gn , X n)).

The new variables k, l , m are introduced as temporary variables to store the different deriva-tives. Next, these factors are used to determine the derivatives at the midpoint of the

50



interval.

k2

= V d

(tn

+ ∆t/2, V dn

+ k1/2, I

gn+ l

1/2, X

n+ m

1/2)

l2 = I g(X n + m1/2) (4.13)

m2 = X (tn + ∆t/2, V dn + k1/2, I gn + l1/2, X n + m1/2, k2).

These intermediate derivatives are used to determine a second intermediate set of deriva-tives.

k3 = V d(tn + ∆t/2, V dn + k2/2, I gn + l2/2, X n + m2/2)

l3 = I g(X n + m2/2) (4.14)

m3 = X (tn + ∆t/2, V dn + k2/2, I gn + l2/2, X n + m2/2, k3).

The last derivatives that are determined are the ones at the end of the time interval(t = tn + ∆t). The second intermediate derivatives (k3, l3, m3) are used for this.

k4 = V d(tn + ∆t, V dn + k3, I gn + l3, X n + m3)

l4 = I g(X n + m3) (4.15)

m4 = X (tn + ∆t, V dn + k3, I gn + l3, X n + m3, k4).

All these derivatives are used to determine the system variables at the next time instanceas follows

V dn+1 = V dn + ∆t6

[k1 + k4 + 2(k2 + k3)]

I gn+1 = I gn +∆t

6[l1 + l4 + 2(l2 + l3)] (4.16)

X n+1 = X n +∆t

6[m1 + m4 + 2(m2 + m3)] .

Combining the different derivatives in this way results in the desired cancelation of thefirst four orders of error terms and only 5th and higher order error terms will exist.

Suppose we use a generator with a certain voltage amplitude,

|V g

|, and single frequency,

f 0. Under these circumstances we can define a diode impedance for all frequencies. Thenonlinearity of the diode creates spectral components at harmonics of f 0 (including DC).Other frequencies will not be excited and will have a zero diode voltage and current, sono meaningful impedance can be formulated at frequencies other than the harmonics of f 0. We can envision the diode as a load Z df 0 for frequency f 0 and as a voltage source withseries impedance Z df n for frequency f n other than f 0.A typical result of the presented model is shown in Figure 4.7, where an Agilent HSMS-2852 [25] Schottky diode has been used. In this figure the wave shape of the diode junctionvoltage V d is shown for a 2.45 GHz input signal with 0dBm power. The properties of thisdiode are described in the next section.

51



Figure 4.7: Time-domain diode junction voltage

Using the presented Runge-Kutta4 routine the diode junction voltage and the generatorcurrent have been determined. By using the impressed generator voltage, the generatorcurrent and the generator impedance the voltage over the diode, D, as in Figure 4.6(a),can be easily determined. Using a Fourier transform (e.g. FFT) of the steady state of thesystem we can transform the time-domain diode current and voltage to a frequency domain

spectrum, giving us phase and amplitude information for each frequency of interest.For each harmonic of f 0 a diode impedance can now be determined as [20]

Z df n =V df nI df n

. (4.17)

The incident power on the diode can be determined from the generator voltage and thegenerator impedance as

P inc = |V g

|2

8Rg , (4.18)

as described in [12].Now, we have presented a method that can determine the input impedance of a diodefor all the harmonics of the input frequency for a certain incident power, P inc. A similarmethod as presented in [2] has been employed to determine the DC voltage when this diodeis used in an extensive circuit. It uses the Thevenin theorem to subtract the nonlinear partof the circuit.

52



Figure 4.8: Diode measurement circuit

4.4 Measurements

To validate the method just presented, a diode was measured. As Schottky diode we havechosen to use the HSMS-2852 from Agilent Technologies. This is a typical RF Schottkydiode for low-power applications, which is widely available. Further, it does not need abias current to operate, which makes it suitable for our application.The properties of this diode can be found in [25]. The properties that are needed todetermine the input impedance with the method presented before are as follows. Thesubstrate resistance, Rs, is 25 Ohm. The saturation current is 3 µA, C j0 is 0.18 pF and the

ideality factor is 1.06. The parasitic package capacitance is 8 · 10−14

F and the parasiticinductance is 2 nH, the barrier potential, φ, is 0.35 V.

To measure the input impedance of this diode, it is mounted directly on a SMA-connector(subminiature version A). This SMA-connector is attached to a network analyzer by aphase-constant coaxial cable. At the network analyzer we measure the complex reflectedpower. To compensate for the coaxial cable the network analyzer was calibrated includingthis cable. So the measurement data that the network analyzer produces is the reflectedpower directly at the diode pins, because the calibration set uses the same SMA-connectors.The generator impedance as well as the characteristic impedance of the coaxial cable is

50 Ohm.

The network analyzer that was used to measure the input impedance of the diode is a HP8753D. This network analyzer (and practically any other network analyzer) can not handleDC voltages at its ports. To prevent this from happening a DC-blocker is placed insidethe network analyzer. The DC signal itself is led to a DC port of the network analyzer.An RF choke is used to make sure only the DC-component is led to the DC-port of thenetwork analyzer. When a 50 Ohm load is placed at this port the model setup of Figure4.6(a) is preserved. The measurement setup resembles the one shown in Figure 4.8.

53



The network analyzer is used to measure the reflection coefficient S 11 at the diode. Imaginethat a transmission line with characteristic impedance Z 0 is terminated with a (complex)

load Z L. From [5] we can derive that the reflection coefficient at the load is

S 11 =Z L − Z 0Z L + Z 0

and (4.19)

Z L = Z 01 + S 111 − S 11

.

In this way, we can determine the diode input impedance using Z d = Z L. The networkanalyzer can only measure the reflection at the fundamental frequency, i.e. at the frequencythat is generated by the network analyzer itself as input excitation. In this way we are able

to determine the fundamental impedance of the diode over a frequency range. However,it is not possible to determine the impedance of the higher harmonics this way, becausethe network analyzer ignores signals at frequencies other than its own generator frequency.A spectrum analyzer has been used for this goal, but the spectrum analyzer measuresonly amplitudes, resulting in missing phase information. A time domain analysis woulddeliver the phase information, however an oscilloscope for the required frequencies was notavailable at that moment.To be able to analyze the harmonic behaviour of the diode we deployed a simulationprogram. We have used Serenade, a RF network simulator, which is well-known for itsreliable simulation results. The results of Serenade can be treated as true values. Serenade

uses a harmonic balance approach to determine the behaviour of nonlinear components.

4.5 Results

The results of the measurement, as described in the preceding section, will now be comparedto the results of the presented model. The fundamental input impedance of the diode hasbeen measured over a frequency range of 100 MHz to 4.1 GHz. In Figure 4.9 the real andimaginary part of the input impedance of the diode are shown for 0 dBm (which equals

1 mW) input power. The noise that appears on the measured impedances is caused bythe fact that the network analyzer is not able to fully suppress the higher harmonics onits input. When a single diode in a HSMS-2852 chip (containing two diodes) is measured,as done here, the second diode ‘floats’. This diode is connected to the shared pin of thechip at one side and is not connected at the other side. However, this floating pin of the second diode results in a capacitive coupling. The measurement data shown here arecompensated with the elimination of a capacitor of 0.3 pF, which roughly corresponds withthe capacitive coupling of the floating diode. We see that the model is able to predict theinput impedance within a few percent at the frequencies we are interested in. The modelhas been used to determine the input impedances at other power levels as well, ranging

54



from -10 dBm to +10 dBm. These input impedances exhibit a similar behaviour as shownin Figure 4.9. The model still reliably determines the input impedance for the other input

power levels. The only result shown here is for 0 dBm input power, because we haveassumed that this is the received power level, as mentioned in Chapter 1. Using these

(a) Real part

(b) Imaginary part

Figure 4.9: Diode input impedance according to model and measurements

55



Figure 4.10: Diode voltage spectrum

input impedances of a single diode the input impedance of a voltage doubler is determinedto be roughly 40-j45 Ω at 2.45 GHz. This impedance is obtained by halving the inputimpedance of a single diode because two diodes are placed in parallel in a voltage doublerresulting in half the impedance.In Figure 4.10 the harmonic components of the diode voltage V D are shown. In this figurethe values based on the results of the Serenade simulations are compared with the valuesobtained with our model. In this figure a 0 dBm input signal at 2.45 GHz is assumed. We

see good agreement between the model and the simulation.As mentioned in Section 4.3 the Runge-Kutta method presented in the previous sectionis compared to the harmonic balance method here. The harmonic balance method issomewhat more complex, where the Runge-Kutta method is more intuitive. The Runge-Kutta method turns out to be slightly computationally more efficient resulting in a fasterprediction of the diode behavior. However, the improvement on the calculation times isnot impressively. Further, the Runge-Kutta method can include the nonlinearity of the junction capacitance more easily. The accuracy of the prediction of the input impedancefor each of the harmonics is comparable. Further, the Runge-Kutta method can be usedmore easily to determine the input impedances for the higher orders, which are necessary

56



58



Chapter 5

Total System

In the preceding chapters the individual parts of the system, which is capable of wirelesspower and data transmission have been described. In this chapter the results are shown when these individual parts are combined.

5.1 Introduction

In this chapter three different systems are described that are composed of the earlierdescribed components and subsystems, which will both be called subsystems. First of all asingle rectifier was designed by combining a patch antenna and a voltage doubler rectifier.In this design a stub has been used that will be described in the next section.A set of eight of these rectennas was combined to increase the received power. Finally acomplete design, including the Wilkinson Splitter is presented.

5.2 Single Rectenna

As a first step in combining the different subsystems a single rectenna is designed. Thiscomes down to combining the patch antenna and rectifier. A typical rectenna setup isshown in Figure 5.1. The received energy is collected by the antenna. This receivedenergy is then rectified and dissipated in a load. A rectenna usually includes an impedancematching circuit. This matching circuit insures a maximum power transfer by matchingthe input impedance of the antenna to that of the rectifier, which differ in general. Inour design we made the matching circuit unnecessary by creating an antenna and rectifier

59



Figure 5.1: Rectenna setup

circuit which match directly.

The absence of the impedance-matching circuit reduces the physical dimension of thesystem. Further, the (minor) losses, mainly reflection losses, in the matching network arenow eliminated. We do not have control over the input impedance of the voltage doublerrectifier. Hence, to match the antenna and rectifier, the impedance of the antenna has tobe adjusted to the impedance of the rectifier. In Section 3.6 it is shown how this inputimpedance is achieved. Since the antenna will not be used in a stand alone way there isno need to have a 50 Ω impedance level at the input port.

The rectifier is a voltage doubler as shown in Figure 4.3. The diodes that have been usedare Agilent HSMS-2852 Schottky diodes. In [2] an inductor is suggested to act as a short

circuit for DC at the rectenna. This inductance can be left out in our design because thesecond diode in the voltage doubler already supplies a short circuit for DC.

In Figure 5.2 two realizations of the entire system without the load are shown. A load(representing an application) can be connected directly at the connector to absorb power.In the left design a stub is included. This stub is part of the rectifier and is included tosuppress the presence of the fundamental frequency, and to a lesser extent higher harmonics,in the rectifier’s output signal caused by its input.

Measurements have shown us that this stub can be omitted without significant decrease

of system performance. This is allowed owing to the fact that the capacitor at the outputof the voltage doubler suppresses the RF components at the output totally for the smallpower levels we are taking into account. This further reduces the size of the rectenna.The size of the rectenna is now hardly larger than the dimensions of the patch antenna, asshown in Figure 5.2(b).

The diode model as presented in Chapter 4 was used to design the voltage doubler. Todetermine the accuracy of this model measurements concerning the rectifier have beenperformed. The voltage doubler is connected directly at the patch resulting in an inputimpedance different from 50 Ohm. Normally to measure a rectifier it is connected directly

60



to a 50 Ω source. Our rectifier requires a source with a different impedance, which could notbe created with the available measurement equipment. Our patch antenna was especially

designed to have this input impedance. Hence, to be able to measure the rectifier reliably, itis measured including the patch antenna. For different levels of input power the unloadedDC output voltage is measured. The input power of the rectifier in the model can beeasily controlled using |V g| in Equation (4.5). On the other hand, the power incidenton the rectifier in the measurement is more difficult to determine, since we do not knowexactly how much power is accepted by the antenna. To determine the power acceptedby the antenna for different transmitted power levels we have used a scaling technique.If we know the antenna’s output power for a given transmitted power generated by atransmitter antenna at a fixed distance from our (receiving) rectenna, we can calculate theantenna’s output power for other transmitted power levels, since this is a linear process.For example, if the antenna’s output power is 0dBm for a transmitted power of 20dBm,the antenna output power will be -10dBm for 10dBm transmitted power. For one powerlevel (-3.75 dBm antenna power output in the model) the model and the measurementshave been matched. The other rectifier input power levels have been determined by thisscaling technique. Now, we are performing a kind of relative measurement.In Figure 5.3 the results of these measurements at 2.45 GHz are shown. We can see agood agreement between the model and the measurements. The predicted and measuredoutput voltages only differ by few percent. This gives us confidence in using the presentedrectifier model as design tool.

This measurement is performed without a load impedance attached to the rectenna. In a

real-life situation a load will always be connected to the output, representing the dissipationof power. However, typical applications for this system will only require power for a small

(a) With stub (b) Without stub

Figure 5.2: A single rectenna

61



Figure 5.3: Voltage doublers DC output voltage for different input power levels

amount of time and then ‘go back to sleep’. To power such applications a capacitor can beused, which is constantly loaded by the rectenna and once in a while the power is extractedfrom this capacitor into the application. In this way the rectenna is still loaded with a highimpedance resulting in a high output voltage, comparable to the results in Figure 5.3. So,

using this technique, the rectenna can deliver a reasonable amount of power at a relativelyhigh voltage with a small duty cycle. When 0dBm of power is incident on the patchantenna, roughly 520 µ W can be dissipated at a voltage of 1.2 V. How these values weredetermined will be explained in the next section.

5.2.1 Efficiency

The efficiency of this rectenna has been measured as well. A linearly polarized horn antennaat 2.42 GHz was used as transmit antenna. This frequency was chosen because the highest

output voltage was observed for this frequency.First, the gain of the transmit horn antenna was characterized by measuring the receivedpower by a standard gain horn in an anechoic chamber. From this measurement the gainof the horn antenna was determined to be 5.95 dB at the specified frequency.Next, the optimal load impedance was determined. We have used a purely resistive load.We have used 100 mW of transmit power. The distance from the transmit antenna to therectenna was chosen such that roughly 1 mW of power was incident on the rectenna. Thismeasurement was not performed in an anechoic chamber, because the load impedance hadto be changed several times, which is a tedious job to do in an anechoic chamber. The

62



Figure 5.5: Schematic layout of the rectennas in series

meter. The dissipated power was 516µW . The efficiency, η, can be determined as

η =

P DC

P rec ∗ 100%, (5.2)

where P rec is the incident power on the rectenna and P DC is the dissipated DC-power. Thisresults in an efficiency of 52 %. This efficiency is an improvement compared to the resultsobtained in [2], where an efficiency of 40 % was achieved. In [26] an efficiency of about80% was achieved for a rectenna operating at 2.45 GHz, but this result was obtained withan input power of 100 mW resulting in a much more efficient rectifier.When we compare our efficiency to the one achieved in [2] the improvement is mainly dueto the elimination of the impedance matching circuit, which causes additional reflectionlosses. Furthermore, our efficiency is more realistic than the high input power efficiencies

reported in [26], since regulations wil not allow these high input powers in most practicalapplications.

5.3 A rectenna-powered wall clock

The rectenna presented in the previous section showed promising results and has beenused in a more extensive system. To further increase the output power eight rectennashave been connected in series, as shown in Figure 5.5. Since the ground planes of the indi-

vidual antennas act as ground for the rectifiers as well, the ground planes of the individualpatch antennas are not connected. This leave space for DC-circuitry.In this way the output voltage and power are increased by a factor of eight. Further, therectenna system can deliver the same amount of power at a larger distance assuming thesame source and transmitted power.As mentioned before this system is particularly suitable for applications with a low dutycycle. A standard electronic wall clock has been used as a demonstrator. Normally a clockis powered by a battery, requiring power only during an instance each second. When weare able to collect the required amount of power at the right voltage (usually 1.5 Volt) wecan supply the power for the clock by the rectenna system.

64



(a) Front (b) Back

Figure 5.6: Wall clock powered by eight rectennas

In Figure 5.6 this clock is shown. On the front side we can see the eight patches anddiodes. At the back we can see the individual ground planes, where it can be seen that the

ground planes are not connected. Further, we can see a small electronics board hangingabove the back of the clock. This is included to make sure that the output voltage of the rectifier never exceeds 2.0 Volts. When the input voltage of the clock exceeds thisvalue, the clock might get damaged. During experiments a clock ‘deceased’ because thesesecurity electronics were not included at that time and the rectenna system received toomuch power resulting in an output level of roughly 15 V.A typical waveform of the supply voltage of the clock is shown in Figure 5.7. Here we cansee the slow charging of the capacitor and the instantaneous power absorption by the clock.This waveform is observed when enough power is collected by the rectennas to power theclock. When more power would be received the output voltage would approach the asymp-totic level of 2 Volt earlier and would stay almost constant until the clock ticks. When lesspower would have been received, the charging would have developed more slowly. Then,the output voltage will not come near 2 V, when the power is absorbed each second. Whenthe output voltage at the moment of power absorption is below 1.2 V the clock will notoperate at all.This system is not suitable to be measured quantitatively since it will let the clock tickor not. The performance of the individual rectennas were already measured in the prece-ding section. What can be measured is the distance for which it works assuming certaincircumstances. We used a horn antenna at 2.45 GHz, which was powered with a 20dBm(100mW) constant wave. The horn antenna was placed in the focal point of a parabolic

65



Figure 5.8: Building blocks of the total system

The building blocks of the prototype are shown in Figure 5.8. The individual blocks willnow be discussed.

RF-source, modulator and data

As RF-source, a network analyzer was used to create a carrier signal with a single fre-quency. The system will be analyzed for carriers varying between 2.25 and 2.65 GHz. Thenetwork analyzer was used to perform a frequency sweep, required for this measurement.

A network analyzer is rather big and for demonstration purposes a more elegant sourcecan be used. There are constant wave sources (without sweeping capabilities) that canproduce a RF carrier between 2 and 3 GHz which are not bigger than 20x10x4 cm.An amplitude modulator using an MSWT-4-20 switch has been made, since no ‘off-the-shelf’ modulator was available to be used at 2.45 GHz. This modulator is shown in Fig-ure 5.9(a). The modulator attenuates or passes an (RF) input carrier signal, dependingon the input voltages of the four transistors. The resistors attenuate the carrier and thecombination of the three resistors ensures that the impedance level is not changed. Bymaking the attenuation small (e.g. 1 dB) it is ensured that power is always delivered tothe system. One could say that the data is superimposed on the carrier. The power in the

transmit carrier mostly determines the transmit power. When FM would have been usedthe instantaneous transmit power would be (almost) independent on the data.To create (dummy) data a function generator is used to generate a block wave. The func-tion generator was used to create a 1 kHz block wave, with an amplitude of 1.25 V and anoffset of -1.25 V. The amplitude and offset have been set in this way to realize the 1 dBattenuation of the amplitude modulator.

67



(a) Modulator (b) Demodulator

Figure 5.9: AM system

Amplifier and transmit antenna

In the prototype design the transmit antenna (Tx Antenna) is a horn antenna, designedfor the 2.45 GHz band, and creates a linearly polarized field. The amplifier is setup insuch a way that the signal fed to the transmit antenna contains 20dBm of power. Thisequals 100 mW and is the maximum output power allowed in this band. A KU 233 BBAamplifier was used, which has a gain of 30 dB over a frequency range from 500 MHz to2500 MHz. After the signal has been sent by the transmit antenna it is received by thereceive antenna (Rx antenna). The main focus of this project has been on the parts afterthe transmit antenna

Receive antenna

As receive antennas (Rx antennas) two patch antennas have been used. This type of antenna has been analyzed in Chapter 3. The antenna that was used in the prototypeis the broadband patch antenna described in Section 3.6. The behaviour of this antennais shown in Figures 3.9-3.11. The important characteristics of this antenna are its dualpolarization and the fact that it is relatively broadband. This results in a larger frequencyrange in which the system can be used and horizontal as well as vertical polarization can be

68



used. It should be noted that the polarization shifts over the frequency band from verticalto horizontal.

Wilkinson splitter

The signals that have been received by the two patch antennas are led to a Wilkinsonsplitter. This splitter is described in Chapter 2. A splitter with K 2 = 1 was used, becausecomparable signal strengths are expected from the two antennas. This splitter requires a100 Ohm resistive element to prevent reflections at the different ports. The rectifier willbe used as resistive element.

Rectifier

A rectifier is used to convert the received RF-power to DC-power. A voltage doubler, asdescribed in Section 4.1, is used. The implementation of this voltage doubler is the sameas described in Section 5.2, so the HSMS-2852 is used here as well. In Chapter 4 the inputimpedance of an diode has been determined. By using the fundamental input impedanceof a single diode, the input impedance at the fundamental frequency of the voltage doublercan be determined. This impedance is around 40

− j45Ω and differs from the 100 Ohm

required by the Wilkinson splitter. This results in a possible mismatch in the system,however the time span of the project did not allow us to improve the match between thevoltage doubler and Wilkinson splitter.

Demodulator

The demodulator is a so-called crystal receiver, as shown in Figure 5.9(b). Initially acrystal was used in such a circuit, but nowadays a diode is used. This receiver is actually

a rectifier itself, however it is not used to subtract power. This demodulator is used at thereceiver side. At the transmitter side the carrier was modulated with the data. Here, thedata is recovered. A series capacitance at the output of the demodulator can be placedto suppress the DC term of the demodulated signal. In the results of the measurementswhich are shown next, the output of the demodulator is filtered with a 50 Hz filter and alow-pass filter, with a cutoff frequency of 10kHz.

69



5.4.1 Measurements

The prototype system described in the preceding section has been measured and comparedto the described models given throughout this report.The same Wilkinson splitter that was measured in Chapter 2 with K 2 = 1 for 2.45 GHzwas used, so it has not been measured separately here. The voltage doubler that was usedin the system was designed with the model given in Chapter 4. The same rectifier was usedin the single rectenna described before so this rectifier has not been measured separatelyeither. The patches that were used have been described and measured in Chapter 3. Themodulation system had not been not measured before so it was measured separately first.The data source was set to create a constant 1 kHz block wave and the amplitude andoffset of the block wave was set in such a way that the output wave of the modulator was

a 10 or 11 dB attenuation of the input wave of the modulator. This was done becauseit turned out that the switch caused some losses itself, so no signal could be transferredthrough the modulator without attenuation. However, the extra 10dB attenuation wasmade up for by the power levels of the input RF source and the amplifier. The amplifier’sgain cannot be set and equals 30 dB, so the RF-source is set to create a 0 dBm signal,resulting in the desired output power of the amplifier of roughly 20 dBm. The modulator’sperformance was measured by means of a network analyzer. The measurements showedthat the modulator modulates the signal up to 100 kHz without any problems, so it couldbe used without any problems in our system. Next, the output of the modulator was leddirectly to the demodulator, so the original shape of the block wave should be recovered.

In this way it was perfectly possible to recover the original (dummy) data stream. To makethis subsystem more perceptible the output of the demodulator was led to a crystal earplug, so the data ‘could be heard’. This earplug creates audible waves for input waves withamplitudes as low as 5 mV.

Once all the subsystems were measured and gave results that were satisfactory enoughto use them further, the subsystems were connected as shown in Figure 5.8. First of all,the output of the demodulator was measured with an oscilloscope. This output voltage isshown in Figure 5.10. The series capacitance, as mentioned before, was used to suppressthe DC-term in the output of the demodulator. A filter has been applied to the receivedsignals to suppress 50 Hz noise. This was necessary because the measurement signal pickedup a 50 Hz noise due to the measurement cables and the surrounding instruments. Thesignal that was measured directly at the output of the demodulator also suffered from noiseat higher frequencies, because the signal levels are quite small. However, as can be seenin the figure the signals can be used to recover the original data. The period time of 1 ms(from the 1 kHz input wave) can be observed clearly. When this signal is led to the earplug, that was described before, the modulator’s input frequency can be heard.

Another measurement was performed to measure the unloaded DC voltage for a varyingfrequency. The network analyzer was used to perform a frequency sweep from 2.25 to

70



Figure 5.10: Demodulated signal

2.65 GHz and an oscilloscope was used to to measure the DC voltage over this frequencyrange. We have used a broadband horn antenna as transmit antenna. As shown in Chap-ter 3 we have used a patch antenna with two perpendicular linear polarizations. To use thepresented model to determine the input impedance of the patch antenna both polarizationsmust be excited equally. Since we only had a linearly polarized horn antenna as transmitantenna at our disposal we placed this antenna in such a way that the polarization tiltangle was 45 degrees compared to the horizontal plane. The received wave can be split upinto two linear perpendicular polarized waves, corresponding to the two polarizations of the patch. Each polarization contains 1/

√2 of the received electromagnetic field and half

of the received power. The measurement was set up in such a way that for each frequencya 0 dBm signal reached the patch antennas. Because of the broadband horn antenna thatwas used, this resulted in a source with constant power over the entire frequency range.The results of this measurement are shown in Figure 5.11. In this figure the voltages thatwere determined from the presented models are shown as well. In the next section it willbe explained how these voltages have been obtained and how reliable the modeling of thesystem is.

5.4.2 Verification

The presented models throughout this report have been used to determine the unloadedDC output voltage of the total system. We have assumed that no power is dissipated inthe demodulator and that all the collected power is transferred to the rectifier.Let us first look at the patch antennas. The horn antenna was used to create a fieldcontaining 0 dBm of power at the patch antennas divided equally over both polarizationsof the patch antennas. The input impedance of the patch antennas over the frequencyrange has been determined from the theory which was presented in Chapter 3.Next, the input impedance of the Wilkinson splitter at port 2 and 3 (which are equal

71



Figure 5.11: DC output voltage of the prototype

72



because a K 2 = 1 splitter was used) has been determined over this same frequency range.Here, the input impedance of the voltage doubler with 0 dBm input power is used as

resistive element in the splitter. This simplification is allowed since the voltage doubler’sinput impedance does not drastically change with input power. Further, our system isdesigned assuming 1 mW input power, so we especially want to compare measurementsand model for this power level. Now we know the complex output impedance of the patchantennas and the complex input impedances at the input ports of the splitter we candetermine the accepted power. In [27] the power reflection coefficient for a complex loadand complex source impedance is determined as

|s|2 =

Z L − Z ∗S Z L + Z S

2

, (5.3)

where Z S is the complex source impedance (patch antenna output impedance) and Z L isthe complex load impedance (splitter input impedance).By using this accepted power the DC voltage of the voltage doubler is determined. TheseDC voltages are shown in Figure 5.11.

5.4.3 Discussion

In Figure 5.11 we see the measured DC-voltage of the total system over a frequency range

from 2.25 to 2.65 GHz. The DC-voltage determined by combining the different models isalso shown in this figure. The distance between transmit and receive antennas was chosensuch that the received power was roughly 1mW, using Equation (5.1). This amount of power could not be determined very precisely because this measurement was not performedin an anechoic chamber. We can see that the voltages that are predicted by the modelsagree with the measured values. This implies that we can combine the different models,which results in accurate predictions of the individual outputs, to determine the totalsystem behaviour. If the system is used at the conditions it is designed for (frequency andpower) the output voltage is predicted within 8%.We clearly see two peaks in the figure, corresponding to the resonance frequency of each

polarization at a different frequency. When we increase the angle of the polarization of thetransmit antenna we see an increase in the hight of one peak and we see the other peakdecreasing. This results from the increase of power in one polarization of the patch and adecrease in the perpendicular polarization. When we compare these voltages to the onesobtained in [2] we can see that this system is much more broadband. This is due to theWilkinson splitter, which still accepts power when the system is operated off resonance,and due to the double resonance of the patch antennas.

The rectenna efficiency of the total system has been measured as described before inSection 5.2.1. An efficiency of 25% was observed with a load impedance of 900 Ω at

73



2.4 GHz and an input power of 0 dBm per antenna. At a first glance this efficiency mightseem poor. However, if we would have used a separate antenna for power conversion and a

separate antenna for data purposes we would have achieved an efficiency of 26% using therectenna described in Section 5.2. Our system can be optimized further to give a higherrectenna efficiency.

1. The input impedance of the rectifier can be transformed to match the desired impedanceof the Wilkinson splitter to accept more power at the input ports of the splitter. Asimple technique to achieve this is to use pieces of microstrip line with certain lengths.However, this is a very narrowband solution and more elegant solutions are possible,as described in [28].

2. Further, a Schottky diode with better rectifying capabilities could be used to optimizethe efficiency. We have used a diode with a substrate resistance of 25 Ω, while in [26]it is reported that a Schottky diode was used with a substrate resistance of only 4 Ω.This diode was not available to us, so the HSMS-2852 Schottky diode was used.

3. Using another substrate can also improve the efficiency since the loss tangent of theused FR4 results in substrate losses. An example of a substrate that could improvethe efficiency is Rogers 4003. We have chosen to use FR4 throughout this projectbecause it can be easily processed using simple etching techniques and it is relativelycheap. It is difficult and time consuming to apply a photosensitive layer on standardmicrowave laminate while FR4 is normally produced with the photosensitive layer

already on the material.

5.5 Conclusion

In this chapter we have combined the subsystems described in the preceding chapters, firstto form a rectenna and second to form a system capable of simultaneous power and datatransfer.

First, a single rectenna was designed, measured and compared to our modeling. Theunloaded DC-output voltage for different input powers was predicted well, within a fewpercent, by combining the different models. Further, the load impedance resulting inan optimal DC-power consumption was determined to be around 900 Ω. The rectennaefficiency was measured to be 52%.

Eight of these same rectennas were combined to make a wireless powered wall clock. Theperformance of this clock was measured quantitatively. The rectenna-powered wall clockoperates even on WLAN.

74



The total system to transfer data and power wireless was presented. This system used somenew building blocks, such as the modulator and demodulator, which have been explained.

A carrier was modulated with a basic digital signal. The system was able to transmit, re-ceive and demodulate the modulated carrier, so the original data was recovered. Further,we have been able to simultaneously transmit and receive power. The rectenna efficiency of this system turned out to be 25%. The modeling was able to predict the system behaviour.Recommendations have been given to increase the efficiency of the single rectenna and thetotal system.

75



76



Chapter 6

Conclusions and Recommendations

The modeling of a rectenna system capable of simultaneous wireless power and data transferhas been investigated. The design of such a system has been presented and the modelinghas been compared to measurements on a realized prototype. It has been shown that theseanalytical models predict the behaviour of the system accurately enough to be used asdesign tool when used appropriately.

The analytical models give insight in the behaviour of the individual parts of the system.

As shown throughout this report the design parameters, such as the dimensions, of theindividual parts of the system are very critical with respect to the behaviour of the totalsystem. A small change in the length or width of a design can result in a huge change inbehaviour. This implies that this system is not suitable to be designed with trial-and-errortechniques and these models can save designers lots of work and time since our models aremuch faster than full wave simulations.

The approach that was followed in this project was to use models that were suitable to bescaled. This means that the presented models can be used for a wide range of frequencies,power levels and substrate materials.

A single rectenna was modeled, designed, produced and measured. This rectenna hassmall dimensions and a relatively high efficiency of 52% under the given circumstances.The small dimensions were especially achieved due to the elimination of the impedance-matching circuit. This miniaturization was allowed because the antenna was matcheddirectly to the rectifying circuit. Compared to results from other rectenna designs thisrectenna performs quite well, especially when the low power levels are taken into account.A voltage doubler was used in the rectenna, resulting in an unloaded DC output voltage of around 1.2 V, with the assumed input power of 1 mW. This voltage is high enough to powerintegrated circuits, which makes the rectenna suitable for a wide range of applications.

77



The individual subsystems have been combined to form the total system capable of wirelesspower and data transfer. This system was manufactured and measured. The results

showed that the system is capable of transmitting and recovering the data. Further, itsimultaneously converts RF power to DC power. The DC output voltage was measuredand compared to the predicted values by the combined models. The models are capableof predicting the behaviour of the DC output voltage, within 8% at the frequencies it isdesigned for. The rectenna efficiency of this system has been determined to be 25% for aload of 900 Ω.

Improvements in the system performance can be achieved by matching the input impedanceof the rectifier better to the required impedance of the splitter. Further, a microwavesubstrate with lower losses will also result in an improved system performance.

The Schottky diode that was used throughout this project is not an optimal rectifier diode.Especially the high substrate resistance results in a poor rectifying efficiency. AnotherSchottky diode with a lower substrate resistance will result in a better system performance.To a lesser extent, a diode with a lower junction capacitance can improve the results.

To improve the reliability of the data transfer the amplitude modulation that was used inthe prototype can be replaced by another modulation technique, e.g. frequency modulation,that is less sensitive to noise.

Further, it could be investigated whether an array of antennas before the rectifier is an

option for rectenna purposes. The array of antennas could collect more energy than onesingle antenna making the rectifying process much more efficient.

In this report the design of a system has been presented that, to our knowledge, has notbeen presented in literature before. Here, an initial modeling and design approach wasproposed which can be used for further improvement and development.

78



Bibliography

[1] .

Note: The number(s) following a bibliography item indicate the page(s) where thereference is used in this report.

W.C. Brown. The history of power transmission by radio waves. IEEE Transactions

on Microwave Theory and Techniques, 1984 5

[2] J.A.G. Akkermans. Design of a rectenna for wireless low-power transmission. Tech-nical report, Eindhoven, University of Technology, 2004. 6 , 20 , 47 , 48 , 52, 60 , 64,73

[3] L.I. Parad and R.L. Moynihan. Split-tee power divider. IEEE Trans. MTT, pages91–95, January 1965. 9 , 11

[4] K.C. Gupta, Ramesh Garg and Rakesh Chadha. Computer Aided Design of MicrowaveCircuits. Artech, 1981. 9 , 24

[5] Sadiku. Elements of Electromagnetics. Saunders College Publishing, 2 edition, 1994.11, 27 , 54

[6] E. Hammerstad and Ø. Jensen. Accurate models for microstrip computer-aided design.Microwave Symposium Digest, 80:407–409, May 1980. 11

[7] W.J. Getsinger. Microstrip dispersion model. IEEE Transactions on Microwave The-ory and Techniques, pages 34–39, January 1973. 12

[8] A. van de Capelle. Transmission Line Model for Rectangular Microstrip Antennas.Peter Peregrinus, 1989. 12

[9] David M. Pozar. Microwave Engineering. New York: John Wiley and Sons, 1997. 14

[10] Keith R. Carver and James W. Mink. Microstrip antenna technology. IEEE Trans-actions on Antennas and Propagaion, pages 3–25, January 1981. 21, 32 , 33

79



[24] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery. Numerical Recipesin C. Cambridge: Cambridge University Press, 2nd edition, 1992. 48 , 50

[25] Agilent Thechnologies. Surface Mount Zero Bias Schottky Detector Diodes, HSMS-2850 Series, 1999. 51, 53

[26] J.H. Suh and K. Chang. A high-efficiency dual frequency rectenna for 2.45 and 5.8 GHz wireless power transmission. IEEE Transactions on Microwave Theory andTechniques, pages 1784–1789, July 2002. 64, 74

[27] Pavel V. Nikitin, K. V. Seshagiri Rao, Sander F. Lam, Vijay Pillai, Rene Martinez and Harley Heinrich. Power Reflection Coefficient Analysis for Complex Impedancesin RFID Tag Design. IEEE Transactions on Microwave Theory and Techniques, pages2721–2725, September 2005. 73

[28] Agilent Technologies. Impedance matching techniques for mixers and diodes. Appli-cation note, 1999. 74

Wireless Clock

Documents

Transcript of Wireless Clock