Low cost precision reflectometer

Department of Science and Technology Institutionen för teknik och naturvetenskap

Linköpings Universitet Linköpings Universitet

Examensarbete

LITH-ITN-ED-EX--06/023--SE

Low cost precision

reflectometer

Fredrik Busck

LITH-ITN-ED-EX--06/023--SE

Low cost precision

reflectometer

Examensarbete utfört i Elektronikdesign

vid Linköpings Tekniska Högskola, Campus

Norrköping

Fredrik Busck

Handledare Bo Franzon

Examinator Shaofang Gong

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URL för elektronisk version

Avdelning, Institution

Division, Department

Institutionen för teknik och naturvetenskap Department of Science and Technology

2006-06-05

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LITH-ITN-ED-EX--06/023--SE

Low cost precision reflectometer

Fredrik Busck

This Master Thesis contains an evaluation of the six-port reflectometer (SPR), an alternative method for measuring reflection. This technique allows the measurement of complex reflection using only scalar detectors. Obtaining both amplitude and phase information gives the possibility to make corrections for systematic errors such as feeder cable loss and directivity error.

The report contains a literature study including the six-port reflectometer technique as well as a

historical overview of reflection measurement techniques. Further more it contains simulation results of parts of the design as well as of the complete system. The calibration algorithm of the SPR is presented step by step with an improvement made in order to reduce the number of calculations. Measurement results of the reflection measurements are presented as a comparison to the result obtained from a network analyzer.

The simulations showed high accuracy when simulating variations in the return loss of the loads as well as variations of the input signal frequency. The simulations also gave some indications on how to affect the accuracy of the reflection measurements.

Measurements showed high accuracy when measuring unknown loads with changing return loss. Variations of the input power also yielded a good result. Measurements performed with the input signal frequency variated failed to show the same high accuracy as in the simulations. However, there are some improvements suggested to increase the accuracy in a presumptive product.

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Master Thesis Project

Low Cost Precision Reflectometer

Fredrik Busck

University of Linköping, ITN Campus Norrköping

Supervisor: Bo Franzon

Low Cost Precision Reflectometer

Abstract

This Master Thesis contains an evaluation of the six-port reflectometer (SPR), an al-ternative method for measuring reflection. This technique allows the measurement of complex reflection using only scalar detectors. Obtaining both amplitude and phase in-formation gives the possibility to make corrections for systematic errors such as feeder cable loss and directivity error.

The report contains a literature study including the six-port reflectometer technique as well as a historical overview of reflection measurement techniques. Further more it contains simulation results of parts of the design as well as of the complete system. The calibration algorithm of the SPR is presented step by step with an improvement made in order to reduce the number of calculations. Measurement results of the reflec-tion measurements are presented as a comparison to the result obtained from a network analyzer.

The simulations showed high accuracy when simulating variations in the return loss of the loads as well as variations of the input signal frequency. The simulations also gave some indications on how to affect the accuracy of the reflection measurements.

Measurements showed high accuracy when measuring unknown loads with changing return loss. Variations of the input power also yielded a good result. Measurements performed with the input signal frequency variated failed to show the same high ac-curacy as in the simulations. However, there are some improvements suggested to increase the accuracy in a presumptive product.

Low Cost Precision Reflectometer

Sammanfattning

Det här examensarbetet innehåller en utvärdering av sexports reflektometern (SPR), en alternativ metod för att mäta reflektion. Den här tekniken tillåter mätning av komplex reflektion med hjälp av enbart skalära detektorer. Erhållandet av både amplitud och fas ger möjligheten att kompensera för systematiska fel som kabelförluster och direk-tivitetsfel.

Rapporten innehåller en litteraturstudie som omfattar sexports reflektometertekniken samt en historisk översikt av olika tekniker för att mäta reflektion. Rapporten in-nehåller även simuleringsresultat från enskilda delar av designen samt från hela sys-temet. Kalibreringsalgoritmen presenteras steg för steg tillsammans med en förbät-tring som gjorts för att minska antalet beräkningar. Mätresultaten från reflektionsmät-ningarna presenteras som en jämförelse med resultaten från en nätverksanalysator. Simuleringarna visade hög noggrannhet under variation av lasternas reflektionsförlus-ter och insignalens frekvens. Simuleringarna gav också en indikation på hur man kan påverka noggrannheten i reflektionsmätningarna.

Mätningarna visade hög noggrannhet vid variation av lasternas reflektionsförluster. Variationer i insignalens effekt gav också goda resultat. Mätningar genomförda med variation i insignalens frekvens resulterade inte i samma höga noggrannhet som under simuleringarna. Det finns emellertid vissa förslag på förbättringar för att öka nog-grannheten i en potentiell framtida produkt.

Low Cost Precision Reflectometer

Acknowledgments

I would like to take the opportunity to thank Bo Franzon, my supervisor at Powerwave Technologies Inc., without whom this master thesis would not have been possible. I would also like to thank the R&AD department and the RF department at Powerwave Technologies. The following people have contributed with knowledge and experience that allowed this master thesis to progress; Leif Ottesen, Torbjörn Olsson, Jarmo Mäki-nen, Mikael Ahlberg, Anders Jansson, Kim Salovaara, Joakim Forsberg, Amir Emdadi, Thomas Baardseth, Martin Szybanow, Claes Lewin and Paul Pettersson. In the same way I would like to thank Shaofang Gong, my examiner at the University of Linköping.

Low Cost Precision Reflectometer

Abbreviations

ANA Automatic Network Analyzer BALUN Balanced- Unbalanced transformer BTS Base Transiever Station

CAD Computer Aided Design CIN Current Injector

CW Continous Wave DAQ Data Acquisition unit DUT Device Under Test

GPIB General Purpose Interface Bus

GSM Global System for Mobile communications IEEE Institute of Electrical and Electronics Engineers PC Personal Computer

PCB Printed Circuit Board RF Radio Frequency RL Return Loss

SPR Six-Port Reflectometer VNA Vector Network Analyzer VSWR Voltage Standing Wave Ratio TEM Transverse Electromagnetic Mode TMA Tower Mounted Amplifier

Contents

Abstract i Sammanfattning ii Acknowledgments iii Abbreviations iv 1 Introduction 1 1.1 Background . . . 1 1.2 Purpose . . . 2 1.3 Method . . . 2 1.4 Notation . . . 2 2 Theory 4 2.1 Historical background on reflection measurements . . . 4

2.2 Six-port reflectometer . . . 6 2.2.1 General description . . . 6 2.2.1.1 Six-port realizations . . . 7 2.2.2 Calibration . . . 9 2.2.2.1 Six-port-to-four-port reduction . . . 10 2.2.2.2 Four-port calibration . . . 14 2.2.2.3 Flatness detection . . . 15 2.2.3 Measuring reflection . . . 16 2.3 Coupler theory . . . 16 2.3.1 Coupler basics . . . 16

2.3.2 Microstrip directional coupler . . . 18

2.3.3 Directivity compensation . . . 20 3 Simulations 21 3.1 Coupler simulations . . . 21 3.1.1 RF simulations . . . 21 3.1.2 EM simulations . . . 22 3.2 Q-point simulations . . . 23 3.3 System simulations . . . 25

Low Cost Precision Reflectometer CONTENTS 4 Measurements 28 4.1 Coupler measurements . . . 28 4.1.1 Measurement description . . . 28 4.1.2 Directivity measurements . . . 29 4.2 Reflectometer measurements . . . 31 4.2.1 Measurement description . . . 31 4.2.2 Measurement setup . . . 32 4.2.3 Detector calibration . . . 33 4.2.4 SPR Calibration . . . 34

4.2.5 Reflection measurement of unknown loads . . . 34

4.2.5.1 Power variation . . . 35

4.2.5.2 Return loss variation . . . 35

4.2.5.3 Temperature variation . . . 36 4.2.5.4 Frequency variation . . . 37 4.2.6 Measurement summary . . . 38 5 Conclusions 40 6 Future work 42 Bibliography 43

A Test board layout 45

List of Figures

1.1 Block diagram showing the principal configuration of a six-port . . . 3

2.1 General test system for reflection measurements . . . 4

2.2 Determination of Γ at the intersection of the three circles . . . 7

2.3 Configuration of the switched SPR . . . 9

2.4 Illustration of the w-plane with the circle formed by loads with con-stant |Γ| . . . 11

2.5 Basic configuration of the ports in a backward-wave directional coupler 16 2.6 Explanation of a backward-wave directional coupler . . . 17

2.7 Cross-section of a microstrip coupler . . . 18

2.8 Capacitor compensated coupler . . . 20

3.1 Simulated coupling and isolation of the microstrip coupler . . . 22

3.2 (a) EM-simulation of the directivity with input parameters from circuit simulator (b) EM-simulation of the directivity with adjusted interdigi-tal capacitors (c) The final coupler design . . . 23

3.3 Model of the SPR from which the q-points were calculated . . . 23

3.4 Signal flow charts used as a basis for the q-point simulations . . . 24

3.5 The simulated placement of the q-points at 1805-1880MHz . . . 25

3.6 Simulated accuracy of calibration when changing the coupling factor . 26 3.7 Simulated accuracy of the reflection measurements when changing the return loss level of the loads used in calibration, (a),(b), and the ampli-tude of the q-points, (c) . . . 27

3.8 Simulated accuracy of calibration when changing the frequency of the input . . . 27

4.1 Test boards before assembly . . . 28

4.2 Measured coupling and isolation on the board 1.16/1/1 with solder mask 29 4.3 Measured coupling and isolation on the board 1.16/1/1 with the solder mask removed . . . 30

4.4 Measured coupling and isolation with the solder mask removed . . . . 30

4.5 Assembled test board . . . 31

4.6 Block diagram of the measurement setup . . . 32

4.7 The measurement setup on the lab bench . . . 33

Low Cost Precision Reflectometer LIST OF FIGURES 4.9 Reflection measurement of the loads used during calibration . . . 34 4.10 Reflection measurements with the input power variated . . . 35 4.11 Reflection measurements with the return loss of the measured load

variated . . . 36 4.12 Reflection measurements with the temperature variated . . . 37 4.13 Reflection measurements with the frequency of the input signal variated 38 A.1 Test board layout . . . 45

List of Tables

4.1 Overview of the directivity measurements made on the test boards . . 31 4.2 Overview of the accuracy of the reflection measurements . . . 39

Chapter 1

Introduction

1.1

Background

Powerwave Technologies Inc. is a telecom company with head quarters in the USA that develops and manufactures a wide range of products for the telecommunication industry. The main products are antenna systems, base station components and cover-age systems.

In the telecom industry reflectometers are used in cellular base stations. The purpose of the reflectometer is to supervise the antenna match to the base station in order to achieve optimal coverage and capacity. A mismatch somewhere in the antenna line will trig an alarm to the operator. The reflectometer function is often integrated in the base station front-end (duplex filter), current injector (bias tee) or tower top amplifier. Today this function is usually solved with traditional scalar reflectometers using am-plitude detectors and directional couplers. Most often the reflectometer is placed at the ground level hence the antenna reflection is masked by feeder cable loss and reflec-tion from components in the antenna line. Scalar reflectometers of this type require directional couplers or similar, having directivity performance that most often cannot be reached without manual tuning of RF hardware. Software correction is not pos-sible since error terms are inherently vector quantities and only scalar information is retrieved from the reflection measurements. If you can gather phase and amplitude in-formation it is possible to make software correction of systematic errors, such as feeder cable loss and directivity error. This would improve the accuracy in the supervision of the antenna line.

At Powerwave the reflectometer is included in the current injector (CIN) or the base station front-end. It has the ability to measure both input and reflected RF-power by switching the forward and backward signals into a single power detector. The design involves high requirements on the couplers used and therefore they sometimes have to be manually tuned during production. Due to increasing demands from the customers a new technique allowing software controlled vector correction would be of great in-terest for Powerwave.

Low Cost Precision Reflectometer

In instrument grade equipment superheterodyne vector receivers are used in conjunc-tion with vector error correcconjunc-tion schemes to correct for limited RF hardware perfor-mance. This solution is usually not possible in the described equipment due to cost constraints. An alternative is the so called six-port reflectometer where the use of a linear six-port network and four amplitude detectors in combination makes it possible to gain vector information from scalar measurements.

1.2

Purpose

The purpose of this master thesis is to evaluate the six-port reflectometer technique with respect to the accuracy of the reflection measurements. This report will then be used as a part of the information which Powerwave will use to evaluate the SPR against other reflectometer concepts.

1.3

Method

In order to understand the background and the problems encountered with the existing reflection measurement technique as well as the six-port reflectometer technique, a literature study was performed. The literature study made upon the existing technique was based on product documents provided by Powerwave along with meetings with the engineers at the company. The literature study concerning the six-port reflectometer technique mainly consisted of articles from the IEEE library.

In parallel with the literature study, simulations of separate parts of the SPR were performed. These simulations constituted the base of the design of the test boards. Once the simulations were completed with satisfactory results a circuit diagram of the test board was drawn. The test board layout was then carried out by the printed circuit board (PCB) design engineers at the company. The test boards were manufactured at an external PCB manufacturer.

The measurements were performed at the lab bench using a vector network analyzer (VNA) (Agilent 8753ES) as reference to the results from the SPR. All data processing was performed using MATLAB R.

1.4

Notation

This section is thought to be an overview of the notation used through out the report. Waves traveling towards the device under consideration (incident waves) will be de-noted by an a and waves traveling outward (transmitted and reflected waves) will be denoted b. The reflection coefficient will be denoted Γ and D1 to D4 are the detectors

used in the six-port, Figure 1.1. 2

Chapter 1. Introduction There is a point in emphasizing that in this report the acronym ANA (Automatic Net-work Analyzer) will be used as a general name for all automatic netNet-work analyzers independent on whether they use vector detectors or scalar detectors. SPR as well as VNA are subsets of ANA.

Two different accuracy measures for the reflection coefficient will be used in the re-port. First the absolute difference between the complex reflection coefficients, called the magnitude of the vector error, which will be denoted by a δ, equation (1.1). Sec-ond the absolute difference in amplitude between the reflection coefficients, called the magnitude error, which will be denoted by an ε, equation (1.2).

δ = |ΓSPR− ΓV NA| (1.1)

ε = ||ΓSPR| − |ΓV NA|| (1.2)

Chapter 2

Theory

2.1

Historical background on reflection measurements

Microwave measurements are associated with difficulties in probing at the measure-ment port of interest. It is generally considered impossible to probe or sample the fields without significantly altering them. Given an arbitrarily chosen measurement port the complex reflection is defined as the incident wave divided by the reflected wave, Γ2= a2/b2.

Figure 2.1: General test system for reflection measurements

Since it is not possible to probe directly at the measurement port of interest, measure-ments have to be made at remote positions as shown in Figure 2.1. If the intervening structure is assumed to be linear, but otherwise arbitrary, it has been shown [1] that the probe response at position 3 and 4 is given by the equations.

b₃ = A1a2+ A2b2

b₄ = A3a2+ A4b2 (2.1)

Where A1...A4are complex constants determined by the intervening geometry.

Chapter 2. Theory It is straight forward to solve for A1...A4given that the detectors provide phase as well

as amplitude response. Since it is hardware expensive to detect phase information at microwave frequencies many of the earlier applications were built around detectors which only provided amplitude response, but instead relied on hardware to fulfill the condition A2, A3' 0. When retrieving information containing only amplitude

informa-tion it is impossible to compensate with software for known imperfecinforma-tions and errors. The slotted line was the first known technique for measuring reflection. Ideally it is constructed such that from equation (2.1) |A1| = |A2| while the phase is adjusted by the

probe position. |Γ| is a function of the ratio of the maximum to minimum response. It is also possible to obtain the phase of Γ by measuring the distance from the extrema to the reference plane.

The reflectometer requires two detectors and the powers are given by the equations p₃ = |b₃|2= |A₁a₂+ A₂b₂|2

p₄ = |A3a2+ A4b2|2 (2.2)

Ideally the reflectometer is constructed such that A2= A3= 0. Knowing |A1/A4|2it is

possible to measure the reflection coefficient Γ.

All of these methods, however, suffer from the same problem. The accuracy obtained is determined upon how well certain design objectives for a particular item of hardware are realized. The search of ways to circumvent this problem gave birth to the tuned reflectometers. The tuned reflectometers permitted the operator to adjust the system parameters such that the ideal response was more accurately approximated. During a limited time of perhaps a decade this became a highly developed art. These schemes later proved to be both frequency sensitive and time consuming.

As to meet the requirements of increased bandwidth the VNA emerged. Meanwhile, a new measurement strategy was introduced. Instead of trying to construct an ideal microwave circuit, e.g. the reflectometer, its imperfections were explicitly recognized, accounted for and finally eliminated from the results. The VNA is based on equations on the form (2.1). As a preliminary step the VNA is calibrated by means of known standards. This leads to obtaining the complex parameters A1...A4. Following this the

system may be solved for a2, b2in terms of b3, b4. Calibrating the VNA leads to

deter-mination of the complex constants A1...A4, or actually only the ratio of three of them

to the fourth. Comparing to earlier systems where only one or two scalar parameters were required and the other two were hardware tuned to zero, the calibration process of the VNA is more complex.The existing VNA’s are a complex piece of equipment. Much of this complexity is due to the requirement imposed by equations (2.1), namely that the detectors must provide phase as well as amplitude.

With this background the SPR was developed. The SPR provides an alternative method of implementing the ANA. In common with previously described designs, hardware

Low Cost Precision Reflectometer

imperfections are taken care of by software compensation. What makes the difference between the SPR and the VNA is that the SPR is based on equations on form (2.2) rather than (2.1). Thus, the requirement for phase information is avoided and simple amplitude or power detectors may be used instead of complex heterodyne schemes.

2.2

Six-port reflectometer

2.2.1

General description

The six-port was originally presented by C. A. Hoer [2] in 1972. Since then a lot of publications has been made considering the general theory and calibration methods [3], [4] and measurement applications [5], [6].

The SPR is a linear passive device that permits the measurement of the complex ra-tio between two signals, Γ. The measurement is performed using four power detec-tors followed by mathematical treatment which makes it possible to retrieve not only amplitude information but also phase information from the signals. To describe the six-port there are a certain number of constants which are determined by a calibration procedure. These constants are then used to find the complex reflection ratio. The amplitude ratio and the phase difference detection can be used to direct modulation of phase modulated signals as well as to measuring the reflection constants of network ports [7]. This is vital in the supervision of the antenna line since this information can be used to make software compensation for the feeder-cable loss and known reflection from components in the antenna line.

A major concern when designing the SPR is the placement of the q-points [5]. The q-points represent the points in the Γ-plane where the detector output is zero. In order to determine complex reflection there is actually only a need of two detectors and con-sequently two q-points. In the reflection plane each q-point together with the reflection of interest represent a circle of possible values for Γ with the radius |Γ − qi|, i = 1, 2, 3.

Using two detectors the unknown reflection is then determined as the intersection of the two circles. The two circles, however, intersect in a pair of points, hence a choice have to be made upon which point is the correct solution. If only passive loads are assumed (|Γ| ≤ 1) one of the intersection points may appear outside the unit circle and may therefore be rejected as a false solution. However, as mentioned before the SPR possess not just two but four detectors. Introducing a third detector will give another q-point creating a new circle of possible values for Γ. This circle will intersect with the two others in one point only and the issue of a choice between two points is solved, Figure 2.2.

The fourth detector is used as a reference detector, ideally but not required this detec-tor is directly proportional to the input signal. In terms of equation (2.2), giving the reference detector number 4 this means that A3= 0. The purpose of this detector is to

compensate for power instabilities of the power source. 6

Chapter 2. Theory

Figure 2.2: Determination of Γ at the intersection of the three circles

This far it has only been established that by using four scalar detectors one can per-form measurements of the complex reflection coefficient of a load. Nothing has been mentioned about the placement of the q-points. One possible placement is shown in Figure 2.2. Another interesting point of view is to place one of the q-points at the ori-gin of the unit circle. The detector corresponding to that q-point will then be a direct measurement of the magnitude of the reflection coefficient which seems very attrac-tive. However, the problems in designing the circuit such that one q-point will appear at the origin seems too large and the deviation from the actual origin will make possi-ble advantages vanish. Disposing the idea of placing one q-point at the circle centre, symmetry considerations make it seem quite obvious that the optimum placement is at the vertices of an equilateral triangle whose center is at the origin. In other words |q1| = |q2| = |q3| and their arguments differ by ±120◦. Considering the choice of

mag-nitude of the q-points, it can be shown that the optimum values lie in the region around |qi| = 0.5 or |qi| = 1.5 [5]. Too large values, e.g. |qi| = 10, which at first glance can

seem interesting, tend to be sensitive to measurement errors. A one per cent error in the measurements will translate into a nominal uncertainty of 10 per cent in the real and imaginary parts of Γ [5]. Similar problems can be experienced if the magnitude becomes too small. Given that the reflection coefficient is determined from the dis-tances from the q-points to the point of reflection it is evident that the system becomes ill-conditioned if those distances are large compared to the distances between the q-points. The optimal choice should be determined based on the application where it is used, but should lie in the region close to 0.5 or 1.5.

2.2.1.1 Six-port realizations

The realization of a circuit comprising the above mentioned design criterias has of course been described in previous literature. The most common configuration is the

Low Cost Precision Reflectometer

one originally described by Engen [8]. It uses one directional coupler to couple the signal from the main line along with a network of quadrature hybrids and 180◦hybrids to determine the q-point placement. Using this technique the differences in the argu-ment of the q-points achieved were 135◦, 90◦, 135◦. The magnitude of the q-points was 2 for two of the q-points and√2 for the third.

Other techniques of implementing the circuit has been described by Collier and El-Deeb [9]. This technique uses a microstrip three-line system to configure the q-points. It has the advantages of a small physical size. However, it seems difficult to achieve a good q-point placement using this technique.

A technique using a five port ring together with a directional coupler has been de-scribed by de Sousa, Huyart and Freire [7]. The design used seems to produce well distributed q-points. It uses four diode based power detectors linearized in the region [−35, +5] dBm.

Common in the described techniques is the use of four detectors. Another technique where the number of detectors are reduced to two has been investigated at Warsaw Polytechnics [10]. The idea is that by using switched amplitude and phase shifters within the structure all combinations of the incident and reflected wave necessary for optimally distributed q-points can be obtained sequentially using only one detector. One more detector is, however, used as a reference detector to the input signal. This is the configuration used in this Master Thesis and will therefore be described more extensively than previous ones. As the number of detectors are decreased from four to two the physical dimensions are naturally also reduced. Since the detectors are the most expensive components used in a microstrip SPR the cost is also lower. The SPR configuration in total using this technique consist of

• Two amplitude detectors

• Two microstrip directional couplers

• One phase shifter with three switched states

and is shown in Figure 2.3. As can be seen the forward signal is coupled at the first directional coupler down to the reference detector, D1. At the second detector, D2, the

forward signal is combined with a phase shifted reflection of the reflected signal. The phase shift is depending on the state of the switch.

Chapter 2. Theory

Figure 2.3: Configuration of the switched SPR

2.2.2

Calibration

The key factor in achieving good measurement results using SPR-techniques is the accurate calibration of the six-port network. The purpose of the calibration is to deter-mine a number of constants describing the six-port circuit. In a standard VNA there are six real constants to be determined and their relation is linear. The six-port re-flectometer on the other hand needs eleven constants in a non-linear relation to fully describe the circuit. This leads to a more complex calibration algorithm to find those eleven constants. A lot of work has previously been performed to find an accurate and effective algorithm where the main focus is on minimizing the number of loads and the computational effort.

Engen developed a calibration algorithm using a minimum of five loads in the first stage and then three standards in the second stage [11]. He showed that the six-port network could be reduced to an imaginary four-port network from which vector de-tection was achieved. This reduction is usually called six-port-to-four-port reduction and is used in many of the now existing calibration methods. This reduction allows the user to use existing and well documented techniques for calibrating the imaginary four-port.

Ghannouchi and Bosisio developed an iterative model which uses four standards [12]. However, there are constraints on the magnitudes and phases for these standards. The algorithm converges rapidly and the relatively few loads used decreases the computa-tional time.

Qian also developed an iterative method using four standards [13]. This method im-plies no constraints on the standards and it converges rapidly as the one developed by Ghannouchi and Bosisio. The method does not need a specific port for normalizing the input power and is easy to calibrate.

The calibration method chosen is a method developed by Wiedmann, Huyart, Bergeault and Jallet [4]. The method is based on the theory described by Engen [11] and further

Low Cost Precision Reflectometer

developed to obtain a method which is robust also in ill-conditioned configurations. It uses five unknown loads for the six-port-to-four-port reduction and then three standards for the calibration of the virtual four-port. The five loads used for the six-port-to-four-port reduction all need to have approximately the same magnitude and a decent angular spacing between them.

2.2.2.1 Six-port-to-four-port reduction

The first step of the calibration is as mentioned above the six-port-to-four-port reduc-tion. With this reduction, three values P1, P2 and P3 of RF power ratios are related to

the complex reflection coefficient w = u + j · v at the input of the imaginary four-port according to

P₁ = |w|2 (2.3) ZP₂ = |w − w1|2 (2.4)

RP₃ = |w − w2|2 (2.5)

Where w1and the power coefficients Z and R are scalars and w2= u2+ j · v2 is

com-plex. The Pi= _pp₄i denote the power measurements p1 to p3, normalized with respect

to p4measured at the reference port 4 of the SPR. The five reduction parameters to be

determined are the values of the real positive variables Z, R and w1 and the complex

variable w2.

The starting point of the six-port-to-four-port reduction is the determination of the con-stants Z and R. Figure 2.4 shows an example of the w plane with the circle formed by loads with a constant absolute value of Γ. In order to obtain the two first constants, Z and R one must first determine the minima Pi minand the maxima Pi maxcorresponding

to the loads connected at the measurement port. Using the law of cosine one can write the following equations for points lying on the circle.

P1 = d02+ r2− 2d0rcos α (2.6)

ZP₂ = d₁2+ r2− 2d1rcos(α − ϕ1) (2.7)

RP₃ = d₂2+ r2− 2d2rcos(α − ϕ2) (2.8)

Where d0, d1and d2are the distances between the circle center and the origin, w1and

w₂ respectively. The radius of the circle is denoted by r, α is the angle between the origin and a point on the circle with respect to to the circle center, and ϕ1and ϕ2are the

angles between the origin and w1and w2with respect to the circle center, respectively.

As shown by Engen [11] it is possible to eliminate α from (2.6) to (2.8), forming three equations of ellipses of the form

X₁P_i2+ 2X2PiPk+ X3Pk2+ 2X4Pi+ 2X5Pk+ 1 = 0 (2.9)

i, k ∈ 1, 2, 3; i 6= k

These equations are linear in their five parameters X1to X5. These parameters are

ob-tained by connecting five different loads with constant absolute value of Γ but different 10

Chapter 2. Theory

Figure 2.4: Illustration of the w-plane with the circle formed by loads with constant |Γ|

phases and solving the corresponding system of linear equations. To obtain more ac-curate results in the presence of noise, more than five loads may be used to find a least-squares solution of the system.

Once X1to X5have been found the extremas of Pimay be calculated by first taking the

derivative of the equation (2.9) with respect to Pktreating Pias a constant.

δPi

δPk

= 2X2Pi+ 2X3Pk+ 2X5 (2.10)

Now by setting the derivative equal to zero it is possible to obtain an expression of Pk

as a function of Pi. Inserting this expression for Pkin (2.9) yields the equation

(X1X3− X22)Pi2+ 2(X3X4− X2X5)Pi+ X3− X52= 0 (2.11)

From this equation an expression for the extremas of Pi containing only the known

constants X1. . . X5can be obtained as follows.

P_iext = 1 X1X3− X₂2 (X2X5− X3X4 ± q X₃(X1X₅2− X1X3+ X₂2− 2X2X4X5+ X3X₄2)) (2.12)

This method tends to become inaccurate for SPR configurations where the correspond-ing ellipse is relatively flat since there is always a certain amount of noise in the power measurements. The situation with a flat ellipse is the one most sensitive to noise, small

Low Cost Precision Reflectometer

inaccuracies in the power measurements can lead to relatively large errors in the Piext

calculated from (2.12). One solution to this problem is to obtain a larger number of estimates for the minimum and maximum of Piand retain the median as the final

so-lution. It can be shown that using the trigonometrical relation in equation (2.13) every linear combination of Pk and Pl will be of the same general form as in (2.6) to (2.7),

namely K1+ K2cos(α − ϕ).

k₁cos(α − θ1) + k2cos(α − θ2) = k cos(α − θ) (2.13)

with

k = q

k₁2+ k2₂+ 2k1k2cos(θ2− θ1) (2.14)

θ = arctank1sin θ1+ k2sin θ2 k₁cos θ1+ k2cos θ2

(2.15) Here K1, K2 and ϕ are functions of r, d0, d1, d2, ϕ1, ϕ2, Z, R and the coefficients of Pk

and Pl in the linear combination, but are independent of α. Each of these linear

combi-nations will form an ellipse together with Pi. Therefore, by substituting various linear

combinations of Pkand Pl(for instance Pk+ Pl, Pk− Pl, Pk+ 2Pletc.) for Pkin (2.9) and

applying (2.12) to the solutions the required extra estimates of the minima and maxima of Pican be obtained.

Another way of avoiding these inaccuracies is described in section 2.2.2.3. Indepen-dent on which method used the solution becomes inaccurate if both ellipses in a pair are flat. However, in cases where the q-points are placed in a satisfactory way the situ-ation with two flat ellipses will not occur.

After the minima and maxima of Pi have been found Z and R may be calculated by

noting that from (2.6) to (2.8) follows that √

P_1max±√P_1min = √ZP_2max±√ZP_2min = pRP_3max±pRP_3min = 2r (2.16) Which leads to Z= √ P_1max±√P_1min √ P_2max±√P_2min 2 (2.17) R= √ P1max± √ P1min √ P3max± √ P3min 2 (2.18) In these equations the minus signs have to be used if di> r for the corresponding di

in (2.6) to (2.8), i.e. if the origin, w1 and w2, respectively, are positioned outside the

circle formed by the loads with constant |Γ|. If the SPR is configured in a way so that the q-points have an absolute value greater than one and only passive loads are used 12

Chapter 2. Theory for calibration, as in most cases, the minus sign will be used.

The second step of the six-port-to-four-port reduction is to determine the constants A, B and C. This is done in a similar way as the determination of Z and R. As mentioned in the section above it would be an advantage to obtain several estimates of these constants to avoid ill-conditioned configurations. This can be achieved if it is possible to find a way to calculate them from the minima and maxima of linear combinations of P1to P3. Fortunately there exists a simple solution to this problem. Using equations

(2.6) to (2.8) and (2.13) to (2.15) as well as the law of cosine, it is possible to show that the following relations hold for the three newly defined quantities QA, QBand QC

QA= RP3− ZP2 = d22− d12− 2r √ Acos(α − ϕA) (2.19) ϕA = arctan d₂sin ϕ2− d1sin ϕ1 d2cos ϕ2− d1cos ϕ1 (2.20) Q_B= P1− RP3 = d20− d22− 2r √ Bcos(α − ϕB) (2.21) ϕB = arctan −d2sin ϕ2 d₀− d2cos ϕ2 (2.22) QC= ZP2− P1 = d21− d02− 2r √ Ccos(α − ϕC) (2.23) ϕC = arctan d1sin ϕ1 d₁cos ϕ1− d0 (2.24) Where A = |w₁− w₂|2 (2.25) B = |w2|2 (2.26) C = |w1|2 (2.27)

Using the same methodology as used to derive the ellipse equations on the form (2.9) one can derive a similar expression including the variables QA, QB and QC.

Y1Q2i + 2Y2QiQk+Yk2+ 2Y4Qi+ 2Y5Qk+ 1 = 0 (2.28)

The extremas of QA, QB and QC can then be determined in a similar way as previously

described for the extremas of P1to P3.

Once the minima and maxima of QA, QB and QC have been found A, B and C can be

calculated as follows A =  QAmax− QAmin 4r 2 (2.29) B =  QBmax− QBmin 4r 2 (2.30) C =  QCmax− QCmin 4r 2 (2.31)

Low Cost Precision Reflectometer Where r follows from (2.16) as

r= √

P_1max±√P_1min

2 (2.32)

At this point the initial values of the five reduction parameters are determined. Engen showed in his article [11] that the variable w can be eliminated from (2.3) to (2.5) yielding the nonlinear constraint equation

AP₁2 + BZ2P₂2+CR2P₃2+ (C − A − B)ZP1P2

+ (B −C − A)RP1P3+ (A − B −C)ZRP2P3

+ A(A − B −C)P1+ B(B −C − A)ZP2

+ C(C − A − B)RP3+ ABC = 0 (2.33)

Equation (2.33) describes the dependencies between the power ratios with the help of the reduction parameters and must be fulfilled for any value of w. An optimization of the reduction parameters may now be performed using equation (2.33).

From the optimized values of the reduction parameters it is possible to determine the complex reflection coefficient w by first determining the positive value of the scalar w1, and the complex value of w2according to

w1 = √ C (2.34) u₂ = B+C − A 2w1 (2.35) v₂ = ± q B− u2 2 (2.36) w₂ = u2+ j · v2 (2.37)

The sign in (2.36) cannot be found without some known loads. Now the value w = u+ j · v of an unknown load is given by

u = P1− ZP2+ w 2 1 2w1 (2.38) v = P1− RP3+ u 2 2+ v22− 2uu2 2w1 (2.39) w = u + j · v (2.40) This step completes the six-port-to-four-port reduction and now the five first of the total eleven calibration constants are determined.

2.2.2.2 Four-port calibration

As previously mentioned one of the great benefits of using the six-port-to-four-port re-duction is that the remaining virtual four-port can be calibrated by well known means. 14

Chapter 2. Theory This part of the calibration is equivalent to the calibration of a one port network ana-lyzer. Calibrating the four-port means determining the real and imaginary parts of the constants a, b and c which together describe the imaginary four-port.

Γ = b− w

cw− a (2.41)

This is the final step and completes the SPR calibration. The complex constants a, b and c may be determined by using three known loads and solving the resulting system of linear equations.

ΓS· a + b − ΓS· wS· c = wS (2.42)

ΓO· a + b − ΓO· wO· c = wO (2.43)

ΓL· a + b − ΓL· wL· c = wL (2.44)

Where ΓS,ΓOand ΓL correspond to the known reflections of the standards Short, Open

and Load. The variables wS, wOand wLcorrespond to the value obtained from equation

(2.40) with the respective standard connected at the measurement port.

The sign in (2.36) may be determined by comparing the values of the reflection for an approximately known load obtained after full calibration. Another possible way of determining the sign is to verify that the loads used to find the initial estimates of the six-port-to-four-port reduction all have the same absolute value and rotate in the correct sense. This method can also be used to determine the sign in (2.16) to (2.18) if necessary.

2.2.2.3 Flatness detection

As described by Wiedmann one way to compensate for ill-conditioned configurations is to create a number of linear combinations of Piand Pkand from those obtain a larger

number of estimates for the extremas of Pi. The idea is then to retain the median of

those estimates as the final solution.

When implementing this algorithm it seemed like a lot of unnecessary calculations were performed. Therefore a new algorithm detecting the most accurate solution for the extremas of Piwas developed. This new algorithm uses the geometry of the ellipses

to choose the best solution. The long and short axes of the six ellipses corresponding to the equations (2.9) are calculated as follows.

d₁ = v u u u u t 2(X1X₅2+ X3X₄2+ X₂2+ 2X2X4X5− X1X3) (X₂2− X1X3) (X3− X1) r 1 + 4X22 (X1−X3)2 − (X3+ X1) ! (2.45) d₂ = v u u u u t 2(X1X₅2+ X3X42+ X22+ 2X2X4X5− X1X3) (X₂2− X1X3) (X1− X3) r 1 + 4X22 (X1−X3)2 − (X3+ X1) ! (2.46)

Low Cost Precision Reflectometer

Then by taking the quotient of the long- and short axes a measure of the ellipses flatness is obtained. d1 d₂ = (X1+ X3) + (X1− X3) r 1 + 4X22 (X1−X3)2 (X1+ X3) − (X1− X3) r 1 + 4X22 (X1−X3)2 (2.47) Comparing each pair of ellipses obtained from equation (2.9) it is now possible to choose the ellipses giving the most accurate estimate of the extremas of Pi (i.e. the

ellipses where the ratio d1/d2is closest to unity).

2.2.3

Measuring reflection

After the lengthy calibration process is completed it is now easy to perform the reflec-tion measurements. As shown in the secreflec-tion 2.2.2.2, w of an unknown load can be calculated as in (2.38) to (2.40). The complex reflection coefficient, Γ, is then given by the equation (2.41).

2.3

Coupler theory

2.3.1

Coupler basics

As a starting point to this section about couplers, the coupler theory will first be de-scribed in a pure TEM-mode type of structure. A coaxial coupled line system and stripline coupled lines are examples of structures which support the TEM-mode. Mi-crostrip couplers do not support a pure TEM-mode, but are instead called quasi-TEM and will be treated in a separate section.

The basic configuration for a directional coupler is shown in Figure 2.5 and the enu-meration of the ports in that figure is the one used through out the report.

Figure 2.5: Basic configuration of the ports in a backward-wave directional coupler As can be seen in the figure the coupled port is the one next to the input port and there-fore it is called a backward-wave directional coupler. It is not obvious why it couples backward and for that reason a discussion of the theory will be briefly presented here. 16

Chapter 2. Theory Considering two coupled lines of TEM-structure with configuration as shown in Fig-ure 2.6(a). A positive voltage is introduced at the right line as showed in the figFig-ure. This voltage excites a current on the left line in order to create a negative voltage on that line (IE1= −IE2). As can be seen this current goes outward from the point of

exci-tation. Now instead considering a current at the right line, this give rise to a magnetic field as shown in Figure 2.6(b). According to Lenz’s law a current is then induced with the direction such as to create a magnetic field which opposes the change of magnetic flux (IM1, IM2). This current will be in the same direction as IE1 and in the opposite

direction of IE2resulting in different performance at the ports. In the direction of port

3 the currents counteract each other leading to isolation in that port. Port 4 is referred to as coupled port since the currents act together in that direction.

In coupler design one strives to design a coupler where IM2 and IE2 counteract each

other to the point where port 3 is totally isolated from port 1. Obviously this is not possible in reality but it can be seen as a goal for the design.

(a) Electric induction (b) Magnetic induction

Figure 2.6: Explanation of a backward-wave directional coupler

As a consequence of the coupling of electromagnetic fields, the coupled lines can sup-port two types of wave propagation with different characteristic impedance. Those two types are called even- and odd-mode due to the way the coupled lines are excited. In the even-mode the coupled lines are excited with signals of equal amplitude and phase. In the odd-mode on the other hand the coupled lines are excited with signals of equal amplitude but opposite phase. In a TEM-mode supporting structure, e.g. cou-pled coaxial lines, the lines are assumed to be surrounded of a homogeneous dielectric medium and therefore the phase velocity of these two modes are equal.

The parameters of interest when designing a directional coupler are • Centre frequency, f_c

• The coupling factor at centre frequency, k = S41= S32

Low Cost Precision Reflectometer • Isolation, i = S31= S42

• The directivity, d = S31/S41 = S42/S32

In general there are three variables which determine the coupler parameters of inter-est. These variables are the line width of the coupled lines and the spacing between them, usually described as w and s respectively. Knowing the even- and odd-mode impedances of the lines these variables can be determined. Therefore as a first step the even- and odd-mode impedances will be determined from

Z0e= Z0 r 1 + k 1 − k (2.48) Z_0o= Z0 r 1 − k 1 + k (2.49)

2.3.2

Microstrip directional coupler

Microstrip directional couplers are used largely because of their low cost and the fact that they can be realized on a two-layer PCB. Microstrip couplers do not support a pure TEM-mode, instead they are described as quasi-TEM [14]. This due to the fact that the dielectric medium surrounding the microstrip is not homogeneous. With a non-homogeneous structure the even-and odd-mode phase velocities are different. This leads to undesirable features in the use of backward wave directional couplers and introduces a need to compensate for these inequalities, see section 2.3.3. However, the even-and odd mode properties are valid also for microstrip couplers. The design parameters sought are shown in Figure 2.7.

Figure 2.7: Cross-section of a microstrip coupler

Due to complexity in calculating the design parameters of a microstrip coupler it is common to use an approximate method to estimate those parameters. These methods are usually based on the fact that there are some variables known from restrictions on the design. The variables usually known are the coupling factor, the system impedance, the substrate thickness and the dielectricity constant of the substrate. Knowing those 18

Chapter 2. Theory one can calculate the even- and odd-mode impedances of the coupler as

Z0e' Z0 r 1 + k 1 − k (2.50) Z0o' Z0 r 1 − k 1 + k (2.51)

It can be useful to check the accuracy of those results by using equation

Z2₀' Z0eZ0o (2.52)

Equations (2.50) and (2.51) tends to become inaccurate as the coupling gets tighter. Since the couplers used in this project have a loose coupling the approximate equation are determined to be sufficient. The forthcoming work is divided into two steps

• Determine the single line shape ratios w_h|so,se

• Determine the coupler shape ratios w_h and _hs The single line impedances are given by the equations

Z0se = Z0e 2 (2.53) Z0so = Z0o 2 (2.54)

The corresponding shape ratios are then given by a graphical model or by using closed-form expressions as follows.

For narrow strips (i.e. Z0≥ {44 − 2εr}Ω):

w h|se,so=  eH 8 − 1 4eH −1 (2.55) where H=Z0se,sop2(εr+ 1) 119.9 + 1 2  εr− 1 εr+ 1   lnπ 2+ 1 εr ln4 π  (2.56) For wide strips (i.e. Z0≤ {44 − 2εr}Ω)

w h|se,so = 2 π n (d_εr− 1) − ln(2d_εr− 1)o + εr+ 1 πεr  ln(dεr− 1) + 0.293 − 0.517 εr  (2.57) where d=59.95π 2 Z0 √ εr (2.58) The shape ratios of the coupler, w/h and s/h is then given by using graphs. A set of graphs suitable for many approximate designs are given by Akhtarzad, Rowbotham

Low Cost Precision Reflectometer

and Jones [15]. Once obtained the shape ratios, it is easy to determine the design pa-rameters w and s since the height of the substrate is already known.

This method among other similar methods are mainly used as a starting point for simu-lations where more exact parameters are obtained. These simulation programmes often provide their own programme which calculates the dimensions of microstrip lines, mi-crostrip coupled lines and other structures.

2.3.3

Directivity compensation

As previously mentioned the even- and odd-mode propagation velocities are different in a microstrip coupler which leads to poor directivity. This section will describe a few techniques to compensate for these differences in the propagation velocities.

The simplest and most obvious form of compensation is introducing input- and output capacitances in the coupled region, Figure 2.8. Those capacitors can in their simplest way be regular lumped capacitors. The disadvantage of using lumped capacitors is that they disrupt the single plane configuration imposed by a microstrip coupler. Another way of realizing these capacitors is with tightly coupled short fingers of microstrip lines, interdigital capacitors. That way the coupler still impose only one plane. Either way the capacitors affect the phases of both transmitted and coupled signals.

The saw-tooth or wiggly line coupler is another commonly used technique for compen-sation. The inner edges of these couplers employ a saw-tooth like configuration [16]. As this configuration too only impose one plane it is a very attractive compensation method and maybe the one most elegant.

Figure 2.8: Capacitor compensated coupler

Chapter 3

Simulations

3.1

Coupler simulations

The SPR configuration chosen for this project consists of two microstrip directional couplers, Figure 2.3. These couplers are used in order to be able to measure both forward- and reflected RF-power. The design criterias for the couplers are based on the fact that the effect on the reference detector from the reflected signal should be minimized and that the power levels at the inputs of the detectors should lie in the region specified by the manufacturer. Given those requirements the desired coupling of the couplers was chosen to be -20dB. A looser coupling would have introduced a need for a power amplifier in order to maintain the desired power levels at the detector inputs. The target for the directivity was chosen to be at least 30dB.

The simulation tools used for simulating the couplers were Eagleware R _{for the circuit}

simulations and ADS Momentum R

for the EM-simulations.

3.1.1

RF simulations

The first simulations were made in a circuit simulator. The starting point of the sim-ulations were calculated using the theory described in section 2.3. These values were compared to the ones obtained from calculations made by the transmission line calcu-lator provided with the circuit simucalcu-lator.

From the starting point a few changes were made in order to obtain -20dB coupling. These simulations showed directivity of only 8dB, Figure 3.1(a). This result correlates well with the theory described in section 2.3 about the even- and odd-mode.

To maintain a coupler structure simple to produce and to simulate while increasing the directivity, input and output capacitors were introduced in the coupled region. The ca-pacitors used are so called interdigital caca-pacitors (tightly coupled fingers of microstrip lines). After compensating the simulated coupler the directivity obtained was 40dB, Figure 3.1(b).

Low Cost Precision Reflectometer

(a) Not compensated (b) Compensated

Figure 3.1: Simulated coupling and isolation of the microstrip coupler

At this point the design criterias were met, the coupling was -20dB and the directivity was greater than 30dB.

3.1.2

EM simulations

The accuracy of the RF simulations, however, were a bit uncertain because of the way the interdigital capacitors were modeled. In the simulations they were shunted with the coupler at the edges of the coupled region. In reality these capacitors are placed inside the coupler, between the two microstrip lines, Figure 3.2(c). This uncertainty started a discussion of whether to perform EM simulations of the couplers or not. At first it was decided not to do these EM simulations but instead to produce multiple variations of the test boards with different lengths of the fingers in the capacitors. At the time when the CAD of the test boards were about to be made the people assigned to do it were busy with another project of higher priority. This gave some extra time and a decision was made to perform the EM simulations.

The simulations made in the EM-simulator showed reasonably good correlation with the result obtained from the circuit simulator. Using the dimensions from the circuit simulations the EM-simulations indicated a directivity of 23dB, Figure 3.2(a). The in-terdigital capacitors were adjusted resulting in a directivity of 35dB, Figure 3.2(b). The length of the fingers in the interdigital capacitors yielding the highest directivity was 1.06mm. Although these simulations made in the EM-simulator should be considered very accurate, the dimensions of the fingers are very small and therefore a decision was made to manufacture three coupler designs. One with the simulated length plus one shorter and one longer. This gave a possibility of choosing the most suitable design after measuring the directivity on the boards.

Chapter 3. Simulations

(a) (b)

(c)

Figure 3.2: (a) EM-simulation of the directivity with input parameters from circuit simulator (b) EM-simulation of the directivity with adjusted interdigital capacitors (c) The final coupler design

3.2

Q-point simulations

In order to design a robust SPR the q-points had to be simulated. Since there was no function available to directly plot the q-points an equation had to be developed. The configuration from which the q-points were calculated can be seen in Figure 3.3. From that figure one can make a signal flow chart, Figure 3.4(a).

Figure 3.3: Model of the SPR from which the q-points were calculated

This flow chart can be simplified by noting that ua4= 0 and by the fact that there is no

Low Cost Precision Reflectometer

(a) (b)

Figure 3.4: Signal flow charts used as a basis for the q-point simulations can form the system of equations shown below.

u_b4 = S41ua1+ S42ua2 (3.1)

u_a2 = Γub2 (3.2)

ub2 = S21ua1+ S22ua2 (3.3)

By inserting equation (3.2) in (3.1) and (3.3) the new system of equations becomes u_b4 = S41ua1+ S42Γub2 (3.4)

u_b2 = S21ua1+ S22Γub2 (3.5)

Eliminating ub2from equation (3.4) using (3.5) gives

u_b4

u_a1 = S41+

S₂₁S₄₂Γ 1 − S22Γ

(3.6) Given that ub4= 0 ∀ ua1when Γ = Γzthe final expression for Γzcan be written as

Γz= −

S₄₁ S21S42− S22S41

(3.7)

This expression assumes ua4 = 0 which can easily be be fulfilled by setting the port

impedance Z04 equal to the detector impedance. Using equation (3.7) in the

simula-tions the angular spacing of the q-points obtained is 135◦, 130◦ and 95◦. A spacing closer to the optimum was possible to obtain, but this was at the cost of greater phase variation over the frequency band. The magnitude of the q-points can be adjusted by adding a pi-attenuator between port 4 and the switched loads. As described in sec-tion 2.2.1 the magnitude should be determined with aspect to the applicasec-tion but lie in the region 0.5 ≥ |qi| ≤ 1.5. The interesting return loss region of the loads in this

application is 7 − 15 dB which can be translated to the inner two thirds of the Smith chart. Considering the desire for minimum frequency variation over the band as well as the interesting return loss region of the loads, the angular spacing of the q-points was chosen to be 135◦, 130◦ and 95◦. The magnitude chosen was qi' 1.15 without

use of the pi-attenuator, Figure 3.5. 24

Chapter 3. Simulations

Figure 3.5: The simulated placement of the q-points at 1805-1880MHz

3.3

System simulations

The system simulations were implemented in order to be able to test the SPR calibra-tion and measurements in a noise free environment. These simulacalibra-tions also provided a simple way of testing variations in parameters known to affect the accuracy of the re-flection measurements. Such parameters could be directivity error, q-point amplitude, q-point separation and frequency variation. The result of these simulations have been used as a verification of the measurements as well as a tool to quantify the sources of error.

The simulations were performed in MATLAB R with input data from the circuit

simula-tor. In the circuit simulator a load simulating the sliding short and the open, short, load was created. The output of the simulations was the scattering parameters correspond-ing to the powers read at the detector inputs. Three different outputs with scattercorrespond-ing parameters were produced. One corresponding to the six-port-to-four-port reduction, one corresponding the virtual four port calibration and one corresponding the reflec-tion measurements. The scattering parameters were loaded into modified versions of the calibration programme and the measurement programme and then the calibration and measurement were performed in the same way as during the measurements. In theory the measured reflection of the loads used during calibration should not differ from the exact value by more than what could be numerical errors appeared during the data processing. During these simulations, however, the maximum absolute difference was simulated to be δ = 0.018, Figure 3.6(a). Further simulations showed that if the coupling was decreased the absolute difference was improved. A coupling factor of -40dB, a value likely to be used in future products, gave a maximum absolute differ-ence δ = 0.002, Figure 3.6(b). As the coupling decreased even more, the accuracy increased to the point where the difference could be considered as numerical errors. The explanation to this phenomena is that when the coupling is tighter the reference detector is affected by the reflections from the switched loads. Ideally the reference detector is not affected at all by those reflections. As the coupling decreases the effect on the reference detector is reduced, hence the accuracy is improved. A loose coupling

Low Cost Precision Reflectometer

is therefore preferred. The relatively tight coupling used was chosen in order to avoid the use of a power amplifier.

(a) 20dB, εmax= 0.007, δmax= 0.018 (b) 40dB, εmax= 0.002, δmax= 0.002

Figure 3.6: Simulated accuracy of calibration when changing the coupling factor Another interesting observation during these simulations was that the accuracy in-creased as the return loss of the loads used in the calibration inin-creased. A calibration was performed with loads having return loss 1dB, close to the border of the Smith chart. Then simulations of unknown loads with return loss 1dB, 5dB and 15dB were performed and the maximum absolute difference was measured to be δ = 0.050 at re-turn loss 1dB, Figure 3.7(a). When the rere-turn loss of the loads used in the calibration increased to 5dB the difference decreased to δ = 0.027, Figure 3.7(b). This result gave a hint of that the amplitude of the q-point is of importance. Up to this point the ampli-tude of the q-points was 1.15. A new simulation was performed where the ampliampli-tude of the q-points was increased to 1.7, the return loss of the loads used was 1dB. As pre-sumed this increased the accuracy of the unknown loads measured. The simulations now gave an absolute difference of δ = 0.018 at 1dB return loss, Figure 3.7(c). The coupling used in these simulations was -20dB.

When simulating the reflection measurements performed at different frequencies over the whole band of interest (GSM1800) the accuracy was measured as the absolute dif-ference in amplitude, ε. There is an obvious phase shift in the measurement points due to the phase shift of the q-points over frequency. As the amplitude of the reflection co-efficients is the parameter of interest for this application it seemed natural to use ε as a measure of accuracy. A calibration was simulated at center frequency, 1842.5MHz, and then the reflection measurements were simulated at the lower and upper band edges, 1805MHz and 1880MHz respectively. The simulations showed continuously high accuracy, the maximum absolute difference in amplitude over the whole band was ε = 0.022, at 1805MHz, Figure 3.8(a). By decreasing the coupling, as described earlier in this section, it was possible to obtain an even higher accuracy. With a cou-pling factor of 40dB the maximum absolute difference in amplitude over the whole band was ε = 0.014, Figure 3.8(b).

Chapter 3. Simulations (a) |qi| ' 1.15; RL=1dB; εmax= 0.042, δmax= 0.050 (b) |qi| ' 1.15; RL=5dB; εmax= 0.024, δmax= 0.027 (c) |qi| ' 1.7; RL=1dB; εmax = 0.015, δmax= 0.018

Figure 3.7: Simulated accuracy of the reflection measurements when changing the return loss level of the loads used in calibration, (a),(b), and the amplitude of the q-points, (c)

(a) 20dB, εmax= 0.025, δmax= 0.094 (b) 40dB, εmax= 0.020, δmax= 0.079

Figure 3.8: Simulated accuracy of calibration when changing the frequency of the input

Chapter 4

Measurements

4.1

Coupler measurements

4.1.1

Measurement description

The first measurements of the test boards were made before any components were mounted, Figure 4.1. The boards used for the coupler measurements were 1.16/1/1, 1.06/1/2 and 0.96/1/1, Figure A.1. The names of the boards correspond to the length of the fingers in the interdigital capacitors. At this point the parameters measured were the coupling and the isolation of the couplers. These measurements were performed in order to choose from the three different coupler designs manufactured. The aim was to find the coupler with the highest directivity. A table with the measurement results is provided at the end of this section.

Figure 4.1: Test boards before assembly

Chapter 4. Measurements

4.1.2

Directivity measurements

The first measurements, made on the board 1.16/1/1, showed very low directivity, around 8dB, even though the couplers were compensated with interdigital capacitors as in the simulations, Figure 4.2.

Figure 4.2: Measured coupling and isolation on the board 1.16/1/1 with solder mask

According to the simulated values the directivity should have been around 35-40dB so this result was a bit worrying. After some discussions and reviews of the boards it occured that what differed the simulations from the reality was that on the test boards there is a solder mask over the couplers and in particular over the interdigital capac-itors. Due to the even-and odd-mode theory described in section 2.3 this will result in different propagation velocities in the even- and odd-mode. In the simulations the material used above the couplers was air with a relative dielectricity constant εr = 1.

The solder mask used has relative dielectricity constant εr = 2.8. This results in that

the compensation made with the interdigital capacitors will not have the wanted effect. In a "wide" microstrip the main part of the capacitance goes from the microstrip to the ground plane and almost nothing goes in the material above the microstrip. Therefore the coupling factor was not largely affected by the solder mask. In the case of the inter-digital capacitors where the distance between the fingers is small a greater part of the capacitance go in the material above the microstrip. When in this case these capacitors were covered with a solder mask the reality became different from the situation in the simulations and therefore the directivity measured was found very low.

The solution to this problem was simply to remove the solder mask from the couplers. This was performed by using a glass fibre brush to cause minimum damage on the microstrip and the substrate. The removal of the solder mask improved the directivity by 7dB and now measured 15dB on the board 1.16/1/1, Figure 4.3.

Low Cost Precision Reflectometer

Figure 4.3: Measured coupling and isolation on the board 1.16/1/1 with the solder mask removed

The following measurements on the boards with shorter finger length were made only with the solder mask removed since it had been determined in the previous measure-ments that the removal improved the directivity. The measuremeasure-ments on the board with finger length 1.06mm gave the same result as the board with finger length 1.16mm, 15dB directivity, Figure 4.4(a). The board with the highest measured directivity, 30dB, was the one with the finger length 0.96mm, Figure 4.4(b). A measurement on the board 1.16/1/1 with the finger length cut to 0.82 mm was also made, but this measure-ment showed lower directivity than the 30dB measured on the board with finger length 0.96mm.

(a) Finger length 1.06mm (b) Finger length 0.96mm

Figure 4.4: Measured coupling and isolation with the solder mask removed

Chapter 4. Measurements Finger length Test board label Solder mask Directivity

1.16 1.16/1/1 Yes 8 1.16 1.16/1/1 No 15 1.06 1.06/1/2 No 15 0.96 0.96/1/1 No 30 0.82 1.16/1/1 No 20

Table 4.1: Overview of the directivity measurements made on the test boards

4.2

Reflectometer measurements

4.2.1

Measurement description

Once the coupler measurements were performed and a coupler design was chosen the test boards were assembled. Two boards with the same coupler design were mounted, Figure 4.5, one with a narrow band balun (1805-1880MHz) and one with a broad band balun (500-2500MHz). The balun is a device which permits the transformation from a single ended signal to a differential signal. It also transforms the system impedance of 50Ω to the desired differential input impedance of 200Ω at the detector circuit. The reason why two different baluns were chosen was that in the manual of the detector circuit used they recommended the use of a narrow band balun in order to assure good performance of the detectors. The broadband balun was chosen because it is the one used in the products today.

Figure 4.5: Assembled test board

The parameter of interest during these measurements was the accuracy of the measured reflection coefficient during variations in frequency, input power, temperature and re-turn loss of the load. All measurements were performed with a CW-signal as input. The SPR was calibrated at center frequency, 1842.5MHz, and then measurements with