department of electronics, mathematics and...

DEPARTMENT OF ELECTRONICS, MATHEMATICS AND NATURAL SCIENCES

Antenna Elements Matching Time domain Analysis

Cristina Condori-Arapa

June 2010

Examiner: Dr. Magnus Isaksson. Master Program in Electronics /Telecommunications

Supervisor: Dr. Torbjörn Olsson.

Abstract Time domain analysis in vector network analyzers (VNAs) is a method to represent the frequency response, stated by the S-parameters, in time domain with apparent high resolution. Among other utilities time domain option from Agilent allows to measure microwave devices into a specific frequency range and down till DC as well with the two time domain mode: band-pass and low-pass mode. A special feature named gating is of important as it allows representing a portion of the time domain representation in frequency domain. This thesis studies the time domain option 010 from Agilent; its uncertainties and sensitivity. The task is to find the best method to measure the antenna element matching taking care to reduce the influence of measurement errors on the results. The Agilent 8753ES is the instrument used in the thesis. A specific matching problem in the antenna electric down-tilt (AEDT) previously designed by Powerwave Technologies is the task to be solved. This is because it can not be measured directly with 2-port VNAs. It requires adapters, extra coaxial cables and N-connectors, all of which influences the accuracy. The AEDT connects to the array antenna through cable-board-connectors (CBCs). The AEDT and the CBCs were designed before being put into the antenna-system. Their S-parameters do not coincide with the ones measured after these devices were put in the antenna block. Time domain gating and de-embedding algorithms are two methods proposed in this thesis to measure the S-parameters of the desired antenna element while reducing the influence of measurement errors due to cables CBCs and other connectors. The aim is to find a method which causes less error and gives high confidence measurements. For the time domain analysis, reverse engineering of the time domain option used in the Agilent VNA 8753ES is implemented in a PC for full control of the process. The results using time-domain are not sufficiently reliable to be used due to the multiple approximations done in the design. The methodology that Agilent uses to compensate the gating effects is not reliable when the gate is not centered on the analyzed response. Big errors are considered due to truncation and masking effects in the frequency response. The de-embedding method using LRL is implemented in the AEDT measurements, taking away the influences of the CBCs, coaxial cables and N-connector. It is found to have sufficient performance, comparable to the mathematical model. Error analysis of both methods has been done to explaine the different in measurements and design.

Acknowledgments Firstly, I would like to thank to my sponsor Powerwave Technologies Sweden AB, to the Antenna group partners and my dear supervisor Dr. Torbjörn Olsson, who was my inspiration and my guide during the hardest time; more than a supervisor a good friend, without his good ideas nothing of this work would have been possible. I am totally grateful to the University of Gävle and the Linuss-Palme agreement, which gave me the opportunity to come to Sweden and take a master in Telecommunications/ Electronics, one of the major passions in my life. I would like to mention also all the people who trust in me, in special my family in Peru and my beloved people here in Gävle. The thanks also go to all my classmates in HiG, who made of these two years of leaving in a foreign country a fruitful and unforgettable experience, thanks for these nice memories, which I will keep as a treasure in my life.

Abbreviations VNA : Vector Network Analyzer CBC : Cable-board-cable. AEDT : Antenna electric down-tilt. LRL : Line-reflect-line. PC : Personal computer. RF : Radio frequency. CZT : Chirp z-transform. ICZT : Inverse CZT. DUT : Device under test. F/FT : Fourier transform. DFT : Discrete Fourier transform. FD : Frequency domain. TD : Time domain. GPC : General precision connector. LPC : Laboratory precision connector. TRL : Thru-reflect-line. TDR : Time domain reflectometry. GPIB : General purpose interface bus. DC : Direct current. WCM : Wave cascading matrix. KW : Windowed gate. NF : Noise figure. BIF : IF bandwidth. SF : Shape factor.

Table of contents

Abstract …………………………………………………………………………………………..…….i Acknowledgments ………………………………………………………………………………ii Abbreviations ……………………………………………………………………………..iii Abbreviations Chapter 1 Introduction ………………………………………………………….1

1.1 Problem statement 2 1.2 Objectives of the thesis 2

General objectives 2 Specific objectives 3

1.3 Outline of the thesis 3 Chapter 2 Theoretical framework .…………………………………………….4

2.1 Introduction 4 2.2 Background 4 2.3 Time domain analysis 4

Time domain vs. frequency domain 5 2.4 Chirp z-transform 6 2.5 Windowing 7

Kaiser-Bessel window 7 2.6 Gating, filtering in time domain 9

Design if Finite Response filter by windowing 10 Rectangular window 11

2.7 Effects of time domain analysis 12 Masking effect 13

2.8 Impedance Matching 14 2.9 Connector 14 2.10 Transmission-lines 15 2.11 Power dividers 15

Chapter 3 Time domain and reverse engineering …………………………….17

3.1 The TD option: Description 17 3.2 Time domain: Transform modes 18 3.3 Test setup 18 3.4 Design and methodology 19

Windowing 19 a).Windowing in band-pass mode 20 b) Windowing in low-pass mode 21

3.5 Gating 22 Gating effects 23 Edge effects 23

3.6 Experimental results 24 FD transform to TD: Band-pass mode 24 FD transform to TD: Low-pass mode 26 Time domain gating 28

Chapter 4 De-embedding Process……………………………………………….33

4.1 Introduction 33 4.2 S-parameters and T-parameters 33 4.3 TRL Method and De-embedding 35 4.4 CBC de-embedding algorithm 36

Chapter 5 Uncertainties and sensitivity of the synthesis response……………40

5.1 Uncertainties due to mismatching 40 5.2 42

Chapter 6 Conclusions …………………………………………………………..43

6.1 Conclusions and discussion References 45 Appendix A Matlab functions: Flow diagrams…………………………………...46 Appendix B Design of FIR filter by windowing…………………………………..53 Appendix C Overcompensation in truncated response…………………………..55

Chapter 1: Introduction

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Chapter 1 Introduction

Nowadays, the reliability of a wireless telephone system is a very important issue to the user, who transmits and receives information through remote and crowded areas. The antenna designers have to design antennas that work under hard conditions to avoid interfering with other cells and to have maximum radiation to a desired area. Using antenna arrays (multiple antennas) gives better performance in terms of directivity and gain for long distances and the total field of this antenna can be modified changing the phase between the elements [1]. Many modern base station antennas consist of vertical arrays of radiating elements and a feed network arrangement, also named manifold, which controls the phase difference between the elements. Powerwave Technologies has developed a moving dielectric phase controller, called Antenna Electric Down-Tilt (AEDT), and slot fed radiating patch elements that need to be matched to the phase controller. The corresponding theory will be explained with more details in the Chapter 2. The phase controller and the patch elements are intended to be perfectly matched according to design. The first problem emerges when the measurement and the simulation do not coincide. This means that there is a mismatch somewhere in the antenna network arrangement, AEDT and slot radiating patch elements. To measure the AEDT or any other element of the antenna with a commercial 2-port vector network analyzer (VNA), it is necessary to use additional elements such as cable-to-board connectors (CBC), cables, adapters, etc. These elements and mutual coupling effects [1] for micro-strip patches produce the mismatches. For this special case, frequency domain measurements do not give enough information because it is not possible to see the location of each element in the frequency domain. Time domain is a useful tool for this kind of analysis when there is a need to locate the effects of each element as a function of time instead of frequency. Time domain analysis is a software implemented in commercial vector network analyzers (VNA’s), with a number of functions such as windowing and gating [2], to ease analysis of various time-dependent (actually distance-dependent) effects. The VNA’s have been considered a useful commercial tool in RF applications since it was developed and patent by Sharrit [3]; it bases its measurements in the travelling voltage waves that are often transformed into scattering parameters (S-parameters). Since the beginning of the VNA many options have been developed, among them the time domain option. Of major importance is to know how accurate this principle and the implemented software are and to what extent the test-development engineer can rely on it. In order to know each of the steps involved in the time domain option, the functionality must be reproduced. The gating function allows cutting unwanted responses and plots the response of a particular part of the system being measured, in frequency domain, normally as the S-parameters. This process causes effects like truncation and masking error, which may be solved by renormalization and windowing [4].


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De-embedding is another method with the same characteristics as gating but it can be implemented directly from the known S-parameters in the frequency domain. It is the second alternative explored in this thesis. In this thesis the existing time domain option in an Agilent VNA, the 8753D [2], is reverse engineered to give understanding of the process and to allow modification of the parameters fixed by the manufacturer. It is also a good starting point to generate a complete PC-based software that emulates this time-domain option with fully reconfigurable parameters. 1.1 Problem statement

In multi element antennas the individual elements are connected to a manifold / power splitter and it is impossible to detect the matching condition through that network without a suitable de-embedding mechanism. The de-embedding can be achieved in frequency domain using knowledge of the s-parameters of the splitter network. Alternative to this de-embedding technique the time resolution offered by modern wide band network analyzers can be used to the advantage. Such measurements are being done and they usually involve studying the step response in low pass filtered mode in conjunction with choosing a gate window and back transformation to the frequency domain, however this is only one option of several. The problem is that there is no knowledge how accurate the impedance can be obtained after the splitter network and if improved methods of measurement and display can be found such that the prototyping is made more expedient. In the end a rigorous method in prototyping leads to improved array performance and to improved yield in production.

1.2 Objectives of the thesis

The main objective of this thesis is to find the correct method to measure the AEDT with the best performance, i.e. with small stochastic and systematic errors. Minimizing the systematic errors helps to improve the design of the model and provides a better method to measure antenna elements before sending them for manufacturing. The uncertainties of a design must be known in advance in order to produce a product. If uncertainty is not taken into account, the products will not necessarily fulfil the given specifications, possibly violating the customer specifications and in the worst case also the regulations. A further objective is to study improved ways of achieving matching of antenna elements to 50 ohm in development and prototyping of antennas.

1.2.1 General objectives

• Study the time domain option used by Agilent VNA 8753. • Write the complete set of functions which emulates the time- domain option in the

commercial Vector Network Analyzer 8753 from Agilent, in software, which allows modifying and improving the same method as a future work.

• Describe in detail the uncertainties after test-development process of the AEDT. • Show the uncertainties produced by the different steps involved in the time domain

transformation. • Describe a method to improve the time domain option. • Compare time domain option with de-embedding function, name advantages and

disadvantages too. • Describe and approximate the uncertainties of the AEDT after it has passed through

the test process.


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1.2.2 Specific objectives

• Gather data from the VNA and process it into the computer to show the time domain representation from data measured in frequency domain.

• Implement low pass and pass band mode with the respective settings as the commercial VNA 8753ES from Agilent.

• Find which mode is more accurate to measure the antenna element inside the system. • Apply windows to the data in order to reduce unwanted effects. • Gate (Filter) the data with three options (different windows) given by the VNA. • Renormalize and explain the gating effects. • Measure the S-parameters of the antenna elements (micro-strip patches), CBC’s,

cables and AEDT with the help of gating and/or de-embedding methods in order to make the comparison between the one port measurements of the complete system with the result of the cascaded S-matrices using the antenna element as the load.

• Find the associated uncertainties in the chosen methods. 1.3 Outline of the thesis

The thesis report is organized as follows: • Chapter 1: Introduction. • Chapter 2: Theoretical Framework

It gives a brief explanation of the used theory, starting from time domain representation, chirp-z transform, windowing, filters and the effects that these cause on the processed data.

• Chapter 3: Time-domain and reverse engineering It describes the reverse engineering applied to time-domain option from Agilent, from the description of the different VNA options in time domain and the applied mathematics. The functions which emulate the time domain option are implemented using MATLAB 2007B.

• Chapter 4: De-embedding process This chapter explains how the de-embedding method models the system and how can this be applied to a general application, not necessarily only the one given in this thesis.

• Chapter 5: Uncertainties and sensitivity of the synthesis response This chapter studies the uncertainties of the measurement and the modelling of the system (i.e. AEDT and antenna element), produced after time domain transform is applied and looks for solutions to reduce the systematic errors.

• Chapter 6: Conclusions This chapter gives the final conclusions and suggests future work.

Chapter 2: Theoretical Framework

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Chapter 2 Theoretical Framework

2.1 Introduction

This chapter explains the basic theory needed to understand the tools used in the analysis and the development of this thesis. The first section gives the fundamental concepts of frequency-time domain analysis and the second section describes the uncertainties in the measurements and the sensitivity of the equipment based on the matching between load and source impedance.

2.2 Background

Vector network analyzers (VNAs) are tools for radio and microwave frequency (RF) device measurements. The VNAs measure incident and reflected travelling waves with both magnitude and phase, and normally displays the results as S-parameters, although the wave magnitude and/or phase can also be displayed as a function of frequency. Their advantages have had many applications due to the wide bandwidth available and the added tools, such as the time domain option. This option mathematically transforms the frequency response in the form of S-parameters to time domain using the inverse chirp Z-transform (ICZT). The analysis in time domain has applications in fault location and detection/finding of mismatches in a system. The time domain tool, in low-pass impulse and band-pass mode, shows the response of a device under test (DUT) in time (position) and its amplitude. The low-pass step mode shows the impedance of the response as a function of the time/position. An added tool in both low-pass and band-pass mode is the gating function, which gives the possibility to remove unwanted segments in time/position of the system. All these processes are explained in detail by [4]. Variations of the methods are used and presented by representatives of other brands, as Anritsu [5] and Rohde & Schwarz [6], but they have the same background. This background is explained in the first part of this chapter taking as a reference the Agilent Vector network analyzer HP8753ES as it is the instrument used in this thesis.

2.3 Time-domain analysis

Time-domain (TD). The time domain represents the response of a system as a function of time, e.g. amplitude and/or impedance. For discrete time, the representation of the continuous response is sampled in separate instants, whereas frequency domain represents the response of a system in each given frequency. Fig.1 shows the representation of a system in time and frequency domain. Both graphs are the S11, or reflection, of the cable and two connectors but represented in different domains.


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Fig.2.1 Reflection (S11) of a cable with two connectors in frequency and time domain

respectively. Reprinted with permission from Agilent. [2]. Fig. 2.1 shows a simple example of a system formed of one cable and two connectors. For real systems, dependent upon more components, the figure may be more complicated. The cable is terminated in a resistive load with the same impedance as the equipment used to measure the DUT, in this case 50 Ohms. The time domain response highlights the reflection in both connectors in contrast to the frequency response. This shows the interference of the two reflections with a free spectral range between undulations determined by the distance between the connectors and the speed of the electro-magnetic waves in the cable, i.e. the speed of light scaled with the velocity factor. Time domain vs. frequency domain Equations 2.1 and 2.2 define the continuous Fourier transform (FT) and the inverse continuous Fourier transform (IFT) respectively for an impulse response, ( )h tΓ .

( ) ( ) 2 j ftH hf t e dtπ∞

−

−∞

Γ = Γ∫ (2.1)

2( ) ( ) j fth Ht f e dfπ

∞

−∞

Γ = Γ∫ (2.2) By the Fourier principle, every periodic continuous signal can be represented as a sum of sine and cosine signals. A periodic signal with finite period leads to a discrete spectrum (DFT). If also the spectrum is periodic with finite frequency period, a discrete periodic time signal may be represented. The Discrete Fourier Transform (DFT) is a mapping from N samples in time domain to N/2+1 unique complex samples representing the frequency domain as shown in Fig. 2.2. Thus, network analyzer data is a representation of a measured and sampled finite piece of


6

frequency space which can be transformed to a finite and emulated sampled piece of the time domain.

Fig.2.2 The direct relationship between time and frequency domain for an N arbitrary sample signal. Either the real or imaginary part of the frequency domain response have

N/2+1 unique complex values. [7]. DFT is used as relationship between both domains, because the total analysis should be finite and discrete for working with discrete equipments. Some approximations are used here as a continuous periodic signal does not have finite energy or power. 2.4 Chirp z transform

In [8] an algorithm which improves the flexibility of the discrete Fourier transform was developed. It is called the Chirp Z-transform (CZT) and took its name due to the characteristics of the transform. This algorithm is a fusion between Discrete Fourier and Z transforms and it is stated in (2.5). In contradistinction to FT, (2.4); the z-plane contour point begins in A, the radio of the spiral contour depends of W and the angular spacing is dependent of the M time domain samples.

1

0

Nn

k n kn

X x z−

−

=

=∑ (2.3)

1

0exp( 2 / ) 0,1,..., 1

N

k nn

X x j k N k Nπ−

=

= − = −∑ (2.4)

1

00,1,..., 1

Nn nk

k nn

X x A W k M−

−

=

= = −∑ (2.5) Notice that (2.5) is equivalent to (2.4) when 1A = and 2 /1* j k NW e π−= . The characteristics of the Chirp z-transform are [8]: 1) The number of time samples (N) in time domain does not have to equal the

number of samples (M) of the transform in frequency domain and vice versa. 2) Neither M nor N, need to be an integer. 3) The angular spacing (distance between each sample) of the kz is arbitrary. 4) Whereas Z-transform, the contour doest not need to be a circle but can spiral in or

out with respect to the origin. In addition, the point 0z is arbitrary, but this is also the case with the FFT if the samples nx are multiplied by 0

nz − before transforming.


7

Chirp z-transform allows zooming into an almost arbitrary piece of time using a large number of points for the representation, increasing in some way the resolution on the display of the Network analyzer. Note that the true resolution does not increase but is dependent on the number of sample points and the frequency span.

2.5 Windowing

By necessity every signal has to be finite to be processed in a computer. According to (2.1), the Fourier Transform is processed over an infinite interval and it is done in a continuous way. In reality the signal must be sampled and truncated to a finite length. The result of a finite recording of a signal causes spectral leakage [9]. Fig. 2.3 shows the comparison between the continuous signal and the same signal after it is truncated with a rectangular window and periodically repeated to fulfil the requirements of the DFT.

Fig. 2.3 Arbitrary periodic signal, truncated from 0 to NT [9].

A window is a function that weights (multiply) a response with a special shape (kind of window). It is a multiplicative, usually real number, weighting, that reduces the ringing effects produced by the sharp rectangular truncation. This truncation is caused by the inherent limits of the equipment and is present in both TD and FD. The discontinuities showed in Fig. 2.3 causes the Gibbs’ effects [10]. Once the response has been sampled and windowed, it needs rescaling in amplitude, see Fig.2.10. In frequency domain the method is implemented using a window function, represented here as ( )W f . The simplest case is the rectangular window defined by:

1/ ( ) 1

( )0

length W for n NW n

otherwise ≤ −

=

(2.6)

Considering scaling effects and that the area under the window function must be one, the factor 1/M must be used, where M is the bandwidth of the window ( )W n . The rectangular window sharply truncates a response between the start and end points. This behaviour causes ringing once it is transformed to the other domain. These ringing are the Gibbs phenomena and cause aliasing in the analysis. These oscillatory effects are to be reduced by the use of another window named the Kaiser-Bessel window that will reduce the kind of steps highlighted in figure 2.3. Kaiser- Bessel Window


8

Kaiser-Bessel window is a special kind of window developed by Kaiser using the studies done by Slepian and Pollak [24]. Kaiser has discovered a simple approximation of the prolate-spheroidal, using the zero order modified Bessel function of the first kind, defined by

( )

2

00

2!

k

k

x

I Xk

∞

=

=

∑

The Kaiser-Bessel window is defined as

[ ]

2

0

0

1.0/ 2

( ) 02

0

nIN Nw n n

Iotherwise

πα

πα

− = ≤ ≤

(2.7)

The Kaiser window is a dynamic function, which means that its length and shape could be modified by varying its length, N, and the shape parameter, .β π α= [9]. The Kaiser-Bessel window is illustrated in Fig. 2.4 for some different beta values.

0 20 40 60 80 100

0.7

0.75

0.8

0.85

0.9

0.95

1

n

V

KB=1KB=0KB=3KB=6

Fig. 2.4 Kaiser-Bessel window for N=100 samples and . 1,0,3,6β α π= = .

The performance of the Kaiser-Bessel window has the ability to keep the maximum energy in the main lobe (i.e. narrow main lobe width) and can reduce side-lobe level, in contrast to the rectangular window with narrow main lobe but high side-lobes at -13 dB for the first side-lobe. Fig. 2.5 shows the frequency response of two Kaiser windows with 6,13β = .


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350 400 450 500 550 600 650

-160

-140

-120

-100

-80

-60

-40

-20

n

Am

plitu

de(d

B)

beta=6beta=13

Fig.2.5. Spectral representation of two Kaiser Bessel windows with beta 6 and 13. Notice the

trade-off between side-lobe level and main lobe bandwidth. From Fig. 2.5 it is important to notice the trade-off between the main-lobe width and side-lobe level. In [9], Harris showed a table with the windows and figures-of merits for design purposes. An extension of the Kaiser-window application is the filter design using window that is given in the next section, the filter design is developed in Appendix B. 2.6 Gating, filtering in time domain

Gating or filtering in time domain is a complementary tool in the current VNA. Time domain gating is a function that removes or selects a portion of signal in time-domain and then displays the remaining signal in frequency domain (FD). The ideal gate can be represented as a brick wall where the multiplicative factor is “1” in the desired region and “0” elsewhere.

1 1 2

( )0

( ) ( ( ). ( ))

t t tG t

elsewheref DFT t G t

<


10

Fig.2.6. Representation of the gating function in time domain (red) with and arbitrary signal

(blue).

The gating process is carried out by windowing the response in time domain as describe by (2.8) and shown in Fig. 2.6. This process is called filtering and is represented in frequency domain by a convolution of the two spectra. By Fourier transform duality, (2.9) and (2.10) state the gating in time and frequency domain respectively. For time domain ( ) ( ) ( ).gated t t G tΓ = Γ (2.9) and for FD ( ) ( ) ( )*gated f f G fΓ = Γ (2.10) The ideal gate also produces Gibbs phenomena from sharp truncation (see windowing) [10]. These effects are minimized by smoothly reducing the edges of the gate. The Kaiser window is preferred for this purpose as it gives more flexibility in the filter design. The parameter πα is commonly represented as β in MATLAB. When β is 0 the rectangular window is obtained and when β is 0.5 the Hanning window is obtained. As mentioned before, there is the trade off between side-lobe level and main-lobe width as well. The Kaiser-Bessel window is attractive for engineering application and claimed to be the top performer of the windows [9] due to the easy generation of the coefficients. Design of Finite Impulse Response filter by windowing [10] FIR filter design using windows is a method to design discrete filters and is restricted to discrete-time implementation. The gate function resembles a band-pass filter. The band-pass filter is defined by the side-lobe level and ripples given by pA and ∂ (delta) respectively.

t1 t2

50 100 150 200

-0.2

0

0.2

0.4

0.6

0.8

1

V

t


11

Fig.2.7. Band-pass filter designed using windows. The gate function, described in time domain by the start and stop points, is transformed using CZT to frequency domain and then windowed with a Kaiser-Bessel window. This smooths the sharp edges of the inherent rectangular window when the product is transformed back to time domain by the ICZT. Fig. 2.7 shows the gate in time domain and its different parameters that define the filter. The Kaiser-Bessel window sets the minimum gate transition slope in time domain. By the duality inherent in time and frequency domain and by convenience [7] the gating process is done only in frequency domain. The rectangular window defined by t1 and t2, is shown in Fig. 2.6. The Fourier transform of a rectangular window is the sinc function. This function can be determined entirely by the width of the gate and no Fourier transform is needed. Appendix B gives the formulas used to design filters using Kaiser Bessel window. Rectangular window The Fourier transform of a rectangular window ( )w n , can be calculated analytically using the equations

( )( ) ( )( )sin ( ) 0

sin ( ). 0cjwtc w n t

F w nc w n e t

== ≠ (2.11)


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When the gate function is not centred in t=0, the sinc function is multiplied by a complex exponential factor as shown in (2.12), where ct is the offset [11]. In Fig.2.8 the rectangular window in TD and its spectra in FD using analytic approximation are shown. Notice that the number of nulls in the spectra is the same number that the width of the rectangular window. The amplitude must be normalized by 1/ D to have 1 in amplitude as maximum, where

( ) ( )2 1D n t n t= − .

Fig. 2.8 Rectangular window and its transform using direct analytic transform. 2.7 Effects of time domain analysis

The time domain option 010 from Agilent HP8753ES is a mathematical approximation from the FD to TD using CZT. This option resembles time-domain reflectometry, commonly done using an oscilloscope and a pulse generator, but here it is totally done in FD. The mathematical approximation, the digital sampling, the windowing and gating with the few amount of points available to work with, are the sources of error. Fortunately it is possible to compensate for some of the errors using renormalization. Fig. 2.9 shows the comparison between the analytic transform and the measured data of a 3-dB filter and the possible sources of error.

Fig. 2.10 Comparison of the analytic transform and the VNA time domain transform of the

same device under test (i.e. 3-dB filter) Reprinted with permission of Agilent. [2].

n(t1) n(t2)

D

n

n


13

Unfortunately not all the errors can be corrected or predicted. There are extra effects cause by the gating process and the same renormalization, and then there are masking and truncation effects [11]. Masking Effect Masking occurs when the response of a discontinuity (reflection) affects the remaining reflections. Fig. 2.11 shows the TD response of 2 capacitors (e.g. similar capacitors) in shunt configuration. It is expected to get the same amplitude for both capacitors but Fig. 2.11 shows the time domain representation of the circuit when the second discontinuity is masked by the first one.

Fig.2.11 Time domain response of two shunt capacitors, in low pass impulse mode.

Reprinted with permission of Agilent [2] Figure 2.12 shows the frequency response of each capacitor gated separately. The masking affects the second reflection. The grey line shows the ideal response of the capacitor without masking. ( )22 1 21 .aρ ρ ρ= − (2.12) Where 2aρ is the apparent reflection of the second discontinuity and 1ρ is the reflection of the first reflection.

Fig.2.12 Frequency domain response after gating the second capacitor. Reprinted with

permission from Agilent [2].


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Masking can be partly compensated for using renormalization. This renormalization must give the real amplitude to the masked discontinuity and follows (2.12). From (2.12) the real amplitude of the second reflection is found.

( )

22 2

11aρρρ

=−

(2.13)

2.8 Impedance Matching

In order to deliver the maximum power to the load in the circuit shown in Fig. 2.13, the input and output impedance must be the same. The capacitor, Cs, is represented as a negative reactance in the Smith chart, therefore a positive reactance, Ls (matching network), must be included in the circuit, in this way compensating the difference on phase added for the capacitor (i.e. -90 degrees).

Fig. 2.13 The input and output impedance of this simple circuit do not match. The matching

network added must be an inductor to compensate the capacitance. The matching network must be lossless in order not to dissipate the input signal as heat. 2.9 Connector

The developing of coaxial connector during the 1940s, led to the introduction of Type-N connector. Years later, the high demands of accurate measurement in transmission lines gave the needed of using precision coaxial connectors. Thus, in 1960s, standards were established for high frequency applications. For high frequency signal, a section of transmission line could represent more than a simple resistance, and this is one of the most often problem in the matching of components. In many RF applications and measurements the role of the connector is very important. In fact the good performance of a device or measurement instrument depends upon the good performance of the connector [12]. The connector are classified in two groups GPC(General Precision Connector) and LPC (Laboratory Precision connector). These differ for the type of dielectric used to support the centre conductor. For GPC, a solid dielectric is used, commonly named ‘bead’ whereas LPC uses air. The Type-N connector is included in both groups, precision and non-precision connectors.


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Connectors have two very important electrical characteristics, the characteristic impedance and the maximum operating frequency (i.e. cut-off frequency). For example, the type-N connector has 19.4 GHz as a theoretical upper frequency limit [12].

2.10 Transmission-lines

Coaxial cables at high frequency become transmission lines. Transmission-lines can be represented as two-port network, where voltages and current can vary in magnitude and phase along its length. For a small section of transmission line, x∆ , the lumped-element circuit is shown in Fig. 2.14, where Ax x= ∆ only for this figure.

LAx

CAxGAx

RAxU1

U2

U3

U4

Fig. 2.14 Lumped-element circuit of a transmission line, x∆

, , ,R L G C represent the resistance, inductance, conductance and capacitance per unit

length, x∆ . The scattering-matrix of a transmission line can be model directly from its electrical length, which is .lθ γ= , where γ is the intrinsic propagation constant. In the lossless case γ reduces to jβ since, .jγ α β= + . Thus, (2.14) states the S-matrix of a transmission-line [13].

.

.

00

i

i

l

li l

eS

e

γ

γ

=

(2.14)

2.11 Power dividers Power dividers or splitters are passive components. They are used to divide an input power into 2 or more output power. Each output is designed in such a way that they are isolated each other. The simplest case for a power divider is the T-junction, which has two outputs and one input. The scattering matrix of a T-junction is given by

11 12 13

21 22 23

31 32 33

S S SS S S S

S S S

=

(2.15)

If all the ports are matched, there are not reflection in any port, then 0iiS = . For passive components, the S-matrix is reciprocal (i.e. ij jiS S= ) . Then, the S- matrix can be reduced to


16

12 13

12 23

13 23

00

0

S SS S S

S S

=

(2.16)

However, it is impossible to construct such a reciprocal, lossless device and match at all ports power divider, as it was indicated in Pozar [14] and in chapter 4.

Chapter 3: Time domain and Reverse engineering

17

Chapter 3

Time domain and reverse engineering As a first step of this thesis in order to understand the processes and performance of the time domain option given by Agilent, reverse engineering is attempted. This chapter explains what is commercially sold in the market and what functionalities the 010 option (Time domain) from Agilent has. A set of programs, which simulates each of the functionalities of the option 010, are described by the flow diagrams in appendix A. The goal in this first task is to find which of the modes low-pass and band-pass is more accurate to measure the antenna element inside the DUT. The second task is to test a second method to compare with, for this purpose a specific technique of de-embedding that was already utilised in a previous master thesis [15] was applied. This is a specific application of a Thru-Reflect-Line (TRL) calibration [16] which will be described in chapter 4. 3.1 The TD option: Description Fig. 3.1 represents the total functions included in the time domain option 010 from the HP8753ES from Agilent. The 010 option adds extra utilities to the VNA to do time domain analysis.

Fig. 3.1 Time domain transform with its different functionalities.

First of all, designing a time domain option with similar characteristics in the MATLAB environment as the one used in the named VNA requires reverse engineering. Given the complete VNA time domain option already designed according Fig. 3.1, the task is to find out how accurately it represents the impedance of a given object (the antenna element) of interest placed inside a specific system, e.g. the cascade of AEDT, cables, CBC’s and antenna patch elements. The time domain option has two modes, low pass and band pass mode. Each of these has window and gate options, likewise compensation and renormalization. These options are designed according to Tables I and II.

Transform

Window:

Gating:

Low-pass

Band pass

Minimun

Normal

Maximum

Minimum

Wide

Normal

Maximum

Time domain option


18

For instance, in windowing there are three types of windows: minimum, normal and maximum and these are defined through the parameter β of a Kaiser window according to [2]. Beta modifies the side-lobe level and main-lobe width [9]. When one chooses the window in the VNA it is this parameter that is changed. The gating process, or filtering in time domain, is designed according to the Kaiser window filter design method described in [10]. It has four options: minimum, normal, wide and maximum shape. As explained in chapter 2, gating is a process done in FD [11]. Gating, unlike windowing, depends of two parameters, β and the window length L . These are chosen following formulas given in Appendix B. 3.2 Time domain: Transform modes There are two modes in TD analysis, low-pass and band-pass mode. Low-pass mode resembles time domain reflectometry (TDR) because the measurements can go down till DC and cover the total span allowed by the VNA, which commonly is wider than an oscilloscope. This increases the accuracy in the measurements but complicate in some way for the devices, as not all devices can work at low frequencies without risk of breaking. The band-pass mode shows the time domain measurements done in a specific frequency region of interest defined by the user. Time domain option in the VNA is made differently than the traditional TDR. Normal TDR works as a traditional radar, sending a short signal (impulse) and measuring the reflections. The transform is made taking data directly from the FD (S-parameters) and then using ICZT. 3.3 Test setup Fig. 3.2 shows the first test setup used to acquire the S-parameters from the VNA using IEEE-488 General Purpose Interface Bus (GPIB), to connect to the computer. The data is gathered and processed in a PC running MATLAB 2007B.

Fig. 3.2 Test setup used to gather the S-parameters towards the PC. A device under test (DUT) is measured and the error-corrected S-parameters are downloaded to the computer for further processing. The maximum number of points of data which can be downloaded from the VNA is 1601 and the directly plot of them in a computer will not show good resolution in (time domain), contrary to the one shown by the VNA.

GPIB

DUT


19

The CZT-algorithm is used to transform the FD to TD response; a rescaling is added to compute the inverse ICZT using the direct transform CZT. ICZT has the advantage over DFT that it can represent the transformation with more points than the original data. This helps the VNA to show the 1601 points in about 10000 points according to the span selected by the user. Although this resolution is really not available it is a useful interpolation. 3.4 Design and methodology Windowing The reflection in the port 1 of the VNA (S11) of a cable ended with a short circuit is measured with the total span (i.e. 30 kHz to 6 GHz) at 801 frequency points. This is gathered in a file and transformed to time domain using ICZT. The data to time domain, the S11 must be windowed to reduce leakage [2]. For windowing, a Kaiser-window is used. Table I shows the values of side-lobe levels and chosen β to obtain the same level as stated in the manual. The values of β are obtained from a table in [9] and modified empirically to obtain the same result as the VNA. The given specification for windowing, according manual are stated in Table I.

Table I. Window design [17] Window

type Side-lobe

level β

Minimum -13 dB 0 Normal -44 dB 6

Maximum -75 dB 13

The Kaiser window is defined by (2.6), where the values of β are replaced in the equation according to Table I. The length is the same as the gathered frequency response data, 801, M=N (cf. windowing ch.2). The Kaiser-window truncates the edges of the total sampled response in a smooth way. The transform to time domain of the windowed S-parameters is shown in Fig. 3.3 for low-pass and band-pass transform respectively using minimum window. The blue colour shows the one downloaded directly from the VNA and the red shows the one made by synthesis using the approximation of the Table I. As there are two different modes in the TD transform, the window is applied differently as well.


20

-5 0 5-80

-70

-60

-50

-40

-30

-20

-10

0

t(ns)

S11

(dB

)

Band-pass transform with minimum window

SynthesisVNA

-1 -0.5 0 0.5 1-80

-70

-60

-50

-40

-30

-20

-10

0

X: 0.1206Y: -13.23

t(ns)

S11

(dB

)

Low-pass transform with minimum window

SynthesisVNA

Fig.3.3 Time domain response of a cable terminated with a SHORT. Minimum window Beta=0,

N=801. Total span≈6 GHz a) Windowing in band-pass mode The windowing process is performed differently in low-pass and band-pass mode. For band-pass mode, the Kaiser-window has the same length as that of the measured frequency response, M N= . In FD, windowing means multiplication of the Kaiser-window with the frequency


21

response. The time resolution, i.e. distance between two points in TD, for band-pass mode is [18]

1 1.

tT f N

∆ = =∆

(3.1)

where N is the length of the gathered data (e.g. 801 points) and 2 1f f f∆ = − with f1 and f2 being the lower and upper frequency limits in the measurement respectively.

However even for band-pass is necessary to arrange the spectrum centered around f=0, the

frequency shift is ( )2 1 2cf ff += .

Supposing that the rectangular window is applied to a frequency response of 5 points as shown in Fig.3.4, Fig. 3.4 Windowing in band-pass mode, the rectangular window is directly multiplied with the

data, M=N. The window function of an arbitrary Kaiser window in band-pass mode is

0( ) ( ) /W n k n M n n N= < ≤ where ( )k n are the Kaiser-Bessel coefficients in each sample point, 0n is the start point and M is the window length (i.e. 801 points). Take into account that the signal has been sampled and digitized, from that the indices ‘n’ in the last equation. After the numerical transformation based in ICZT, the time domain response must be multiply by the factor 2 cj f te π to compensate the frequency shift, cf f− [18]. b) Windowing in low-pass mode Windowing in low-pass mode is not that straight as for band-pass mode. In fact, low-pass mode resembles TDR, this means that it could measure down till DC (at least theoretically with some extrapolation). Most VNAs for RF cannot measure at really low frequencies. The Agilent 8753ES allows measurements at lowest 30 kHz. Following the last example, the response must be extrapolated and mirrored around DC as shown in Fig. 3.5.

f f1 f2

( )fΓ


22

Fig.3.5 Windowing in a low pass mode, the pink magnitude is extrapolated and the data was

mirrored around DC before window is applied. M=2N The extra point is added using the method of extrapolation. For linear phase responses the amplitude of the point is the real number of the closest point, ( ) ( )( )0 1realΓ = Γ . But if the phase between points varies, the method might get problems [18]. In this thesis, to avoid the extrapolation problem, the first discrete sampled point is located at n=0, like this ( ) ( )0 1Γ = Γ and the negative frequency response is the complex conjugated of ( )2 : endΓ since its second sample, and then mirrored. One more requirement to take into account for low-pass mode is that the number of points must be equally spaced through all the frequencies, at so called harmonic grid points defined by

21ffn

= (3.2)

For instance, in the first measurements the total range of frequencies was taken from 30 kHz to

6 GHz with 801 points, the 16 7.49801GHzf MHz= ≈ once the renormalize button is pushed. Fig.

3.5 shows the windowing in low-pass mode of 5 points, where they are mirrored and the pink data is extrapolated to fulfil the TDR approximation. The 8753ES has a minimum working frequency of 30 kHz; it means that DC must be extrapolated from the measurement. This process has to be considered in chapter 5 as a possible error source due to non-periodicity of the frequency response of the DUT. The time step for low-pass mode is half of band-pass mode, using the same principle as (3.1) [18].

1 1.2

tT f N

∆ = =∆

(3.3)

3.5 Gating The gating, or filtering in TD, can be represented as a rectangular window in TD. This multiplies the time response in order to gate the signal and then the response will be transformed to FD. To avoid doing redundant transformations and truncation effects all of this process is done solely in FD but the representation in TD is needed in order to choose the region of interest [11] ( 1t and 2t , see Fig. 2.6).

n 1 2 3 4 -4 -3 -2 –1

( )fΓ


23

This rectangular window, defined by 1 2,t t , is transformed analytically to FD and is there windowed with the Kaiser-Bessel window defined by the parameters given in Table II. The advantage, having a soft-design filter, is that it is possible to change the parameters which define the filter thus extending the applications further than only comparison (i.e. better windows, lower side-lobes, etc). Appendix B gives the complete design of the filters using Kaiser windows, Table II helps in the design according to the manual [17]. Appendix A shows the flow diagram of the process.

Table II Gate design [17] Gate

Shape Pass band

Ripple Side-lobe levels

Cut-off time

Minimum 0.10± dB -48 dB 1.4 /Freq Span Normal 0.10± dB -68 dB 2.8 /Freq Span Wide 0.10± dB -57 dB 4.4 /Freq Span

Maximum 0.10± dB -70 dB 12.7/Freq Span The gating process or filter in TD, is a very useful tool in time domain applications because it allows the removal of sections of the response that are irrelevant for the measurements. The DUT, in general, needs connectors and cables to be measured. These added cables and connectors may be gated away from the AEDT response using gating. Gating effects The time domain gating effects has been widely studied, [2], [19] and are referred to in [11]. The gating effects can be produced from the way the filter is located till the method the supplier used to design the function and can yield considerable errors if they are not taken into account. Edge effects The convolution between the spectra of the gate and the gathered frequency response, gives edge effects of about 6 dB of roll-off in one side for low-pass mode and in both sides in band-pass mode to the final gated frequency response. This is produced by the circular convolution [dig Signal book]. Fortunately there is a simple method to renormalize the final response. A unitary frequency response, ( )F w , is carried through the same convolution process. The original final frequency response, ( )( )G wΓ , is divided by the gated unitary frequency response, ( )( )G F w and in this way compensated for the problem of the convolution [11]. This is described by the equations

( ) 1F w = ( )( ) ( ) ( )*G F w G w F w=

( )( ) ( )( )( )( )compG w

G wG F w

ΓΓ = (3.4)

The G in front of each response means gate in FD and ( )* convolution.


24

This renormalization only works if the gate is located at the centre of the response being gated. If not the response is going to be overcompensated as shown in Fig. 3.6 measured using low pass mode. If masking is presented, an extra renormalization must be done [11]. As this project does not experience significant issues with high masking effect before a response is gated, this renormalization is not considered any further. Although masking effects were considered and the renormalization is propose in [17], it did not give any good compensation. The re-scaling is done only in magnitude.

Fig.3.6 Consequences of offset in the gate location of three time gates in low pass mode. Reprinted with permission from Agilent [2].

3.6 Experimental results In this section, the results and the comparison of the time domain analysis are shown. The intention of this study is to compare between time domain analysis done directly with the Agilent 8753ES and the analysis done with the developed functions, according to Appendix A. In order investigate the accuracy of the method the comparison is done step by step with each option as stated in Fig. 3.1. FD transform to TD: Band-pass mode The band-pass mode transform option is measured directly using the setup shown in Fig. 3.2, i.e. a coaxial cable with an ended short connection. The reflection, S11, is measured using minimum, normal and maximum window. The number of points is 801 and the dB- scale is compared. The same measurements were download from the VNA with 801 points and the frequency range from 30 kHz to 6 GHz. To give the same apparent resolution to the plots the ICZT is used in this way displaying the 801 points in about 10000 points. The windows are applied before the frequency response (e.g. S11) is transformed.


25

The function ‘bptprocess.m’ given in Appendix A is a function in charge to transform the S-parameters stored in the computer to time domain using the CZT algorithm. The comparison between the synthesized response and the one download directly from the VNA are similar for minimum and normal windows as shown in Fig 3.7 a) and b).

-5 0 5-80

-70

-60

-50

-40

-30

-20

-10

0

t(ns)

S11

(dB

)

Band-pass transform with minimum window

SynthesisVNA

a)

-5 0 5-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

t(ns)

Band-pass transform with normal window

S11

(dB

)

SynthesisVNA

b)


26

-5 0 5-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

t(ns)

Band-pass transform with Maximum window

S11

(dB

)SynthesisVNA

c)

Fig. 3.8 Time-domain transform of a coaxial connector and a female short in band-pass mode. The comparison is done with the response after synthesis and the response directly from the VNA. a) Minimum window 0β = . b) Normal window 6β = . c) Maximum window 13β = .

That is not the case with the Fig. 3.8 c) were the transform in the VNA shows non-steady response at lower values of -70 dB. These differences could be due to the quantization error made by the processor inside the VNA. Remember that the time-domain response is just an approximation of the Table I where the design was done using the side-lobe level. The amplitude re-scaling is done directly over the transformed response and it is done empirically (i.e. by comparison).The Kaiser-window is formed with the values of β given in Table I. For band-pass mode the length of the window depends upon the length of the S-parameter, this case 801 points. FD transform to TD: Low-pass mode The low-pass mode, unlike to band-pass mode, needs to adjust the signal before it is processed. The gathered data is mirrored and conjugated to fulfil the requirements and get the DC-point, cf. Fig. 2.2. The frequency response that is to be processed must follow

( ) ( )( ) ( )1)2 : 1, 2,3,...,LPn N n n N Γ = Γ Γ = (3.5) where 1) means mirror and complex conjugate of the ( )nΓ data. The total length of the S-parameter before process, is 2* 1LPN N= − , this is due to there is no extrapolation applied to the response before the transform to TD. Windowing is carried out using the same parameters as for band-pass mode but the length of the window depends on the


27

( ) ( )2* 1LP Nn −Γ data. Fig. 3.9 shows the low-pass mode transform with minimum, normal and maximum window. The apparent resolution is 2:1 compared with band-pass mode if (3.3) is taken into account.

-1 -0.5 0 0.5 1-80

-70

-60

-50

-40

-30

-20

-10

0

X: 0.1206Y: -13.23

t(ns)

S11

(dB

)Low-pass transform with minimum window

SynthesisVNA

a)

-1 -0.5 0 0.5 1-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

t(ns)

Low -pass transform with normal window

S11

(dB

)

SynthesisVNA

b)


28

-1 -0.5 0 0.5 1-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

t(ns)

Low-pass transform with maximum window

S11

(dB

)

SynthesisVNA

c)

Fig. 3.9 Time-domain transform of a coaxial connector and a female short in low-pass mode. The comparison is done with the response after synthesis and the response directly from the VNA. a) Minimum window 0β = . b) Normal window 6β = . c) Maximum window 13β = .

For maximum window the synthesis response is not quite similar to the VNA download data with the same amount of points and high apparent resolution. This difference is considered in section Uncertainties and sensitivity of the synthesis response in chapter 5. Time domain gating The gating is solely done in FD due to reasons explained in section 2.1. See the gate function developed in Appendix A under the name ‘gatef.m’. The Fig. 3.10 shows the comparison between the frequency response of a coaxial cable and a short termination with no-gated and gated with minimum shape. It is measured from 1-2 GHz in band-pass mode. The gate starts at

1 7.472t ns= and ends at 2 13.47t ns= . Both responses were downloaded directly from the VNA and no synthesis is applied yet. In order to download data directly from the VNA a routine is done and it is explained in appendix A as well under the name of ‘readdata.m’.


29

Fig. 3.10 Linear scale representation of S11 of a cable with a short end before and after is

gated. Minimum-shape. X axis with 801 points represents 1 GHz of bandwidth. 1 1.25801GHzf MHz∆ = ≈ .

Fig. 3.10 shows the frequency response of the terminal without the influences of other block of components, e.g. connections, cable. The synthesis for gating process is done directly applying the theory from Appendix B. Notice that the total gating synthesis is done based on table II and the possibility of errors in the design has influences in the process. To design the gate function, two methods were implemented. The principle of gating a signal is not more than filter it. If only one portion of the time domain response is needed, then filter away the remaining. This process can be done either in time or frequency domain. In time domain, gating is just the multiplication between the time response and the gate. In frequency domain gating is the convolution between the original response and the spectra of the gate. Fig. 3.11 shows the frequency response of a cable and a terminal gated by the VNA, in time domain and in frequency domain. The roll off in both edges is evident, especially if the gate is done in time domain. The first step is to decide in which domain it is convenient to apply the gate function to avoid adding errors. Reasons why gating is done solely in frequency domain: • The time domain response after transformed from S-parameter in frequency domain to the

time domain has about 10000 samples.


30

• From Chapter 2, the gate is held using Kaiser window and in order to apply gate to a response in TD, these must have the same length, M=N.

• With more than 10000 points the gate must be transformed using CZT to frequency domain into only 801 points and then back using ICZT to TD and spread to 10000 points again before it is multiplied with the time response.

• The gated response in TD must be transformed into frequency domain using CZT into 801 points.

• There are many processes involved in the time gated response which is the cause of the errors roll off, in both sides for band-pass mode.

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time gatedgated VNAfreq gated

Fig. 3.10 dB-scale representation of S11 of a cable with a short end before and after it is gated.

Minimum shape. X axis with 801 points represents 1GHz of bandwidth. 1 1.25801GHzf MHz∆ = ≈ . Each n sample is f∆ .

The gating in frequency domain simplifies the gate function to a linear convolution between the original response and the spectra of the filter. The roll off, in both sides for band-pass and in the right side for low pass mode, is due to the linear convolution process. For further details, see [7]. The renormalization is done using the theory explained in section Edge effects and (3.4).For the representation of the response in time domain it is important to define the starting and ending points of the filter. As all the process is done in MATLAB, talking about time in seconds is talking about samples as well. The gate in time domain is in fact the rectangular window, see (2.7). As one objective of the thesis is to have high performance comparable with the VNA, the function ‘gatef.m’ has inputs in nanoseconds and the gate has four shapes: minimum, normal, wide and maximum. The transform of the rectangular window is done only analytically using (2.11). The spectra of the gate has the same length as the original response, i.e. use (3.5) for low-pass mode.


31

At this point the sinc-function is windowed with a Kaiser window defined by the gate shape utilizing Table II and appendix B for the design. The product of both windows if it is transformed using ICZT to time domain should look like Fig. 2.7 and fulfil the side-lobe level and cut-off time values from the mentioned table. Fig. 3.11 a)-d) shows the gated frequency response made by synthesis and compares it with the same gated response but from the VNA in both modes. The DUT is a long coaxial cable ended with a 1:2 adapter and both outputs are loaded with 50 ohms. From a) till c) the plots are comparable and, but there is a kind of overcompensation notorious in band-pass mode made by the VNA. Such a detail will be explained briefly in Appendix C.

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

n

S11(

V)

bptlptbpt(VNA)lpt(VNA)

a)

0 200 400 600 8000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

n

S11(

V)

bptbpt(VNA)lptlpt(VNA)

b)


32

0 200 400 600 8000.02

0.04

0.06

0.08

0.1

0.12

n

S11(

V)


c)

0 200 400 600 8000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

n

S11(

V)


d)

Fig. 3.11 Linear S11 gated in frequency domain with 801 points in low-pass mode. BW≈6 GHz. a) minimum shape, b) normal shape, c) wide shape, d) maximum shape.

Rather in Fig. 3.11 d) the gated frequency responses in both modes are quite similar. Notice that masking and truncation errors are not considered in this section but they are going to be considered in Chapter 5 as a part of uncertainties. The gate was located in the centre of the time domain response to avoid overcompensation, cf. Fig. 3.6. The time domain option from the Agilent 8753ES is a good option for fault location of components in time domain. It also shows the impedance level as a function of distance. But after the complete time-analysis is done and recognized the errors added due to many renormalizations, the option is not considered to have good accuracy. De- embedding is the next option to work with and this is developed in Chapter 4.

Chapter 4: De-embedding Process

33

Chapter 4

De-embedding Process 4.1 Introduction The S-parameters represent the reflection and transmission coefficient of an arbitrary system. A system conformed by many DUTs can be thought as cascade network and it may represented by the scattering transfer parameters or T-parameters. Fig. 4.1 shows the equivalence to measure a system with the total S-parameters and T-parameters.

Fig. 4.1 The black box formed by three DUTs as a cascade network.

Fig. 4.1 can be seemed as a cascade network where the total S parameter of the black box is the multiplication of each sub-DUT T-parameter. Generalizing, the S-parameters of a black box, formed from ‘ n ’ number of DUTs, can be found using (4.1). [ ]1 2 3* * *...*DUT DUT DUT DUT DUTnS S T T T T= (4.1) The S-parameters are complex because both the amplitude and phase of the input signal are changed by the network. 4.2 S-parameters and T parameters The S-parameters defined by the incident and reflected waves at the two port network is the primary measured quantity of the VNA, for a 2-port VNA, these are defined by

[ ] 11 1221 22

S SS

S S

=

(4.2)

These are dependent of frequency and to guarantee maximum power transmission they must be measured using a matched termination according to the equations in (4.3). For instance, to measure 11S the output port 2 must be terminated with the characteristic impedance of the system, 0 50Z = Ω , that is to say the incident power wave at port 2 is zero, 2 0a = . The S-parameters for a linear system are represented as follows

DUT 1 DUT 2 DUT 3 Port 1 Port 2


34

2

2

1

1

111

1 0

221

1 0

222

2 0

112

2 0

a

a

a

a

bSa

bSa

bSa

bSa

=

=

=

=

=

=

=

=

(4.3)

Fig. 4.2 a) shows the relationship of the incident and reflected power in each port while the Fig. 4.2 b) shows the network representation of the S-parameters for a 2-port VNA.

a)

b)

Fig.4.2 S-parameters of 2 port network. Transmission parameters, or commonly named T-parameters are closely related to the S-parameters. The advantage of the T-parameters is that they are used to determine the effect of cascading networks. The conversions from S to T-parameters and reverse are given in (4.4) and (4.5) respectively; according to definition described in [13]. From S to T:

S11 S22

S21

b1

b2

a2 S12

a1

Two-port network

Port1 Port2

a1(x)

b1(x)

a2(x)

b2(x)


35

( )11

21

1112

21

2221

21

2221

det

1

ST

SSTS

STS

TS

−=

=

−=

=

(4.4)

From T to S:

( )

1211

22

1222

2122

2122

22

det

1

TST

TS

T

ST

TST

=

=

=

−=

(4.5)

4.3 TRL Method and De-embedding The S-parameter of the AEDT can not be measured directly with a 2-port network analyzer. It needs extra adapters and cables that being reciprocals and linear should be easy to remove with the help of de-embedding. The Fig. 4.3 shows how the S-parameters of one branch of the AEDT are measured with the 2-port VNA 8753ES from Agilent.

Fig. 4.3 Measurement setup of the AEDT using HP8753ES from Agilent.

Port2

Port1Zo

Zo

Zo

Zo


36

Added N connectors and (f)-(f) adapters are needed to measure the S-parameters. Cables are indeed added as transmission lines. The conclusion of the chapter 3 led to look for another method which implies more accuracy in the measurements and does not add error itself. The transmission lines, adapters and connectors are extra components, which by principle add phase and modify the amplitude creating a wrong representation of the DUT response. The task of this chapter is to improve the measurements of the AEDT S-parameter, so indeed the remaining blocks (cf. 3.7) must be de-embedded. To measure an AEDT (i.e. splitter 1:5 with β able to modify electrically) with 2-port network analyzer; the remaining ports must be matched with a load the closest to the characteristic impedance of the AEDT, i.e. 50 ohms. The (f)-(f) adapters, which connect the AEDT to the load, must be de-embedded as well because if there exists mismatch in the auxiliary ports, the measurement will not be a good representation of the AEDT. That is why special attention must be taken in every element into the system [20]. Task The task of this chapter is to implement an algorithm which is able to measure the AEDT S-parameter, despite there are blocks which interfere in the measurements. This algorithm must model the error network and then subtract it from the total measurement. The model of each network error is found using an algorithm previously developed by [15]. The algorithm uses TRL (through-reflect-line) method and is used commonly in de-embedding applications [16]. The Fig. 4.3 can be represented with wave cascading matrix (WCM), or a serial of T matrix representation of each element. The equation (4.1) becomes (4.6) for the measurement setup described in Fig. 4.3. [ ]1 1 1 * 2 2 c 2* * * * *DUT N connector TL CBC AEDT CBC TL N conne torS S T T T T T T T− −= (4.6) 4.4 CBC de-embedding algorithm

The CBC or cable-board-cable adapter is a component of vital importance in the measurement process and can not be measure directly. The total S-parameters of the Fig. 4.3 are measured using as before 801 points in low-pass and band-pass mode. It is expected that low-pass mode gives better resolution, cf. (3.3). In order to measured the CBC S-parameters the cable and the N connector must be de-embedded first.

a) Coaxial cable and N-connector: De-embedding As it was stated before, to measure the S-parameters from a C-B-C with 2-port VNA, the use of cable and N-connectors are needed. Then, the first issue is to model the coaxial cable and the N-connector using the algorithm developed in [15]. Thus, they can be removed from the measurement once they are characterized. The Fig. 4.4 can be seen as a WCM of the two N-connectors and the cable and it is represented by (4.7). 1 1 1 2. .c L cT T R T= (4.7)


37

Fig.4.4 WCM representation of the error block: N-connector and transmission line. The S-parameters of a coaxial cable are transformed from (2.15) using (4.4) to T-parameters. The WCM representation of the transmission line is stated in (4.7), where γ is the propagation constant and iL is the line length.

0

0

i

i

L

L L

eR

e

γ

γ

− =

(4.8)

As stated in [15], to perform the measurements, two setups are needed. The first one, the connector-cable-connector with line-length 1L and the second with line-length equal to

2 1L L L= + ∆ . A third setup is needed for the N-connector measurements, and this must be totally reflected, see Fig. 4.

Fig. 4.5 Cables with two line length and one without transmission line, for LRL de-embedding

algorithm. The L∆ must be chosen carefully, this algorithm is applicable only if 4L

λ∆ = and not

2Lλ∆ = .

L1 Tc1 Tc2

L2 Tc1 Tc2

L Tc1 Tc2


38

For UMTS, the band 4 is chosen, f=1.7 GHz to 2.2 GHz, then the line length must fulfil the equations in (4.9).

1.7

2

2.22

4

4

GHz

GHz

L L

L L

λ

λ

= +

= + (4.9)

From (4.9) the allowed values are 3.09L cm∆ = and 2.38L cm∆ = for 1.7 GHz and 2.2 GHz respectively. To avoid wrong measurements the 4.77L cm∆ < and 6.17L cm∆ < , which represent the 2

λ in those frequencies.

The measurements were done using L1=4.78 cm. and L2=8.50 cm. and the N-connector total in reflection: open. If the WCM of each component is represented by R, the equation (4.6) may be represented by the equations (4.10) and (4.11) for each setup, mi. 1 1 1 2. .m C L CR R R R= (4.10) 2 1 2 2. .m C L CR R R R= (4.11) After this algorithm was done, the model of the N-connector and the coaxial cable can be de-embedded using cascading matrices. The S-parameters for both, the N-connector and cable can be represented by (2.14) as a transmission line. From measurements, the 1CS and 2CS do not show reciprocity, e.g. 1 211 11C CS S≠ . The de-embedding is also applied to the Fig. 4.6, where the aim is to find the CBC S- parameters. This is using the same algorithm, LRL.

Fig. 4.7 Board with two CBC soldered in the micro-strip.

The de-embedded reflection response of the CBC is shown in Fig. 4.8.

CBC1 CBC2


39

1400 1600 1800 2000 2200 2400

-25

-24

-23

-22

-21

-20

-19

X: 1703Y: -21.73

X: 2198Y: -19.87

X: 2198Y: -19.91

X: 1703Y: -22.15

f(MHz)

S11

(dB

)

Real S11 of CBC

Fig. 4.8 S11(dB) of a 2 CBC connectors using de-embedding algorithm. Frequency range of

interest (1.7 GHz to 2.2 GHz). The CBC model was developed previously by Powerwave Technologies with a series of equations, which describes the CBC behaviour in a specific region. For the frequency of interest, 1.7 GHz to 2.2 GHz, the S11 is shown in Fig. 4.9.

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3x 109

-25.5

-25

-24.5

-24

-23.5

-23

-22.5

-22

-21.5

-21

X: 1.7e+009Y: -24.29

f(Hz)

S11

(dB

)

Ideal S11 of CBC

X: 2.2e+009Y: -21.98

Fig. 4.9 Ideal representation of CBC connector modelled in ADS, using previous model [23].

Chapter 5: Uncertainties and Sensitivity

40

Chapter 5

Uncertainties and sensitivity of the synthesis response

The uncertainty is related to the level of confidence in a measurement system. In order to quantify the uncertainties of any process, all the factors that could have influence in the result must be taken into account. There are many factors that can influence in the real measurements of a system. For instance, when the frequency response of a filter is measured using an RF equipment; the uncertainties in the equipment, the systematic errors, the resolution of the electronics inside the equipment, the sensitivity and environmental effects will influence the final measurement accuracy. Being awarded of the source of uncertainties in the VNA, it will give more reliable results. In fact, for the time domain analysis and de-embedding methods developed in Chapter 3 and 4 respectively, the uncertainties are not defined until all the processes are known in detail. An important issue that it is aimed in this chapter is to find out if the measurements show the true value or only an approximation of a measurement system. For the evaluation of the uncertainty, the systematic errors, is used in this chapter. A further statistical analysis goes beyond the matter of this thesis but it can be referenced to [21] for further information. 5.1 Uncertainties due to mismatching Harris and Warner in [22] describe the mismatch effects using the ‘U-shape’ the probability distribution. The expression to find the standard uncertainty for U-shape distribution is defined in (5.1).

( )2i

iau x = (5.1)

To translate the uncertainty in length units, the sensitivity coefficient is needed. The sensitivity describes how sensitive the result changes to a particular influence and it is described by the partial derivative of the desired variable to be analyzed, f, in function of the input iX , this is set according (5.2).

i

fSensitivityX∂

=∂

(5.2)

At RF and microwave frequencies the mismatch of components is one of the most important source of errors, because the phases of the reflection are unknown and can not be corrected easily [21]. For a circuit shown in Fig. 5.1, the input reflection depends directly from the load reflection.

12 2111221

Lin

L

S SSSΓ

Γ = +− Γ

(5.3)

If there is not mismatch at port 2, 0LΓ = and 11in SΓ = (ideal case). The motive of this thesis is to show the sensitivity of the antenna network implemented with both methods: time domain analysis and de-embedding. The multi-patch antenna acts as the load in (5.3) and a little reflection will modify significantly the input reflections. The sensitivity of the mentioned system, i.e. AEDT and antenna, can be found using (5.4) based on (5.2).


41

( )

21 122

221in

L L

S SSensitivityS

∂Γ= =∂Γ − Γ

(5.4)

Fig. 5.1 Block diagram of a 2-port network: source and load. The inΓ represents the input

reflection and LΓ the load reflection. The time domain analysis requires gating to filter the influences of undesired components in the final measurement. The equation (5.5) evaluates the sensitivity of the gated response when the reflections is being measured, (5.3) becomes (5.5) when it is gated. To gate a frequency response, this must be convolved with the spectrum of the windowed gate, KW, see (2.9). *gated inKWΓ = Γ (5.5) Replacing (5.3) in (5.5)

( )

21 1211

22

*1

Lgated

L

S SKW SS

ΓΓ = + − Γ

(5.6)

The sensitivity of the gated input reflection is stated in (5.7)

( )* *gated in inL L L

KWKW

∂Γ ∂ Γ ∂Γ= =

∂Γ ∂Γ ∂Γ (5.7)

( )

21 122

22

*1

gated

L L

S SKWS

∂Γ=

∂Γ − Γ (5.8)

The sensitivity input reflection shown in (5.4) must be normalized due the random measurement errors. The new sensitivity depends of the parameters of the instrument, LN. ( ) ( ) ( )20log 20log 20log( / )NL KTB NF dB SF BIF Hz= + + + (5.9)

where, NF : Noise figure of the instrument. BIF: IF bandwidth. SF: Shape factor.

Γin ΓL

[S]


42

Thus, the equation (5.4) might be modified to (5.10), which depends of the random measurement errors.

22 1

in in

L L

Ninin

in

sensitivityL

SNRP

∂Γ ∂Γ∂Γ ∂Γ

= =Γ +Γ +

(5.10)

0 1000 2000 3000 4000 5000 60000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

LoadGated Load

Fig. 5.2 Sensitivity of the gated load.

The sensitivity of the measurement system used in this thesis depends of multiple variables, which have influences in the graphs shown in the chapter 3 in time domain analysis. The gating that generally reduces the measured reflection allows the operator to increase the screen sensitivity to the point where either the trace goes outside the range or the fluctuating noise makes any further expansion of the scale pointless. Measurement at an expanded scale is perceived as an increased sensitivity, but that is a matter having to do with screen and eye resolution. In the non-gated measurement a mathematical treatment of the trace data amplifying the changes experienced upon material changes could work as nice. Also it is noticed that this expression did not show that sensitivity was necessarily larger in the gated case, however the screen effect is real.

Chapter 6: Conclusions

43

Chapter 6 Conclusions

The time domain option 010 from Agilent VNA 8753ES was emulated using reverse engineering. Reverse engineering is used to understand, emulate and, as future work, improve the different functions given in the mentioned option. The emulated time domain option shows good comparison with the time domain option from Agilent. A series of compensation is done in order to approximate the most to the studied VNA. These compensations are source of errors and will have influences in the final screen shoot. The windowing process is carried out using Kaiser window to mitigate the truncation influence. The Kaiser Bessel properties are modified empirically to reach the established by theory. The limited number of samples gathered from the VNA (i.e. 1601) is a source of uncertainty and must be normalized before is processed. The gating function is designed using Kaiser-Bessel window to taper the Fourier transform of the gate and in order to mitigate the Gibbs phenomenon and control the level of side-lobes. This method named filter using windows is chosen for the easy design and the good performance in this application. Unfortunately gating process has two important disadvantages, which are the compensation method used to relieve the convolution. These are the masking effects and the edge effects. The edge effects have big influences on the response and might be solved with the wise decision of the measurement-test engineer. There are edge effects when the gate is located asymmetrically on the wished time domain response. There exists different compensation for both modes transform in time domain analysis. For band-pass mode the overcompensation is in both band edges and for low-pass mode it is in the upper limit. If the operator chooses to use a wider band than the region of interest he may discard the errors in the band edges. But the interested region might have influenced by this overcompensation, so especial care must be taken. Masking effects are not into consideration on this thesis. The results show no compensation errors, but it exist still doubt if those solutions are included in the 010 option since there were some remaining deviation between the synthetic result and the output of the VNA. This compensation is not reached in the present thesis and may be taken into account as future work. A sensitivity metric was developed as a derivative augmented by an ad hoc addition of the noise floor to the denominator. The input reflection measurements depend strongly of the output impedance. The sensitivity of the measurement process even in the gated case depends still on the load reflection. The de-embedding algorithm is proposed as a second method for de-embedding purposes. It is implemented using LRL method for finding by steps (2 by 2) the S-parameter of each component. N-connectors are not reciprocal and must be modelled separately for each system. There exists high relationship with the quality of the connector.


44

CBC connector is an important element into a microwave system. It is made a comparison between the model and the de-embedded response of the CBC and the difference is about 2 dB. Coaxial cables were modelled using the L-R-L algorithm, its S-parameters depend of the line-length and the propagations constant. As a transmission line, it is frequency dependent. The de-embedding is a more cost-time process because each element of the system must be modelled and with the help of T-parameter, subtracted from the system. In this case there are more probabilities to commit systematic errors. Gating a frequency response will give higher uncertainty values, making the time domain analysis a method, even if easy to use, not very accurate. The MATLAB routines developed in this work may help the operator to analyse at what load reflection (impedance) the visual result is showing perfect match. The problem is not so much that the time domain method is associated with error, rather the problem is our ability/inability to describe those errors. The findings of this thesis are at least a partial success in this direction although some deviations remains and some inconsistencies in the result of the network analyzer could be found.


45

References [1] C. Balanis, Antenna Theory: Analysis and design, John Wiley & sons, 2005. [2] “Time-Domain Analysis using a network analyzer,” Agilent Technologies Inc., Santa Rosa, CA.,

Applicat. Note 1287-12 , 2007. [3] Sharrit, D, “Vector network analyzer with integral processor”, US Patent No.4, 703,433. [4] J. Dunsmore, “Time domain network analyzer measurements,” presented at the ARFTG

Microwave measurement Symp., Portland, OR, 9-12 Dec. 2008. [5] “Time domain for vector network analyzers,” Anritsu, Morgan Hill, Applicat. Note 11410, 2003. [6] “Time domain Measurements using Vector network analyzer ZVR,” Rohde & Schwarz, Applicat.

Note 1EZ44_0E, May 1998. [7] S. Smith, The Scientist and Engineer's Guide to Digital Signal Processing, for more information

visit the book's website at: www.DSPguide.com" [8] L. Rabiner et al., ”The Chirp z-Transform Algorithm,” IEEE Trans. Audio and Electro acoustics,

vol. 17, pp. 86 – 92, 1969. [9] F. J. Harris, “On the use of windows for harmonics analysis with the discrete Fourier transform,”

in Proc. of the IEEE, vol. 66, pp. 51-83, Jan. 1978. [10] A. Oppenheim and R. Schafer, Discrete-Time Signal Processing, Prentice Hall, 1999, ch. 7. [11] J. Dunsmore, “Gating Effects in time domain transforms,” in ARFTG Microwave measurement

Symp., Portland, OR, 9-12 Dec. 2008. [12] N. Ridler, ”Connectors, air lines and impedances,” in The 14th IEEE Microwave measurements

training course, 9-13 May 2005 [13] G. Gonzalez, Microwave transistor amplifiers: Analysis and design, Prentice Hall, 1997. [14] D. Pozar, Microwave Engineering, John Wiley & Sons, 1999. [15] B. El-Banna, “Passive inter-modulation from printed circuit boards,” Master thesis, Royal

Institute of Technology, Stockholm, 2006. [16] C. Hoer, G. Engen, “Calibrating a dual six-port or four-port for measuring two-ports with any

connectors,” Microwave Symp. Digest, MTT-S International, 1986. [17] “User’s Guide: Agilent technologies 8753ES Network analyzers”, Printed in USA, 2002. [18] M. Heibel, Fundamental of vector network analysis, Rohde & Schwarz, 2005, ch.5. [19] K. Lu, T. Brazil, “A systematic error analysis of HP8510 time-domain gating techniques with

experimental verification,” Microwave Symp. Digest, IEEE MTT-S International, 1993. [20] J. Collier and A. Douglas, Microwave measurements, Athenaeum Press, 1985, ch. 11. [21] J. Hurll, “Uncertainty and confidence in measurements,” in The 14th IEEE Microwave

measurements training course, 9-13 May 2005. [22] I. Harris, F. Warner, “Re-examination of mismatch uncertainty when measuring microwave

power and attenuation,” Microwaves, Optics and antennas, IEE Proceedings, 1981, pp.35-41. [23] “Cable-board-cable connector: mathematical model”, developed by Powerwave technologies,

2009. [24] D. Slepian et al., “Prolate spheroidal wave functions, Fourier analysis and uncertainty”, Parts I

and II, Bell System Technical Journal, Vol. 40, No 1, pp.43-84, Jan.1961.

Appendix A

46

Appendix A

Matlab functions: Flow Diagrams. In an effort to understand the functions explained in previous chapters, a flow chart for each function is explained in this appendix for the next tasks. 1. Time domain transform: band-pass mode. 2. Time domain transform: low-pas

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