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
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].
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.
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
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.
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, Γ
h( ) t .
( ) ( )
2 j ftH
f
ht e
πdt
∞ −
−∞
Γ = Γ ∫ (2.1)
( ) ( )
j2 fth
t
Hf e
πdf
∞
−∞
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.
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 N
n
k n k
n
X x z
− −
=
= ∑ (2.3)
1
0
exp( 2 / ) 0,1,..., 1
N
k n
n
X
−x j π k N k N
=
= ∑ − = − (2.4)
1
0
0,1,..., 1
N
n nk
k n
n
X x A W k M
− −
=
= ∑ = − (2.5)
Notice that (2.5) is equivalent to (2.4) when A = 1 and W = 1* e
−j2πk N/. 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 z is arbitrary.
k4) 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 z is arbitrary, but this is also
0the case with the FFT if the samples x are multiplied by
nz
0−nbefore
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 N
W 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
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
0
0
2
!
k
k
x I X
k
∞
=
=
∑
The Kaiser-Bessel window is defined as
[ ]
2
0
0
1.0 / 2
( ) 0
2 0
I n
N N
w n n
I
otherwise 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=1 KB=0 KB=3 KB=6
Fig. 2.4 Kaiser-Bessel window for N=100 samples and β α π = . = 1, 0, 3, 6 .
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 β = 6,13 .
350 400 450 500 550 600 650 -160
-140 -120 -100 -80 -60 -40 -20
n
A m pl it ude (dB )
beta=6 beta=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 t G t elsewhere
f DFT t G t
< <
=
Γ = Γ
(2.8)
The spectral response of the ideal gate is the sinc function.
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 t
Γ = Γ (2.9)
and for FD
( ) ( ) ( ) *
gated
f f G f
Γ = 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 A
pand ∂ (delta) respectively.
t1 t2
50 100 150 200
-0.2 0 0.2 0.4 0.6 0.8 1
V
t
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 ( ).
jwtc0 c w n t F w n
c w n e t
=
= ≠ (2.11)
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 t is the offset [11]. In Fig.2.8 the rectangular
cwindow 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
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.
(
2)
2a
1
1.
2ρ = − ρ ρ (2.12)
Where ρ is the apparent reflection of the second discontinuity and
2aρ
1is the reflection of the first reflection.
Fig.2.12 Frequency domain response after gating the second capacitor. Reprinted with
permission from Agilent [2].
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.
(
2)
2 2
1
1ρ
aρ = ρ
− (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.
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, ∆ , the lumped- x element circuit is shown in Fig. 2.14, where Ax = ∆ only for this figure. x
LAx
GAx CAx RAx
U1
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, ∆ . The scattering-matrix of a transmission line can be model directly from its x electrical length, which is θ γ = .l , where γ is the intrinsic propagation constant. In the lossless case γ reduces to j β since, γ α = + j . β . Thus, (2.14) states the S-matrix of a transmission-line [13].
.
.
0 0
i
i
l
li l
S e 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 S
S S S S
S S S
=
(2.15)
If all the ports are matched, there are not reflection in any port, then S
ii= . 0
For passive components, the S-matrix is reciprocal (i.e. S
ij= S
ji) . Then, the S- matrix can be
reduced to
12 13
12 23
13 23
0 0
0
S S
S 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
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
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
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 1
3
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.
-5 0 5 -80
-70 -60 -50 -40 -30 -20 -10 0
t(ns )
S 11( dB )
Band-pas s trans form with m inim um window
Synthes is VNA
-1 -0.5 0 0.5 1
-80 -70 -60 -50 -40 -30 -20 -10 0
X: 0.1206 Y: -13.23
t(ns )
S 11( dB )
Low-pas s trans form with m inim um window
Synthes is VNA
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
response. The time resolution, i.e. distance between two points in TD, for band-pass mode is [18]
1 1
t .
T f N
∆ = =
∆ (3.1)
where N is the length of the gathered data (e.g. 801 points) and ∆ = f f 2 − f 1 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)
2
c
f f
f = + .
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 ( ) ( ) /
0W n = k n M n < ≤ n N
where k n are the Kaiser-Bessel coefficients in each sample point, ( ) n is the start point and
0M 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 e
j2πf tcto compensate the frequency shift, f − [18]. f
cb) 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.
f1 f2 f
( ) f
Γ
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, Γ ( ) 0 = real ( Γ ( ) 1 ) . 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 Γ ( ) 0 = Γ ( ) 1
and the negative frequency response is the complex conjugated of Γ ( 2 : end ) 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
2 1
f f
= n (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 7.49
801
f = GHz ≈ 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
t .2
T 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] ( t 1 and t 2 , see Fig. 2.6).
1 2 3 4 n -4 -3 -2 –1
( ) f
Γ
This rectangular window, defined by t t , is transformed analytically to FD and is there
1,
2windowed 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 ( Γ ( ) w ) , 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
( ) 1
F w =
( ( ) ) ( ) ( ) *
G F w = G w F w
(
comp( ) ) G ( ( ( ) ( ) w ) )
G w
G F w
Γ = Γ (3.4)
The G in front of each response means gate in FD and ( ) * convolution.
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.
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 )
S 11( dB )
Band-pas s trans form with m inim um window
Synthes is VNA
a)
-5 0 5
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
t(ns )
Band-pas s trans form with norm al window
S 11( dB )
Synthes is VNA
b)
-5 0 5 -100
-90 -80 -70 -60 -50 -40 -30 -20 -10 0
t(ns )
Band-pas s trans form with Maxim um window
S 11( dB )
Synthes is VNA
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 β = 0 . b) Normal window β = 6 . c) Maximum window β = 13 . 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
( ) n
LP ( ( 2 : N ) )
1)( ) n n 1, 2, 3,..., N
Γ = Γ Γ = (3.5)
where
1)means mirror and complex conjugate of the Γ ( ) n data.
The total length of the S-parameter before process, is N
LP= 2 * N − , this is due to there is no 1
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
( ) n
LP(
2*N−1)
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: 0.1206 Y: -13.23
t(ns )
S 11( dB )
Low-pas s trans form with m inim um window
Synthes is VNA
a)
-1 -0.5 0 0.5 1
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
t(ns )
Low -pas s trans form with norm al window
S 11( dB )
Synthes is VNA
b)
-1 -0.5 0 0.5 1 -100
-90 -80 -70 -60 -50 -40 -30 -20 -10 0
t(ns )
Low-pas s trans form with m axim um window
S 11( dB )
Synthes is VNA
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 β = 0 . b) Normal window β = 6 . c) Maximum window β = 13 . 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
