University of Southern Queensland

Faculty of Health, Engineering and Sciences School of Mechanical and Electrical Engineering

Course Number: ELE4605 Course Name: Fields and Waves

Internal ?

Assessment No: 2

External ? This Assessment carries 250 of the 1000 marks total for this Course.

Examiner: Dr Andrew Maxwell Moderator: Dr Tony AhFock

Assignment: Field Analysis

Date Given:

Date Due: Week 4 2016

7th June 2016 Penalty for Late Submission: Loss of 5% of total marks for this assignment per working day late.

Assignments are to be submitted electronically, using the link provided on StudyDesk. Marked assignments will be returned via the StudyDesk feedback system.

Please use PDF format to submit your assignment. Please use the naming convention

StudentNumber-LastName.pdf, where StudentNumber is your 10-digit student number, and LastName is your last (family) name.

By submitting this assignment, you agree to the following Student Declaration:

I hereby certify that no part of this assignment has been copied from any other student’s work or from any other source except where due acknowledgement is made in the assignment. No part of this assignment has been written for me by any other person except where such collaboration has been authorised by the Examiner concerned.

Any non USQ copyright material used herein is reproduced under the provision of Section 200(1)(b) of the copyright Amendment Act 1980.

This document is best used as electronic only. Consider before you print.

Published by

ELE4605 – Fields and Waves

University of Southern Queensland Toowoomba Queensland 4350

www.usq.edu.au

c Andrew Maxwell, University of Southern Queensland 2016.1

Parts sourced from Jim Ball, University of Southern Queensland 2008.1.

Copyrighted material reproduced herein are used under the provision of the Copyright Act 1968 as amended, or as a result of application to the copyright owner.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without prior permission.

Produced using LATEX.

This document is best used as electronic only. Consider before you print.

(Release 16.1)

…/ 3

Contents

i Submission Notes 5

ii Assignment Overview 6

1 Finite differences solution to determine TEM transmission line parameters 7

1.1 Summary of Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Report Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Hints on program design 9

2.1 Relevant sections of text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Hints and Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Numerical Considerations 11

4 Configurations and balanced versus unbalanced operation 13

A Marking Scheme 15

B Transmission line cross-sections 17

B.1 Useful Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

List of Figures

1 Graphical estimation of the correct capacitance value. . . . . . . . . . . . . . 12

2 Graphical representation of configuration excitation methods. . . . . . . . . 14 4

List of Tables

1 Marking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5

i Submission Notes

• All assignments for this course are to be submitted electronically via StudyDesk: http://uconnect.usq.edu.au/

• Paper submissions (i.e. non electronic/eSubmitted) will not be marked. See the Introductory book for this course under “Assessment policies” for full details.

• PDF submissions are the ONLY accepted format.

• This assignment requires a written report detailing your design approach and discussing your findings whilst at the same time addressing the marking rubrics ( See Section A). Your report should include diagrams, figures, waveforms and/or images as appropriate for this assignment.

• Your eSubmission will consist of the following only: (It is your responsibility to identify and upload the correct file. The examiner will not follow up on incorrect submissions)

1. Report in .PDF format

(including all aspects of the marking rubric, and embedded code)

• Late assignments after the due date without extenuating circumstances will incur a penalty of 5% of the assigned mark for each working day late up to a maximum of ten working days at which time a mark of zero will be recorded for that assignment.

If you wish to apply for consideration for late submission without penalty, it must be done at least one week prior to the due date in writing or via email. Include documentary evidence of illness (a medical certificate) or additional work commitments (a written confirmation of changed work circumstances from your supervisor). For extension applications for other reasons, please contact the examiner at least 2 weeks in advance of the due date.

• Students are reminded of the penalties applying to plagiarism. Copying all or part of an assessment from another student, or from the web, is unacceptable. Plagiarism may result in loss of marks, or other penalties as determined by the Academic Misconduct Policy:

http://policy.usq.edu.au/portal/custom/detail/student-academic-misconduct/

Further helpful hints on how to correctly reference (and how to avoid plagiarism) may be found at: http://www.usq.edu.au/library/referencing/what-is-plagiarism

6

ii Assignment Overview

The objectives addressed in this assignment are:

Objective 4. model static and dynamic field problems numerically.

Please consult the course specification for details.

(http://www.usq.edu.au/course/synopses/2015/ELE4605.html)

7

1 Finite differences solution to determine TEM transmission line parameters

1.1 Summary of Task

You are required to write a MatLab(tm) computer program to perform a finite differences solution of a transmission line cross-section using a grid with a square cell.

All node voltages are to be determined simultaneously by solution of a matrix equation as per Section 10.2.2 of the Study book.

You will be notified by week 4 which one of the transmission line cross-sections you have been allocated.

Your computer program must produce the following outputs, (a) through to (c), for at least three (3×) grid spacings, and (d) & (e) for one (1) additional grid spacing:

(a) Contour plots showing equipotentials at 1 volt intervals.

(b) An estimate of the capacitance per-unit-length in the presence of an air dielectric, obtained by means of Gauss’s Law using a contour integration (actually a summation) as per Module 10 of the Study book.

(c) From the capacitance value, a calculation of:

• characteristic impedance (Zo) at high frequencies. (as per Section 2.3.4 of Module

3 in the Study book); AND

• inductance (L) per-unit-length.

(d) You will then repeat this process one more time (using, for convenience, your smallest grid spacing – h, i.e. the largest number of nodes) for your secondary modified geometry

(very closely related to the first ), and again calculate: (continued over the page…)

8

• an estimate of the capacitance per-unit-length in the presence of an air dielectric;

• characteristic impedance (Zo) at high frequencies; and

• inductance (L) per unit length.

(e) You will then determine the design of a quarter wave transformer (?/4) to join these two transmission lines.

A reminder: you must solve the initial transmission line problem ((a), (b), and (c) above) for at least three (3×, and preferably more) significantly decreasing grid spacings and hence examine and clearly demonstrate how the capacitance changes as the number of nodes increase. (See Figure 1, appearing much further on in this document)

You must attach an electronic copy of your program code to allow the examiner to verify:

(i) that you have done the work yourself; and

(ii) that you have used the required method of solution.

Please only embed (submit as a text block inside your assignment) electronic versions of your code into your submission. Do not print-out your program code and post to USQ. Please see the Introductory book for details regarding eSubmission.

1.2 Report Structure

The structure of your report will obviously be largely dictated by the marking scheme (Appendix A of this document). It should follow the guidelines established in ENG1001 Principles of Professional Engineering and Surveying which is available on the course page for this course.

Poor presentation and organisation of the report will be penalised, particularly essay-style ramblings.

Any additional samples of output which are not critical to the main body of the report but which you wish to present as supplementary information may be included as Appendices.

9

2 Hints on program design

2.1 Relevant sections of text

The principle behind the finite differences method is that each node voltage should be the average of the voltages at the four nearest neighbours.

4V (i,j) ~= V (i,j + 1) + V (i,j – 1) + V (i + 1,j) + V (i – 1,j) (1)

This is deduced in Section 10.2.2 in the Study book and should be completely understood before proceeding.

2.2 Hints and Tips

Always use a grid with equal distances between nodes in the horizontal and vertical directions, i.e. each cell of the grid should be a square, with sides of length h. This will lead to some problems in accommodating curved boundaries, but the errors will reduce as the number of nodes increases (i.e. cell size decreases).

If your transmission line cross-section has two-fold symmetry, then your program should only need to solve for the node voltages over one quarter of the cross section. On either side of a line of symmetry the voltages of comparable nodes will be equal. On the centre line of a cross section with balanced voltages there will be a zero-volt equipotential, and comparable nodes on either side of this will have voltages which are equal in magnitude but opposite in sign.

The numbers of nodes required in each direction must be chosen to fit the dimensions of the cross-section. To get your computer program working initially, start with a minimum set of nodes. Example 10.1 in the Study book shows that as few as one hundred nodes is sufficient to obtain an answer of the right magnitude, even if it is not very accurate.

To get a reasonably accurate solution you will probably need several thousand of nodes in each direction. Therefore you will end up with millions of nodes in your solution, possibly even more (+10M nodes). It will not be possible for you to manage this many nodes unless your program is designed to create the problem matrices automatically. You will be able to quickly determine the physical limitations of your computer/software.

Because contour plots of equipotentials and manipulation of significantly large matrix data sets are required, it is recommended that on-campus day students use MatLab(tm), and externals use the same, or GNU Octave(tm). See Appendix B.1 for useful information, but note that existing field plotting packages are only mentioned here on the understanding that they could be used to verify your own program.

10

2.3 Getting Started

To help you get started, here is a summary of the steps which must be accomplished within your program:

1. Decide on a number of nodes in a grid pattern of square cells which fits the geometry of the cross-section of the transmission line. Take full advantage of symmetry. Grid lines (rows or columns of nodes) should coincide with conductor boundaries. All signal conductors which are flat strips can then be represented by fixing the voltages of a row of nodes. Remember that you will need to change the cell size at least thrice (3×) so make provision for this. Nodes may be numbered, or referred to by their location in the grid.

2. Nodes defining the conductor perimeters must be initialised permanently to the conductor voltages, e.g. 0, ±5 or +10 volts as appropriate (See Section 4 for important information regarding this). In the case of curved boundaries the nodes will not lie exactly on the boundaries and you will have to select the nearest nodes and initialise these, i.e. your effective boundaries will be a bit jagged. Nodes which are in the interior of a conductor may be set to the conductor voltage (i.e. there is no skin effect, and the voltage is the same at all points within the conductor).

3. Set up and solve the finite differences matrix equation. Matlab(tm) and GNU Octave(tm) have special facilities for sparse matrices.

4. These packages can then produce contour plots for you, e.g. at 1 volt intervals.

5. Calculate the charge q by performing a contour integral (actually a summation) as described in Module 10 of the Study book. This is a fairly crude technique, and you will probably get more accurate answers when the contour is far away from the signal conductors. The results will vary somewhat with contour position. If your configuration has a rectangular shield then the outer contour could coincide with the nodes defining the shield, so that one of the summations will be zero. Then calculate the capacitance as :

6. Calculate the characteristic impedance AND inductance.

7. Reduce the node spacing to create more nodes, and repeat steps 1 to 6.

You may use this as the basis for an initial flowchart.

11

3 Numerical Considerations

When creating numerical analysis software it is always difficult to verify the results or to estimate their accuracy. Sometimes a theoretical solution or an accurate set of measurements can be used as a check, but often these are not available.

Common sense should suggest whether the result is of about the right magnitude. For example, most practical transmission lines used for high frequency applications have characteristic impedances in the range 25? to 200?.

With an air dielectric this corresponds to a capacitance per unit length range of 133.3pF/m down to 16.7pF/m.

If your program produces values well outside this range they are almost certainly wrong. Likewise, the equipotential plots should be scrutinised to see if they look physically reasonable. Compare them with pictures from textbooks. Do they look sensible? It is best to take a pragmatic approach to questions like:

• How many nodes are required for a given accuracy?

• Where is the best location for the contour around which the numerical integration is performed to estimate charge and hence capacitance?

Vary these parameters in your program and note the changes in capacitance values and equipotential distributions.

To estimate the errors in the capacitance result due to the number of nodes being limited to a finite value, a good method is to plot a graph. The calculated capacitance should be shown on the vertical axis, versus the cell dimension (h) plotted on the horizontal axis, as in Figure 1 below (next page). If three or more results are available the graph will probably show a trend line or curve which can be projected back to cut the vertical axis, to give an estimate of the true result.

12

Figure 1: Graphical estimation of the correct capacitance value.

13

4 Configurations and balanced versus unbalanced operation

The conductor configurations which follow (found in Appendix B) represent cross sections of various possible TEM mode transmission lines.

Note that these configurations vary according to the following taxonomy.

Single Signal Conductor

A transmission line comprising a single signal conductor enclosed within a shield or between a pair of ground planes will only ever be operated in unbalanced mode with the shield ( or ground planes) grounded, i.e. at zero volts. To analyse this type of cross-section, assume the centre conductor is held at +10 volts with respect to ground.

Dual/Two Signal Conductors

A transmission line which consists of two conductors plus a shield may be operated in either balanced or unbalanced mode.

The shield (or ground planes) will always be grounded, i.e. at 0 volts.

Balanced mode

In balanced mode (differential mode) operation the voltages on the two conductors will be equal and opposite, so if you have been allocated a configuration of this type then assume the conductor voltages are +5 and -5 volts. If the cross-section has symmetry about the vertical axis, this will coincide with the zero volt equipotential. Comparable nodes on either side of this axis will have voltages which are equal in magnitude but opposite in sign. The integration contour should enclose only the positive conductor. It would be simplest if the outer contour coincided with the shield, and the 0 volt equipotential on the centre line.

Unbalanced mode

In unbalanced mode both conductors are at the same voltage with respect to ground, e.g. +10 volts. If the cross section has symmetry about the vertical axis, comparable nodes on either side of the axis will have the same voltages. The integration contour should enclose both conductors, and is best positioned against the shield so that the outer summation is zero.

14

The above taxonomy can be represented as follows in Figure 2.

Figure 2: Graphical representation of configuration excitation methods.

15

A Marking Scheme

Marking scheme appears on next page in Table 1.

NB: If you copy diagrams, text or program code from another source and this is discovered, you will lose most of the marks for this assignment.

16

Table 1: Marking Scheme

Criterion Requirement to pass Marks

Theoretical knowledge A concise presentation of the relevant theory and how it is implemented 20

Program documentation Flowchart and brief description of your program, together with a full program listing. The program listing should be included as an appendix in your report, and provide sufficient commentary, that is be largely self documenting. DO NOT send a disk, NOR physically-post any material. 20

Equipotentials or node voltages As proof that your program is working, and to allow the examiner to check that your solution is physically sensible, you are required to supply plots of the equipotentials for at least 3 node size configurations. These plots should clearly show the conductor geometry, node configuration, the number of nodes, and clearly show the equipotentials. In terms of example node size configurations, examples might be 10,000 nodes, 100,000 nodes, and 1,000,000 nodes. 90

Numerical results and convergence The calculated capacitance-per-unit-length should be of the right order for the configuration you have been allocated (You may have to estimate it in order to compare). As noted above, you should solve the problem at least thrice (3×) using significantly different cell sizes so that convergence can be demonstrated. 40

Characteristic impedance,

and induc-

tance The characteristic impedance and also the inductance per unit length ( for every node number size you calculate) should be calculated from your capacitance results, assuming an air dielectric. Again, these values are expected to be close to previous results for the configuration allocated to you. 20

Second char-

acteristic

impedance, inductance,

and ?/4 transformer The characteristic impedance and also the inductance per unit length for the secondary geometry, assuming an air dielectric. Quarter wave transformer design to join these two transmission lines. 20

Critical evaluation An assessment of the accuracy of your results. Do they demonstrate convergence to a final value? What is the residual error, as estimated from a graph such as Figure 1 (over page). Where possible, make a comparison with known values for similar transmission lines (for obscure geometries, you may have to estimate). Discuss in your report whether your results are adequate, and how they might be improved. 30

Report presentation Your report should be logically organised, and divided into subsections with appropriate headings. The report will be enhanced by including relevant diagrams and illustrations. Extra supporting material which is not essential should be included as appendices. The report should comply with Faculty guidelines established in ENG1001 Principles of Professional Engineering and Surveying. 10

Submit Submit electronically using StudyDesk on or before the due date Total 250 marks

17

B Transmission line cross-sections

NB. Strip conductors with negligible thickness should be represented by a simple row ( or column) or nodes.

Below appears your geometries to solve as part of this assignment. Each geometry has a secondary design, described as accompanying text.

1. Stripline in a box

Secondary geometry: Ground Box is 4 units wide instead of 5 units.

2. Edge-coupled stripline in a box

(a) Balanced & (b) Unbalanced

3. Broadside stripline in a box Secondary geometry: Ground Box is 5 units wide instead of 6 units.

(a) Balanced & (b) Unbalanced

continued… Secondary geometry: Ground Box is 4 units wide instead of 5 units.

18

4. Offset coupled stripline in a box

(a) Balanced & (b) Unbalanced

Secondary geometry: Ground Box is 5 units wide instead of 6 units.

(NB: this cross-section is not symmetrical about either axis. Instead, nodes in diagonally opposite quadrants will be related. You will need to solve for at least half the node voltages. You may however wish to solve for the entire geometry without any symmetry lines.) 5. Ribbon line

6. Coupled ribbon line Secondary geometry: Ground Box is 5 units wide instead of 4 units.

(NB: useful in switches.)

(a) Balanced & (b) Unbalanced

continued… Box is 4 units wide instead of 5 units.

19

7. Channel line

Secondary geometry: Ground Box is 5 units wide instead of 4 units.

8. Coupled channel line, type 1

(a) Balanced & (b) Unbalanced

9. Coupled channel line, type 2 Box is 7 units wide instead of 6 units.

(a) Balanced & (b) Unbalanced

continued… Box is 7 units wide instead of 6 units.

20

10. Square conductor slab line

Box is 6 units wide instead of 5 units.

11. Coupled square conductor slab line

(a) Balanced & (b) Unbalanced

Box is 7 units wide instead of 6 units.

12. Coupled square conductors, in a square shield

(a) Balanced & (b) Unbalanced

…conclusion of geometries. izontal “1 unit” width spacings increased to 2. Vertical spacing should hence be 2, 1, 1, 1, 2 units. Vertical spacing remains unchanged.

21

B.1 Useful Links

• MatLab ?http://www.mathworks.com.au/

• Octave ?http://www.gnu.org/software/octave/

• Arbitrary Transmission Line Calculator ?http://atlc.sourceforge.net/

• QuickField ?http://www.quickfield.com

• Vector Fields ?www.vectorfields.com

• PDFill ?http://www.pdfill.com/

Click here to order this paper @Superbwriters.com. The Ultimate Custom Paper Writing Service