**Non-Foster Metamaterials
**

The objective of this project is to do 3D simulation of non-Foster metamaterials.

NOTE: Use the Project Report Template and keep answers to questions on consecutive sheets of paper with all plots at the end.

IN NO CASE may code or files be exchanged between students, and each student must answer the questions themselves and do their own plots, NO COPYING of any sort! Nevertheless, students are encouraged to collaborate in the lab session.

- In the following:

- You will investigate wideband metamaterials, and observe negative relative permeability and negative relative permittivity
- The structure of the wideband negative
permittivity metamaterial consists of an I-beam with a negative capacitor in series with the
post, replacing the inductor that would normally tune the
center frequency of operation. The structure of the wideband negative permeability
metamaterial consists of an split ring with a negative inductor and negative capacitor
in the gap, replacing the capacitor that would normally tune
the center frequency of operation.

- The I-beam structure is closely related to capacitively loaded strips (CLS as described in Design, fabrication, and testing of double negative metamaterials
- The negative capacitors and negative inductors are examples of circuit components that violate Foster's reactance theorem
- Non-Foster circuits are often implemented using an NIC
(negative impedance converter) and/or implemented with current
conveyors that were introduced by Sedra and Smith around
1968

- Is a negative capacitor a non-Foster element? Hint: see http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5729685 ( Q1 )
- Is a negative inductor a non-Foster element? Hint: see http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5729685 ( Q2 )

- NOTE: The project below uses a more accurate logarithm method to recover propagation constant k, as described in A Review of the Scattering-Parameter Extraction Method with Clarification of Ambiguity Issues in Relation to Metamaterial Homogenization The HFSS variable functions did not permit an easy solution to the more thorny issues of determining branches of possible solutions, so extraction results below must be taken with a grain of salt, but are useful as a "first look." This is used in computing real parts of relative permeability "mur" and relative permittivity "epsilonr" below.

**Wideband non-Foster Negative Permittivity**- In this part, you will simulate a simple wideband metamaterial
(having wideband negative permittivity) in a parallel-plate
waveguide

- The structure is very similar to the structures and theory
given in A
Wideband Microwave Double-Negative Metamaterial with
Non-Foster Loading

- Download the HFSS file rfcourseFi8p12.aedt and put it in your hfss directory
- In older versions of HFSS, download the HFSS file rfcourseS15p12.hfss and move it to your hfss
directory

- Open the file in HFSS simulator, and you should see the design 00splitRingIbeamInParPlate_IbeamOnlyNonFos as follows:

Fig 001

- Orient and zoom in on the Ibeam as shown in the red circle
above. Highlight the "Rectangle" RLC sheet (red arrows
above). Save a snapshot of the design as above and
paste it into your report.

- Make sure that your plots, component values, legends, axes, and fonts are legible in your report!
- Determine the negative capacitance in the gap in the post of the I-beam by selecting Boundary::LumpedRLC (blue arrows above) that is placed in the gap (Rectangle at red arrows above ). This "RLC" sheet is used as a negative capacitor in the gap, to enable wideband negative permittivity using the I-beam.
- What is the capacitance (blue arrows above) of the RLC sheet
in the gap of the post of the I-beam?

- Run the simulation (magenta arrow above)
- Open the plot for S-parameters (blue circle above), and
paste a snapshot into your report.

- Open the plot for ReImMuEps (yellow circle above)
(real and imaginary parts of permittivity and permeability),
and paste a snapshot into your report.

- Non-zero imaginary parts of permittivity and permeability
indicate losses or gain in the material. Are the
imaginary parts of permittivity and permeability nearly zero
(lossless condition)? yes/no

- What is the value of the real part of the permeability from 0.1 to 4 GHz, to
within +/- 0.1?

- What is the value of the real part of the permittivity at 1 GHz?

- What is the band of frequencies where the real part of the
permittivity is more negative than than -2.0, ... from 0.1 GHz
to ??? GHz?

- What is the band of frequencies where the real part of the
permittivity is more negative than than -1.0, ... from 0.1 GHz
to ??? GHz?

- This demonstrates that the non-Foster element has a profound effect on metamaterial
performance

- The theory for the negative permittivity is given in
equation (19) of "A
Wideband Microwave Double-Negative Metamaterial with
Non-Foster Loading." Copy this equation to your
report, and explain why the permittivity should be
wideband.

- Save your work!

**Wideband non-Foster Negative Permeability**

- In this part, you will simulate a simple wideband metamaterial
(having wideband negative permeability) in a parallel-plate
waveguide

- The structure is very similar to the structures and theory
given in A
Wideband Microwave Double-Negative Metamaterial with
Non-Foster Loading

- Open the file in HFSS simulator, and you should see the design 00splitRingIbeamInParPlate_ringOnlyNonFos as follows:

- Orient and zoom in on the split ring as shown in the red
circle above. Highlight the "Rectangle2" RLC sheet (red
arrows above). Save a snapshot of the design as above
and paste it into your report.

- Determine the negative capacitance and negative inductance in the gap in split ring by selecting Boundary::LumpedRLC (blue arrow above) that is placed in the gap (Rectangle at red arrows above ). This "RLC" sheet is used to enable wideband negative permeability in the split ring.
- Note: to provide best wideband
performance, split rings should have both negative inductance
and negative capacitance in the gap. The negative
capacitance is necessary to compensate for inherent stray
capacitance in the gap (more details on this are given below).

- What is the capacitance and
inductance (blue circle above) of the RLC sheet in
the gap of the split ring?

- Run the simulation

- Open the plot for ReImMuEps (yellow circle above)
(real and imaginary parts of permittivity and permeability),
and paste a snapshot into your report.

- What is the value of the real part of the permeability at 1 GHz?

- What is the band of frequencies where the real part of the
permeability is more negative than than -2.0, ... from ??? GHz
to ??? GHz?

- This example again demonstrates that the non-Foster element has a profound effect on metamaterial performance

- If there was no stray capacitance in the gap, the "ideal
theory" for the negative permeability is given in equation (9)
of "A
Wideband Microwave Double-Negative Metamaterial with
Non-Foster Loading." Copy this equation to your
report, and explain why the permeability should be
wideband.

- When stray capacitance in the gap is included, the
theory for the negative permeability is given in equation (5)
of "Wideband
Negative Permeability Metamaterial with Non-Foster
Compensation of Parasitic Capacitance." Copy this
equation to your report, and comment on why the general form
of this equation can result in narrowband permeability.

- Save your work!

- Exit the program, File->Exit

- ============================
WARNING !! ====================================

- **** WARNING ****
**YOU MUST USE THE**__PROJECT__R__EPORT__TEMPLATE__Below__: - See the Project
Report Template at bottom of this page

- A well-written report/paper is
EXPECTED

- STRONGLY RECOMMEND that you read IEEE authorship series: How to Write for Technical Periodicals & Conferences
- Clearly describe everything, including:
- variables in block diagrams
- variables in formulas
- units of variables kHz, pF, nH, m, s,

- all traces on plots
- all curves on plots
- all results in any tables

- Minimum
**required data content**for your report and demos - Required theory/formulas numbered as below:
- (1) formula for theoretical mu of non-Foster wideband SSRR (singe split ring resonator)

- (2) formula for theoretical epsilon of non-Foster wideband EDR (electric disk resonator)
- Required figures:
- Any illegible plots receive zero credit (must be able to read all numbers, axes, labels, curves, grids, titles, legends)
- All plots must of professional quality as in IEEE papers

- LEGIBLE snapshot of the non-Foster Ibeam as in Fig 001 above (
**in red circle**), with appropriate caption.

- LEGIBLE plot of S11 and S21 of non-Foster Ibeam as in Fig 001 above (
**in blue circle**) - LEGIBLE plot of Re{mur} and Re{epsr} of non-Foster Ibeam as in Fig 001 above (
**in yellow circle**) - LEGIBLE snapshot of the non-Foster split ring as in Fig 002 above (
**make sure split is visible**) - LEGIBLE plot of S11 and S21 of the non-Foster split ring as in Fig 002 above
- LEGIBLE plot of Re{mur} and Re{epsr} of the non-Foster split ring as in Fig 002 above (
**in yellow circle**) - Required tabular data content:
- Table with 2 columns: parameter, value

- Row 1: capacitance of RLC sheet in Ibeam gap, pF
- Row 2: real part of the permittivity at 1 GHz
- Row 3: inductance of RLC sheet in the split ring gap, nH
- Row 4: capacitance of RLC sheet in the split ring gap, pF
- Row 5: value of the real part of the permeability of the split ring at 1 GHz

**See report template below**

NOTE ReportTemplate: Use the Project Report Template

YOU MUST ADD CAPTIONS AND FIGURE NUMBERS TO ALL FIGURES!!

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