Development of a Predictive Shielding Effectiveness Model for Carbon Fiber/Nylon Based Composites

By

Nicholas B. Janda

Bachelor of Science, Case Western Reserve University, 2003

A Thesis

Submitted to the Graduate Faculty

of

Michigan Technological University

In partial fulfillment of the requirements

For the degree of

Master of Science

In

Chemical Engineering

Houghton, Michigan

August 2004

© Nicholas B. Janda

This thesis, “Development of a Predictive Shielding Effectiveness Model for Carbon Fiber/Nylon Based Composites,” is hereby approved in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in the field of Chemical Engineering.

DEPARTMENT – Chemical Engineering

Signatures:

Thesis Advisor: ____________________________

Dr. Julia A. King

Thesis Co-advisor: ___________________________

Dr. Jason M. Keith

Department Chair: ______________________________

Dr. Michael Mullins

Date: ________________________________________

i

Abstract

“Development of a Predictive Shielding Effectiveness Model for Carbon Fiber/Nylon Based Composites”

The need for electromagnetic interference (EMI) shielding materials has increased recently due to the more prevalent use of personal communications devices (cell phones, pda’s). Metals have typically been used the material of choice for shielding applications. Design weight limitations for highly portable devices, however, has limited the applicability of metals for these applications. A need for light weight materials capable of providing EMI shielding exists. Through the addition of conductive fillers to normally electrically insulating polymer resins, electrically conductive composites can be used for shielding applications, providing light weight shielding materials. Shielding theory for composite materials, however, is largely undeveloped, unlike for metals. These models developed for metals cannot be used to accurately predict the shielding effectiveness provided by a composite containing a wide range of conductive fillers . The shielding effectiveness (SE) of two different carbon fiber/nylon based composites was studied over the radio frequency range (300 to 1000 MHz). The effects of incident electromagnetic wave (EM) frequency, filler volume percent, filler size (radius), and filler orientation on the measured SE were examined. The objective of this analysis is to characterize the factors involved in determining the SE of a composite from first principles. From this analysis, a model predicting shielding effectiveness for carbon fiber/nylon based composites is developed. The model is expected to perform well at low filler loadings, but also can be used to accurately predict shielding effectiveness at filler loadings above the percolation threshold, as seen from comparisons of the model to experiments with ThermalGraph™ and Fortafil carbon fibers in nylon 6,6.

ii

Acknowledgements

I must first acknowledge the Michigan Technological University Graduate School for

providing me the opportunity to further pursue my degree. I have traveled a long road to get

to this point and feel fortunate to complete my degree at Michigan Tech.

I also need to thank Dr. Warren Perger for providing the guidance, knowledge and focus

needed to complete the project. Thank you for teaching me how to “think/see like a wave”

and providing such truisms as “in the land of the blind the one eyed man is king.”

I must thank my co-advisors, Dr. Julia King and Dr. Jason Keith. Your input and aid was

always valued. I truly appreciated your consistent enthusiasm and support when the project

left the realm of typical Chemical Engineering. Thank you for your patience in dealing with

an atypical situation, project and student.

I would be remiss if I did not acknowledge the MATLAB assistance of Troy Oxby and

Dr. Jason Keith. Thank you for helping me rediscover my programming skills and showing

me that the program has more to offer than just Simulink.

The generosity of the National Science Foundation must be acknowledged. The funding

provided through Award Number DMI-9973278 allowed for prior fabrication of the samples

investigated in this study.

I must thank Brian Ott and Chris Copeland for providing a nearly endless amount of

distractions. Carrie Majkrzak, you deserve a medal of honor for sharing an office with me for

the past year.

Finally, I need to thank my Mom and Dad. Thank you for putting up with my educational

pursuits and never losing faith when things did not go smoothly. Your help and support along

the way has never gone unappreciated. A special thank you to my Mom: thanks for never

letting your level of frustration reach a point to where you felt it was necessary to strangle

your son.

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Table of Contents Abstract.......................................................................................................................... i Acknowledgements ...................................................................................................... ii Table of Contents ........................................................................................................ iii List of Figures................................................................................................................v List of Tables .............................................................................................................. vii CHAPTER 1: Introduction..........................................................................................1

1.1 Electromagnetic Radiation and Interference...................................................1 1.2 Polymer Based Composite Materials..............................................................3 1.3 Predicting Shielding Effectiveness in Composite Materials...........................3 1.4 Project Outline ................................................................................................5

CHAPTER 2: Project Materials and Sample Formulation ......................................6

2.1 Introduction.....................................................................................................6 2.2 Materials .........................................................................................................6 2.3 Sample Preparation .........................................................................................8

2.3.1 Extrusion.................................................................................................8 2.3.2 Injection Molding..................................................................................10

2.4 Formulations .................................................................................................12 CHAPTER 3: Experimental and Characterization Methods.................................13

3.1 Introduction...................................................................................................13 3.2 Electrical Resistivity .....................................................................................13

3.2.1 Transverse Electrical Resistivity Test Method .....................................13 3.2.2 Longitudinal Electrical Resistivity Test Method ..................................13

3.3 Shielding Effectiveness.................................................................................15 3.4 Balance of Power Analysis ...........................................................................18 3.5 Fiber Volume Fraction, Fiber Length and Aspect Ratio...............................19 3.6 Orientation ....................................................................................................20

3.6.1 Fiber Orientation...................................................................................20 3.6.2 Transmission Orientation Dependence .................................................20

CHAPTER 4: Experimental Results.........................................................................22

4.1 Introduction...................................................................................................22 4.2 Shielding Effectiveness Results....................................................................22

4.2.1 Pure Nylon 6,6 ......................................................................................22 4.2.2 ThermalGraph™ DKD X......................................................................23 4.2.3 Fortafil 243............................................................................................25

4.3 Balance of Power Results .............................................................................26 4.4 Orientation Results........................................................................................29

CHAPTER 5: Electromagnetic Theory ....................................................................32

5.1 Introduction...................................................................................................32 5.2 Shielding Theory...........................................................................................32

iv

5.3 Scattered Field Theory..................................................................................33 5.4 Scattered Field Equation Derivation.............................................................35

5.4.1 Maxwell’s Equations ............................................................................35 5.4.2 Permittivity - Absorption Loss..............................................................36 5.4.3 Phasor Notation.....................................................................................38 5.4.4 Wave Equation Solution – Incident Field.............................................39 5.4.5 Wave Equation Solution – Scattered Field ...........................................42

5.5 Scattering Width ...........................................................................................49 CHAPTER 6: Shielding Effectiveness Model Design..............................................52

6.1 Introduction...................................................................................................52 6.2 Review of Problem Description and Focus ..................................................52 6.3 Analysis of Scattering Equations ..................................................................53

6.3.1 Dependence on Frequency, Optical Radius and Distance From Scatterer to Observer............................................................................................................53 6.3.2 Deterministic Nature of Scattering Equations ......................................55

6.4 Accounting for Collision Probability............................................................57 6.5 Scaling Factor Analysis ................................................................................59 6.6 Shielding Effectiveness Model Results ........................................................62 6.7 Scaling Factor - Linear Fit ............................................................................65 6.8 White Model Comparison.............................................................................68

CHAPTER 7: Conclusions and Future Work..........................................................71

7.1 Thesis Goal ...................................................................................................71 7.1.1 Conclusions from Electrical Resistivity/Conductivity Experiments ....71 7.1.2 Conclusions from Shielding Effectiveness Experiments......................71 7.1.3 Conclusions from Power Balance Analysis ..........................................72 7.1.4 Conclusions from Fiber Orientation Studies.........................................72 7.1.5 Conclusions from Model Development and Analysis ..........................72

7.2 Future Work ..................................................................................................74 CHAPTER 8: References ...........................................................................................76 Appendix A: Formulation Summary .......................................................................78 Appendix B: Shielding Effectiveness Experiment Results .....................................81 Appendix C: Balance of Power Results (mW) ........................................................88 Appendix D: Reflected, Absorbed and Transmitted Signal Results in dB...........95 Appendix E: Scaling Factor Analysis.....................................................................102 Appendix F: Shielding Effectiveness Model Results ............................................109 Appendix G: Shielding Effectiveness Model Results............................................116 Appendix H – White Model Derivation ..................................................................123 H.1 Introduction ....................................................................................................123 H.2 Absorption Term Derivation ..........................................................................124 H.2 Reflection Loss Term Derivation ...................................................................125 Appendix I: Proposed Model Comparison to White Model ................................128

v

List of Figures Figure 1.1-1: IEEE Standard for Safety Limits on Human Exposure to RF Fields (3) .2 Figure 2.3-1: Leistritz Extruder Used for Compounding of Composites ......................8 Figure 2.3-2: Extruder Screw Design, Note Flow is From Right to Left ......................9 Figure 2.3-3: Niigata Injection Molder........................................................................10 Figure 2.3-4: Four Cavity Mold...................................................................................11 Figure 2.3-5: Shielding Effectiveness Disk .................................................................11 Figure 3.2-1: Bar From Which Longitudinal Electrical Resistivity Samples Were Cut

...............................................................................................................14 Figure 3.2-2: (A) Experimental Set-up for Four Probe Test Method, .........................15 Figure 3.3-1: Shielding Test Fixtures With Support................................................16 Figure 3.3-2: Transmission Holder Without Sample...................................................16 Figure 3.3-3: Cross Sectional View of Transmission Holder (24) ..............................17 Figure 3.3-4: Reference and Load Shielding Effectiveness Disks (24).......................17 Figure 3.3-5: Reference Disk Alignment on Trasmission Fixture (25) .......................18 Figure 3.4-1: Shielding Test Apparatus Schematic (25)..............................................19 Figure 3.6-1: Dipole Antenna and Sample Holder ......................................................21 Figure 4.2-1: Shielding Effectiveness for Pure Nylon 6,6...........................................23 Figure 4.2-2: Shielding Effectiveness As a Function of Filler Volume Percent At

Select Frequencies ................................................................................24 Figure 4.2-3: Shielding Effectiveness Results for ThermalGraph™ DKD X .............25 Figure 4.2-4: Shielding Effectiveness Results for Fortafil 243 ...................................26 Figure 4.3-1: Balance of Power Results (mW) for NCN20 (ThermalGraph™)..........28 Figure 4.3-2: Balance of Power Results (mW) for NDN20 (Fortafil 243) ..................28 Figure 4.4-1: NDN40 Fiber to Incident Wave Orientation Dependence for

Transmitted Signal Strength ...............................................................29 Figure 4.4-2: Depictions of Perpendicular and Parallel Fiber to Wave Orientations ..30 Figure 4.4-3: Carbon Fiber/Epoxy Sheet Fiber to Incident Wave Orientation

Dependence for Transmitted Signal Strength ......................................31 Figure 5.2-1: Representation of Shielding Phenomena for Plane Waves Passing

Through a Homogeneous Barrier (10)..................................................33 Figure 5.3-1: A cylinder Impinged by a Uniform Plane Wave....................................34 Figure 5.4-1: Electromagnetic Frequency Spectrum (25)............................................37 Figure 5.4-2: Cross Sectional View of Transmission Holder (24) ..............................39 Figure 5.4-3: Cylindrical Coordinate System ..............................................................41 Figure 5.4-4: Block diagram Depicting the Two Step Process for Solving for the

Radiated Fields Given a Current and Charge Source (32)..............43 Figure 5.4-5: Diagram of the Position Vectors. The vector potential A at ρ is

obtained by integrating the current J at 'ρ . (3) .................................46 Figure 5.4-6: Uniform Plane Wave of TMz Orientation Impinging a Single

Cylindrical Scatterer With Radius a (32)............................................48 Figure 5.5-1: Cross Sectional View of Transmission Holder (24) ..............................50 Figure 6.3-1: Near Zone (ρ = 1.0 x 10-4 m) Scattering Width for Both Fibers............54 Figure 6.3-2: Far Zone (ρ = 50 m) Scattering Width for Both Fibers .........................55

vi

Figure 6.3-3: Theoretical Shielding Effectiveness of a Single Carbon Fiber Scattering an Incident Wave ..................................................................................56

Figure 6.4-1: Sample Wavelength Sized Window For Shielding Disk .......................58 Figure 6.5-1: Scaling Factor Analysis for NCN05 ......................................................59 Figure 6.5-2: Scaling Factor Analysis for NDN05 ......................................................60 Figure 6.6-1: Model Predicted and Experimentally Determined Shielding

Effectiveness for NCN05......................................................................63 Figure 6.6-2: Model Predicted and Experimentally Determined Shielding

Effectiveness for NDN05......................................................................63 Figure 6.6-3: Model Fit Quality Analysis for ThermalGraph™ Based Composites ...64 Figure 6.6-4: Model Fit Quality Analysis for Fortafil Based Composites ..................65 Figure 6.7-1: Linear Fit Applied to ThermalGraph™ Scaling Factor Data.................66 Figure 6.7-2: Linear Fit Applied to Fortafil Scaling Factor Data ................................66 Figure 6.7-3: Model Predicted and Experimentally Determined Shielding

Effectiveness for ...................................................................................67 Figure 6.7-4: Model Predicted and Experimentally Determined Shielding

Effectiveness for ...................................................................................68 Figure 6.8-1: White Model and Proposed Model Comparison and Experimentally

Determined Shielding Effectiveness for NCN10..................................69 Figure 6.8-2: White Model and Proposed Model Comparison for NDN40 ................70

vii

List of Tables Table 2.2-1: Properties of DuPont Zytel 101 NC010 (15).............................................6 Table 2.2-2: Properties of BP/Amoco ThermalGraph DKD X (16)..............................7 Table 2.2-3: Properties of Akzo Nobel Fortafil 243 PAN based 3.2mm Chopped and

Pelletized Carbon Fiber (17)......................................................................7 Table 2.3-1: Extrusion Conditions for Nylon 6,6 (19)...................................................9 Table 2.3-2: Injection Molding Conditions for Conductive Nylon .............................11 Table 2.4-1: Loading Levels for Composite Samples Studied ....................................12 Table 6.5-1: ThermalGraph™ Scaling Factor Results.................................................61 Table 6.5-2: Fortafil Scaling Factor Results ................................................................61

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CHAPTER 1: Introduction

1.1 Electromagnetic Radiation and Interference

In today’s electronic age, electromagnetic (EM) fields are radiated from numerous

sources. EM waves with frequencies in the range of approximately 0.3 to 1000 MHz (Radio

Frequency - RF range) are used for communications signals (radio, television, cellular

telephones). The emitted fields from these communications devices can interfere with the

operation of other nearby electronic equipment. This situation is known as electromagnetic

interference (EMI). Some adverse effects of EMI are connectivity problems in cellular

phones, interrupted television signals and even data corruption on computer hard drives.

Along with interfering with the operation of electronic devices, EMI in the RF band may have

harmful biological effects. Some studies have found a correlation between length of exposure

time to the EM fields emitted from power lines and leukemia occurrences (1). There is also

increasing concern that EMI might adversely affect the operation of biological devices such as

pacemakers (2). IEEE currently provides a standard for safety limits on exposure to RF

electromagnetic waves, shown in Figure 1-1.

As the number of communications devices in use has drastically increased over the recent

past decades, stringent regulations controlling the field strength emitted by electronic devices

have been instigated by the Federal Communications Commission, producing a need for EMI

controlling materials (4). Along with external interference concerns, the trend of personal

electronics miniaturization has resulted in devices containing densely packed electronic

components. Due to the close proximity, the EM fields generated by the internal components

may interfere with each other, resulting in electromagnetic incompatibility problems. A

material capable of controlling the amount of EMI radiated between the components is

essential.

2

Figure 1-1: IEEE Standard for Safety Limits on Human Exposure to RF Fields (3)

EMI radiation control is known as shielding. Materials with known shielding ability are

used to encase an electronic product to prevent it from emitting or receiving unwanted

electromagnetic energy. The ability of a material to resist the passage of an EM signal is

quantified as shielding effectiveness (SE). The SE of a material is ratio of the power received

with and without a material present for the same incident signal power. It is expressed in units

of decibels (dB), as shown in Equation 1.1-1

2

110PP

log SE 10dB = [1.1-1]

Where: 1P = received power with the material present (watts)

2P = received power without the material present (watts)

3

1.2 Polymer Based Composite Materials

Until recently, electrically conductive metals were most commonly used to provide EM

shielding. For example, plastic computer cases are usually lined with a thin metal shroud to

control EMI emissions. Weight considerations decrease the viability of metal shields in

portable electronics. For example it is disadvantageous to use internal metal shrouds lap-top

computer cases. The demand for low cost, low weight shielding materials has shifted the

focus to plastics. Most polymer resins are electrically insulating, and therefore, typically

incapable of providing EM shielding. Through the addition of conductive fillers, such as

conductive metal fibers or carbon fibers, the electrical conductivity of these resins is increased

and acceptable shielding ability is obtained (4-7). An electrically conductive composite can

be used for computer cases and cell phone housings without the need for an extra metallic

shield. These devices retain the light weight desired by consumers and meet the FCC

guidelines.

1.3 Predicting Shielding Effectiveness in Composite Materials

The utility of different types of fillers for shielding applications has been thoroughly

researched (2,4-7). Bigg has experimentally studied composites based on: carbon black,

carbon fibers, metal fibers, metal flakes and metal-coated glass fibers (4-7). The long standing

reliance on shielding metals has produced a void in shielding theory for composite materials.

In contrast, the shielding behavior for metals is well understood (8). The work of White is

typically referenced in the EMI composite shielding literature (2,4-7,9) as a viable model for

shielding effectiveness in composites. The White model, however, is predicated on

assumptions that decrease its applicability and validity for composite materials. The model

proposed by White was derived for a homogeneous planar metallic barrier (10). Applying it

to a composite assumes that the filler particles behave similar to a uniform pure metal. Unlike

4

a pure metal, the composite are not homogeneous and the material will not present uniform

resistance to the EMI signal.

Theoretical work focusing on shielding effectiveness of composite materials is currently

limited to the plane wave shielding characteristics of periodic, anisotropic laminated

composites (11-14). Both Chen and Krohn developed theory to model the shielding behavior

of laminated composites made of several plies of fiber-reinforced panels with various fiber

orientation patterns. The panels were assumed to be composed of regularly spaced,

unidirectional collimated fibers imbedded in a polymer resin. The resin was modeled as a

dielectric material, translucent to electromagnetic waves. Chen and Krohn both predicted a

direct relationship between impinging wave frequency and shielding effectiveness (11-14).

Also noticed was a preferred fiber orientation for the strength of signal reflected from the

composite. Fibers oriented parallel with the electric field were predicted to reflect 20 dB more

than fibers oriented perpendicular (13).

The complexity of non-periodic, non-laminar composites has typically discouraged

researchers from focusing on composites formed via injection molding. The works of Chen

and Krohn provide some insight into the shielding behavior of composite materials but the

usefulness of their proposed models for non-laminar composites is quite limited. The

materials analyzed by Krohn consisted of only 5 large filaments, spaced widely apart

(Filament radius: 317 µm, Spacing: 5.69 cm) (14). These conditions are unrealistic for

injected molded parts typical used in personal communications devices. Injected molded parts

commonly utilize densely packed conductive fillers with radii several orders of magnitude

smaller.

Because of the simplistic nature of the analyses conducted by Chen and Krohn, the

probability of the incident electric field colliding with a fiber was not investigated. Since the

unwanted signal will only be impeded when it encounters a conductive filler within the

composite, determining the probability of a signal/filler interaction is key. Both researchers

5

instead focused on developing direction dependent, effective expressions to describe the

properties of the material as a whole (11-14). For complex anisotropic composites containing

non-uniform fiber spacing, tensor expressions describing the effective properties of the

composite are difficult to obtain.

The probability of a wave/fiber collision is dependent on a multitude of factors: the

apparent size of the filler (scattering width), the wavelength of the incident wave, the

orientation of the filler to the wave and the volume fraction of filler in the resin. The amount

of shielding provided by a single strand of filler is dependent on the electrical conductivity of

the filler and frequency of the incident wave. Pursuant to the goal of predicting shielding

effectiveness in conductive composite materials, each factor will be characterized and

investigated in this study. A model capable of predicting shielding effectiveness in complex

injection molded composites will be proposed.

1.4 Project Outline

Chapter 2 discusses the creation of carbon fiber/nylon based composites. The properties

important in determining the shielding effectiveness of a material (electrical conductivity,

filler orientation, filler volume fraction) and the shielding effectiveness of the composites are

measured using techniques shown in Chapter 3. The results from the analyses are detailed in

Chapter 4. A theory to describe the shielding effectiveness of a material is developed in

Chapter 5. A basic discussion of electromagnetic wave theory is presented, starting with the

basic laws of electricity and magnetism and culminating with the derivation of an equation

describing a reflected electric field. The equations developed in Chapter 5 will be applied to

the data accumulated during the shielding experiments, resulting in the creation of a predictive

model for the shielding effectiveness of nylon 6,6 based composites in Chapter 6. Chapter 7

will present the conclusions obtained throughout the analysis and possible future areas of work

to investigate.

6

CHAPTER 2: Project Materials and Sample Formulation

2.1 Introduction

This chapter discusses the methods used in the fabrication of the polymer composite

samples. These samples were produced by Quinton Krueger (25) and Jessica Heiser (20).

The properties of both the matrix and filler materials are also given.

2.2 Materials

The thermoplastic matrix used was DuPont Zytel 101 NC010, an unmodified semi-

crystalline nylon 6,6 polymer of medium viscosity. The properties are listed in Table 2.2-1

below.

Table 2.2-1: Properties of DuPont Zytel 101 NC010 (15)

Melting Point 262°C

Tg (Glass Transition Temp, DAM) 60°C-70°C (approx.)

50% Relative Humidity 23°C (approx.)

Melt Flow Rate 12.35 g/10 min

Shear Viscosity at 1000 sec–1 shear rate and 280°C 137 Pa-sec

Tensile Strength at 23°C (DAM) 82.7 MPa

Flexural Modulus at 23°C (DAM) 2,827.0 MPa

Tensile Elongation at Break at 23°C (DAM) 60%

Notched Izod Impact, 23°C 53.0 J/m

Density at 23°C 1.14 g/cm3

Electrical Conductivity at 23°C 10-15 S/cm

Electrical Resistivity at 23 oC 1015 ohm-cm

Thermal Conductivity at 23°C 0.25 W/mK

DAM = Dry As Molded

7

Two different carbon fiber fillers were employed in this project: BP/Amoco’s

ThermalgraphTM DKD X and Akzo Nobel’s Fortafil 243 PAN (polyacrylonitrile) based fiber.

ThermalGraph DKD X is a milled, 200 µm long, petroleum pitch-based carbon fiber that is

both highly anisotropic and graphitized. This particular fiber was used due to its ability to

improve thermal and electrical conductivity of the conductive resin. Table 2.2-2 lists the

properties below. Akzo Nobel’s Fortafil 243 PAN based 3.2 mm chopped, surface treated and

pelletized carbon fiber was also used to improve the electrical and thermal conductivity of the

resin. A proprietary polymer was used as a binder for the pellets that also promoted adhesion

with nylon. Table 2.2-3 lists the properties for this fiber.

Table 2.2-2: Properties of BP/Amoco ThermalGraph DKD X (16)

Tensile Strength >1.39 GPa Tensile Modulus 687-927 GPa

Electrical Resistivity 2.2 µohm-m Thermal Conductivity 400-700 W/m K

Fiber Density 2.15 to 2.25 g/cm3 Bulk Density 0.25 to 0.55 g/cm3

Fiber Diameter 10 microns Filament Shape Round

Average Filament Length 200 microns Filament Length Distribution <20 % less than 100 microns

<20% greater than 300 microns Carbon Assay 99+ wt% Surface Area 0.4 m2/g

Table 2.2-3: Properties of Akzo Nobel Fortafil 243 PAN based 3.2mm Chopped and Pelletized

Carbon Fiber (17)

Tensile Strength 3800 MPa Tensile Modulus 227 GPa

Electrical Resistivity 16.7 µohm -m Thermal Conductivity 20 W/m K (axial direction)

Bulk Density 356 g/liter Fiber Diameter 7.3 microns Filament Shape Round

Fiber Mean Length 3.2 mm (entire range is 2.3 mm to 4.1 mm) Carbon Assay 95%

Binder Content 2.6 wt% proprietary polymer that adheres pellet together and promotes adhesion with nylon matrix

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2.3 Sample Preparation

For this project, the fillers were used as received. The Zytel 101 NC010 was dried in an

indirect heated dehumidifying drying oven (dewpoint of the recirculating air = -40oC). After

drying, the polymer was stored in moisture barrier bags.

2.3.1 Extrusion

An American Leistritz Extruder Corporation Model ZSE 27 was used for all polymer

extrusion throughout the course of the project. The extruder, shown in Figure 2-1, has a 27

mm co-rotating intermeshing twin screw with 10 zones and a length/diameter ratio of 40. The

screw design used produced minimal filler degradation while still providing adequate dispersal

of the filler within the polymer. This screw design is shown in Figure 2-2

Figure 2-1: Leistritz Extruder Used for Compounding of Composites

9

The polymer pellets (Zytel) were introduced in Zone 1. The second side stuffer, located

at Zone 7, was used to introduce the carbon fibers into the polymer melt. Two Schenck

AccuRate gravimetric feeders were used to accurately control the amount of each material

added to the extruder. A complete list of all formulations extruded is provided in Appendix A

and the extrusion conditions for each are discussed in detail by Weber (18), Clingerman (19)

and Heiser (20). Typical extrusion conditions are listed in Table 2.3-1.

Atmospheric Vent

Atmospheric Back Vent Side Stuffer Side Stuffer Main Feed

40D 36D 32D 28D 24D 20D 16D 12D 8D 4D

GFA

2-3

0-30

GFA

2-3

0-90

GFA

2-4

0-90

GFA

2-3

0-60

GFA

2-4

0-90

KB

5-2

-30-

60

KB

5-2

-30-

30

KB

5-2

-30-

90

KB

5-2

-30-

60

KB

5-2

-30-

30

KS

1-2-

10 E

GFA

2-3

0-60

GFA

2-4

0-90

GFA

2-4

0-90

KB

5-2

-30-

90

KB

5-2

-30-

60

KB

5-2

-30-

30

KB

5-2

-30-

60

GFA

2-2

0-30

GFA

2-3

0-90

GFA

2-4

0-90

KS

1-2-

10 A

0D

For Screw Type Elements

GFA-d-ee-ff G = co-rotating F = conveying A = Free-Meshing d = number of threads

ee = pitch (length in millimeters for one complete rotation)

ff = length of screw elements in millimeters Kneading disks

KBj-d-kk-ll KB = kneading block J = number of kneading segments d = number of threads

k = length of kneading block in millimeters l = twisting angle of the individual kneading segments

Kneading disks

KS1-d-hh-i KS1 = Kneading disc d = number of threads h = length of kneading disc in millimeters i = A for initial disc and E for end disc Zones

0D to 4D is Zone 1 (water cooled, not heated) 4D to 8D is Zone 2/Heating Zone 1 8D to 12D is Zone 3/Heating Zone 2 12D to 16D is Zone 4/Heating Zone 3 16D to 20D is Zone 5/Heating Zone 4 20D to 24D is Zone 6/Heating Zone 5 24D to 28D is Zone 7/Heating Zone 6 28D to 32D is Zone 8/Heating Zone 7 32D to 36D is Zone 9/Heating Zone 8 36D to 40D is Zone 10/Heating Zone 9 Nozzle is Heating Zone 10

Figure 2-2: Extruder Screw Design, Note Flow is From Right to Left

Table 2.3-1: Extrusion Conditions for Nylon 6,6 (19)

Zone 1 Temperature (by feed hopper) 210oC Zone 2 Temperature 250oC

Zone 3 to Zone 5 Temperature 270oC Zone 6 to Zone 7 Temperature 275oC

Zone 8 to Zone 10 Temperature 280oC Total Throughput 19.0 kg/hr

Screw rpm 300 rpm

10

2.3.2 Injection Molding

The test specimens were molded using a Niigata injection molding machine, model

NE85UA4, (Figure 2-3). Implementing a 40 mm diameter single screw with a length/diameter

ratio of 18, the lengths of the feed, compression and metering sections of the single screw

were 396 mm, 180 mm and 144 mm, respectively. Two different molds were used for this

project. The four cavity mold shown in Figure 2-4 was used to produce 3.2 mm thick ASTM

Type I tensile bars (end gated) and 6.4 cm diameter disks of 3.2 mm thickness. The tensile

bars were used for longitudinal electrical conductivity measurements while the disks were

used for transverse electrical conductivity tests. Figure 2-5 shows the mold from which the

shielding disks of 130 mm diameter and 3.2 mm thickness were created. The molding

conditions for each formulation using the four-cavity mold are discussed in detail in

Clingerman, Weber and Heiser (18-20). The typical operating conditions for the injection

molding machine can be found in Table 2.3-2.

Figure 2-3: Niigata Injection Molder

11

Table 2.3-2: Injection Molding Conditions for Conductive Nylon (20)

Zone 1 Temperature (by feed hopper) 285oC Zone 2 Temperature 290oC Zone 3 Temperature 299o C

Zone 4 Temperature (die nozzle heater) 310 oC Mold Temperature 88oC

Screw rpm 54 rpm Injection Pressure 154 MPa

Hold Pressure 109 MPa Back Pressure 3 MPa Injection Time 15 seconds Cooling Time 15 seconds Interval Time 2 seconds

Figure 2-4: Four Cavity Mold

Figure 2-5: Shielding Effectiveness Disk

Mold

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2.4 Formulations

Test specimens were labeled according to the material, weight percent filler, and the

order that the specimen came out of the injection molder using the following nomenclature:

N – W – X – Y - ##

N = National Science Foundation Project W = Filler used X = Polymer Y = Weight percent of conductive fiber ## = Sample Number, indicating the order that the sample came out of the injection molder

All formulations were designated with an N as the first letter to denote that they were

from a previous NSF project (Award Number DMI-9973278). Following was a multi-letter

combination to denote the filler (W). “C” denoted the ThermalGraph carbon fiber, while “D”

referred to Fortafil 243. X was used to designate the polymer matrix used, with “N” referring

to nylon 6,6. The Y in the above formula was the weight percent of the conductive filler.

Following the above naming convention, a sample labeled NCN15-3, refers to the third

composite sample from the mold containing 15 wt% ThermalGraph DKD X carbon fiber in a

nylon 6,6 matrix.

Table 2.4-1 shows the concentrations of the resins produced for use in this project.

Table 2.4-1: Loading Levels for Composite Samples Studied

Fiber Loading Levels, wt%

ThermalGraph DKD X 5.0, 10.0, 15.0, 20.0, 30.0, 40.0 Fortafil 243 5.0, 7.0, 10.0, 15.0, 20.0, 30.0, 40.0

13

CHAPTER 3: Experimental and Characterization Methods

3.1 Introduction

In this section, the techniques used to determine the properties of the composite samples,

which were used to test the shielding effectiveness model developed in this thesis, are

discussed. These properties include: transverse (through-plane) and longitudinal (in-plane)

electrical resistivity (inverse of the electrical conductivity), shielding effectiveness, filler

volume fraction, filler orientation, filler length and aspect ratio.

3.2 Electrical Resistivity

3.2.1 Transverse Electrical Resistivity Test Method

For samples with an electrical resistivity greater than 104 ohm-cm, a through-plane (also

called transverse), volumetric electrical conductivity test was conducted on the as molded test

specimen. In this method, a constant voltage (typically 10 V or 100 V) was applied to the test

specimen and the resistivity was measured according to ASTM D257 using a Keithley 6517A

Electrometer/High Resistance Meter and an 8009 Resistivity Test Fixture (21). The Keithley

6524 High Resistance Measurement Software was used to automate the conductivity

measurement. For each formulation, a minimum of six specimens were tested. Each test

specimen was an injection molded disk that was 6.4 cm in diameter and 3.2 mm thick. Since

the presence of water can affect a sample’s conductivity, all samples were tested dry as

molded (DAM).

3.2.2 Longitudinal Electrical Resistivity Test Method

The volumetric longitudinal electrical resistivity (in-plane) was measured on all samples

with an electrical resistivity less than 104 ohm-cm. Test specimens cut from the center gauge

portion of a tensile bar, Figure 3-1, were surface ground on all sides and cut into sticks 2 mm

14

wide, 2 mm thick and 25.4 mm long. As with the transverse electrical resistivity test method,

the presence of water can have a marked effect on the measured conductivity of the sample,

so all were tested dry as molded.

Typically for each formulation, a total of six specimens were cut from a single tensile

bar, with four tensile bars generally used to obtain a total of twenty four test specimens.

After machining, test specimens were dried in a vacuum oven at 660 mmHg and 60°C for

two hours and then sealed in moisture barrier bags. These samples were then tested using a

four probe technique, as shown in Figure 3-2(a). This technique measures resistivity by

applying a constant current (typically 5 to 10 mA) with a Keithley 224 Programmable

Current Source and measuring the voltage drop over the center 6 mm of the sample with a

Keithley 182 Digital Sensitive Voltmeter. The electrical resistivity is then calculated from

Equation 3.2.1 (22):

Li

twVER

∆

= [3.2-1]

Where: ER = electric resistivity (ohm-cm) V∆ = voltage drop over center 0.6 cm of sample (volts) w = sample width (cm) t = sample thickness (cm) i = current (amps) L = length over which V∆ is measured (0.6 cm)

Figure 3-1: Bar From Which Longitudinal Electrical Resistivity Samples Were Cut

Flow & Measurement

15

2mm

2mm

25mm

∆V fromcenter 6mm

Constant current in through sample

Constant current out of

sample

SampleVolt

Meter

Current Source

(A) (B)

2mm

2mm

25mm

∆V fromcenter 6mm

Constant current in through sample

Constant current out of

sample

SampleVolt

Meter

Current Source

2mm

2mm

25mm

∆V fromcenter 6mm

Constant current in through sample

Constant current out of

sample

2mm

2mm

25mm

∆V fromcenter 6mm

Constant current in through sample

Constant current out of

sample

SampleVolt

Meter

Current Source

SampleVolt

Meter

Current Source

(A) (B) Figure 3-2: (A) Experimental Set-up for Four Probe Test Method,

(B) Sample Dimensions and Longitudinal Current Flow (19)

3.3 Shielding Effectiveness

The electromagnetic shielding effectiveness of each formulation was measured

according to ASTM D 4935-89 (Reapproved 1994), for planar materials using a plane-wave,

far-field EM wave. Although it provides a method of measuring far-field SE, the nature of

the shielding test apparatus used in this study allowed for measurement of near-field shielding

effectiveness values (23). To be able to measure near-field power values, one must be able to

fully characterize the impinging wave directly before it collides with the shielding media.

The method is valid over a frequency range of 30 MHz to 1.5 GHz.

An Electro-Metrics, Inc. shielding effectiveness test fixture (model EM-2107A) was

used to hold the sample with a HP 8752C network analyzer generating and receiving the EM

signals. Figure 3-3 and Figure 3-4 show the shielding test apparatus and sample holder.

Figure 3-5 shows a cross-sectional view of the test fixture.

16

Figure 3-3: Shielding Test Fixtures With Support

Figure 3-4: Transmission Holder Without Sample

For each formulation, one reference sample and at least 5 load samples were tested over a

frequency range of 30 MHz to 1.0 GHz. A reference sample consists of a large ring and a smaller

inner disk as shown in Figure 3-6. The shielding effectiveness (SE) of a material is the ratio of

the power received with and without a material present for the same incident power. For these

experiments, therefore, it is the difference ratio of the load sample to the reference sample. It is

expressed in units of decibels (dB), as shown in Equation 3.3-1 (4).

2

110PP

log SE 10dB = [3.3-1]

Where: 1P = received power with the material present (watts)

2P = received power without the material present (watts)

The input power used was 0 dBm, corresponding to 1 mW. The dynamic range (difference

between the maximum and minimum signals measurable by the system) of the system was 80 dB.

17

Figure 3-5: Cross Sectional View of Transmission Holder (24)

Figure 3-6: Reference and Load Shielding Effectiveness Disks (24)

18

Figure 3-7: Reference Disk Alignment on Trasmission Fixture (25)

Figure 3-7 shows the placement of the reference sample on the transmission fixture. The

small disk and larger outer ring must be precisely aligned on the fixture to obtain accurate

readings. The nylon 6,6-based samples were tested DAM. The results from the analysis are found

in Appendix B and discussed in Chapter 4.

3.4 Balance of Power Analysis

The shielding effect test apparatus was also used to determine the contribution of reflection

(scattered) and absorption to the overall shielding effectiveness of a sample. The HP 8752C

Network analyzer is capable of measuring the transmitted power from test fixture and reflected

power from the top of the sample holder. Accounting for cable loss for both the input and output

cables from the fixture, as seen in Figure 3-8, the amount of signal reflected and transmitted

through the sample can be directly measured. The absorbed signal power can then be calculated

using a conservation of power analysis:

Absorbed (W) = Incident (W) – Reflected (W) – Leakage (W) [3.4-1]

Transmission Fixture

Small Reference Disk

Large Reference Ring

Hollow Coupling Area

19

Figure 3-8: Shielding Test Apparatus Schematic (25)

The transmitted and reflected power was measured for at least 6 load samples for each

formulation. Results from the balance of power analysis are discussed in Chapter 4 and listed in

Appendices C and D.

3.5 Fiber Volume Fraction, Fiber Length and Aspect Ratio

A solvent digestion method was used to determine the weight percent of the filler in the

composite sample. As described in ASTM Standard D5226, this method completely dissolves the

polymer, leaving only clean filler particles (26). A 0.2 g sample cut from the center of a

transverse ER disk was used. Formic acid was used to dissolve the nylon 6,6 based composites at

23 oC. The filler was separated from the solvent/polymer mixture through vacuum filtration. The

mass of the dried filler particles was then compared to the weight of the original mass of the

composite/filler sample to determine the weight percent of the filler within the sample. 2 to 4

samples were tested per formulation. These filler volume fraction results are shown in detail

elsewhere (20,25). In all cases, the actual filler content of each formulation matched the target

amount within acceptable tolerances.

HP Analyzer

Shielding Apparatus

Regulator

Input Cable

Output Cable

Air Cylinder

20

After the fibers were extracted frm the nylon 6,6 matrix, they were dispersed onto a glass

slide and viewed using an Olympus SZH10 optical microscope with an Optronics Engineering

LX-750 video camera. The images (at 60x magnification) were collected using Scion Image

version 1.62 software and then processed using Adobe Photoshop 5.0 and the Image Processing

Tool Kit v. 3.0. The length of each fiber was measured and the aspect ratio (AR), Diameter

LengthAR = ,

was calculated. For each formulation, between 200 and 3000 individual fibers were measured

(18-19,27). These results are shown in Appendix A.

3.6 Orientation

3.6.1 Fiber Orientation

The orientation of the carbon fibers within the composite was determined by viewing a

polished sample with an optical microscope. For each formulation, a 12.7 mm x 12.7 mm section

was cut from a SE test disk. The sample was mounted in epoxy and positioned such that the

depth of the sample could be viewed (3.2 mm). The samples, in the epoxy plug, were polished

and then viewed via an Olympus BX60 reflected light microscope at a magnification of 200x.

Scion Image version 1.62 software was used to collect the images, which were later processed in

Adobe Photoshop 5.0 using Image Processing Kit v. 3.0. The average orientation of 1000 to 4000

fibers per formulation was determined (28). Appendix A shows the results of this analysis.

3.6.2 Transmission Orientation Dependence

The effect of fiber orientation on the transmitted signal strength was investigated using the

fixture shown below in Figure 3-9. A large circular metal plate was affixed to a sheet of

plexiglass that was held in place by slots cut into a PVC pipe. The plate reduced the possibility of

wave diffraction around the composite sample interfering with the measured transmitted signal

strength. The shielding disk samples were placed in the remaining slot on the sample holder, in

front of the antenna and large metal plate. A dipole antenna was positioned directly behind the

metal plate with a transmission cable connecting it to the HP 8752C network analyzer. An

21

electric field of known orientation (parallel to the plane of the floor) and strength was sent

through the sample to be received by the antenna. The transmitted signal strength was measured

over a frequency range of 500 to 2000 MHz. The shielding disk sample was then rotated 90

degrees and the process repeated. All measurements were conducted in an anechoic chamber to

reduce the error inducing effects of outside interference and incident field reflection.

Figure 3-9: Dipole Antenna and Sample Holder

22

CHAPTER 4: Experimental Results

4.1 Introduction

The results from the balance of power analysis and shielding effect experiments, described

in Chapter 3, are discussed in this chapter. Also presented are the results from the fiber

orientation studies.

4.2 Shielding Effectiveness Results

From the data measured using the techniques discussed in Chapter 3, the shielding

effectiveness for each individual sample for each formulation was calculated using Equation

4.2.1.

2

110PP

log SE 10dB = [4.2-1]

Where: 1P = received power with the material present (watts)

2P = received power without the material present (watts) The SE results compiled in this investigation compared favorably to the work of Krueger (25) and

Heiser (20).

4.2.1 Pure Nylon 6,6

As expected, the pure matrix of only nylon 6,6 showed essentially no ability to shield

electromagnetic fields due to its dielectric nature. Ideally, an impinging electromagnetic wave

should encounter no resistance when passing through a dielectric material. Assuming the

material exhibits no conversion of the incident energy into heat while the wave travels through

the dielectric (condition known as a non-lossy dielectric), the shielding effectiveness should be

zero. Figure 4.2-1 shows the pure nylon 6,6 matrix following this behavior. Little shielding

effectiveness was measured, approximately 0.1 dB at the higher frequencies. This corresponds to

shielding only 2% of the incident field strength.

23

Figure 4-1: Shielding Effectiveness for Pure Nylon 6,6

The solid line in Figure 4.2-1 represents the mean shielding effectiveness for the

formulation. For each formulation, the mean was calculated from at least 4. Typically, 6 samples

were measured. The upper dashed line corresponds to the highest SE value recorded for any of

these samples. Similarly, the lower dashed line refers to the lowest SE value recorded. It is

possible for a single trial to produce a maximum at one frequency and a minimum at another.

This, however, was frequently not the case. A single specific specimen of a formulation typically

would produce SE values that were either high, average or low.

4.2.2 ThermalGraph™ DKD X

The introduction of ThermalGraph™ carbon fiber into the nylon 6,6 matrix resulted in

enhanced EM shielding characteristics. Increasing the amount of filler within the sample resulted

in decreased electrical resistivity (ER) and increased shielding effectiveness. Also observed was

the effect of increased frequency on the measured SE values. This trend is expected and has been

300 400 500 600 700 800 900 1000 -0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Frequency (MHz)

24

reported elsewhere (9-14). As frequency is increased, the wavelength of the EM wave decreases

and becomes for comparable to the size of the fiber. Thus, higher frequency waves are more

likely to encounter fiber embedded in the polymer matrix. Similarly, as the weight percent of

fiber is increased, there is an improved probability that the wave will collide with a fiber. The

fibers, as opposed to the polymer rich areas, are more likely to scatter or absorb the wave, as the

nylon is virtually invisible to the wave. Hence, SE increases as frequency increases.

For all formulations studied, listed in Table 2.4-1, SE increased at higher frequencies. The

SE results for the ThermalGraph™ DKD X composites at 300, 500 and 800 MHz are shown in

Figure 4-2. Figure 4-3 shows both how shielding effectiveness directly increased as a function of

filler weight percent and frequency.

Figure 4-2: Shielding Effectiveness As a Function of Filler Volume Percent At Select Frequencies

0 5 10 15 20 25 30 0

2

4

6

8

10

12

14

Volume Percent Fiber (%)

300 MHz 500 MHz 800 MHz

25

Figure 4-3: Shielding Effectiveness Results for ThermalGraph™ DKD X

4.2.3 Fortafil 243

The addition of Fortafil 243 fibers into the matrix produced similar SE trends. Like the

ThermalGraph™ DKD X, the SE for the Fortafil samples increased with both frequency and filler

weight percent. The Fortafil samples, however, showed markedly better shielding behavior. For

example, NCN40 was found to have the best shielding effect performance among the

ThermalGraph™ samples, approximately 14 dB at 1.0 GHz. In comparison, NDN40 was found

to have a maximum shielding effectiveness of 72 dB at 1.0 GHz. This disparity between the

behaviors of the two fillers tracked with the ER results (Appendix A). As shown in Tables 2.2-2

and 2.2-3, both the Fortafil and ThermalGraph™ fibers have similar electrical resistivities. When

both fibers, however, were introduced into the nylon 6,6 matrix in equal weight percents, the

Fortafil sample was found to be two orders of magnitude more conductive than the

ThermalGraph™. Thus, improved shielding for the Fortafil samples was observed.

300 400 500 600 700 800 900 1000 0

2

4

6

8

10

12

14

Frequency (MHz)

5 wt% 10 wt% 15 wt% 20 wt% 30 wt% 40 wt%

26

A corresponding trend was noticed in the percolation thresholds for Fortafil and

ThermalGraph™ electrical resistivity. A prior investigation determined the thresholds to be 9.5

and 3.4 volume percent, respectively (19). Previous work has suggested that the increased

shielding effectiveness afforded by the Fortafil 243 filler may be due in part to the increased

heteroatoms present on the surface of the individual fibers. Fortafil 243 results in improved

adhesion with the nylon matrix material which might explain increased composite SE (20,28-29).

The Fortafil based formulations are listed in Table 2.4-1. The SE results for the Fortafil based

composites are shown in Figure 4-4. Again, as frequency and filler weight percent were

increased, shielding effectiveness increased.

300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

(dB

)

5 wt%7 wt%10 wt%15 wt%20 wt%30 wt%40 wt%

Figure 4-4: Shielding Effectiveness Results for Fortafil 243

4.3 Balance of Power Results

From the frequency dependent transmitted and reflected power data accumulated from the

balance of power experiments, the relative effects of electric field reflection (scattering) and

27

absorption on the SE performance of a the composite samples was determined. Although the

experimental apparatus does not allow for direct measurement of the absorption power loss, a

simple power balance accounting for all methods of signal degradation allows for indirect

calculation of the absorption term, Equation 4.3-1.

Absorbed (W) = Incident (W) – Reflected (W) – Leakage (W) [4.3-1]

No noticeable change in transmitted or reflected signal strength was noticed when the

flanges of the shielding test fixture were wrapped with an EM insulator (aluminum foil).

Therefore, the effect of leakage on the behavior of the system was assumed to be negligible. The

HP 8752C network analyzer did not provide consistent incident signal power over the frequency

range investigated. The source EM signal was found to decrease monotonically with increased

frequency (from 1.0 mW at 30 MHz to approximately 0.9 mW at 1.0 GHz). To normalize the

input power at 1.0 mW across the frequency range, the measured transmitted and reflected power

was scaled-up according to the discrepancy between the desired set value and actual applied

signal.

Figure 4-5 and Figure 4-6 show the results of the analysis for NCN20 and NDN20.

Graphs for the remaining formulations can be found in Appendix C. The dashed lines again

indicate the maximum and minimum measured value during the course of the experiment. The

graphs are also expressed in dBm in Appendix D using the following equation to convert:

( )PdBm 10log10P = [4.3-2] Where: PdBm = Power (dBm)

P = Reflected or Absorbed Power (mW)

28

Figure 4-5: Balance of Power Results (mW) for NCN20 (ThermalGraph™)

Figure 4-6: Balance of Power Results (mW) for NDN20 (Fortafil 243)

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

29

The reflected power was equal to or greater than the absorbed power for each formulation

(with the exception of the pure nylon 6,6 sample). As the weight percent of both fillers was

increased, the reflection term became more dominant, indicating that it is the prevailing form of

signal loss. The reflection term also showed significantly more frequency dependence. Over the

frequency range under investigation, the absorbed power was relatively constant while the

reflected power varied greatly.

4.4 Orientation Results

From the transmission orientation dependence analysis, it was determined that for the

injected molded shielding disks the transmitted signal strength from an incident plane wave is

independent of disk orientation. Figure 4.4-1 shows the result from the analysis of NDN40.

500 1000 1500 2000-95

-90

-85

-80

-75

-70

-65

-60

-55

Frequency (MHz)

Tra

nsm

issi

on (

dB)

Figure 4-7: NDN40 Fiber to Incident Wave Orientation Dependence for Transmitted Signal

Strength

Perpendicular

Parallel

30

This result, however, did not conclusively prove that transmitted signal strength was

independent of fiber orientation. To further investigate fiber orientation dependence, the

experiment was repeated replacing the injection molded samples with a thin sheet of

unidirectional carbon fiber/epoxy (Hexcel Carbon Fiber/Epoxy AS4/3501-5A - 35 wt% carbon

fiber). The carbon fiber/epoxy sheet was analyzed with the fibers oriented in the plane of the

electric field and again with the fibers oriented transversely to the field. Figure 4.4-2 shows these

orientations.

Figure 4-8: Depictions of Perpendicular and Parallel Fiber to Wave Orientations

With this material, definite fiber orientation dependence was observed as the transmitted

signal strength differed by an average of 10 dB. Figure 4.4-2 shows this directional dependence.

The results are in partial agreement with the work of Chen. Chen also noticed that the best

shielding occurred when fibers were aligned in the plane of the impinging field (11-13). Casey

has suggested that the tensor constitutive parameters of the system can be estimated to

mathematically model the system (30). Casey, however, analyzed time-domain behavior while

this project has focused on the frequency domain response of the polymer composites.

E E

Perpendicular Orientation

Parallel Orientation

31

500 1000 1500 2000-80

-75

-70

-65

-60

-55

-50

-45

-40

Frequency (MHz)

Tra

nsm

issi

on (

dB)

Figure 4-9: Carbon Fiber/Epoxy Sheet Fiber to Incident Wave Orientation Dependence for

Transmitted Signal Strength

Although the orientation image analysis, results in Appendix A, found a general orientation

to the carbon fibers within the disk, the fibers were not sufficiently oriented to show any

significant dependence. Therefore, sample orientation with the electric field is not a dominant

factor in determining the shielding effectiveness of the composite. When theoretically modeling

the system, a fiber/wave orientation, however, must be chosen. Because parallel alignment of the

electric field and fiber was found to produce the greatest shielding, this orientation will be

selected for use in later analyses.

Perpendicular

Parallel

32

CHAPTER 5: Electromagnetic Theory

5.1 Introduction

A theory to describe the shielding effectiveness of a material is developed in this chapter. A

basic discussion of electromagnetic wave theory is presented, starting with the basic laws of

electricity and magnetism and culminating with the derivation of an equation describing a

scattered field.

5.2 Shielding Theory

When an electromagnetic field is passed through an object, there are two specific

phenomena that determine how the field strength is lost as it interacts with the object: attenuation

(from absorption) and scattering (also known as reflection). As the wave impinges the surface of

the object, it forces charges in the object to oscillate at the same frequency of the incident wave.

This forced oscillating charge behaves as an antenna and reradiates the incident signal. The entire

signal is not reradiated in the direction of the incident wave, resulting in measured signal loss.

The field is emitted in many directions in a pattern associated with a signal charge oscillating

antenna, and hence the field is scattered. As the charge is forced to vibrate in the medium, energy

is lost in the form of heat. This mode of signal loss is known as attenuation due to absorption. It

is important to note that depending on the properties of the object the wave may not encounter

enough free and mobile charges to result in any marked signal loss.

Figure 5-1 below, depicts both loss phenomena for an incident field as it passes through a

medium. In the figure, an incident field collides with an object known to exhibit some shielding

behavior. Only a fraction of the incident field passes into the object as a portion of the signal is

scattered at the media interface. The amplitude of the entering signal, represented by the dashed

line within the object, then decreases with increased depth of penetration. Internal reflections are

33

possible, especially in the case of non-homogeneous materials, but typically play only a minor

role in determining the extent of transmission power loss (10).

Figure 5-1: Representation of Shielding Phenomena for Plane Waves Passing Through a

Homogeneous Barrier (10)

5.3 Scattered Field Theory

Although the results from the electrical resistivity and shielding effectiveness experiments

show correlation, it has been argued that resistivity tests alone cannot provide enough information

to predict SE (9). The resistivity experiments do not take into account all of the filler present,

only the fibers aligned in a conductive network. The fibers not connected in the network,

however, still have the potential of scattering or absorbing electromagnetic fields.

From the balance of power analysis described in Chapter 3 and discussed in Chapter 4, it

was determined that absorption played only a minor role in determining the shielding

effectiveness of the composite disks, leaving field scattering (reflection) as the largest SE

contributor. Because the pure nylon samples showed an inability to shield, it is proposed that the

scattering behavior of the system can be attributed singularly to the presence of the fibers. Thus,

a single cylindrical fiber will serve as the focus for the derivation of the relevant equations.

A

B

Scattered Fields

Transmitted Fields

34

Figure 5-2 shows a simple plane wave traveling in the x-direction and the electric field

oscillating in the z-direction colliding with a cylinder. The magnitude of the incident plane wave

is E0. Although Figure 5-2 shows a single point of scattering, the plane wave will strike the

cylinder in multiple places, and thus, produce multiple points of scattering. Each point of impact

will emit a scattered wave of the same frequency and wavelength of the incident wave. The

amplitude of the scattered wave, indicating the strength of the wave, however, will be less than

E0.

Figure 5-2: A Cylinder Impinged by a Uniform Plane Wave

xz

yz

H H

ρ

z

θ Point of Scattering E0

35

5.4 Scattered Field Equation Derivation

5.4.1 Maxwell’s Equations

To solve for the scattered fields, an understanding of basic electromagnetics is required.

Maxwell’s equations provide the fundamental basis.

tB

E∂∂

−=×∇ [5.4-1]

tD

JH∂∂

+=×∇ [5.4-2]

0=⋅∇ B [5.4-3]

vD ρ=⋅∇ [5.4-4]

Where:

E = electric field strength (volts/meter)

D = electric flux density (coulombs/meter2)

H = magnetic field strength (amperes/meter)

B = magnetic flux density (webers/meter2)

J = electric current density (amperes/meter2) vρ = electric charge density (coulombs/meter2)

To solve Maxwell’s equations, the following constitutive relations, Equations 5.4-5 and 5.4-6, are

required, which relate electric field and magnetic field strength to electric flux density and

magnetic flux density, respectively. The constitutive relations provide information about the

environment in which the electromagnetic fields propagate. (3)

ED∧

∈= [5.4-5]

HB µ= [5.4-6]

Where:

∧

∈ = complex permittivity (Farads/meter) µ = permeability (Henrys/meter)

36

5.4.2 Permittivity - Absorption Loss

Permittivity is a measure of how much a medium changes to absorb energy when subject

to an electric field. As seen in Equation 5.4-5, it is defined as the ratio ED

. Complex permittivity

is further defined by:

ωσ

j−=∈∈∧

[5.4-7]

Where:

∈ = permittivity (real part) (Farads/meter) σ = conductivity (Siemens/meter) ω = angular frequency (radians/second)

j = 1− (imaginary number)

The following equation relates angular frequency to frequency, f :

fπω 2= [5.4-8]

fc λ= [5.4-9]

Where: c = speed of light (Faradays/meter) λ = wavelength (meters)

The imaginary part of Equation 5.4-7 describes the absorption loss at a given frequency. The

relationωσ

is termed the loss factor (31). The ratio of the loss factor to the real part of the

permittivity indicates whether or not a material will exhibit large absorption losses. If the ratio is

large for a given frequency, the material is regarded as a good conductor (31).

∈ω

σ ⟩⟩ 1 [5.4-10]

Although the real part of the permittivities for ThermalGraph™ and Fortafil are not known,

both have electrical conductivity values within two orders of magnitude of copper

37

(ThermalGraph™ 4.55 x 105 S/m, Fortafil 0.60 x 105 S/m), a material known for exhibiting good

electrical conduction and little absorption losses. Thus, both fillers are assumed to demonstrate

similar properties. For copper, with a conductivity of 5.80 x 107 Siemens/m, the frequency at

which the above ratio (Equation 5.4-10) approaches unity is in the optical range (1014 Hz) (31).

Therefore, it can be expected that both ThermalGraph and Fortafil will display little to no

absorption losses up to infrared frequencies (1012 Hz). The frequency range covered in this

analysis peaked at 109 Hz, hence, the filler particles can be assumed to be perfect conductors and

only the real part of the permittivity (∧

∈) will be considered. Figure 5-3 shows the frequency

spectrum.

Figure 5-3: Electromagnetic Frequency Spectrum (25)

38

5.4.3 Phasor Notation

Substituting the constitutive relations, Equation 5.4-5 and 5.4-6, into the original Maxwell

expressions, Equations 5.4-1 - 4, yields the following modified Maxwell’s equations:

tH

E∂

∂−=×∇ µ [5.4-11]

tE

JH∂∂

∈+=×∇∧

[5.4-12]

0=⋅∇ H [5.4-13]

vE ρ=⋅∇∈∧

)( [5.4-14]

The time derivatives in Equations 5.4-11 and 5.4-12 can be placed into phasor notation by using

the rule of equivalence for time-harmonic quantities. A phasor is a complex quantity that

represents a time-harmonic physical quantity. Phasor notation is a more convenient method of

representing the equations associated with electromagnetics.

The following sinusoidal, time-harmonic real physical quantity, )(tV ,

)cos()( 0 φω += tVtV [5.4-15]

can be expressed as a complex quantity using Euler’s Identity.

xjxe jx sincos += [5.4-16]

In phasor notation )(tV can be written as:

)Re)( 0tjj eeVtV ωφ= [5.4-17]

Where: Re = denotes taking the real part

39

For convenience, the Re symbol and frequency-time dependence term, tje ω , are generally

omitted in the literature and will not be written for the phasors listed in the remainder of this

chapter.

φjeVtV 0)( = [5.4-18]

Therefore, ωj can be used to replace a time derivative when representing a time-harmonic

function as a complex quantity.

)Re)Re)( 00tjjtjj eeVjeeV

ttV

tωφωφ ω=

∂∂

=∂∂

[5.4-19]

φω jeVjtVt 0)( =

∂∂

[5.4-20]

Finally, Equations 5.4-11 and 5.4-12 can be expressed as:

HjE ωµ−=×∇ [5.4-21]

EjJH∧

∈+=×∇ ω [5.4-22]

5.4.4 Wave Equation Solution – Incident Field

Although the plane waves created by the HP 8752C network analyzer and transmitted

through the shielding apparatus do not propagate through free space, it was assumed the conical

wave guide of the test fixture (Figure 3-5) allowed the EM fields to behave as if they are in a

source free media. This assumption was made to simplify the mathematics required to

characterize the incident electrical field.

Figure 5-4: Cross Sectional View of Transmission Holder (24)

40

For a source free media, no source currents or charges are present to interact with the

plane wave, 0=J and 0=vρ . This stipulation simplifies Equations 5.4-14 and 5.4-22 as

follows:

0=⋅∇ E [5.4-23]

EjH∧

∈=×∇ ω [5.4-24]

Using the well known tensor indentity listed in Equation 5.4-25, further simplification of

Equation 5.4-21 is possible (3).

aaa 2)()( ∇−⋅∇∇=×∇×∇ [5.4-25]

Applying the identity with the phasor form of the Maxwell equation describing the electric field

produces Equation 5.4-26.

HjE ωµ−=×∇ [5.4-21]

)()(2 EHjE ⋅∇∇−×∇−=∇− ωµ [5.4-26]

Substituting Equations 5.4-23 and 5.4-24 into Equation 5.4-26 yields a general second order

differential equation describing the electric field, Equation 5.4-28, which is typically referred to

as the wave equation (3)

)0()(2 ∇−∈−=∇−∧

EjjE ωωµ [5.4-27]

0ˆ22 =∈+∇ EE µω [5.4-28]

The collection of terms, ∈2µω , can be lumped together under a single variable 2β .

∧

∈= µωβ [5.4-29] β is called the wavenumber. For free space, the wavenumber, 0β , is the number of waves in a

distance of 2π wavelengths. This relation becomes more apparent after substituting the free space

values of µ and ∧

∈ into Equation 5.4-29.

41

0µ = permeability of free space ( 7104 −×π Henrys/meter)

0∈ = permittivity of free-space (0

2

1µc

≡ = 91036

1 −×π

Farads/meter)

20

12c

cλπ

β = [5.4-30]

λπ

β2

0 = [5.4-31]

Since the geometry of the system in question is cylindrical in nature (fiber shape) the electric

equation should be solved in cylindrical coordinates of the following form:

),,( zE φρ [5.4-32]

Where ρ , φ and z are cylindrical coordinates diagrammed in Figure 5-5.

Figure 5-5: Cylindrical Coordinate System

External to the fiber, the plane wave introduced in Section 5.3 and shown in Figure 5-2 travels in

the x-plane with an electric field oscillating in the z-plane. Therefore, only the partial derivatives

with respect to the x-direction for the z component of the field, given by equation 5.4-28, are of

concern.

0002

2

2

=∈+∂

∂z

z ExE

µω [5.4-33]

42

Finally, Equation 5.4-34, describing the electric field of the simple plane wave can be found as a

solution to the simplified wave equation, Equation 5.4-33.

φρββ cos00

00 jxj eEzeEzE −∧

−∧

== [5.4-34]

Where: z = unit vector pointing in the direction of increasing z

The incident field,i

E , can further be expressed as:

∑∞

=

∧

−=0

00 )cos()()(n

nnni

nJjEzE φρβε [5.4-35]

Where:

nε = 0 20 1

≠=

n n

The summation arises from the representation of the plane wave as an infinite sum of cylindrical

wave functions (32).

5.4.5 Wave Equation Solution – Scattered Field

A solution for the incident electric field can be found with relative ease because of the no

source/ free space assumption made at the beginning of the derivation. For an electromagnetic

signal to propagate through or interact with an object, however, oscillating currents must exist

within the object. The creation of a scattered electric field requires the induction of a current

source on the scattering object. Thus, the Maxwell’s equations must include the current density

term, J .

HjE ωµ−=×∇ [5.4-21]

EjJH∧

∈+=×∇ ω [5.4-22]

Solving the two differential equations is challenging when the value of J is not known. Because

of the presence of the curl operator, the electric field “wraps” around the current source. Thus,

43

the math required for solving Equations 5.4-21 and 5.4-22 can be quite challenging. Introduction

of two intermediate auxiliary functions A and Φ allows for easier determination of a solution.

Figure 5-6 illustrates this concept. A is defined as having the same vector direction as

J (traveling in the same direction). The solution for A can then be used to determine E .

A and Φ are defined by the following relationships (3):

AB ×∇= (definition of A ) [5.4-36]

Φ∇−−= AjE ω (definition of Φ ) [5.4-37]

Figure 5-6: Block diagram Depicting the Two Step Process for Solving for the Radiated Fields Given a Current and Charge Source (32)

Using the tensor identity described in 5.4-25 with Equation 5.4-36 gives a second order

differential equation for A in terms of magnetic flux. The scattered fields are produced by a

current source, J . Therefore, the equations must be solved for in terms of J .

AAAB 2)()( ∇−⋅∇∇=×∇×∇=×∇ [5.4-38]

The curl of the magnetic flux density, B×∇ , can be found using the constitutive relation given in

Equation 5.4-6.

HB µ= [5.4-6]

Sources

J , ρv

Vector Potentials

A , Φ

Radiated Fields

E , H

Integration Path 1

Integration Path 2 Differentiation Path 2

44

)()( HHB ×∇=×∇=×∇ µµ [5.4-39a]

Inserting this new expression, Equation 5.4-39a into Equation 5.4-38 gives:

AAH 2)()( ∇−⋅∇∇=×∇µ [5.4-39b]

The divergence of A is defined by the Lorentz condition (3):

0=Φ∈+⋅∇∧

ωµjA (Lorentz Condition) [5.4-40]

AjH 2)()( ∇−Φ∈−∇=×∇∧

ωµµ [5.4-39c]

The curl of the magnetic field, expressed in terms of a current source J and electric field E ,

(Equation 5.4-22) produces the following equation:

AjEjJ 2)()( ∇−Φ∈−∇=∈+∧∧

ωµωµ [5.4-39d]

Rearranged, Equation 5.4-39d becomes:

)(2 Φ∈−∇+∈−=∇+∧∧

ωµωµµ jEjAJ [5.4-39e]

The definition of Φ can then be used to express the electric field:

)()(2 Φ∈−∇+Φ∇−−∈−=∇+∧∧

ωµωωµµ jAjjAJ [5.4-39f]

A simple rearrangement of Equation 5.4-39f produces the second order differential equation for

the vector potential, A , in terms of a current source ( J ).

)()(22 Φ∇∈−Φ∇∈+∈−=∇+∧∧∧

ωµωµµωµ jjAAJ [5.4-39g]

JAA µµω −=∈+∇∧

22 [5.4-41]

Similarly, a second-order differential equation for Φ can be found using the definition

of Φ , Equation 5.4-14 and the Lorentz condition. The divergence of the definition of Φ ,

Equation 5.4-37, can be used to represent the expression in known terms.

Φ∇−−= AjE ω [5.4-38]

45

Taking the divergence gives:

)( Φ∇−−⋅∇=⋅∇ AjE ω [5.4-42a]

)()( Φ∇⋅∇−⋅∇−=⋅∇ AjE ω [5.4-42b] The divergence of the electric field has previously been related in Equation 5.4-14.

)()( Φ∇⋅∇−⋅∇−=∈∧ Ajv ω

ρ [5.4-42c]

The Lorentz condition again defines the quantity A⋅∇ :

)()j(jv Φ∇⋅∇−Φ∈−−=∈

∧

∧ωµω

ρ [5.4-42d]

Upon rearrangement, a second order differential equation for Φ in terms of surface charge ( vρ )

is realized.

∧

∧

∈−=Φ∈+Φ∇ vρ

µω 22 [5.4-43]

An infinitesimal antenna is an extremely short and thin wire driven by a current source (3).

This theoretical antenna is a good approximation for the tiny antenna produced from the induced

oscillating charge on the surface of the fiber. Assuming that the antenna oscillates in the z-plane

over an infinitesimal length ( z∆ ), a current density ( J ) multiplied by the cross-sectional area

( A∆ ) equal to I with the origin is set at the center of the antenna ( 'ρ = 0), the vector potential

generated by the antenna is given by (3):

πρ

µ βρ

4ˆ jzeIz

A−∆

= [5.4-44]

One can clearly see the similarity of the above equation and the equation for a scalar potential of

a point charge, Equation 5.4-45.

Scalar Potential = ρπε04

q [5.4-45]

Where: q = point electric charge (Coulombs)

46

Using this definition for an infinitesimal antenna, the general solutions for A and Φ can be

found and are listed in Equations 5.4-46 and 5.4-47.

∫∫∫ −=

−−

v

jeJ

dVA'

)'('

4)(

'

ρρ

ρπµ

ρρρβ

[5.4-46]

∫∫∫ −∈=Φ

−−

∧v

j

v edV

'

)'('

4

1)(

'

ρρ

ρρ

πρ

ρρβ

[5.4-47]

Where:

ρ = vector indication the position of the potentials

'ρ = position vector of the sources

'ρρ − = distance between observation point ρ and 'ρ

Figure 5-7 shows the position vectors ρ and 'ρ . Equations 5.4-46 is integrated over all points

where the source, )'(ρJ , is not zero (3).

Figure 5-7: Diagram of the Position Vectors. The vector potential A at ρ is obtained by

integrating the current J at 'ρ . (3)

'ρρ −

ρ 'ρ

)'(ρJ

47

Through use of the solutions for A and Φ and their associated definitions, an expression

for the scattered field, i

E , can be developed. Since the scattered waves travel outward from the

cylindrical fibers, the E-field solution must be expressed by cylindrical wave functions (32).

∑∞

−∞=

=n

nn

sHcEzE )(ˆ )2(

0 βρ [5.4-48]

Where: nc = unknown amplitude coefficients

)2(nH = Hankel Fuction of the second kind

given by:

nn)(

n jYJH −=2 [5.4-49]

Where: nJ = Bessel Function of the first kind

nY = Bessel Function of the second kind

Equation 5.4-48 includes only the z component of the scattered field and ignores the ρ and φ

directions. The results of the orientation analysis, discussed in Chapter 4, made this

simplification possible. From the fiber orientation shielding preference study, it was found that

aligning a length of a carbon fiber parallel to the electric field produced the most shielding.

Therefore, to model the maximum amount of scattering, we can narrow our focus to this

orientation, known as TMz mode. Figure 5-8 shows a wave traveling in the x-plane with the

electric field (E) pointing in the z-plane and the magnetic field pointing in the y-plane. This

orientation allows for the electric field to oscillate charges over the greatest distance in the fiber,

(the length of the fiber) and thus produce the largest scattered field intensity.

48

a

E

x

z H

y

Figure 5-8: Uniform Plane Wave of TMz Orientation Impinging a Single Cylindrical

Scatterer With Radius a (32)

To solve for the unknown amplitude coefficients, nc , the wave must be expressed in its

entirety, T

E , including both the scattered and incident fields (s

E and i

E ).

siT

EEE += [5.4-50]

For a perfect conductor, no charge or electric field can exist within the material. Therefore, at the

surface of the fiber, ρ = a, there is no electric field.

0) ,20 ,( =≤≤= zaET

πφρ [5.4-51]

Coupled with the boundary condition listed below that states that the tangential electric fields on

both sides of a surface (within the fiber and outside) must be equal, nc is found to be:

0)( 21 =−×∧

EEn (Boundary Condition) [5.4-52]

φ

ββ jn

n

nnn e

aHaJ

jc)(

)()2(

−−= [5.4-53]

49

The scattered field, Equation 5.4-48, can now be expressed as [Equation 11-85a in Balanis] (32):

∑∞

−∞=

−−=n

jn

n

nnnse

aHHaJ

jEzE φ

ββρβ)(

)()(ˆ

)2(

)2(

0 [5.4-54]

∑∞

−∞=

−=n n

nnn

n naH

HaJjEz )cos(

)()()(

)(ˆ)2(

)2(

0 φβ

βρβε [5.4-55]

Where:

nε = 0 20 1

≠=

n n

5.5 Scattering Width

The radar cross section,σ , is defined as the measure of a target's ability to reflect (scatter)

incident radar signals in the direction of the radar receiver, i.e. it is a measure of the ratio of

backscatter power. Backscatter is defined as reflecting the signal from the target directly back to

the source. Simply, radar cross section provides an indication of how well a given target reflects

incident electric field energy. A small radar cross section indicates that the object in question

backscatters little of the incident signal. The less power backscattered, the less power received by

the source, meaning the object appears small to the source. This is the fundamental principle

behind stealth technology.

Scattering width, in principle, is very similar to shielding effectiveness. Shielding

effectiveness is a measure of how much energy from an impinging wave transmits through an

object while radar cross section indicates how much power is backscattered. In theory, the two

should show similar trends and, in fact, be related. Equation 5.4-55 shows the mathematical

definition of scattering width (32).

∞→

=⋅ 2

2

2 2limi

s

D

E

Eπρσ

ρ

[5.5-1]

50

Where: ρ = radial distance from center of target to the observer (meters)

s

E = scattered electric field strength (volts/meter)

i

E = incident electric field strength (volts/meter) Substitution of Equation 5.4-45 into Equation 5.5-1 yields the following expression for the

scattering width:

∑

∞+

=⋅ −=

∞→

2

0

)2()2(2 )cos()(

)()(

)( 2lim φβρβ

βεπρσ

ρ

nHaH

aJj

nn

n

nn

nD [5.5-2]

Application of the limit, ∞→ρ , produces the far-field scattering width. Because of the

design of the shielding test apparatus, near-field measurements were made, instead. Although it

appears that the source and receiver are 34.4 cm apart, as shown in Figure 5.4-2, they are actually

very close together. Since the test fixture is essentially a conical wave guide, it behaves as a

transmission line, conducting the signal right up to the shielding disk sample. Therefore, the

source and receiver are very close to the scattering event.

Figure 5-9: Cross Sectional View of Transmission Holder (24)

Because the investigation was concerned with the near-field scattering width (small values

of ρ ), the limit was removed. Further simplification of the equation was also made by selecting a

specific phase angle. When using the shielding apparatus to measure the intensity of the scattered

signal the orientation of the incident wave and fibers in the shielding disks was such that the

back-scattered wave (φ = 180±) was measured. Equation 5.5-2 then reduces to:

51

2

0

)2()2(2 )(

)()(

)( 2 ∑∞+

=⋅ =

nn

n

nn

nD H

aHaJ

j βρβ

βεπρσ [5.5-3]

Where: a = optical radius of object (meters)

β = 0β = λπ2

(meters-1)

nε = 0 20 1

≠=

n n

As can be seen in Equation 5.5-3, the scattering width is frequency dependent. The amount

of power backscattered from the impinged object will depend upon the frequency of the incident

field. Since the scattering width is the width of an object as it appears to the wave, at low

frequencies, the wavelength of the impinging wave is too large to discern the exact detail of an

object. Thus, to the wave, the width of the object at these low frequencies will appear to be larger

than at higher frequencies (such as visible light). It is this concept that provides the foundation

for the development of the shielding effectiveness model in Chapter 6.

52

CHAPTER 6: Shielding Effectiveness Model Design

6.1 Introduction

This chapter will present the methods and supporting logic utilized in the development of the

shielding effectiveness model. The equations for the scattering width and scattered electric field

strength developed in Chapter 5 will be applied to the data accumulated during the shielding

experiments described in Chapter 3, resulting in the creation of a predictive model for the

shielding effectiveness of nylon 6,6 composites containing ThermalGraph™ and Fortafil carbon

fibers.

6.2 Review of Problem Description and Focus

From the orientation analysis described in Chapter 4, it was determined that the composite

shielding disks are comprised of somewhat uniformly oriented cylindrical carbon fibers in a

nylon 6,6 matrix. Thus, the disks are complex non-homogeneous, non-isotropic systems.

Because of the dielectric nature of the nylon 6,6 matrix, the impinging wave sees only a

collection of fibers, some of which are in a conductive network arrangement. It is the interaction

of the wave with these fibers that determines the shielding effectiveness of the composite.

As discussed in Chapter 5, the scattered electric field produced by a plane wave impinging a

carbon fiber can be modeled with Equation 5.4-55.

∑∞

−∞=

−=n

)(n

)(nn

nns

)ncos()a(H

)(H)a(J)j(EzE φ

ββρβ

ε 2

2

0 [5.4-55]

Where: a = optical width of object, fiber diamter (meters)

β = 0β = λπ2

(meters-1)

ρ = distance from scatterer to observer (meters)

nε = 0 20 1

≠=

n n

From this equation, the scattering width of the fiber can be calculated using Equation 5.5-3:

53

2

0

)2()2(2 )(

)()(

)( 2 ∑∞+

=⋅ =

nn

n

nn

nD H

aHaJ

j βρβ

βεπρσ [5.5-3]

Like shielding effectiveness, both measurements give a sense as to how much power from the

incident signal is lost when colliding with an object. Because it was proposed that

scattering/reflection was the dominant loss mechanism for the composite disks, the scattering

width equation for a perfect conductor should be applicable to the carbon fibers within the nylon

6,6 matrix.

6.3 Analysis of Scattering Equations

6.3.1 Dependence on Frequency, Optical Radius and Distance From Scatterer

to Observer

The scattered field and scattering width equations are functions of the impinging wave

frequency (from β ), the optical radius of the object and the distance from the center of the

scatterer and scattered signal receiver (observer), known as ρ. Figure 6-2and Figure 6-1 show the

result of the comparison. For the frequency range investigated (30 to 1000 MHz), in both near

(ρ = 0.1 mm) and far zone (ρ = 50 m), the scattering width of a single fiber of ThermalGraph™

was found be comparable to that of Fortafil. In the near zone, the scattering widths of

ThermalGraph™ and Fortafil were calculated to be 290 µm and 280 µm at 1000 MHz. This

result was expected as both fibers have similar diameters measured in the optical range

(ThermalGraph™ 10.0 µm and Fortafil 7.0 µm). The 2 orders of magnitude difference between

scattering width and respective optical fiber diameter was expected. Consistent between both

near and far zones was the effect of frequency on the scattering width of the fiber. Both fibers

appeared to be much larger at low frequencies than high frequencies.

Also noticeable is the large dependence on ρ . In the near zone, the fibers appear almost 4

orders of magnitude smaller than in the far zone. For the remainder of the analysis, near zone

scattering will be investigated using ρ = 1.0 x 10-4 m. This value was chosen as a best estimate.

54

The actual distance between the scatterer (fiber) and observer should be a fraction of the sample

thickness, 3.2 mm. Nevertheless, the choice of ρ will not affect the quality of the final model

results. It will, however, influence the numerical values of the derived model parameters.

Figure 6-1: Near Zone (ρ = 1.0 x 10-4 m) Scattering Width for Both Fibers

0 100 200 300 400 500 600 700 800 900 1000 2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7 x 10 -4

Frequency (MHz)

ThermalGraph™

Fortafil

55

Figure 6-2: Far Zone (ρ = 50 m) Scattering Width for Both Fibers

6.3.2 Deterministic Nature of Scattering Equations

The deterministic nature of the scattered electric field and scattering width equations is a

major drawback in the applicability of the equation for non-homogeneous materials. The

equations include no prediction of whether or not the wave actually hits the object in

question. They simply give an indication of the power of the field scattered when impinged

with a plane wave. The wavelengths investigated ranged from 10 to 0.3 m (30 MHz – 1.0

GHz). The fibers are on average 6 orders of magnitude smaller than the impinging wave.

This huge discrepancy in size produces a high probability that the wave will never see a fiber.

Thus, the reflected power portion of shielding effectiveness cannot be directly modeled with

the scattered field equation. This can best be seen by plotting the shielding effectiveness of a

single fiber of ThermalGraph™ due solely to scattering by using the solution for the scattered

electric field, Equation 5.4-55 and the definition of SE.

0 100 200 300 400 500 600 700 800 900 1000 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency (MHz)

ThermalGraph™

Fortafil

56

−

=∑

∞

−∞=

0

)2(

)2(

0

10

)cos()(

)()()(

log20E

naH

HaJjE

SE n n

nnn

n φβ

βρβε

[ 6.3-1]

Figure 6-3: Theoretical Shielding Effectiveness of a Single Carbon Fiber Scattering an Incident

Wave

The results above presumably indicate that a shielding disk containing only a single carbon fiber

would produce a shielding effectiveness of at least 9 dB. From the experiment results shown in

Appendix B, one can clearly see this is completely unrealistic. A minimum fiber volume percent

of 18.52 was required to obtain a shielding effectiveness of 9 dB for ThermalGraph™. The result

is not wrong; it is just not representative of the actual situation. There is no guarantee that the

incoming signal collides with the lone fiber.

Also, the inverse frequency dependence at first might appear to not agree with the results

from the Shielding Effectiveness tests. Increased frequency was found to increase shielding

0 200 400 600 800 1000 1200 9.5

10

10.5

11

11.5

12

12.5

13

Frequency (MHz)

ThermalGraph™

Fortafil

57

effectiveness. Increased frequency, however, results in decreased scattered field strength, and

therefore, decreased shielding effectiveness for a single fiber and decreased scattering width

(Figure 6-1). One would correctly assume that since a larger object produces a stronger back-

scattered field, it would be a better shielding material. This fact must be accounted for in the

shielding effectiveness model.

6.4 Accounting for Collision Probability

The probability of a wave collision with a fiber is dependent on a multitude of factors: the

apparent size of the fiber (radar cross section), the wavelength of the incident wave and the

volume fraction of fibers within the nylon 6,6 matrix. Prior assumptions in this analysis have

eliminated the influence of other factors such as fiber length and fiber orientation on the

probability of a collision. Because of the multiplicative behavior of the factors, it is quite

challenging to single out the direct effect of each component. This research will focus on

quantifying the cumulative effect of the factors.

As previously mentioned and shown in Figure 6-1 and Figure 6-3, the scattering width of

the cylindrical carbon fibers reduces in size as frequency in increased. The chance of a collision

with a fiber, however, increases with increased frequency (reduced wavelength). Both effects can

be accounted for by dividing the scattering width ( D⋅2σ ) by wavelength (λ) to form a new term,

λσ D⋅2 , known as the bistatic scattering width. This ratio gives an indication of the size of the

fiber in a window one wavelength long. It shows the relative importance of the scattering width

and fiber visibility due to the incident wavelength. The scattering width of a ThermalGraph™

fiber varies from 3.7 to 3.0 x 10-4 m from 30 MHz to 1.0 GHz. The size of the incoming wave,

however, shows greater frequency dependence (10 to 0.3 m). Therefore, even though the

scattering width of the fiber decreases slightly, the relative size of the fiber ( D⋅2σ ) in a unit cell of

length λ increases greatly with respect to frequency. Figure 6-4 shows the unit cell/window.

58

Figure 6-4: Sample Wavelength Sized Window For Shielding Disk

Inside the window, it is assumed that there is a single fiber surrounded by the wave

transparent nylon 6,6. The fiber has an actual diameter of d which is smaller than its scattering

width. It is hypothesized that the shielding effectiveness of this window is proportional to the

scattering width of the fiber. As the scattering width is increased within the window, the

remaining area available for nylon 6,6 is decreased and an increase in shielding effectiveness will

occur. Thus, it is proposed that the bistatic scattering width is related to shielding effectiveness

by a constant factor, Equation 6.4-1.

ttanconsData Effect Shielding

D

=⋅

λσ 2

[6.4-1]

d

λ

σ2 D

59

6.5 Scaling Factor Analysis

The constant was calculated for each formulation by using Equation 6.4-1 and the respective

D⋅2σ (calculated from Equation 5.5-3) for either ThermalGraph™ or Fortafil and is called the

Scaling Factor in this work. The Scaling Factor is defined as the average of the Shielding

Effectiveness data divided by the ratio λ

σ D⋅2 over the range from 300 MHz to 1000 MHz. A

separate Scaling Factor was determined for each material formulation. Figure 6-5 and Figure 6-6

show the analysis for NCN05 and NDN05. Following the same convention from the shielding

effectiveness plots, the upper and lower dotted lines indicate the maximum and minimum

“constants” calculated at that given frequency. The solid line represents this average value. The

remaining scaling factor graphs can be found in Appendix E.

Figure 6-5: Scaling Factor Analysis for NCN05

300 400 500 600 700 800 900 1000 500

1000

1500

2000

2500

3000

Frequency (MHz)

Scaling Factor = 861.503

60

Figure 6-6: Scaling Factor Analysis for NDN05

Although the formulations have been referred to by their respective weight percents

throughout this document, shielding effectiveness is a filler volume fraction dependent

phenomenon. It is the volume of space occupied by the fibers that determines the amount of

obstruction the wave will encounter. By calculating a scaling factor for each individual

formulation, volume percent is essentially accounted for even though the original equations were

derived for a single fiber. The scaling factor relates the volume percent of filler in the composite

to the cumulative effect of a large amount of wavelength sized windows containing a single fiber.

300 400 500 600 700 800 900 1000 6500

7000

7500

8000

8500

9000

9500

10000

10500

11000

11500

Scaling Factor = 8.1833 x 103

Frequency (MHz)

61

Table 6.5-1 and Table 6.5-2 list the results of the analysis for both ThermalGraph™ and

Fortafil, respectively.

Table 6.5-1: ThermalGraph™ Scaling Factor Results

Weight Percent Volume Percent Scaling Factor (dB)

5.0 2.71 861.5

10.0 5.56 1,309.3

15.0 8.55 3,282.0

20.0 11.70 5,686.0

30.0 18.52 11,562

40.0 26.12 20,266

Table 6.5-2: Fortafil Scaling Factor Results

Weight Percent Volume Percent Scaling Factor (dB)

5.0 3.24 8,183.3

7.0 4.56 13,635

10.0 6.58 26,082

15.0 10.08 57,668

20.0 13.69 70,235

30.0 21.29 97,111

40.0 29.49 111,480

Due to the higher SE experimental data for Fortafil, the scaling factors for Fortafil are

much larger than for ThermalGraph™ when compared at similar filler volume percents. At low

volume percents, the quality of the mean scaling factor fit is relatively good for both fibers. As

volume percent is increased, however, the shielding effectiveness data/bistatic scattering width

ratio appears to no longer approximate to a constant value. This occurs around a volume percent

of 18 for ThermalGraph™ and 6 for Fortafil. Because this non-constant behavior is noticed at

different fiber volume percents for the two fibers, the direct cause is unknown. It is however

assumed that because the physical model is founded on the shielding ability of a single fiber, the

theory of Equation 6.4-1 is only valid at low volume percents. Deviations from this linear theory

62

arise at higher volume percents due to the greater likelihood that the unit cell shown in Figure 6-4

contains more than one fiber, most likely arranged in a conductive network.

6.6 Shielding Effectiveness Model Results

Combining the scaling factors collected in the analysis above with the bistatic scattering

width equation produces an equation capable of predicting the shielding effectiveness of a nylon

6,6 sample containing either ThermalGraph™ or Fortafil carbon fibers, Equation 6.6-1 below:

λ

σ D Factor Scaling(dB) ivenessEffect Shielding ⋅⋅= 2 [6.6-1]

Where:

2

0

22

2 2 ∑∞+

=

⋅ =n

)(n)(

n

nn

nD )(H)a(H

)a(J)j( βρ

ββ

ελπρ

λσ

[5.5-3]

a = radius of object (optical range) (meters)

= 5.0 x 10-6 (ThermalGraph™)

= 3.65 x 10-6 (Fortafil)

β = 0β = λπ2

(meters-1)

ρ = distance from scatterer to observer (meters)

nε = 0 20 1

≠=

n n

Figure 6-7 and Figure 6-8 show the model predicted shielding effectiveness for NCN05 and

NDN05. Comparisons of the shielding effectiveness predicted by the model (shown in Equation

6.6-1) to the actual shielding effectiveness data compiled during the experiments are found in

Appendix F.

63

Figure 6-7: Model Predicted and Experimentally Determined Shielding Effectiveness for NCN05

Figure 6-8: Model Predicted and Experimentally Determined Shielding Effectiveness for NDN05

300 400 500 600 700 800 900 1000 2

3

4

5

6

7

8

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

SE Data

Model

64

The model accurately predicts shielding effectiveness values for low fiber volume percent

composites that exhibit low shielding effectiveness (majority of the ThermalGraph™ samples).

The ability of the model to accurately estimate shielding effectiveness decreases rapidly as the

volume percent is increased. The range of utility is from NCN05 to NCN30 and NDN05 to

NDN07. By plotting the actual shielding effectiveness data divided by the model predicted value,

the deviation of the model can be seen giving a sense to the quality of the model fit. Ideally, the

plot for each formulation should equal 1. Figure 6-9 and Figure 6-10 below show the results.

NCN10 shows a different trend than the other formulations. The quality of the fit is reversed.

The model predicts values too high at low frequencies and too low at high frequencies. It is

assumed that this behavior is a

300 400 500 600 700 800 900 10000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Frequency (MHz)

Fit

Qua

lity

5 wt%10 wt%15 wt%20 wt%30 wt%40 wt%

Figure 6-9: Model Fit Quality Analysis for ThermalGraph™ Based Composites

65

300 400 500 600 700 800 900 10000.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (MHz)

Fit

Qua

lity

5 wt%7 wt%10 wt%15 wt%20 wt%30 wt%40 wt%

Figure 6-10: Model Fit Quality Analysis for Fortafil Based Composites

6.7 Scaling Factor - Linear Fit

As discussed in Section 6.5, the scaling factors are functions of only filler volume percent.

Ideally, applying a fit to the scaling factor/volume percent data should produce a master equation

(Equation 6.7-1) for predicting shielding effectiveness for both ThermalGraph™ and Fortafil™.

A linear fit was chosen to follow the prior assumption that the scaling factor linearly relates the

volume percent of filler to the cumulative effect of a large amount of wavelength sized windows

containing a single fiber. Equation 6.7-2 and Figure 6.7-1 and Equation 6.7-3 and Figure 6.7-2

show the results from the linear fit for ThermalGraph™ and Fortafil, respectively.

ThermalGraph™

λ

σ DFit Factor Scaling .Approx(dB) SE ⋅⋅= 2 [6.7-1]

3188.9) % Vol.(848.8285 (dB) SEFit −×= [6.7-2]

66

Fortafil

λ

σ DFit Factor Scaling .Approx(dB) SE ⋅⋅= 2 [6.7-1]

3007.1) % Vol.(4085.7 (dB) FactorScaling +×= [6.7-3]

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

2

2.5x 10

4

Volume Percent Filler

Sca

ling

Fac

tor

(dB

)

Figure 6-11: Linear Fit Applied to ThermalGraph™ Scaling Factor Data

0 5 10 15 20 25 300

2

4

6

8

10

12

14x 10

4 Linear Scaling Factor Fit

Volume Percent Filler

Sca

ling

Fac

tor

(dB

)

Figure 6-12: Linear Fit Applied to Fortafil Scaling Factor Data

R2 = 0.9737

R2 = 0.9348

67

Although both scaling factors appear to show some semblance of linear filler volume percent

behavior, the equations derived from the fit include enough error to significantly reduce the

ability of the model (Equation 6.7-1) to accurately predict shielding effectiveness. The fit for

ThermalGraph™ is quite good between 5 and 20 filler volume percent. The scaling factor

equation for ThermalGraph™, however, predicts a negative value (corresponding to negative SE)

for composites with filler volume percents less than 4. The scaling factor equation for Fortafil

does not estimate negative values at any volume percent but is based on a lower quality fit.

Deviations from perfect linear behavior are again assumed to be a result of focusing on only the

interaction of the incident signal and a single fiber, ignoring the effects fibers oriented in a

conductive network. Figures 6.7-3 and 6.7-4 show the model fit using the linear scaling factor

equations for NCN05 and NDN05. The accuracy of these model results are noticeably worse

than those shown in Figures 6.6-1 and 6.6-2. The remainder of the model results are shown in

Appendix G.

Figure 6-13: Model Predicted and Experimentally Determined Shielding Effectiveness for NCN05 using Linear Scaling Factor Fit Equation

300 400 500 600 700 800 900 1000 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (MHz)

Model

SE Data

68

Figure 6-14: Model Predicted and Experimentally Determined Shielding Effectiveness for NDN05 using Linear Scaling Factor Fit Equation

6.8 White Model Comparison

This new model is a significant improvement over models proposed by White and Bushko

for predicting shielding effectiveness in composite materials having low electrical conductivities.

As shown in Appendix H, the White model equation was derived for homogeneous, isotropic

materials (9-10).

−+=

r

rrrdB

fft

σµ

σµ 10log 1016834.3SE [6.8-1]

Where: t = thickness of material (inches) f = frequency (Hertz)

rσ = conductivity relative to copper

rµ = magnetic permeability relative to copper

300 400 500 600 700 800 900 1000 2

4

6

8

10

12

14

16

Frequency (MHz)

Model

SE Data

69

The assumption of homogeneity produces the greatest error when applying the White model

to a complex composite system. As discussed in Section 6.3.2, a shielding effectiveness model

for a media containing a collection of both shielding (fiber) and non-shielding (nylon 6,6)

materials must include a method for predicting the occurrence of shielding material/wave

collisions to be capable of accurately predicting shielding effectiveness.

The White model relies on effective electrical conductivity of the sample to determine the

shielding effectiveness. As previously mentioned, Bushko has proposed that electrical

conductivity does not provide enough information to be the main SE predictor (9). This becomes

readily apparent when applying the White model to the NCN10 formulation. NCN10 was found

to have a conductivity of 1.31 x 10-16 Siemens/cm. If the shielding disk was completely and

uniformly comprised of a material with this conductivity, one would expect it to provide little

shielding. The shielding disk, however, was found exhibit to some shielding ability, Figure 6-15.

Figure 6-15: White Model and Proposed Model Comparison and Experimentally Determined

Shielding Effectiveness for NCN10

300 400 500 600 700 800 900 1000 -140

-120

-100

-80

-60

-40

-20

0

20

Frequency (MHz)

Proposed Model

SE Data

White Model

70

The theory proposed by White does give a good estimate of shielding effectiveness for nylon

6,6 based composites exhibiting high electrical conductivity. It is assumed that as the

conductivity of the sample increases due to the development of aligned fiber networks, the

impinging wave no longer sees an ensemble of individual fibers. Instead, the wave encounters a

homogeneous-like collection of fibers behaving as a single object. For NDN40, the formulation

found to have the highest electrical conductivity, the White model approximates the SE with

decent accuracy, Figure 6.8-2. The White model is only applicable to highly conductive

composites. It uses the conductivity of the disk to describe the system. Conductivity is a

homogeneous quantity and the sample only approaches homogeneity when the fibers are present

in sufficient quantity (high volume percent). By focusing on the ability of the individual fibers to

shield the model proposed in this text is, however, generally more capable of predicting shielding

effectiveness in composites.

Figure 6-16: White Model and Proposed Model Comparison for NDN40

300 400 500 600 700 800 900 1000 30

40

50

60

70

80

90

100

110

Frequency (MHz)

Proposed Model

SE Data

White Model

71

CHAPTER 7: Conclusions and Future Work

7.1 Thesis Goal

The goal of this investigation was to develop a model to predict the shielding effectiveness

of nylon 6,6/carbon fiber based composites. It has been shown that prediction of SE for

composite materials depends on the electrical conductivity of the filler, filler orientation, filler

size, filler volume percent and frequency of the impinging electric field. This chapter will briefly

summarize the results and conclusions obtained during the development of the shielding

effectiveness model. Future work considerations will also be presented.

7.1.1 Conclusions from Electrical Resistivity/Conductivity Experiments

For both ThermalGraph™ and Fortafil based composites, electrical resistivity increased with

increased filler weight percent. The Fortafil based composites were found to have dramatically

better conductivities than the ThermalGraph™ composites at similar weight percents. Overall

effective values for electrical conductivity of the composite samples were found to be

inconclusive when used to predict SE. The electrical resistivity for the composite as a whole

ignores the shielding contribution of fibers not aligned in a conductive matrix.

7.1.2 Conclusions from Shielding Effectiveness Experiments

The introduction of ThermalGraph™ carbon fiber into the nylon 6,6 matrix resulted in

enhanced EM shielding characteristics. Increasing the amount of filler within the sample also

produced increased shielding effectiveness. Also observed was the effect of increased frequency

on the measured SE values.

The addition of Fortafil 243 fibers into the matrix produced similar SE trends. Like the

ThermalGraph™ DKD X, the shielding effectiveness for the Fortafil samples increased with both

frequency and filler weight percent. The Fortafil samples, however, showed markedly better

72

shielding behavior. This disparity between the behaviors of the two fillers tracked with the ER

results.

7.1.3 Conclusions from Power Balance Analysis

The reflected power was equal to or greater than the absorbed power for each formulation

(with the exception of the pure nylon 6,6 sample). As the weight percent of both fillers was

increased, the reflection term became more dominant, indicating that it is the prevailing form of

signal loss. The reflection term also showed significantly more frequency dependence. Over the

frequency range under investigation, the absorbed power was relatively constant while the

reflected power varied greatly. From these results, it was proposed that focusing solely on signal

loss due to reflection would sufficiently describe the system.

7.1.4 Conclusions from Fiber Orientation Studies

Although the orientation image analysis found a general orientation to the carbon fibers

within the disk, the fibers were not sufficiently oriented to show any significant dependence.

Therefore, sample orientation with the electric field was concluded to not be a dominant factor in

determining the shielding effectiveness of the composites investigated in this study. For highly

aligned composite materials, it was found that fiber orientation does influence transmitted field

strength. Fibers aligned in the plane of the wave were found to reflect more signal than fibers

aligned perpendicular to the electric field. It was decided that the theoretical shielding

effectiveness model would use a parallel fiber/wave orientation to maximize the predicted

shielding effectiveness.

7.1.5 Conclusions from Model Development and Analysis

The shielding disks are complex non-homogeneous, non-isotropic systems. Because of the

dielectric nature of the nylon 6,6 matrix, the impinging wave sees only a collection of fibers,

73

some of which are in a conductive network arrangement. It is the interaction of the wave with

these fibers that determines the shielding effectiveness of the composite.

It has been shown that scattering width, in principle, is very similar to shielding effectiveness.

Shielding effectiveness is a measure of how much energy from an impinging wave transmits

through an object while scattering width indicates how much power is backscattered. The two

should show similar trends and, in fact, be related. Because it was proposed that

scattering/reflection was the dominant loss mechanism for the composite disks, the scattering

width equation for a perfect conductor should be applicable to the carbon fibers within the nylon

6,6 matrix.

Determining or at least providing a method to account for the probability of a wave/fiber

collision is paramount when developing a shielding effectiveness model. The probability of a

wave/fiber collision is dependent on a multitude of factors: the apparent size of the filler

(scattering width), the wavelength of the incident wave and the volume fraction of filler in the

resin. The chance of a collision increases with increased frequency (reduced wavelength). This

effect can be accounted for by dividing the scattering width ( D⋅2σ ) by wavelength (λ) to form a

new term, λ

σ D⋅2 , known as the bistatic scattering width. This ratio gives an indication of the size

of the fiber in a window one wavelength long. It was hypothesized that the shielding

effectiveness of this window is proportional to the scattering width of the fiber. It was proposed

that the bistatic scattering width is related to shielding effectiveness by a constant factor.

Scaling factors were calculated for each formulation. The scaling factors were shown to

relate the volume percent of filler in the composite to the cumulative shielding effect of a large

amount of wavelength sized windows containing a single fiber.

The proposed model, an extrapolation of the shielding behavior of a single fiber was found

to accurately predict shielding effectiveness at low filler volume percents. Deviations from the

experimentally compiled SE data were noticed at high filler volume percents. It was

74

hypothesized that the formation of conductive networks of fibers within the sample, which occur

at higher loading levels, produced non-linear shielding effectiveness behavior with respect to

filler volume percent.

7.2 Future Work

To strengthen the linearity hypothesis serving as the basis of the proposed model, it is

suggested that composites of different filler loading levels be formulated and analyzed. With the

increased number of data points, trends will be further elucidated. The analysis of more

ThermalGraph™ based composites with low loading levels will probably serve as most useful in

determining the validity of the hypothesis.

The model proposed in this thesis is relatively simplistic in approach. Most noticeably, the

effects of fiber to wave orientation, wave absorption, multiple scattering within the composite and

conductive fiber network formation have either been neglected or assumed to be insignificant in

this analysis. Ideally, a shielding effectiveness model for the composites analyzed should be able

to account for these factors.

A statistical simulation such as a Monte Carlo analysis should be implemented to better

predict the probability of fiber/wave collisions. A similar analysis could be used along with the

probability function obtained during the orientation analysis to better quantify the effect of the

non-uniformly oriented fibers. The scattering equations can be derived for different orientations,

including oblique. Theoretically, given the orientation of fibers within the composite, one should

be able to determine the cumulative shielding effect of the collection of fibers with orientation

dependence.

If the continuation of this research investigates shielding at higher frequencies, multiple

scattering will become more dominant. A Monte Carlo analysis could also be used to model the

multiple scattering that occurs within the composite. Similar studies have used Monte Carlo

simulations to study multiple light scattering of polymers (33-34).

75

The greatest challenge and possibly the key to fully understanding shielding effectiveness in

composites lies in determining the effect of the conductive network fiber alignment. The

percolation threshold gives an indication as to the minimum volume percent of fibers needed to

form a network so prediction of network formation is relatively simple. Further refinements need

to be made to determine the extent of the network.

Also, what the wave sees when it encounters the network needs to be investigated. An

impinging electric field will induce currents in the network. Because of the conductivity and

large size of the network, the currents will have a high probability of occurring and will be large.

The scattered field from a network comprised of a known number of fibers will be much greater

than that from an ensemble of fibers containing the same number but not connected. It is the

conductive networks that most likely produce the disparity between the shielding performance of

ThermalGraph™ and Fortafil. The fibers are similar in shape, size and electrical properties.

Modeling the composite based on these characteristics alone does not fully describe the system

behavior. The propensity of Fortafil to form the networks at low filler volume percents (lower

than ThermalGraph™) must be included in the model.

The model proposed in this thesis meets the original goal of the investigation. It predicts

shielding effectiveness for carbon fiber/nylon based composites as a function of frequency and

filler volume percent. The analysis leading to the creation of the proposed model has produced

more questions than answers. This thesis will serve as both an end result and a foundation for

future work.

76

CHAPTER 8: References

1. Thomas, P., Harvard Health Letter., 18, 9, p.1 (1993).

2. Das, N.C., Khastgir, D., Chaki, T.K., and Chakraborty, A., J. Elastomers and Plastics, 34, p.199 (2002).

3. Liang Chi Shen, Jin Au Kong, Applied Electromagnetism, 3rd ed. PWS Publishers, Boston, (1995).

4. Huang, J., Adv. Polym. Tech., 14, 2, p.137 (1995).

5. Bigg, D. M., Polym.Comp., 8, 1, p.1 (1987).

6. Bigg, D. M., Polym.Comp., 7, 2, p.89 (1986).

7. Bigg, D. M., Adv. Polym. Tech., 4, 3/4, p.255 (1977).

8. Schulz, R. B., Plantz, V. C., and Brush, D. R., IEEE Trans. Elect. Compat., 30, 5, p.187 (1988).

9. Bushko, W. C., Stokes, V. K., Wilson, J., ANTEC ’99, p.1499 (1999).

10. White, D.R. J., EMI/EMC Handbook Series, Germantown, MD: Don White Consultants, Inc., 4 (1971).

11. Chu, H. C., Chen, C. H., IEEE Trans. Elect. Compat., 38, 1, p.1 (1996).

12. Lin, M. S., Lin, C. M., Wu, R. B., Chen, C. H., IEEE Trans. Elect. Compat., 35, 3, p.357 (1993).

13. Lin, M. S., Chen, C. H., IEEE Trans. Elect. Compat., 35, 1, p.21 (1993).

14. Krohn, T. L., Medgyesi-Mitschang, L. N., IEEE Trans. Antenna Propagat., 37, 2, p.219 (1989).

15. DuPont Zytel Nylon Resin Product and Properties, Version 95.9, Printed in USA. (2001).

16. Amoco Performance Products: High Thermal Conductivity Pitch Based Graphite Fibers, Amoco Polymers; Alpharetta, GA 30005. (2001).

17. Akzo Nobel Electrically Conductive Fortafil 243 Product Literature, Akzo Nobel Chemicals Inc., 300 S. Riverside Plaza, Chicago, IL, 60606.

18. Weber, E., PhD Dissertation, “Development and Modeling of Thermally Conductive Polymer/Carbon Composites”, (2001).

19. Clingerman, M. L., PhD Dissertation, “Development and Modeling of Electrically Conductive Composite Materials”, (2001).

77

20. Heiser, J. A., MS Thesis, “Conductive, Shielding, Tensile and Impact Properties of Carbon Filled Nylon 6,6 Based Resins”, (2003).

21. “Standard Test Methods for D-C Resistance or Conductance of Insulating Materials,” ASTM Standard D257-91, American Society for Testing and Materials, Philadelphia. (1998).

22. King, J.A., Tucker, K.W., Meyers, J.D., Weber, E.H., Clingerman, M.L., and Ambrosius, K.R., Polym. Compos., 22, 1, p.142 (2001).

23. Annual Book of ASTM Standards, D4935-89. Standard Test Method for Measuring the Electromagnetic Shielding Effectiveness of Planar Materials. D4935-89. Copyright ASTM, (1989).

24. Shielding Effectiveness Test Fixture Electro-Metrics Model EM-2107-A Manual, Jan, (1999).

25. Krueger, Q. J., M.S. Thesis, "Electromagnetic Interference and Radio Frequency Interference Shielding of Carbon-Filled Conductive Resins", (2002).

26. “Standard Practice for Dissolving Polymer Materials”, ASTM Standard D5226-98, American Society for Testing and Materials, Philadelphia, Pennsylvania, (1998).

27. Konell, J. P., Ph.D. Dissertation "Characterization and Tensile Modulus Modeling of Conductive Resins", (2002).

28. Krueger, Q. J., and King, J. A., Adv.Polym. Tech, 22, 2, p.96 (2003).

29. J. A. Heiser, J. A. King, J. P. Konell, and L. L. Sutter, Polym. Comp., 25, 4, p.407 (2004).

30. Casey, K. F., IEEE Int. Symp. Electr. Compat., p.228 (1978).

31. Ramo, Whinnery and Van Duzer., Fields and Waves in Communication Electronics, Wiley, New York, (1965).

32. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, New York, (1989).

33. Cipelleti, L., Physical Review E., 55, 6, p.7733 (1996).

34. Alvarez, A., Wang, C., Ye, Z., J. Comput. Physics, 154, p.231 (1999

78

Appendix A: Formulation Summary

Composite Formulation

Name Description Wt% Vol%

Volumetric Electrical Resistivity (ohm-cm) *

Fiber Length (µm)

Aspect Ratio

Through Plane Orientation (degrees)

NN Zytel 101 NC010 100 100 Avg 1.45E+16 - - - s 7.70E+15 - - - n 11 - - - NCN05 Zytel 101 NC010 95 97.29 Avg 1.06E+16 88.5 8.85 72.477 ThermalGraph DKD X 5 2.71 s 4.00E+15 60.99 6.099 21.310 n 12 2750 2750 831 NCN10 Zytel 101 NC010 90 94.44 Avg 7.65E+15 95.326 9.5326 71.609 ThermalGraph DKD X 10 5.56 s 2.90E+15 59.907 5.9907 21.679 n 12 415 415 2058 NCN15 Zytel 101 NC010 85 91.45 Avg 5.24E+15 105.8 10.58 66.455 ThermalGraph DKD X 15 8.55 s 1.23E+15 70.01 7.001 24.635 n 12 6183 6183 2595 NCN20 Zytel 101 NC010 80 88.30 Avg 5.04E+08 96.54 9.654 68.558 ThermalGraph DKD X 20 11.70 s 2.27E+08 52.62 5.262 22.477 n 11 397 397 4183 NCN30 Zytel 101 NC010 70 81.48 Avg 120.74 94.726 9.4726 63.374 ThermalGraph DKD X 30 18.52 s 53.15 44.99 4.499 23.810 n 19 713 713 4405

79

Table A-1: Overall Summary (continued)

Composite Formulation

Name Description Wt% Vol%

Volumetric Electrical Resistivity (ohm-cm)*

Fiber Length (µm)

Aspect Ratio

Through Plane Orientation (degrees)

NCN40 Zytel 101 NC010 60 73.88 Avg 10.08 89.364 8.9364 65.113 ThermalGraph DKD X 40 26.12 s 3.63 40.983 4.0983 22.446 n 18 634 634 4142 NDN05 Zytel 101 NC010 95 96.76 Avg 2.65E+15 140.95 19.308 79.941 Fortafil 243 5 3.24 s 2.96E+14 126.25 17.294 12.082 n 6 1147 1147 299 NDN07 Zytel 101 NC010 93 95.44 Avg 2.23E+07 149.899 20.534 63.87 Fortafil 243 7 4.56 S 2.22E+07 114.317 15.660 24.86 N 6 1095 1095 1125 NDN10 Zytel 101 NC010 90 93.42 Avg 4.13 149.997 20.547 63.53 Fortafil 243 10 6.58 s 1.94 115.661 15.844 21.14 n 12 1276 1276 850 NDN15 Zytel 101 NC010 85 89.92 Avg 1.19 130.310 17.851 70.799 Fortafil 243 15 10.08 s 0.31 100.261 13.734 20.261 n 18 1798 1798 1520 NDN20 Zytel 101 NC010 80 86.31 Avg 0.47 120.718 16.537 64.038 Fortafil 243 20 13.69 s 0.11 95.557 13.091 22.686 n 24 1519 1519 826

Table A-1: Overall Summary (continued)

80

Composite Formulation

Name Description Wt% Vol%

Volumetric Electrical Resistivity (ohm-cm)*

Fiber Length (µm)

Aspect Ratio

Through Plane Orientation (degrees)

NDN30 Zytel 101 NC010 70 78.71 Avg 0.14 114.415 15.673 61.183

Fortafil 243 30 21.29 s 0.01 87.399 11.972 23.243 n 24 419 419 2311

NDN40 Zytel 101 NC010 60 70.51 Avg 0.093 86.291 11.821 65.71

Fortafil 243 40 29.49 s 0.005 59.767 8.187 22.90 n 24 2714 2714 951

* Transverse Electrical Resistivity Measured: NN, NCN05, NCN10, NCN15, NCN20, NDN5 and NDN7 Longitudinal Electrical Resistivity Measured: NCN30, NCN40, NDN10, NDN15, NDN20, NDN30 and NDN40

81

Appendix B: Shielding Effectiveness Experiment Results

300 400 500 600 700 800 900 1000-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-1: NN Shielding Effectiveness

300 400 500 600 700 800 900 10000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-2: NCN05 Shielding Effectiveness

82

300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-3: NCN10 Shielding Effectiveness

300 400 500 600 700 800 900 10001

1.5

2

2.5

3

3.5

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-4: NCN15 Shielding Effectiveness

83

300 400 500 600 700 800 900 10002

2.5

3

3.5

4

4.5

5

5.5

6

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-5: NCN20 Shielding Effectiveness

300 400 500 600 700 800 900 10004

5

6

7

8

9

10

11

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-6: NCN30 Shielding Effectiveness

84

300 400 500 600 700 800 900 10009

10

11

12

13

14

15

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure B-7: NCN40 Shielding Effectiveness

300 400 500 600 700 800 900 10002.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-8: NDN05 Shielding Effectiveness

85

300 400 500 600 700 800 900 10004

5

6

7

8

9

10

11

12

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-9: NDN07 Shielding Effectiveness

300 400 500 600 700 800 900 100011

12

13

14

15

16

17

18

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-10: NDN10 Shielding Effectiveness

86

300 400 500 600 700 800 900 100031

31.5

32

32.5

33

33.5

34

34.5

35

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-11: NDN15 Shielding Effectiveness

300 400 500 600 700 800 900 100035

36

37

38

39

40

41

42

43

44

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-12: NDN20 Shielding Effectiveness

87

300 400 500 600 700 800 900 100050

52

54

56

58

60

62

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-13: NDN30 Shielding Effectiveness

300 400 500 600 700 800 900 100055

60

65

70

75

80

Frequency (MHz)

Shi

eldi

ng E

ffect

iven

ess

Figure B-14: NDN40 Shielding Effectiveness

88

Appendix C: Balance of Power Results (mW)

Figure C-1: NN Balance of Power

Figure C-2: NCN05 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Transmitted

Absorbed

Reflected

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Transmitted

Reflected

Absorbed

89

Figure C-3: NCN10 Balance of Power

Figure C-4: NCN15 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

90

Figure C-5: NCN20 Balance of Power

Figure C-6: NCN30 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

91

Figure C-7: NCN40 Balance of Power

Figure C-8: NDN05 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

92

Figure C-9: NDN07 Balance of Power

Figure C-10: NDN10 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

93

Figure C-11: NDN15 Balance of Power

Figure C-12: NDN20 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

94

Figure C-13: NDN30 Balance of Power

Figure C-14: NDN40 Balance of Power

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

0 100 200 300 400 500 600 700 800 900 1000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

Absorbed

Reflected

Transmitted

95

Appendix D: Reflected, Absorbed and Transmitted Signal Results in dB

Figure D- 1: NN dB results

Figure D- 2: NCN05 dB results

0 100 200 300 400 500 600 700 800 900 1000 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted

Absorbed Reflected

0 100 200 300 400 500 600 700 800 900 1000 -20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted

Absorbed

Reflected

96

Figure D- 3: NCN10 dB results

Figure D- 4: NCN15 dB results

0 100 200 300 400 500 600 700 800 900 1000 -16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted

Absorbed

Reflected

0 100 200 300 400 500 600 700 800 900 1000 -16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted

Absorbed

Reflected

97

Figure D- 5: NCN20 dB results

Figure D- 6: NCN30 dB results

0 100 200 300 400 500 600 700 800 900 1000 -15

-10

-5

0

Frequency (MHz)

Transmitted

Absorbed

Reflected

0 100 200 300 400 500 600 700 800 900 1000 -12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted

Absorbed

Reflected

98

Figure D- 7: NCN40 dB results

Figure D- 8: NDN05 dB results

0 100 200 300 400 500 600 700 800 900 1000 -16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted Absorbed

Reflected

0 100 200 300 400 500 600 700 800 900 1000 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted Absorbed

Reflected

99

Figure D- 9: NDN07 dB results

Figure D- 10: NDN10 dB results

0 100 200 300 400 500 600 700 800 900 1000 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (MHz)

Transmitted

Absorbed

Reflected

0 100 200 300 400 500 600 700 800 900 1000 -30

-25

-20

-15

-10

-5

0

Frequency (MHz)

Transmitted

Absorbed Reflected

100

Figure D- 11: NDN15 dB results

Figure D- 12: NDN20 dB results

0 100 200 300 400 500 600 700 800 900 1000 -40

-35

-30

-25

-20

-15

-10

-5

0

Frequency (MHz)

Transmitted

Absorbed Reflected

0 100 200 300 400 500 600 700 800 900 1000 -40

-35

-30

-25

-20

-15

-10

-5

0

Frequency (MHz)

Transmitted

Absorbed Reflected

101

Figure D- 13: NDN30 dB results

Figure D- 14: NDN40 dB results

0 100 200 300 400 500 600 700 800 900 1000 -70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

Transmitted

Absorbed Reflected

0 100 200 300 400 500 600 700 800 900 1000 -80

-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

Transmitted

Absorbed Reflected

102

Appendix E: Scaling Factor Analysis

Figure E-1: Scaling Factor for NCN05

Figure E-2: Scaling Factor for NCN10

300 400 500 600 700 800 900 1000 200

400

600

800

1000

1200

1400

1600

1800

2000

Scaling Factor = 1309.3

Frequency (MHz)

300 400 500 600 700 800 900 1000 500

1000

1500

2000

2500

3000

Frequency (MHz)

Scaling Factor = 861.503

103

Figure E-3: Scaling Factor for NCN15

Figure E-4: Scaling Factor for NCN20

300 400 500 600 700 800 900 1000 2500

3000

3500

4000

4500

5000

5500

Scaling Factor = 3282.0

Frequency (MHz)

300 400 500 600 700 800 900 1000 4500

5000

5500

6000

6500

7000

7500

8000

Scaling Factor = 5686.0

Frequency (MHz)

104

Figure E-5: Scaling Factor for NCN30

Figure E-6: Scaling Factor for NCN40

300 400 500 600 700 800 900 1000 0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7 x 10 4

Scaling Factor = 11562

Frequency (MHz)

300 400 500 600 700 800 900 1000 1

1.5

2

2.5

3

3.5 x 10 4

Scaling Factor = 20266

Frequency (MHz)

105

Figure E-7: Scaling Factor for NDN05

Figure E-8: Scaling Factor for NDN07

300 400 500 600 700 800 900 1000 6500

7000

7500

8000

8500

9000

9500

10000

10500

11000

11500

Scaling Factor = 8.1833 x 103

Frequency (MHz)

300 400 500 600 700 800 900 1000 0.8

1

1.2

1.4

1.6

1.8

2

2.2 x 10 4

Scaling Factor = 1.3635 x 104

Frequency (MHz)

106

Figure E-9: Scaling Factor for NDN10

Figure E-10: Scaling Factor for NDN15

300 400 500 600 700 800 900 1000 1.5

2

2.5

3

3.5

4

4.5

5 x 10 4

Scaling Factor = 2.6082 x 104

Frequency (MHz)

300 400 500 600 700 800 900 1000 3

4

5

6

7

8

9

10

11 x 10 4

Scaling Factor = 5.7668 x 104

Frequency (MHz)

107

Figure E-11: Scaling Factor for NDN20

Figure E-12: Scaling Factor for NDN30

300 400 500 600 700 800 900 1000 4

5

6

7

8

9

10

11

12

13

14 x 10 4

Scaling Factor = 7.0235 x 104

Frequency (MHz)

300 400 500 600 700 800 900 1000 0.6

0.8

1

1.2

1.4

1.6

1.8 x 10 5

Scaling Factor = 9.7111 x 104

Frequency (MHz)

108

Figure E-13: Scaling Factor for NDN40

300 400 500 600 700 800 900 1000 0.6

0.8

1

1.2

1.4

1.6

1.8

2 x 10 5

Scaling Factor = 1.1148 x 105

Frequency (MHz)

109

Appendix F: Shielding Effectiveness Model Results

λσ D Factor Scaling (dB) SE ⋅⋅= 2

Figure F-1: SE Model Fit for NCN05

Figure F-2: SE Model Fit for NCN10

300 400 500 600 700 800 900 1000 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Frequency (MHz)

SE Data

Model

110

Figure F-3: SE Model Fit for NCN15

Figure F-4: SE Model Fit for NCN20

300 400 500 600 700 800 900 1000 1

1.5

2

2.5

3

3.5

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Frequency (MHz)

SE Data

Model

111

Figure F-5: SE Model Fit for NCN30

Figure F-6: SE Model Fit for NCN40

300 400 500 600 700 800 900 1000 3

4

5

6

7

8

9

10

11

12

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 6

8

10

12

14

16

18

20

22

Frequency (MHz)

SE Data

Model

112

Figure F-7: SE Model Fit for NDN05

Figure F-8: SE Model Fit for NDN07

300 400 500 600 700 800 900 1000 2

3

4

5

6

7

8

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 4

5

6

7

8

9

10

11

12

13

Frequency (MHz)

SE Data

Model

113

Figure F-9: SE Model Fit for NDN10

Figure F-10: SE Model Fit for NDN15

300 400 500 600 700 800 900 1000 8

10

12

14

16

18

20

22

24

26

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 15

20

25

30

35

40

45

50

55

Frequency (MHz)

SE Data

Model

114

Figure F-11: SE Model Fit for NDN20

Figure F-12: SE Model Fit for NDN30

300 400 500 600 700 800 900 1000 20

25

30

35

40

45

50

55

60

65

70

Frequency (MHz)

SE Data

Model

300 400 500 600 700 800 900 1000 30

40

50

60

70

80

90

100

Frequency (MHz)

SE Data

Model

115

Figure F-13: SE Model Fit for NDN40

300 400 500 600 700 800 900 1000 30

40

50

60

70

80

90

100

110

Frequency (MHz)

SE Data

Model

116

Appendix G: Shielding Effectiveness Model Results

λσ D

FitFit SF(dB) SE ⋅⋅= 2

Figure G-1: SE Model Fit for NCN05

Figure G-2: SE Model Fit for NCN10

300 400 500 600 700 800 900 1000 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (MHz)

Model

SE Data

300 400 500 600 700 800 900 1000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Frequency (MHz)

Model

SE Data

117

Figure G-3: SE Model Fit for NCN15

Figure G-4: SE Model Fit for NCN20

300 400 500 600 700 800 900 1000 1

1.5

2

2.5

3

3.5

4

4.5

Frequency (MHz)

Model

SE Data

300 400 500 600 700 800 900 1000 2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Frequency (MHz)

Model

SE Data

118

Figure G-5: SE Model Fit for NCN30

Figure G-6: SE Model Fit for NCN40

300 400 500 600 700 800 900 1000 4

5

6

7

8

9

10

11

12

13

Frequency (MHz)

Model

SE Data

300 400 500 600 700 800 900 1000 6

8

10

12

14

16

18

20

Frequency (MHz)

Model

SE Data

119

Figure G-7: SE Model Fit for NDN05

Figure G-8: SE Model Fit for NDN07

300 400 500 600 700 800 900 1000 2

4

6

8

10

12

14

16

Frequency (MHz)

Model

SE Data

300 400 500 600 700 800 900 1000 4

6

8

10

12

14

16

18

20

22

Frequency (MHz)

Model SE Data

120

Figure G-8: SE Model Fit for NDN10

Figure G-9: SE Model Fit for NDN15

300 400 500 600 700 800 900 1000 8

10

12

14

16

18

20

22

24

26

28

Frequency (MHz)

Model

SE Data

300 400 500 600 700 800 900 1000 10

15

20

25

30

35

40

45

Frequency (MHz)

Model

SE Data

121

Figure G-10: SE Model Fit for NDN20

Figure G-11: SE Model Fit for NDN30

300 400 500 600 700 800 900 1000 15

20

25

30

35

40

45

50

55

Frequency (MHz)

Model

SE Data

300 400 500 600 700 800 900 1000 20

30

40

50

60

70

80

90

Frequency (MHz)

Model

SE Data

122

Figure G-12: SE Model Fit for NDN40

300 400 500 600 700 800 900 1000 30

40

50

60

70

80

90

100

110

120

Frequency (MHz)

Model

SE Data

123

Appendix H: White Model Derivation

H.1 Introduction

Investigations into shielding effectiveness theory have previously been conducted by

others such as White (10) and Bushko (9). The work of White is frequently cited in shielding

effectiveness literature. The White model serves as the basis for the work of Bushko, the

most recent reference to composite SE modeling. The model is an extrapolation of the

absorption (skin depth) and transmission coefficient concepts of basic electromagnetics.

Shown below is Equation H-1 that was derived to estimate shielding effectiveness of planar

materials (9).

−+=

r

rrrdB

fft

σµ

σµ 10log 1016834.3SE [H-1]

Absorption Loss: rrft. σµ343 [H-1a]

Reflection Loss:

−

r

rflog

σµ

1010168 [H-1b]

Where:

t = thickness of material (inches) f = frequency (Hertz)

rσ = conductivity relative to copper

rµ = magnetic permeability relative to copper

This equation was originally developed for solid homogeneous metals assuming the same

loss mechanisms previously discussed (absorption and reflection). Each portion of the

equation has been calculated and represented in reference to a 0.001” (2.54 x 10-5 m) thick

sheet of copper. The derivations for the absorption and reflection loss terms in Equation H-1

are shown below.

124

H.2 Absorption Term Derivation

For a highly conductive material, the wavenumber, β, can be approximated by (3):

( )jf −

≅ 1

22

σµπβ [H-2]

As stated in Chapter 5, the loss due to absorption is mathematically represented by the

imaginary portion of the wavenumber. The portion is defined as the skin depth, δ (3).

µσπ

δf1

≡ [H-3]

The skin depth is the distance within the medium where the amplitude of the wave has

decreased to e1

times its original value, where e is the natural number (e ≈ 2.718). A large

skin depth value indicates that the wave will experience little attenuation due to absorption

while passing through the medium. The absorption coefficient,α , is the inverse of the skin

depth (3).

σµπδ

α 0

1f=≡ [H-4]

Where: 0µ = magnetic permeability of free space

= 4 π x 10-7 (Henrys/meter)

f = frequency (Hertz) σ = conductivity (Siemens/meter)

For a thin copper sheet, 0.001” thick with conductivity of 5.82 x 107 S/m at a frequency of

1.0 MHz, the absorption coefficient is =α 15158 m-1, resulting in an transmitted field

strength amplitude of -3.34 dB.

125

343200

010 .

EeE

logEt

dB −=

=

−α

[H-5]

White approximates the absorption loss for an arbitrary metallic material to be linearly

dependent on thickness relative to the original thin copper sheet and proportional to the ratio

of the absorption coefficients for the new material and copper. The ratio includes a frequency

term to retain absorption frequency dependence.

Absorption Term: rrft. σµ343 (with f [MHz] – Bushko)

Absorption Term: rrft. σµ310343 −× (with f [Hz] – White)

H.2 Reflection Loss Term Derivation

The reflection loss is related to the mismatch of impedances that occurs when the wave

passes from free space to the shielding medium. It is this mismatch of impedances that

determines how much of the impinging wave actually enters the material; the bigger the

discrepancy between the impedances, the greater the amount of reflected signal. The

impedance of a material is defined as:

∈

=ˆµ

η [H-6]

In Chapter 6, it was shown that permittivity (∈) is complex and is defined by Equation 6.4-7:

ωσ

j−=∈∈∧

[6.4-7]

Because copper is a conductor, the real part of the permittivity is not defined and is ignored

when calculating the impedance. The impedance for the copper sheet at 1.0 MHz, Cuη ,

therefore is 2.604 +2. 604j x 10-4 Ω and the impedance of free space, 0η , 377 Ω. The

transmission coefficient is the ratio of impedances at the interface of the two different

126

mediums. It must be calculated at both interfaces, where the signal enters and leaves. For the

first interface, it is given by (13):

Cu

CuTηη

η+

=0

12

2 [H-7]

)j(.T += 1382112 x 10-6

and for the second interface, the coefficient is (13):

Cu

Tηη

η+

=0

021

2 [H-8]

=21T 2.0 – (1.382 x 10-6 )j

Multiplying 12T and 21T together yields the fraction of the original signal strength that exits

the shielding media. Squaring this result gives the power loss due to the mismatched

impedances. The power loss can be represented in SE dB using Equation H-9.

( )211210 TTlogLossdB = [H-9]

108−=dBLoss

The negative loss value indicates that the strength of the signal is reduced as it passes through

the media. To represent it as a portion of total shielding effectiveness the sign must be

changed to a positive. To use the above value to determine the reflection loss for a material

other than copper at a given frequency, the relative reflection loss as compared to the copper

sheet must be calculated. This relative loss is given by:

r

rflogσµ

10 [H-10]

Where: rσ = conductivity relative to copper

rµ = magnetic permeability relative to copper

127

A negative value again indicates that less signal passes through the new media as compared

to copper. Therefore, a negative value means a greater shielding effectiveness and the

equation should be subtracted from the other portions of the shielding effectiveness equation.

If the frequency ( f ) is expressed units of [Hz] instead of [MHz], an extra 60 dB must be

added to the loss calculated in H-9.

( ) 600110110 6 ==× MHz.Hzlog dB [H-11]

The total shielding effectiveness of a material is the sum of the absorption and reflection

terms (9):

−+=

r

rrrdB

flog ft.SE

σµ

σµ 1010168343 [H-1]

Where: t = thickness of material (inches) f = frequency (Hertz)

rσ = conductivity relative to copper

rµ = magnetic permeability relative to copper

128

Appendix I: Proposed Model Comparison to White Model

Figure I-1: White Model and Proposed Model Comparison for NCN05

Figure I-2: White Model and Proposed Model Comparison for NCN10

300 400 500 600 700 800 900 1000 -140

-120

-100

-80

-60

-40

-20

0

20

Frequency (MHz)

Proposed Model

SE Data

White Model

300 400 500 600 700 800 900 1000 -140

-120

-100

-80

-60

-40

-20

0

20

Frequency (MHz)

White Model

SE Data Proposed Model

129

300 400 500 600 700 800 900 1000-140

-120

-100

-80

-60

-40

-20

0

20

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure I-2: White Model and Proposed Model Comparison for NCN15

300 400 500 600 700 800 900 1000-70

-60

-50

-40

-30

-20

-10

0

10

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure I-3: White Model and Proposed Model Comparison for NCN20

SE Data

White Model

Proposed Model

130

300 400 500 600 700 800 900 10000

2

4

6

8

10

12

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure I-4: White Model and Proposed Model Comparison for NCN30

300 400 500 600 700 800 900 10006

8

10

12

14

16

18

20

22

Frequency (MHz)

Shi

eldi

ng E

ffect

(dB

)

Figure I-5: White Model and Proposed Model Comparison for NCN40

Proposed Model

White Model

SE Data

White Model

Proposed Model

SE Data

131

Figure I-6: White Model and Proposed Model Comparison for NDN05

Figure I-7: White Model and Proposed Model Comparison for NDN07

300 400 500 600 700 800 900 1000 -60

-50

-40

-30

-20

-10

0

10

20

Frequency (MHz)

Proposed Model SE Data

White Model

300 400 500 600 700 800 900 1000 -140

-120

-100

-80

-60

-40

-20

0

20

Frequency (MHz)

White Model

SE Data Proposed Model

132

Figure I-8: White Model and Proposed Model Comparison for NDN10

Figure I-9: White Model and Proposed Model Comparison for NDN15

300 400 500 600 700 800 900 1000 15

20

25

30

35

40

45

50

55

Frequency (MHz)

Proposed Model

SE Data

White Model

300 400 500 600 700 800 900 1000 8

10

12

14

16

18

20

22

24

26

Frequency (MHz)

Proposed Model SE Data

White Model

133

300 400 500 600 700 800 900 100020

25

30

35

40

45

50

55

60

65

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Frequency (MHz)

Shi

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ng E

ffect

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Figure I-10: White Model and Proposed Model Comparison for NDN20

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Frequency (MHz)

Shi

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Figure I-11: White Model and Proposed Model Comparison for NDN30

SE Data

White Model

Proposed Model

SE Data

White Model

Proposed Model

134

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Frequency (MHz)

Shi

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Figure I-12: White Model and Proposed Model Comparison for NDN40

SE Data

White Model

Proposed Model