luis r.j. costa - defmin.fi · luis r.j. costa matine seminar 18.11.2015 rad department. sphere of...

Scattering from a sphere of nonlinear material

Luis R.J. Costa

Department of Radio Science and EngineeringAalto University School of Electrical Engineering

MATINE seminar 18.11.2015

Contents

Why study scattering from a sphere of nonlinear material?

Model of the problem to be simulated

Simulations

What do the data tell us?

What next?

Conclusions

Why study scattering from a sphere of nonlinearmaterial?

Nonlinearity generates harmonics

How much energy is shifted from the fundamental frequency tothe harmonics?Type of nonlinearity used: that of a pn-junction (exp. function)

A real material; it has the most extreme nonlinearity possible

0 50 100 150 200 250 300

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

Diode voltage

APLAC Simulator User: TKK/RAD Circuit Theory Group

Wav

efo

rm/V

Time/ps

Waveform

0 10 20 30 40 50 60

0

1.5

3

4.5

6

Diode voltage


Sp

ectr

um

/V

f/GHz

Spectrum (V)

0 10 20 30 40 50 60

−50.00

−38.75

−27.50

−16.25

−5.00

Diode voltage


Sp

ectr

um

/dB

f/GHz

Spectrum (dB)

Scattering from a sphere of nonlinear material 3/25

Luis R.J. Costa MATINE seminar 18.11.2015RAD department


Nonlinearity generates harmonicsHow much energy is shifted from the fundamental frequency tothe harmonics?

Type of nonlinearity used: that of a pn-junction (exp. function)A real material; it has the most extreme nonlinearity possible

0 50 100 150 200 250 300

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

Diode voltage


Wav

efo

rm/V

Time/ps

Waveform

0 10 20 30 40 50 60

0

1.5

3

4.5

6

Diode voltage


Sp

ectr

um

/V

f/GHz

Spectrum (V)

0 10 20 30 40 50 60

−50.00

−38.75

−27.50

−16.25

−5.00

Diode voltage


Sp

ectr

um

/dB

f/GHz

Spectrum (dB)




Nonlinearity generates harmonicsHow much energy is shifted from the fundamental frequency tothe harmonics?Type of nonlinearity used: that of a pn-junction (exp. function)

A real material; it has the most extreme nonlinearity possible

0 50 100 150 200 250 300

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

Diode voltage


Wav

efo

rm/V

Time/ps

Waveform

0 10 20 30 40 50 60

0

1.5

3

4.5

6

Diode voltage


Sp

ectr

um

/V

f/GHz

Spectrum (V)

0 10 20 30 40 50 60

−50.00

−38.75

−27.50

−16.25

−5.00

Diode voltage


Sp

ectr

um

/dB

f/GHz

Spectrum (dB)




Why sphere?it is a canonical benchmark whose radar cross-section (RCS) iswell known (Mie scattering)

Can nonlinear materials be exploited in stealth technology?



Model of the problem to be simulated

Problem is simulated using the FDTD method

green region: plane wave source (TFSF region)

yellow surface: Huygens surface

grey surface: computational space terminated inCPMLsphere of nonlinear material

The Huygens surface is required to obtain the far field



Modelling the nonlinear materialModelling a lumped element in FDTD

Il(Vl)Vl

i, j, b

i, j, a

Ez → IEM Rg Il(Vl)Vl

∮CHn+ 1

2·dl=∫

S

(ε∂En+ 1

2

∂t+σEn+ 1

2+Jn+ 12

l

)·ds

Ampere’s law

Update equation:

Ez|n+1i,j,k+ 1

2=(

1− σ∆t2ε

1 + σ∆t2ε

)Ez|ni,j,k+ 1

2+( ∆t

ε

1 + σ∆t2ε

)×

{Iz|ni,j,k∆x∆y

−Il(V

n+ 12

l )∆x∆y

}

Iz|ni,j,k = (Hy|n+ 1

2i+ 1

2 ,j,k+ 12−Hy|

n+ 12

i− 12 ,j,k+ 1

2)∆y − (Hx|

n+ 12

i,j+ 12 ,k+ 1

2−Hx|

n+ 12

i,j− 12 ,k+ 1

2)∆x

Lumped elements can be directed only in the direction of the primary axes



Modelling the nonlinear materialModelling a diode spanning one or many Yee cells

Diode current-voltage equation: Id = Is

(eαVd − 1

), α = q/(kT)

Diode voltage Vd:spanning one cell from i, j,a to i, j,a + 1: Vn+ 1

2d = −Ez|

n+ 12

i,j,a ∆z

spanning cells from i, j,a to i, j,b: Vn+ 12

d = −∆z

k=b−1∑k=a

Ez|n+ 1

2i,j,k

Iterate Vn+ 12

d using, e.g., Newton-Raphson:

Vn+ 12 ,l+1

d = Vn+ 12 ,l

d −Vn+ 1

2 ,ld /Rg + Is

(e

qkT V

n+ 12 ,l

d − 1)− IEM

1/Rg + qkT Ise

qkT V

n+ 12 ,l

d

,Vn+ 12 ,0

d = Vn− 12

d

Compute Id and plug in Il = −Id via the update equations in thecolumn of cells in which it exists



Sphere of nonlinear material

Sphere is filled with z-directed diodes

A diode current passes through every Yee cell within the sphere



Encountered problems and work-arounds

The presence of many diodes in close proximity makes thesimulation susceptible to instability

the material is strongly nonlinear

The time step may have to be reduced from the largest possiblestep obtained via the Courant-Friedrichs-Lewy conditionThe plane-wave amplitude cannot be arbitarily large

the strong nonlinearity in the current-voltage dependence causesinstability

the larger the field strength, the larger the voltage implying largerchanges in current even for relatively small changes in the largevoltage



SimulationsDifferent spheres are simulated for

wavelength much larger than the spherewavelength of similar size as the sphere

Simulation parameters

Grid size: 120× 120× 120Discretisation: ∆x = ∆y = ∆z = 0.5 mm,time step damping factor: 0.5⇒ ∆t = 0.963 ps

Plane wave propagates along x-axis with amplitude 5000 V/m

Diode saturation current: 1 pA (typical value for a silicon diode)

Sphere diameter: 32 mm (64 cells at equatorial plane)

Scattered-field observation point: 15,60,60 (between CPML andthe Huygens surface)



Simulations: first a PEC sphereCase 1a: f = 1 GHz, λ ≈ 300 mm, time steps n = 16 384

The plane-wave frequency is obvious.The model errors are also obvious

the error can be reduced but not eliminated by increasing nScattering from a sphere of nonlinear material 11/25


Simulations: first a PEC sphereCase 1a: f = 1 GHz, λ ≈ 300 mm (arbitrary scale)

RCSθ

RCSφ

f 2f 3fScattering from a sphere of nonlinear material 12/25


Simulations: diode sphereCase 1b: f = 1 GHz, λ ≈ 300 mm, n = 16 384, εr = 1

A very strong DC component (for electromagnetic rectification) ispresentHarmonic peaks are distinguishable



Simulations: diode sphereCase 1b: f = 1 GHz, λ ≈ 300 mm, εr = 1 (arbitrary scale)

RCSθ

RCSφ

f 2f 3f



Simulations: diode sphereCase 1c: f = 1 GHz, λ ≈ 300 mm, n = 16 384, εr = 11.68

The fundamental frequency is clearly visible



Simulations: diode sphereCase 1c: f =1 GHz, λ≈300 mm, εr =11.68 (arbitrary scale)

RCSθ

RCSφ

f 2f 3f



Simulations: again a PEC sphereCase 2a: f = 10 GHz, λ ≈ 30 mm, n = 16 384

The plane-wave frequency and model error are obvious here too.Scattering from a sphere of nonlinear material 17/25


Simulations: PEC sphereCase 2a: f = 10 GHz, λ ≈ 30 mm (arbitrary scale)

RCSθ

RCSφ

f 2f 3f



Simulations: diode sphereCase 2b: f = 10 GHz, λ ≈ 30 mm, n = 16 834, εr = 1

A very strong DC component is presentThe fundamental frequency and harmonics are visible



Simulations: diode sphereCase 2b: f = 10 GHz, λ ≈ 30 mm, εr = 1 (arbitrary scale)

RCSθ

RCSφ

f 2f 3fScattering from a sphere of nonlinear material 20/25


Simulations: diode sphereCase 2c: f = 10 GHz, λ ≈ 30 mm, n = 16 834, εr = 11.68

The fundamental frequency is clearly visible



Simulations: diode sphereCase 2c: f =10 GHz, λ≈30 mm, εr =11.68 (arbitrary scale)

RCSθ

RCSφ

f 2f 3f



What do the data tell us?εr = 1

Harmonics are generated as expected, but they are weakCharge accumulates at the nodes of the diodes since they haveno discharge path available

the scattered field stays positive in the back-scattered directiona prominent DC component emerges in the spectra as a result

εr = 11.68Effects of nonlinearity are no longer apparent

harmonics are not visibleThe fundamental frequency is now significant

Other remarksSeveral cycles of the plane wave are required for thenonlinearities to kick inA relatively wide modulated pulse is necessary for thenonlinearities to have an effect



What next?

Is it possible to arrange the diodes so as to enhance the effectsof nonlinearities?

Can the incident polarisation current be manipulated through asuitable arrangement of the diodes?

Study how different pulses interact with spheres of suchnonlinear materials



Conclusions

A model for a nonlinear material is successfully implemented andused to compute scattering from a sphere made of a nonlinearmaterialSimulation results show interesting behaviour in the spheremodelled

harmonics are created, but not always

Energy is shifted primarily to DC, but to other frequencies as well

Application to stealth technology requires further study



luis r.j. costa - defmin.fi · luis r.j. costa matine seminar 18.11.2015 rad department. sphere of...

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