radio frequency engineering lecture #6 engineering approach stepan lucyszyn...

Radio Frequency EngineeringLecture #6 Engineering Approach

Stepan Lucyszynステファン・ルシズィン　インペリアル・カレッジ・ロンドン准教授

Lecture #6

Engineering Approach for Analytical Electromagnetic Modelling of THz

Metal StructuresThe main objective of this lecture is to teach a useful tool for enabling a student to derive analytical expressions for a number of electromagnetic problems relating to THz metal structures. The traditional approach when using the classical relaxation-effect model can be mathematically cumbersome and not insightful. This lecture briefly introduces various interrelated electrical engineering concepts as tools for characterizing the intrinsic frequency dispersive nature of normal metals at room

temperature. This Engineering Approach dramatically simplifies the otherwise complex analysis and allows for a much deeper insight to be gained into the

classical relaxation-effect model. Problems with worked-though solutions will be given in class.



OVERVIEW Drude’s frequency dispersion model The Engineering Approach

Equivalent transmission line model Q-factor for metals Kinetic inductance Complex skin depth Boundary resistance coefficient

Applications Metal-pipe rectangular waveguides Cavity resonators Single metal planar shield

Conclusions



ModelEffectSkinClassical

ModelEffectRelaxationSimple

jj

o

oRo

RRR

2)(1

'

1'''

Classical relaxation-effect frequency dispersion model:

Drude’s Model



]['''

'''''' 1

mjjjjj

rorororoI

]['

'''''

'''

roro

roroI

j

j

j

j

Compare vector Helmholtz equations (for an unbounded plane wave in one-dimensional space) with those of the telegrapher's equations.

Electromagnetic Characterisation of Homogenous Materials

#1 Line Modelling



Free space Normal material at room temperature

RL

G C

LR

G C Zo

o o

o o

Dz << l

Generic Equivalent Transmission Line Model



Sccc

SSSro

S

rSSS

r

o

rSSS

jandandjZ

jj

sradj

j

jjXRZ

1'''

11

/10

0

1500

with

Generic equations:

Intrinsic Modelling for Normal Metals at Room Temperature



Classical skin-effect model:

Normal metal at room temperature

o

Dz << lo

o o

o

LSo

RSo

]/[)(

][12

squarejXRZ

jj

SoSoSo

o

o

o

oIo



RR

RRR CjG

LjRZo

RRRRR

RRR CjGLjR

Zo

LjR

)(

11 22

SRSRSR

oo

oIRR jXRZ

j

jZo

1So

SRSR

RZR

1SoSRSR

RZL

1124 uuuu u

Classical relaxation-effect model:



Classical relaxation-effect models:

21

o 22

21

zo D

21

o

22

21

zo D

Dz<< lR

/}{ SRSR ZL

}{ SRSR ZR

oo

Normal metal at room temperature

o o LSR

RSR 21

o

21

o

21

o

21

o

Dz<< lR



Distributed-element parameters for the classical relaxation-effect model

222 11;

1;;0

oo

oR

oRoRR CGLR

Distributive shunt inductance

]/[

1122

2

22_ mHzzC

LoR

RSHUNT D

D



Propagation Delay Per Unit Wavelength

RRSHUNT

RRRRRR

RpR

RRR

pR

R

pRpR

jzL

jGLjCjGLj

vwheresfvv

l

lll

D

2_

1)(

}{;

}{

2/

1



Circuit-based Simulation Results for Gold at Room Temperature

Equivalent Transmission Line model

Elementary Lumped-element Circuit Values ZIN [](Theory:

0.4608+j1.1124)pR

[fs/R]

(Theory:170:494)

RRz[]

LRz[fH]

GRz[mS]

CRz[fF]

LSHUNT_Rz[pH]

ZT = ZSR

Extracted model

1.341 24.1 -0.654 0.4607 + j1.137 170.491

Alternative Extracted

model

1.341 24.1 1.126 0.4607 + j1.137 170.491

with = 1, depth l = R, z = R / 400 = 1.067 [nm]



]/[1 squareHLLLL SokSoSR

2/1

2

So

SoL 2/1

SookL

k

SR

SoSR

L

L

LR

k

So

SRSR

L

L

LX

and

and

20539.0 fora

Kinetic inductance is created from the inertial mass of a mobile charge carrier distribution within an alternating electric field (i.e. classical electrodynamics).

#2 Kinetic Inductance



0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

2.5

Classical skin-effect RSo

Simple relaxation-effect RSR '

Classical relaxation-effect RSR

Surf

ace

Res

ista

nce,

Ohm

s/sq

uare

Frequency, THz

Gold at room temperature

Intrinsic classical skin-effect model overestimates ohmic losses and Detuning. Also,underestimates extrinsic losses.



0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

2.5

Surf

ace

Rea

ctan

ce, O

hms/

squa

re

Frequency, THz

Classical skin-effect XSo

Simple relaxation-effect XSR '

Classical relaxation-effect XSR


Intrinsic classical skin-effect model underestimates perturbation detuning



0 1 2 3 4 5 6 7 8 9 10 11 1210-3

10-2

10-1

100

f = 5.865 THz

a = 0.539

LSO

LSK

LSR

(LSO

+ a LSK

)

Surfa

ce R

eact

ance

s, [O

hms/

squa

re]

Frequency, THz

0 1 2 3 4 5 6 7 8 9 10 11 12

2.5x10-3

10-2

10-1

f = 5.865 THz

LSO

= (oo) / 2 L

SK= R

So

LSR

= XSR

/ L

SO+ a L

SK

Surfa

ce In

duct

ance

s, [p

H/sq

uare

]Frequency, THz

a = 0.539



Material Q-Factor

effectiver

effectiverm tanQ

1

mR

mo

S

S

equivalent

equivalentm Qfor

Qfor

Z

Z

n

nQ

0

02

2

2

2

2

2

#3 Q-factor



Component Q-Factor

c

cc R

XQ

cR

co

S

Sc Qfor

Qfor

Z

Z

n

nQ

1

1

c

cm Q

QQ

2

1 2

21 mmc QQQ


)(mRQ 21 mRcR QQ

and



0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Qmo

QmR

Qco

Q-F

acto

rs

Frequency, THz

Qc

Qm

f = 5.865 THz

QcR




]['''1

mj ccc

Complex Skin Depth: c

ccQ

#4 Complex Skin Depth




cRSRcR

cR

SoSR

cRcR

cR

cRcocR

jQZQjQ

ZZ

fc

jQQ

jQ

1}{1

}{

..

}{1

}{



1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50

60

70

80

90

SO

SR

Im{ CO

}

Im{ CR

}

Re{ CR

}

lo/10

lR/10

Leng

ths,

nm

Frequency, THz


3/1



s

o

Rk

Boundary resistance coefficient:

Surface dissipative power density per unit area:

k

PP oS Po = Power flux density per unit area in air

k1tan Forward E-field and average power tilt angle:

1 0.74~

6

2 kkQ

mnlTEU

For a cube in any mode (with perturbation theory only):

#5 Resistance Coefficient



21

o

22

21

zo D

21

o

22

21

zo D

Dz<< lR

/}{ SRSR ZL

}{ SRSR ZR

oo

Elementary Lumped-element Circuit

)cosh()sinh(

)sinh()cosh(][

llYo

lZol

DC

BAABCD

Line Model Validation



When the terminating impedance is equal to the complex conjugate of the characteristic impedance of the transmission line:



Parameters for

l = R , = 1, ZT = Z*SR

TheorySynthesized Transmission Line

ABCD Parameter Matrix Calculations

Microwave Office®

No. of Sections, N --- 400 800 400 800ZSR[] 0.46079043

+j1.112446500.46079964

+j1.112468750.46079043

+j1.11244650--- ---

RR[R-1] 15.1689

+j6.283215.1681-j0.0013

15.1685-j0.0004

--- ---

S11 0 0.0006+j0.0271

0.0002+j0.0135

0.0006+j0.0270

0.0001+j0.0133

Return Loss [dB] -247.51 -31.35 -37.41 -31.37 -36.9S11= e-2(1+2)

[1+j(1+2)]107

2.583+j6.237

2.579+j6.242

2.582+j6.240

2.564+j6.238

2.567+j6.239

GMAX [dB] -123.4126 -123.4093 -123.4099 -123.41 -123.4Absorption Loss [dB] -131.756 -131.748 -131.752 -131.9 -131.9S11= tan-1(1+2)[o] 67.50 67.55 67.52 67.58 67.56

Comparison of modelled parameters for gold at room temperature, at 5.865 THz, determined from theory, direct ABCD parameter matrix calculations and synthesized equivalent transmission line

models using commercial circuit simulation software



22_

11

f

fc

o

c

o

oidealg

l

ll

ll

]/[)()(

1010

mNpGRs

oTEWG

]/[)( squarej

jZs

o

ro

][

1

21

)( 1

2

2

10

10

10

m

b

a

b

G

TEc

TEc

d

x

y

z

b

a

and

THz MPRWGs



New Waveguide Standard for Terahertz Frequencies

• No extension of industry standards above 325 GHz (WR-3)

• No compatibility among hardware• Impractical transitions• Defined in inches, rounding errors

Why New Standard ?

5 THz horns and coupler (C. Walker U. Arizona)

1.5 THz Frequency Multiplier Chain (JPL)

John Ward, JPL 2006



• Widely used global industry standard• Logarithmic scale• Infinitely extendable• Repeats every decade• Decision tree with range

of coarse and fine spacing

New Waveguide Standard for Terahertz Frequencies:ISO 497 Preferred Metric Sizes



Proposed air-filled MPRWG and cavity resonator definitions and specifications



Variational Methods for CalculatingPropagation Constant for the TEmo Mode



0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.00.0

0.1

0.2

0.3

0.4

0.5

0.6

100 m

125 m

160 m

Classical skin-effect o

Simple relaxation-effect R '

Classical relaxation-effect R

Atte

nuat

ion

Cons

tant

, dB/

mm

Frequency, THz

200 m

2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

25 m

32 m

50 m

Classical skin-effect o

Classical relaxation-effect R '


Atte

nuat

ion

Con

stan

t, dB

/mm

Frequency, THz

75 m

Calculated attenuation constants for the dominant TE10 mode



0 1 2 3 4 5 6 7 8 9 10 11 120

50

100

150

200

250

300

350

400

Erro

r in

Atte

nuat

ion

Con

stan

t, %

Frequency, THz

Classical skin-effect Simple relaxation-effect

Resulting errors in attenuation constant calculations

%10011

1%100

2

2'

'

R

RRR

E

%10011%100 2

R

Roo

E

Classical skin-effect: 108% error at 12 THz

Simple relaxation-effect: 373% at 12 THz

Power Loss Method (for simplicity)

2

2

0

21

1f

f

a

b

f

fb

R c

c

sc

dc



%10011%100 2

WGR

WGRWGoo

E

%100

SR

SRSo

R

RRE

o

1

1

1jRZ SoSR

20

%100%1001 mRcR QQEo

with

for

Classical skin-effect: 108% error at 12 THz

%100539.0 mRQEo

e.g. Calculated error using this approximation is 110% at 12 THz

Errors in attenuation constant



HFSSTM electromagnetic simulations of attenuation constant for JPL 100 m band

1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.90.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60 Classical skin-effect

o

Simple relaxation-effect R '


Effective relaxation-effect using R' &

R'

Atte

nuat

ion

Con

stan

t, dB

/mm

Frequency, THz

1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.90.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

Calculated classical skin-effect o

Calculated simple relaxation-effect R '

Calculated classical relaxation-effect R

Classical skin-effect in HFSSTM

All relaxation-effect in HFSSTM

Atte

nuat

ion

Con

stan

t, dB

/mm

Frequency, THz



With Perturbation model, unloaded Q-factor in the mnl mode : mnl

oS

mnlI

TEoU RQ

mnl

'' _

Geometric factor :

][Hmnlomnl

][

2

2

8 2233

2/322_ m

daaddab

dabmnlImnl

l

where

' 0.74~ ' 6

2' cube, aFor

'' where,'

22'' 2/1

2233

2/322

ooTEoU

oS

ooooTEoU

kkQ

Rk

daaddab

dabkQ

mnl

mnl

For all modes!

THz Cavity Resonators



mnlImnlI f

c

__ l

22_

222

_mnlI

mnlI d

l

b

n

a

mcf

badcommonmosta

badeiheighthalffora

da

adI

2223/8

2..2222101_l

badeicommonmosta

badeiheighthalffora

badeicubefora

adforaVolume/Are

daaddab

daabd

222..)]102(2/[3

2..8/

..6/

)]()(2[2

)(2233

22

101



Complex Resonant Frequency

modelon perturbati with 2

~~

22

11'''~

2

ou

ooo

ou

o

ouoooo

Qj

Qj

Qj

As determined by Eigenmode solution in HFSSTM

Overall frequency detuning )'(' oIo Dtakes into account both perturbation (represented by Xs) and detuning due to Ohmic losses (represented by Rs)

)( oII D

Io'o

Due to 0SR 0SXDue to0

ID'oD



I ISR Iu

IIo Q

j

2

~~

Over-simplified Solution(O. Klein et al.,1993, and M. Dressel & G. Gruner, 2002)

IS

IIU R

Q

Simplified Solution(J. C. Salter, 1946)

Iu

IIIo Q

j

2

)(~~ D where ID is due to perturbation



Exact Solution

Io

oUQ ou

o

ouoooo Q

jQ

j

22

11'''~

2

'o 'oSR 'oUQ

E QE

Self-consistent analytical solution for all dispersion models

Self-consistent analytical solutionWith and1r 0

Using oc Im instead of S

Change in Surface Impedance:

Cavity Perturbation Formula



oRo

Derivation of Driven Frequency of Oscillation

After equalizing equations for surface reactance:

'oRo

ooo

badeicommonmostfora

Kwhere

K

r

oo

oRI

oRoRoR

222..)102(2

38

0)(1 2

0

)(1

'

2''

KoRI

oRoR

0

KooI

oo

2

22

2

1

KWWW IIIoo where



Self-consistent Solution for General Case of Metal

2233

2

322

2

2

8

)(

daaddab

dabQ

o

I

ocr

IIou

l

badeicommonmostfora

Qo

I

ocrou 222..

1254

231



0 1 2 3 4 5 6 7 8100

150

200

250

300

350

400

450

500

550

600

650

700

750

Classical skin-effect Simple relaxation-effect Classical relaxation-effect

Classical skin-effect in HFSSTM

25 m32 m

50 m

75 m

100 m

125m

160m

200 m

Frequency, THz

Fre

qu

en

cy

De

tun

ing

, GH

z

Un

loa

de

d Q

-facto

r fo

r T

E 101 C

avit

y M

od

e

0

5

10

15

20

25

30




0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

8025 m

32m

50m

75 m

100 m125 m

160 m

200 m

Classical skin-effect Simple relaxation-effect

Err

or

in Q

Fac

tor,

%

Frequency, THz

0

5

10

15

20

25

30

35

40

45

Erro

r in F

requ

ency D

etun

ing

, %




mnloo

mnlI

ooSorTEooUo

mnlQ

''

2' _

mnloR

mnlI

oRcRrTEoRUR

mnlQ

''

1' _

oomR

TEUo

TEooUoQ

QQ mnl

mnl '

1'

oRmR

oRcRTEUoTEoRUR Q

QQQ

mnlmnl '

'1'



Errors in Unloaded Q-factor

%1001

''

'%100

)'(

)'()'('

oRcRoomR

oRmR

oRUR

oRURooUooRQo QQ

Q

Q

QQE

%1001'

1'

oRcR

oRQoQ

E

oooR ''

%100'8553.11

1'

1

oRmR

oRQoQ

E

20

Classical skin-effect: 41% error at 7.3 THz

e.g. Calculated error using this approximation is 40% at 7.3 THz



• HFSSTM current versions (v.10 & 11) cannot accurately predict the performance of structures operating at terahertz frequencies

• Intrinsic dispersion models: Classical skin-effect & simple relaxation-effect inflate the attenuation. Therefore, extrinsic loss effects (e.g. surface roughness) may be underestimated



RF metal shielding is found in many applications; ranging from:

• Construction of high isolation subsystem partitioning walls

• Efficient quasi-optical components (e.g. planar mirrors and parabolic reflectors for open resonators and antennas)

• Creating guided-wave structures that have (near-)zero field leakage (e.g. metal-pipe rectangular waveguides and associated closed cavity resonators) • Embedding ground planes within compact 3D multi-layer architectures.

THz Metal Shielding



Metal shields should be made as thin as possible, while meeting the minimum values for figures of merit within the intended bandwidth of operation, in order to reduce weight and cost.

For reasons of structural integrity, thin metal shielding can be deposited onto either a solid plastic/ceramic or even honeycomb supporting wall.

Thin metal shielding embedded between dielectric layers (e.g. to create conformal ground planes or partition walls) can avoid issues of poor topography when integrating signal lines within 3D multi-layered architectures.

Shielding effectiveness, return loss and absorptance (or absorptivity) are important figures of merit that are quoted to quantify the ability to shield electromagnetic radiation.

An electrical engineering approach, which can include network analysis and the synthesis of predictive equivalent transmission line models, can accurately solve specific EM problems.



Uniform plane wave at normal incidence to an infinite single planar shield in air

Physical representation Equivalent 2-port network model

TS

TS

ZZ

ZZ

1 12

TS

ST

ZZ

ZZ

TS

S

ZZ

Z

21 11

TS

T

ZZ

Z

21 22

Voltage transmission coefficients:

Voltage-wave reflection coefficients:



2

2

02121

i

i

TT eeS

22

1

1211

T

T

e

eS

2

2

022

21111

i

i

TT eeS

2

1

2

1111

1

T

T

e

eS

Transient response solution

Steady-state solution

]..2,1,0[ i

11 Te

S-Parameters Analysis for Single Planar Shield



cS Q

jaaT 1

]5[15}{

55 1

SR

cRcRR

RSRR Q

j

Q

Boundary resistance coefficient 1 cS

o QR

k

11

11

c

c

jQk

jQk

k

jQ

jQk

jQ

jQk

jQ c

c

c

c

c

1212

1

121 2

2

1

22

cc jQk

k

jQk

k

SRRaT and

Therefore,

e.g. )21()1( cRQ ]5[21

15)1(5)1( 1

SRSRR

j

S-Parameters Analysis for Single Planar Shield

a = thickness normalized to normal skin depth



cRRR

cR

Q

ja

cRRcRR

Q

ja

cRRR

Q

jak

jQ

ejQkjQk

ejQkS

cRR

cRR

1sinh

12

11

14

1222

1

21

cRR

cRR

cRR

cRR

RQ

ja

Q

ja

RR

jQk

jQk

Q

ja

Q

ja

e

eS

cRR

cRR

1

1ln1sinh

1sinh

1

12

1

1

12

111

Approximation error : < 0.4% up to aR = 10, frequency less than 12 THz

S-Parameters Analysis: Classical Relaxation-effect Model



jak

j

ekjkj

ekjS

ooja

oo

jao

oo

o

2sinh

22

22

242222

2

21

o

oo

o

oja

ja

oo

kj

kjja

ja

e

eS

o

o

2

2ln2sinh

2sinh

1

12

12

22

111

So

SRR

Soo a

Ta

S-Parameters Analysis: Classical Skin-effect Model

Approximation error : < 0.6% up to aR = 10, frequency less than 12 THz



dBdBdBRTj

RdB MARSeSSE o 21102110 log20log20

Analysis of Shielding Effectiveness (SE)

][log10 10 dBP

PSE

i

tdB

cR

R

SR

o

oSR

oSRRRdB jQ

k

ZZ

ZR

14log20

4log20~

4log20log20 1010

2

102110

RaTT

dB aeeeA RSRR 686.8log20log20log20 10/

1010

cRR

RR Q

ja

TR

TdB eeeM

12

102

10

2

110 1log201log20~1log20

Cross-boundary reflections

Absorption

Multiple reflections



Correction Factor for Multiple Reflections Calculations

Classical-relaxation Model Error using Approximation

%100_

__

exactdB

exactdBionapproximatdBMdB M

MME 21101log20 R

TdB

ReM



Screening Effectiveness Calculations

Classical Relaxation-effect Model

)/2cos()2cosh(

/18log10

22

10cRRR

RcRdBR Qaa

kQSE

)2cos()2cosh(

/16log10

2

10oo

odBo aa

kSE

Classical Skin-effect Model

Less shielding as frequency increases and/or thickness decrease



%100

dBR

dBRdBodBSE SE

SESEE

Due to absorption, a peak value

3/1

).

Beyond the peak region

Linearly increase,11.7% at T = lm = 1.6SR

Screening Effectiveness Calculations-Error using Approximation

046.2

7.8% at aR = 10



Reflection Characteristics: Return Loss Calculations

][log20log10log10 11101010 dBSP

PRL

i

rdB

32

22

1

2

11

RR

RR

RR

R afork

kaallforS

RL

Less reflected power as frequency increases and/or thickness decrease



%100

dBR

dBRdBodBRL RL

RLRLE

Reflection Characteristics: Return Loss CalculationsError using Classical Skin-effect

109% at T = 10SR

046.2Thickness invariant above

3Ra



Absorptance Calculations

RLSE

P

P

P

P

P

PAB

i

r

i

t

i

adB

1log10

1log10log10

10

1010

34

1

12

1

2

11

2

21

RSRR

R

RRR

aforRk

aallforSSAB

More absorbed power as frequency increases and/or thickness decrease



Absorptance Calculations: Error using Classical Skin-effect

%100

dBR

dBRdBoABdB AB

ABABE

14% at T = 10SR

046.2Thickness invariant above

3Ra



Synthesis Modelling of Metal Shielding Walls

21

o

22

21

zo D

21

o

22

21

zo D

o o o o o o

R1

o o o o

R2

R1 R2

2N

Rz lD

Classical Relaxation-effect Model: Equivalent transmission line model

][341.1 fHzLR D ][1.24 mSzGR D ][126.1_ pHzL RSHUNT DFor example, at 5.865 THz, gold at room temperature



Comparison of modelled parameters for gold at room temperature, at 5.865 THz, determined from theory, direct ABCD parameter matrix calculations and synthesized equivalent transmission line

models using commercial circuit simulation software



Conclusions• An electrical engineering approach to the modelling of specific electromagnetic problems at THz frequencies is introduced, using 5 interrelated concepts (transmission line modelling, kinetic inductance, Q-factor, complex skin depth and boundary resistance coefficient).

• The Engineering Approach has 5 main advantages:– Excellent pedagogical tool– Greatly reduces otherwise lengthy mathematical derivations– Reduces risk of introducing mistakes– Avoids the need for poor approximations– Can replace the need for slow numerical computations (with simple structures)– Gives new perspectives and deeper insight

• While the focus has been on the characterization of normal metals (magnetic and non-magnetic) at room temperature, it is believed that the same methodology may also be applied to metals operating in anomalous frequency-temperature regions, superconductors, semiconductors, carbon nanotubes and metamaterials.

radio frequency engineering lecture #6 engineering approach stepan lucyszyn...

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