transmission line (tl) approach of left-handed (lh)...
Post on 06-Aug-2020
18 Views
Preview:
TRANSCRIPT
Microwave Electronics Lab
Transmission Line (TL) Approach ofLeft-Handed (LH) Materials
Christophe Caloz, Hiroshi Okabe, Taisuke Iwai and Tatsuo Itoh
Electrical Engineering DepartmentUniversity of California, Los Angeles
Microwave Electronics Lab
LH-TL as the Dual of the Conventional TL (1)Conventional RH-TL (lossless) LH-TL (lossless)
LCjZYjβγ ω===
linear→= LCωβ
>==
>==
01
01
LCddv
LCv
g
p
βω
βω
distortion no→== cstevv gp
( )LCjZYjβγ ω−===
( ) nonlinear1 →−= LCωβ
>+=
<−=
0
02
2
LCv
LCv
g
p
ω
ω
distortion)( →=−= ωfctvv gp
dzLjZ )( ω=
dzCjY )( ω=pass-low
( )dzCjZ ω1=
( )dzLjY ω1=pass-high
( ) ( ) CLCjLjYZ === ωωη ( ) ( ) CLLjCj == −− 11 ωωη
Microwave Electronics Lab
LH-TL as the Dual of the Conventional TL (2)Conventional (RH) TL LH-TL
( )LCωβ 1−=LCωβ =
β
0slope >= gv 0slope <= pv
prop. z+(energy)
prop.z−(energy)
ωprop.z+prop. z−
β
pv=slope0>= gv
ω
[ ]matrix thefrom S21 ABCDϕ
(MHz) f
p-βS ⋅=21ϕ [ ]matrix thefrom S21 ABCDϕ
(MHz) f
0→ϕ∞→ωas
0 as →ω∞→ϕ
= m 10:length
p
Microwave Electronics Lab
Determination of LH Material Parameters ε & µ
)1(, ωεωωµω jCjYjLjZ ====• Relations TL / Field for a plane wave in an (RH) medium:∞
( ) ( ) )2(1,1 LjYCjZ ωω ==• Impedance / admittance in terms of LH-TL parameters:
( )
( )C
C
21!0
1
ωµµ
µωωµω
−=<
⇒
−==−
• Equating (1) and (2) yields the dispersive ε & µ:
( )
( )L
L
21!0
1
ωεε
εωωεω
−=<
⇒
−==−
→ Demonstration of negative ε and µ in the TL-LH material→ Explicit expressions for ε and µ in this material
Microwave Electronics Lab
Demonstrations for ε, µ and n
• ε & µ satisfy entropy conditions for dispersive materials: (L. D. Landau, E. M. Lifshitz and L.P. Pitaevskii, “Electrodynamics of Continuous Media”, Pergamon, 1984)
( )
( )
>=∂
∂
>=∂
∂
01
01
C
L
ωωµωωε
√
√
( ) ( )
( ) ( )
∀>∂
∂>
∂∂
⇒
>∂
∂+
∂∂
=
ωωωµ
ωωε
ωωµ
ωωε
,0and0
0 :energy Total 22 HEW→
• Demonstration of negative index of refraction& determination of its explicit expression:
( )( ) LCcn
n
cjjjZY
LCjZY
rr
rr2
0
0
!0
:Material
1j :TL-LH
ω
εµ
εµωωεωµ
ωβ
−=
<=⇒
+==
−==
Microwave Electronics Lab
Lumped-Element Approx. of the Physical (RH) LinePhysical Line Lumped-el. Approx.Infinitesimal model
( )( )( ) ( ) ωγωβ
ωωγ
βω
Im 2)
1)
:diagram
=
++=
−•
CjGLjR
pGGpRRpCCpLL
p
S
FH
⋅=⋅=⋅=⋅=
•
Ω ,,,
:length of line
line coaxial :e.g. •
pδRR roi ,,tan,,, :parameters physical
σε•
iRoRrε
δtan
σ
p
,ln2
:parameters TL
i
o
RRL
πµ
=
•
( ) etc. ,ln
'2
io RRC πε=
[ ]( ) ( ) p
Sωϕωβ
βω
21S )2matrix scattering 1)
:diagram
−=
−•
:circuitladder •…1 2 N
:cellunit •uR uL
uG
Nppu =
≡
( ) etc. ,NpLpLL uuH ⋅=⋅=
uC
0→dz
:model TL •( )mHL
( )mSG
( )mR Ω
( )mFC
Microwave Electronics Lab
Lumped-Element Approx. of the LH LinePhysical Line Lumped-el. Approx.Infinitesimal model
( )( )( ) ( ) ωγωβ
ωωγ
βω
Im 2)
1)
:diagram
=
++=
−•
CjGLjR
!!!naturally existingNot •
C,L,R,G
:yarbitraril prescribed becan
length)unit (per parameters TL •
[ ]( ) ( ) p
Sωϕωβ
βω
21S )2matrix scattering 1)
:diagram
−=
−•
:circuitladder •…1 2 N
:cellunit •uR uC
uG
Nppu =
≡
( ) etc. ,NpCpCC uuF ==
uL
0→dz
:model TL •( )mFC ⋅
( )mSG
( )mR Ω
( )mHL
⋅
:length of line p•
pGGpRR S ⋅=⋅=Ω ,,, pCCpLL FH ==
Microwave Electronics Lab
Results for the Physical Model of a RH-TL:line coaxial Physical •
(MHz)frequency
parameters-S of Phase Cumulative
β
pv=slope0>= pv
parameters-S of Magnitude
(MHz)frequency
(some mismatch was intentionallyintroduced to create S11 peaks:
Zin = 71 Ω with ports Zin = 50 Ω ).
prop.z+ω
unwrapping phase
( ) ( ) pωϕωβ 21S−=
pS 10,m 128pF, 470H, 56.2 =Ω=Ω
==S
GRF
CH
L µ
:parameters TL ingCorrespondpS/m 1/m,m 812pF/m, 47nH/m, 256 =Ω=== G.RCL
P1 P2mm, 5.9 mm, 6.2 == ei RR
m 10 ,2.1 == prε
Microwave Electronics Lab
parameters-S of Magnitude
(MHz)frequency
ionapproximat)(cutoff cf
1−= NN peaks
Lumped-Element Approx. (RH-TL)
↓⇒↑∝approx.f
Nfc
20=N
(MHz)frequency
parameters-S of Phase Cumulative
ionapproximat)(cutoff cf
dependency (linear) ωLCωβ =→
PASS-LOW
)(unmatched
Microwave Electronics Lab
Phys. Model vs L.E. Approx. (RH-TL) / Matched TLparameters-S of Magnitude
(MHz)frequency (MHz)frequency
parameters-S of Phase Cumulative
parameters-S of Phase Cumulative
(MHz)frequency
TL-RHMatched
Comparison
parameters-S of Magnitude
(MHz)frequency
ωλ 1∝
Microwave Electronics Lab
(MHz)frequency
parameters-S of Phase Cumulative
ionapproximat)(cutoff cf
dependency 1 ω( )LCωβ 1−=→
Lumped-Element Realization of a LH-TL (1)
20=NPASS-HIGH
parameters-S of Magnitude
(MHz)frequency
1−= NN peaks
ionapproximat)(cutoff cf
(matched)
↓⇒↓∝approx.
1f
Nfc
!!!ωλ ∝
Microwave Electronics Lab
parameters-S of Magnitude
Lumped-Element Realization of a LH-TL (2)
behaviorsfrequencie
higher
( ) ( )LCββω 1−=
)(cutoff cf approx. el. lu.
diagram ω-β
> lossless> unlimited BW
(MHz)frequency
parameters-S of Phase
0→ϕ∞→ωas
0 as →ω∞→ϕ
ω1∝
> unlimited BW> moderate dispersion
unwrappingphase
(MHz)frequency
s velocitiegroup and Phase
(MHz)frequency
LCvg2ω+=
LCvp2ω−=
cutoff
β
Microwave Electronics Lab
Possible Distributed Realization of the LH Linemicrostrip
lineseries
interdigitalcapacitor
shuntspiral
inductor
T-junction
via toground
unit cell
Features:• LH bandwidth limited by Q-factor of C-L, but still broad• still very low losses and moderate dispersion
Microwave Electronics Lab
ϕ↑ω
( ) 000 :start==
=ωλω
p↑ω
ϕ↑ω
( ) 0 :start
=∞=∞=
ωλω
p
↓ω
The “Phase Paradox” of LH Materials
Conventional (RH) TL LH-TL
( )tjpjj eeSeSS ωβϕ +⋅−== 212121 :generalIn
"negatively increases" ϕωωϕωβ
↑⇒−=→= pLCLC
... variationphase oppositeexpect might we,0 Since :Paradox <LHβ
( ) ( )idem! :"positively decreases"
1ϕω
ωϕωβ↑⇒
+=→−= LCpLC
( ) ωωπβπλ 122 ∝== LCbecause
!!!22 ωπωβπλ ∝== LCbecause
In a LH material, phase rotatesin the same sense as in a RH material.
Microwave Electronics Lab
General Considerations
• many other circuits exhibit a LH effect (e.g. periodic)• low losses, broad bandwidth, moderate dispersion
→ potential interest for microwave applications• possible transposition of the distributed LH-TL to a
2D/3D LH “material”• constitutive parameters different than in other approaches:
• possible link with the split-rings / wires structure:
( ) ( ) 20
2
2
2
2
1,1ωω
ωωµωω
ωε−
−=−=Fp ( ) ( )
CL 22
1,1ω
ωµω
ωε −=−=↔
≡ TL-LH
E
Hβ
Microwave Electronics Lab
Plane Wave Propagation at Interface RH-LH (1)Normal Incidence
===
MHz 20 nH/m,256
pF/m,47
0fLC
β
ω
0ω
LHβ RHβ
( ) LCRH ωωβ =( ) LCLH ωωβ 1−=
media LH and RH for the diagram -ωβ
↓↑
⇒↑LH
RH
ββ
ω
distance offct a as amplitude field−E
0>RHpv 0<LH
pv
medium RH medium LH( )44.0=RHβ ( )39.2−=LHβ
RHβ LHβ
Microwave Electronics Lab
Plane Wave Propagation at Interface RH-LH (2)
Oblique Incidence (Γ=0)) law, s(Snell'18,45 rrtransinc n µεϑϑ ±=°−=°=
( )34.20
1−=−=
LCLH ωβ( )65.00 =−= LCRH ωβ
MHz) 30 nH/m,256 pF/m,47( 0 === fLC
CL rr == 11 , µε )/(1),/(1 21
21 CL rr ωµωε −=−=
LHβRHβ
Microwave Electronics Lab
Plane Wave Propagation at Interface RH-LH (3)
Oblique Incidence (|Γ|>0)) law, s(Snell'18,45 rrtransinc n µεϑϑ ±=°−=°=
( )34.20
1−=−=
LCLH ωβ( )65.00 =−= LCRH ωβ
MHz) 30 nH/m,256 pF/m,47( 0 === fLC
CL rr == 11 , µε )/(1),/(1 21
21 CL rr ωµωε −=−=
LHβRHβ
top related