ece 598 js lecture 06 multiconductorsjsa.ece.illinois.edu/ece598js/lect_06.pdf · 2013-03-23 ·...
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Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 1
ECE 598 JS Lecture ‐ 06Multiconductors
Spring 2012
Jose E. Schutt-AineElectrical & Computer Engineering
University of [email protected]
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 2
TELGRAPHER’S EQUATION FOR NCOUPLED TRANSMISSION LINES
V and I are the line voltage and line current VECTORSrespectively (dimension n).
V ILz t
∂ ∂− =∂ ∂
I VCz t∂ ∂
− =∂ ∂
V3(z)
V2(z)
V1(z)
...
L, C
z=0 z=l
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 3
N-LINE SYSTEM
L and C are the inductance and capacitance MATRICES respectively
1
2
n
II
I
I
⎡ ⎤⎢ ⎥⎢ ⎥
= ⋅⎢ ⎥⎢ ⎥⋅⎢ ⎥⎢ ⎥⎣ ⎦
1
2
n
VV
V
V
⎡ ⎤⎢ ⎥⎢ ⎥
= ⋅⎢ ⎥⎢ ⎥⋅⎢ ⎥⎢ ⎥⎣ ⎦
11 12
21 22
nn
L LL L
L
L
⋅ ⋅⎡ ⎤⎢ ⎥⋅ ⋅⎢ ⎥=⋅ ⋅ ⋅ ⋅⎢ ⎥
⎢ ⎥⋅ ⋅ ⋅⎣ ⎦
11 12
21 22
nn
C CC C
C
C
⋅ ⋅⎡ ⎤⎢ ⎥⋅ ⋅⎢ ⎥=⋅ ⋅ ⋅ ⋅⎢ ⎥
⎢ ⎥⋅ ⋅ ⋅⎣ ⎦
V3(z)
V2(z)
V1(z)
...
L, C
z=0 z=l
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 4
N-LINE ANALYSIS
In general LC ≠ CL and LC and CL are not symmetric matrices.
GOAL: Diagonalize LC and CL which will result in atransformation on the variables V and I.
Diagonalize LC and CL is equivalent to finding the eigenvalues ofLC and CL.
Since LC and CL are adjoint, they must have the sameeigenvalues.
2 2
2 2
V VLCz t
∂ ∂=
∂ ∂2 2
2 2
I ICLz t∂ ∂
=∂ ∂
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 5
MODAL ANALYSIS
LC and CL are adjoint matrices. Find matrices E and H such that
is the diagonal eigenvalue matrix whose entries are the inverses ofthe modal velocities squared.
2 21
2 2
EV EVELCEz t
−∂ ∂=
∂ ∂
2 21
2 2
HI HIHCLHz t
−∂ ∂=
∂ ∂
1 1 2mELCE HCLH− −= = Λ
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 6
MODAL ANALYSIS
Second-order differential equation in modal space
2 22
2 2m m
mV Vz t
∂ ∂= Λ
∂ ∂
2 22
2 2m m
mI Iz t
∂ ∂= Λ
∂ ∂
Vm = EV Im = HI
V3(z)
V2(z)
V1(z)
...
L, C
z=0 z=l
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 7
Vm and Im are the modal voltage and current vectors respectively.
EIGENVECTORS
1
2
m
m
m
mn
VV
V
V
⎡ ⎤⎢ ⎥⎢ ⎥
= ⋅⎢ ⎥⎢ ⎥⋅⎢ ⎥⎢ ⎥⎣ ⎦
1
2
m
m
m
mn
II
I
I
⎡ ⎤⎢ ⎥⎢ ⎥
= ⋅⎢ ⎥⎢ ⎥⋅⎢ ⎥⎢ ⎥⎣ ⎦
Im = HI
Vm = EV
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 8
EIGEN ANALYSIS
* The eigenvalues λ of A satisfy the relation |A - λI| = 0 where Iis the unit matrix.
* A scalar λ is an eigenvalue of a matrix A if there exists a vector X such that AX = λX. (i.e. for which multiplication by a matrix is equivalent to an elongation of the vector).
* The vector X which satisfies the above requirement isan eigenvector of A.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 9
EIGEN ANALYSIS
Assume A is an n x n matrix with n distinct eigenvalues, then,
* A has n linearly independent eigenvectors.
* The n eigenvectors can be arranged into an n × n matrix E;the eigenvector matrix.
* Finding the eigenvalues of A is equivalent to diagonalizing A.
* Diagonalization is achieved by finding the eigenvector matrix
E such that EAE-1 is a diagonal matrix.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 10
EIGEN ANALYSIS
For an n-line system, it can be shown that
* Each eigenvalue is associated with a mode; the propagationvelocity of that mode is the inverse of the eigenvalue.
* LC can be transformed into a diagonal matrix whose entriesare the eigenvalues.
* LC possesses n distinct eigenvalues (possibly degenerate).
* There exist n eigenvectors which are linearly independent.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 11
EIGEN ANALYSIS
* Each normalized eigenvector represents the relative lineexcitation required to excite the associated mode.
* Each eigenvector is associated to an eigenvalue and thereforeto a particular mode.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 12
E : Voltage eigenvector matrix
H : Current eigenvector matrix
EIGENVECTORS
11 12 13
21 22 23
31 32 33
e e eE e e e
e e e
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
11 12 13
21 22 23
31 32 33
h h hH h h h
h h h
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
1 2mELCE− = Λ
1 2mHCLH − = Λ
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 13
1
2
3
1 0 0
10 0
10 0
m
mm
m
v
v
v
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
Λ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1
2
3
1 0 0
10 0
10 0
m
mm
m
v
v
v
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
Λ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
11 12 13
21 22 23
31 32 33
h h hH h h h
h h h
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
11 12 13
21 22 23
31 32 33
e e eE e e e
e e e
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
Eigenvalues and Eigenvectors1 2
mELCE− = Λgives
1 2mHCLH − = Λ
gives
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 14
MATCHINGNETWORK
+++
---e31
e32
e33 MODE C
MATCHINGNETWORK
+++
---e11
e12
e13 MODE A
MATCHINGNETWORK
+++
---e21
e22
e23 MODE B
Modal Voltage Excitation
Voltage EigenvectorMatrix
11 12 13
21 22 23
31 32 33
e e eE e e e
e e e
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 15
Modal Current Excitation
Current EigenvectorMatrix
MATCHINGNETWORK
h11
h12
h13 MODE A
MATCHINGNETWORK
h21
h22
h23 MODE B
MATCHINGNETWORK
h31
h32
h33 MODE C
11 12 13
21 22 23
31 32 33
h h hH h h h
h h h
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 16
Line variables are recovered using
V = E−1Vm
I = H−1Im
EIGENVECTORS
Special case: For a two-line symmetric system, E and H are equal and independent of L and C. The eigenvectors are [1,1] and [1,-1]
E : voltage eigenvector matrix
H : current eigenvector matrix
E and H depend on both L and C
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 17
MODAL AND LINE IMPEDANCE MATRICES
Zm = Λm−1ELH−1
By requiring Vm = ZmIm, we arrive at
Zm is a diagonal impedance matrix which relates modal voltagesto modal currents
By requiring V = ZcI, one can define a line impedance matrix Zc.Zc relates line voltages to line currents
Zm is diagonal. Zc contains nonzero off-diagonal elements. We can show that
Zc = E−1ZmH = E−1Λm−1EL
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 18
1
2( )
m
m
mn
j uv
j uv
j uv
e
eW u
e
ω
ω
ω
−
−
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥•⎢ ⎥
⎢ ⎥⎢ ⎥⎣ ⎦
1
2
1
1
1
m
mm
mn
v
v
v
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥Λ = ⎢ ⎥
•⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 19
MATRIX CONSTRUCTION
[ ] siiL L= [ ] ( )m
ijijL L=
( )
1
[ ]n
mii s ij
j
C C C=
= +∑ ( )[ ] mij ijC C= −
[ ]ii sR R= [ ] mij ijR R=
( )
1
[ ]n
mii s ij
j
G G G=
= +∑ ( )[ ] mij ijG= −G
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 20
Vm = EV Im = HI
Vm (z) = W(z)A +W(- z)B
Im (z) = Zm- 1 W(z)A - W(- z)B[ ]
1 1m mZ ELH− −= Λ
1 1 1c m mZ E Z H E EL− − −= = Λ
Modal Variables
General solution in modal space is:
Modal impedance matrix
Line impedance matrix
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 21
N-Line Network
Zs : Source impedance matrix
ZL : Load impedance matrix
Vs : Source vector
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 22
Reflection Coefficients
Source reflection coefficient matrix
Γs = [1n + EZsL−1E−1Λm ]−1[1n − EZsL−1E−1Λm ]
Load reflection coefficient matrix
ΓL = [1n + EZLL−1E−1Λm ]−1[1n − EZLL−1E−1Λm ]
* The reflection and transmission coefficient are n x nmatrices with off-diagonal elements. The off-diagonal elements account for the coupling between modes.
* Zs and ZL can be chosen so that the reflection coefficient matrices are zero
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 23
TERMINATION NETWORK
yp11
yp22
yp12
V1
V2
I1 = yp11V1 +yp12(V1-V2)
I2 = yp22V2 +yp12(V2-V1)
In general for a multiline system
I = YV⇒V = ZI Z = Y−1
Note : yii ≠ ypii, yij = -yji , (i≠j)ij
1Also, z ijy
≠
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 24
Termination Network Construction
To get impedance matrix Z• Get physical impedance values ypij• Calculate yij's from ypij's• Construct Y matrix• Invert Y matrix to obtain Z matrix
Remark: If ypij = 0 for all i≠j, then Y = Z-1 and zii = 1/yii
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 25
1) Get L and C matrices and calculate LC product
2) Get square root of eigenvalues and eigenvectors of LC matrix → Λm
3) Arrange eigenvectors into the voltage eigenvector matrix E
4) Get square root of eigenvalues and eigenvectors of CL matrix →Λm
5) Arrange eigenvectors into the current eigenvector matrix H
6) Invert matrices E, H, Λm.
7) Calculate the line impedance matrix Zc → Zc = E-1Λm-1EL
Procedure for Multiconductor Solution
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 26
8) Construct source and load impedance matrices Zs(t) and ZL(t)
9) Construct source and load reflection coefficient matrices Γ1(t) and Γ2(t).
10) Construct source and load transmission coefficient matrices T1(t), T2(t)
11) Calculate modal voltage sources g1(t) and g2(t)
12) Calculate modal voltage waves:
a1(t) = T1(t)g1(t) + Γ1(t)a2(t - τm)
a2(t) = T2(t)g2(t) + Γ2(t)a1(t - τm)
b1(t) = a2(t - τm)
b2(t) = a1(t - τm)
Procedure for Multiconductor Solution
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 27
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−•−−
=−
−−
−−
−−
)(
)()(
a
mod
22mod
11mod
mi
mnnei
mei
mei
ta
tata
t
τ
ττ
τ )(
τmi is the delay associated with mode i. τmi = length/velocity of mode i. The modal volage wave vectors a1(t) and a2(t) need to be stored since they contain the history of the system.
13) Calculate total modal voltage vectors:
Vm1(t) = a1(t) + b1(t)Vm2(t) = a2(t) + b2(t)
14) Calculate line voltage vectors:
V1(t) = E-1Vm1(t)V2(t) = E-1Vm2(t)
Wave‐Shifting Solution
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 28
ALS04 ALS240Drive Line 1
Drive Line 2
z=0 z=l
Drive Line 3
Sense Line 4
Drive Line 5
Drive Line 6
Drive Line 7
ALS04
ALS04
ALS04
ALS04
ALS04
ALS240
ALS240
ALS240
ALS240
ALS240
7-Line Coupled-Microstrip System
Ls = 312 nH/m; Cs = 100 pF/m;
Lm = 85 nH/m; Cm = 12 pF/m.
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 29
20010000
1
2
3
4
5Drive line 3 at Near End
Time (ns)2001000
-1
0
1
2
3
4
5Drive Line 3 at Far End
Time (ns)
Drive Line 3
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 30
2001000-1
0
1
2Sense Line at Near End
Time (ns)2001000
-1
0
1
2Sense Line at Far End
Time (ns)
This image cannot currently be displayed.
Sense Line
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 31
Multiconductor Simulation
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 32
LOSSY COUPLED TRANSMISSION LINES
Solution is best found using a numerical approach(See References)
V IRI Lz t
∂ ∂− = +∂ ∂
I VGV Cz t∂ ∂
− = +∂ ∂
V3(z)
V2(z)
V1(z)
...
L, C
z=0 z=l
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 33
1 2 3
31 1 211 12 13
VI V VC C Cz t t t
∂∂ ∂ ∂∂ ∂ ∂ ∂
− = + +
32 1 221 22 23
VI V VC C Cz t t t
∂∂ ∂ ∂∂ ∂ ∂ ∂
− = + +
3 31 231 32 33
I VV VC C Cz t t t
∂ ∂∂ ∂∂ ∂ ∂ ∂
− = + +
31 1 211 12 13
IV I IL L Lz t t t
∂∂ ∂ ∂∂ ∂ ∂ ∂
− = + +
32 1 221 22 23
IV I IL L Lz t t t
∂∂ ∂ ∂∂ ∂ ∂ ∂
− = + +
3 31 231 32 33
V II IL L Lz t t t
∂ ∂∂ ∂∂ ∂ ∂ ∂
− = + +
Three‐Line Microstrip
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 34
11 13( )V IL Lz tα α∂ ∂
∂ ∂− = −
11 13( )I VC Cz tα α∂ ∂
∂ ∂− = −
Subtract (1c) from (1a) and (2c) from (2a), we get
This defines the Alpha mode with:
1 3V V Vα = − 1 3I I Iα = −and
The wave impedance of that mode is:
11 13
11 13
L LZC Cα
−=
−
and the velocity is( )( )11 13 11 13
1uL L C C
α =− −
Three‐Line – Alpha Mode
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 35
In order to determine the next mode, assume that
1 2 3V V V Vβ β= + +
1 2 3I I I Iβ β= + +
( ) ( ) ( ) 31 211 21 31 12 22 32 13 23 33
V II IL L L L L L L L Lz t t tβ∂ ∂∂ ∂β β β
∂ ∂ ∂ ∂− = + + + + + + + +
( ) ( ) ( ) 31 211 21 31 12 22 32 13 23 33
I VV VC C C C C C C C Cz t t tβ∂ ∂∂ ∂β β β
∂ ∂ ∂ ∂− = + + + + + + + +
By reciprocity L32 = L23, L21 = L12, L13 = L31
By symmetry, L12 = L23
Also by approximation, L22 ≈ L11, L11+L13 ≈ L11
Three‐Line – Modal Decomposition
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 36
( ) ( )31 211 12 13 12 112
V II IL L L L Lz t t tβ∂ ∂∂ ∂β β
∂ ∂ ∂ ∂⎛ ⎞− = + + + + +⎜ ⎟⎝ ⎠
In order to balance the right-hand side into Iβ, we need to have
( ) ( ) ( )12 11 2 11 12 13 2 11 12 22L L I L L L I L L Iβ β β β β+ = + + ≈ +
212 122L Lβ=
or 2β = ±
Therefore the other two modes are defined as
The Beta mode with
Three‐Line – Modal Decomposition
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 37
The Beta mode with
1 2 32V V V Vβ = + +
1 2 32I I I Iβ = + +
The characteristic impedance of the Beta mode is:
11 12 13
11 12 13
22
L L LZC C Cβ
+ +=
+ +
and propagation velocity of the Beta mode is
( )( )11 12 13 11 12 13
1
2 2u
L L L C C Cβ =
+ + + +
Three‐Line – Beta Mode
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 38
The Delta mode is defined such that
1 2 32V V V Vδ = − +
1 2 32I I I Iδ = − +
The characteristic impedance of the Delta mode is
11 12 13
11 12 13
22
L L LZC C Cδ
− +=
− +
The propagation velocity of the Delta mode is:
( )( )11 12 13 11 12 13
1
2 2u
L L L C C Cδ =
− + − +
Three‐Line – Delta Mode
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 39
1 0 1
1 2 1
1 2 1
E
−⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟−⎝ ⎠
1 2 3
Alpha modeBeta mode*
Delta mode*
Symmetric 3‐Line Microstrip
In summary: we have 3 modes for the 3-line system
*neglecting coupling between nonadjacent lines
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 40
1 2 3
εr = 4.3
113 17 5( / ) 16 53 16
5 17 113C pF m
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
346 162 67( / ) 152 683 152
67 162 346L nH m
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
0.45 0.12 0.450.5 0 0.50.45 0.87 0.45
E⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟− −⎝ ⎠
0.44 0.49 0.440.5 0 0.50.10 0.88 0.10
H⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟− −⎝ ⎠
Coplanar Waveguide
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 41
73 0 0( ) 0 48 0
0 0 94mZ
⎛ ⎞⎜ ⎟Ω = ⎜ ⎟⎜ ⎟⎝ ⎠
56 23 8( ) 22 119 22
8 23 56cZ
⎛ ⎞⎜ ⎟Ω = ⎜ ⎟⎜ ⎟⎝ ⎠
0.15 0 0( / ) 0 0.17 0
0 0 0.18pv m ns
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
1 2 3
εr = 4.3
Coplanar Waveguide
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 42
WS' S'WS
εrh
2Sk
S W=
+
( ) : Complete Elliptic Integral of the first kindK k
'( ) ( ')K k K k=
2 1/ 2' (1 )k k= −
30 '( ) (ohm)( )1
2
ocpr
K kZK k
πε
=+
1/ 22
1cpr
v cε
⎛ ⎞= ⎜ ⎟+⎝ ⎠
Coplanar Waveguide
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 43
S WW
εrh
120 '( ) (ohm)( )1
2
ocsr
K kZK k
πε
=+
Coplanar Strips
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 44
Characteristic Microstrip Coplanar Wguide Coplanar strips
εeff* ~6.5 ~5 ~5
Power handling High Medium Medium
Radiation loss Low Medium Medium
Unloaded Q High Medium Low or High
Dispersion Small Medium Medium
Mounting (shunt) Hard Easy Easy
Mounting (series) Easy Easy Easy
* Assuming εr=10 and h=0.025 inch
Qualitative Comparison
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 45
3-Line Matching Network
• Extend matching network synthesis to multiconductor transmission line (MTL) systems, specifically coupled microstrip
• Characterize MTL systems by their “normal mode parameters”– Mode configurations– Propagation constants– Characteristic impedance matrices
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 46
Longitudinal Immittance Matrix Functions (LIMFs)
• Given terminations, the immittance matrices are expressed as “input”longitudinal functions looking toward the load
m,c m,cin,out
m,c m,cin,out in,out
, ,
,
ΓS
Y Z
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 47
Calibration Methods1st order calibration: 6 distinct TRL calibrations were required
to deembed the fan-in and fan-out sections 2nd order calibration: 1 multimode TRL calibration required to
deembed all fan-ins and fan-outs
Multiconductor transmission line matching network device under test
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 48
Bias Networks and Multiline/Multimode TRL
• Self-biasing for DC power done using air bridges
• Regulator circuit may be built on board with surface mount/chip components
• Calibration must include all modes and allow for the 6-port discontinuity S-parameter measurement
Proposed calibration fixture for transistor amplifier device discontinuity with self-biasing network
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 49
3-line Transistor Amplifier
The general amplifier structure with generic matching networks
Note that a transistor seating hole may be present for convenience, but is not necessary
Photograph of a three-line transistor amplifier autobiased w/o matching networks
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 50
Transistor Amplifier Matching
S-parameter comparison between unmatched and matched three-line amplifier. Dotted line - unmatched; dashed line -matched. Design frequency: 3.1 GHz.
Matching• 2 - 0.8 pF chip cap.• 2 - 0.4 pF chip cap.
Gains• 3.38 dB @ 3.1 GHz
m=5.14 / um= 1.76• 6.22 dB @ 3.22 GHz
m=6.54 / um= 0.32
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 51
2p
h
w
α
dielectric
ground
signal strip
V Transmission Line
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 52
0
50
100
150
200
250
0 0.1 0.2 0.3 0.4 0.5 0.6w/h
CHARACTERISTIC IMPEDANCE
30º37.5º45º52.5º
60ºMicrostrip
0.7 0.8 0.9 1
Zo (o
hms)
Calculated values of the characteristic impedance for a single-line v-strip structure as a function of width-to-height ratio w/h. The relative dielectric constant is εr = 2.55.
V‐Line Characteristic Impedance
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 53
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
2.2
0 0.1 0.2 0.3 0.4 0.5 0.6
w/h
EFFECTIVE PERMITTIVITY
30º
37.5º
45º
52.5º
60º
Microstrip
0.7 0.8 0.9 1
Effe
ctiv
e R
elat
ive
Die
lect
ric C
onst
ant
Calculated values of the effective relative dielectric constant for a single-line v-strip structure as a function of width-to-height ratio w/h. The relative dielectric constant is εr = 2.55
V‐Line – Effective Permittivity
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 54
h
y
x
Region 2
Region 3
Region 4
Region 6
Region 1
Region 5
2pw
dielectricground surface
signal strip
s
Three‐Line – V Transmission Line
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 55
74.00 -0.97 -0.23C (pF/m) = -0.97 74.07 -0.97
-0.23 -0.97 74.00
425.84 20.35 7.00L (nH/m) = 20.36 422.85 20.36
7.00 20.35 425.84
52.49 -5.90 -0.57C (pF/m) = -5.90 53.27 -5.90
-0.57 -5.90 52.49
609.74 113.46 41.79L (nH/m) = 113.54 607.67 113.54
41.79 113.46 609.74
Microstrip V-line
Comparison of the inductance and capacitance matricesbetween a three-line v-line and microstrip structures. The parameters are p/h = 0.8 mils, w/h = 0.6 and εr = 4.0.
h
y
x
Region 2
Region 3
Region 4
Region 6
Region 1
Region 5
2pw
dielectricground surface
signal strip
s1 2 3
V‐Line and Microstrip
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 56
0
50
100
150
200
250
300
350
0 0.2 0.4 0.6 0.8 1 1.2s/h
Mutual Inductance
V-linemicrostrip
Lm
(nH
/m)
0
5
10
15
20
25
30
35
40
0 0.2 0.4 0.6 0.8 1 1.2s/h
Mutual Capacitance
V-linemicrostrip
C m(p
F/m
)
Plot of mutual inductance (top) and mutual capacitance (bottom) versus spacing-to-height ratiofor v-line and microstrip configurations. The parameters are w/h = 0.24, εr = 4.0.
V‐Line: Coupling Coefficients
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 57
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1 1.2s/h
Coupling Coefficient
V-linemicrostrip
Kv
Plot of the coupling coefficient versus spacing-to-height ratio for v-line and microstrip configurations. The parameters are w/h = 0.24, εr = 4.0.
V‐Line vs Microstrip: Coupling Coefficients
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 58
0.5
0.6
0.7
0.8
0.9
1
1 3 5 7 9 11 13 15 17
microstrip
V-60
V-30
Frequency (GHz)
Linea
r Atte
nua ti
on
Propagation Function
2p
h
w
α
dielectric
ground
signal strip
Advantages of V Line* Higher bandwidth
* Lower crosstalk
* Better transition
ground reference
ground reference
dielectric
substrate
signal strip
Advantages of V‐Line
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 59
0.5
0.6
0.7
0.8
0.9
1
1 3 5 7 9 11 13 15 17
microstrip
V-60
V-30
Frequency (GHz)
Linea
r Atte
nua ti
on
Propagation Function
2p
h
w
α
dielectric
ground
signal strip
Advantages of V Line* Higher bandwidth
* Lower crosstalk
* Better transition
ground reference
ground reference
dielectric
substrate
signal strip
Advantages of V‐Line
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 60
-8
-7
-6
-5
-4
-3
-2
-1
0
0 5 10 15 20 25
Insertion Loss
VlineMicrostrip20
*log1
0*|S
21|
Frequency (GHz)
V‐Line vs Microstrip: Insertion Loss
Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 61
• J. E. Schutt-Aine and R. Mittra, "Transient analysis of coupled lossy transmission lines with nonlinear terminations," IEEE Trans. Circuit Syst., vol. CAS-36, pp. 959-967, July 1989.
• J. E. Schutt-Aine and D. B. Kuznetsov, "Efficient transient simulation of distributed interconnects," Int. Journ. Computation Math. Electr. Eng. (COMPEL), vol. 13, no. 4, pp. 795-806, Dec. 1994.
• D. B. Kuznetsov and J. E. Schutt-Ainé, "The optimal transient simulation of transmission lines," IEEE Trans. Circuits Syst.-I., vol. cas-43, pp. 110-121, February 1996.
• W. T. Beyene and J. E. Schutt-Ainé, "Efficient Transient Simulation of High-Speed Interconnects Characterized by Sampled Data," IEEE Trans. Comp., Hybrids, Manufacturing Tech. vol. 21, pp. 105-114, February 1998.
References