TRANSMISSION LINE SYSTEMS (Soustavy s přenosovými vedeními)
Lecture in terms of Ph.D. Study
L. Brančík
Dept. of Radio Electronics FEEC BUT in Brno
Brief survey of techniques for the simulation of
simple transmission lines
a uniform transmission line as a part of more complex system
a transmission line can be excited from both sides
it can be a part of a feedback structure
generally, input and output voltages (currents) are unknown
Basic parcial differential equations (telegraphic) for voltage and current on a transmission line
u x t
xRi x t L
i x t
t
i x t
xGu x t C
u x t
t
( , )( , )
( , )
( , )( , )
( , )
where x is a length from the TL’s beginning
L, R, C, G are primary parameters (per-unit-length)
We are usually interested in conditions on both ends of the TL, and it is therefore considered as a two-port. Using a Laplace transformation and further arrangements we can get cascade equations for the input voltage and current
U s U s s l I s Z s s l
I s U ss l
Z sI s s l
v
v
1 2 2
1 2 2
( ) ( ) cosh ( ) [ ( )] ( ) sinh ( )
( ) ( )sinh ( )
( )[ ( )]cosh ( )
where
Z sR sL
G sCv( )
is a characteristic (wave) impedance
( ) ( )( )s R sL G sC is a propagation constant A steady-state harmonic analysis is simple on principle - s is replaced by j, and a complex admittance matrix is determined
YZ l Z l
Z l Z l
c c
c c
1 1
1 1
tanh sinh
sinh tanh
which is then used for the analysis in the frequency domain by means of a nodal analysis, or a modified nodal analysis.
The above procedure can be used even in special cases, for example: lossless transmission lines, R = 0, G = 0 distortionless transmission lines, L/R = C/G transmission lines with a negligible leakage, G = 0
transmission lines with a negligible inductance, L = 0 A general analysis in the time domain is much more complicated. Some basic possibilities:
1. Solution of basic partial differential equations (telegraphic) by a
finite element method or a finite difference method (e.g. FDTD, Lax-Wendroff, etc.),
2. Replacement of a transmission line with a finite number m of sectional networks composed of discrete elements,
3. Application of the numerical inverse Laplace transformation (NILT),
4. Approximation of Zv(s) and (s) by lumped-parameter circuits, and separation of the TL‘s basic delay τ, 5. Description of a transmission line by its impulse response and
the application of convolution integral.
1. Solution of basic partial differential equations by a finite element method or a finite difference method
Difficulties are in the fact we generally do not know boundary
conditions, The method is not easily compatible with the methods used in
programs for the analysis of electronic circuits. The application of the Wendroff method, see later.
2. Replacement of a transmission line with a finite number m of sectional networks composed of discrete elements
It holds
LLl
mR
Rl
mC
Cl
mG
Gl
md d d d , , , .
A substitute circuit is a low-pass filter with the basic delay
LC l m L Cd d , and with a cut-off frequency
1 1
24
c
d ddd
mf
C L CL
.
Beacuse a cut-off frequency fc must be much more higher than maximal frequency fmax in the spectrum of processed signals, a sufficient number of networks must be used
m f max , in practice e.g.
m f10 0 05 . ,
where f0 05. is the highest frequency on which a modulus of spectral function reaches 5 of its maximal value. Example 1:
We consider 3 test signals of the same unit length on a 50 % level of their amplitudes:
Odezva na výstupu náhradního obvodu přizpůsobeného
bezeztrátového vedení s 50 články a zpožděním 5
SPECTRAL FUNCTIONS MODULI
a) rectangular b) sin 2 ⋅ 𝑥 3 ⋅ 𝑡 c) sin2 𝑥 2 ⋅ 𝑡
Response on the output of the substitute circuit of the matched lossless transmission line with 50 cells and time delay = 5
3. Application of the numerical inverse Laplace transformation Circuit equations are formulated and respective images F s( ) derived by means of the admittance parameters in the s domain, an original f t( ) is then found by a numerical way.
It holds (with an error e a2 , or e a2 )
f t f t ae
tF
a
tF
a
tjn
ts
an
n
( ) ( , ) ( ) Re
1
21
1
or
f t f t ae
tF
a
tj n
tc
an
n
( ) ( , ) ( ) Im
11
21
In case of
f t f t a f t a f t aa s c( ) ( , ) ( , ) ( , ) 1
2 ,
an error e a4 can be achieved. Real or imaginary parts of the image F s F j( ) ( ) evaluated on chosen complex frequencies s are summed. In fact, a finite number of terms in the infinite series are considered (e.g. 50), and convergence is accelerated by Euler transformation.
Advantages:
Z(s) and γ(s) do not require any approximation, The procedure is applicable even for transmission lines with
frequency-dependent parameters (skin effect).
Problems:
An inversion of images of time delayed signals with wide frequency spectrum.
Example 2: DELAYED UNIT STEPS
ABSOLUTE ERRORS (DELAYED UNIT STEP, = 1)
Application of the numerical inverse Laplace transformation on signals from Exanple 1, for time delay = 5
4. Approximation of Zv(s) and (s) by lumped-parameter circuits, and separation of the TL‘s basic delay τ
We follow basic equations for voltages and currents at both sides
of the transmission line, and derive
U s Z s I s E s
U s Z s I s E s
v
v
1 1 1
2 2 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
where
E s U s E s e
E s U s E s e
s l
s l
1 2 2
2 1 1
2
2
( ) [ ( ) ( )]
( ) [ ( ) ( )]
( )
( )
The transmission line can therefore be replaced by a circuit, in which voltage sources 𝑒1 𝑡 and 𝑒2 𝑡 are dependent on voltages of opposed ends of the TL at the time instance 𝑡 − 𝜏.
Similarly it could be possible to derive a model with current sources and characteristic admittances.
R E M A R K S
0 0
0 0
( , ) ( , )( , )
( , ) ( , )( , )
u x t i x tR i x t Lx t
i x t u x tG u x t Cx t
∂ ∂− = +
∂ ∂∂ ∂
− = +∂ ∂
0
0
( , ) ( ) ( , )
( , ) ( ) ( , )
dU x s Z s I x sdx
dI x s Y s U x sdx
− =
− =
0 0 0 0 0 0( ) , ( )Z s R sL Y s G sC= + = +
22
2
22
2
( , ) ( ) ( , )
( , ) ( ) ( , )
dU x s s U x sdx
dI x s s I x sdx
γ
γ
=
= , 0 0( ) ( ) ( )s Z s Y sγ =
Characteristic equation:
2 21,2( ) 0 ( )s sλ γ λ γ− = ⇒ = ±
Solution: ( ) ( )1 2( , ) s x s xU x s K e K eγ γ−= + → 0
( , ) ( ) ( , )dU x s Z s I x sdx
− =
( )( ) ( )2 1
1( , )( )
s x s x
v
I x s K e K eZ s
γ γ−= − , 0
0
( )( )( )v
Z sZ sY s
=
Determination of K1,2 a) 0 :x = b) :x l=
1 1 2(0, ) ( )U s U s K K= = + ( ) ( )2 1 2( , ) ( ) s l s lU l s U s K e K eγ γ−= = +
( )1 2 11(0, ) ( )( )v
I s I s K KZ s
= = − ( )( ) ( )2 2 1
1( , ) ( )( )
s l s l
v
I l s I s K e K eZ s
γ γ−= − = −
1 1 2( )U s K K= +
1 2 1( ) ( )vI s Z s K K= − ⇒ 1 1 1 1( ) ( ) ( 2) ( )v KU s I s Z s E s= =− 1 1 2( ) ( ) ( ) 2vU s I s Z s K+ =
( ) ( )2 1 2( ) s l s lU s K e K eγ γ−= +
( ) ( )2 1 2( ) ( ) s l s l
vI s Z s K e K eγ γ−= − ⇒ ( )2 2 22( ) ( ) ( ) (2 )v
s lK eU s I s Z s E sγ−=− = ( )
2 2 1( ) ( ) ( ) 2 s lvU s I s Z s K eγ+ =
L.T.
R E M A R K S
[ ][ ]
1
( )2
1 1
1 1( )
1 2 2
1 2
( ) ( ) ( )( ) ( ) 2
( ) ( ) ( ) ( ) ( )
( ) ( )
( )
2 ( ) ( ) s l
v
vs l
v v
v
U s I s Z sI s Z s K
I s Z s U s I s Z
E s
U s E s
s e
s s eI Z
γ
γ
−
−
= += +
= +
−
+
= +
[ ][ ]
2
( )
2 2
( )2 2
( )2 1 1
2 1 1
( ) ( ) ( )
( ) ( ) 2
( ) ( ) ( ) ( ) ( )
( )
( )
2 ( )( ) ( )
v
s lv
s lv v
vs l
U s I s Z s
I s Z s K e
I s Z s U s I s Z s e
I s Z s
E s
U s E s e
γ
γ
γ
−
−
−
= +
= +
= +
−
+
= +
Practical realization of the method requires: a) replacement of the wave impedance
Z sR sL
G sCv( )
with an impedance of the lumped-parameter circuit RC or RL
b) separation of a basic delay from a propagation constant ( )s l . c) replacement of the rest of the propagation constant with a transfer of the type of a rational function. A characteristic impedance can be expanded into the form
Z s Z sR
GH yv v k
k
kn( ) ( ) exp ( )
01
,
or
Z s Z ss
L
CH zv v k
k
kn( ) ( ) exp ( )
1
1
,
and an exponential propagation constant into the form
e e es
e e H zs l sk
k
k
sn
( ) exp ( )0 01
1
where the exponential functions have been approximately replaced by rational functions Hn (in a sence of so-called Padé approximation).
LC l
This approximation enables to ensure zero errors at s 0 and s . Example 3: ERRORS OF MODULUS Zv(s) FOR APPROXIMATION DEGREES n = 1 AND n = 2,
AND EXPANSIONS AT ZEROS
EXTREME Zv AS A FUNCTION OF q = C/L FOR EXPANSION AT ZERO
Advantages:
The method is useful even for nonlinear circuits, Problems:
Difficult approximation for special cases of TLs (G = 0, L = 0, nonuniform, with frequency-dependent parameters),
Standard integration algorithms should be adapted with respect to delayed signals.
Example 4:
Possible variants of 𝑒−𝛾 𝑠 𝑙 synthesis
Possible variants of 𝑍𝑣 𝑠 synthesis
RELATIONSHIP BETWEEN SOLUTIONS OF TELEGRAPHIC
EQUATIONS AND MODELING VIA T- OR -NETWORKS
1. Basic formulae
Let’s consider a uniform transmission line of a length l , with primary parameters L, R, C, and
G, excited from both sides as shown in Fig. 1.
The TL’s secondary parameters:
( )v
R sLZ s
G sC
(1) ( ) ( )( )s R sL G sC (2)
The solution can be based on a superposition theorem. Fig. 2 shows a respective two-port
model where only the left source iLU is in action.
A cascade matrix xA of the TL part of a length x is derived as
2 ( ) 2 ( )
( ) 2 ( ) 2 ( )
cosh ( ) ( )sinh ( ) 1 ( )[ 1]1( )
sinh ( ) ( ) cosh ( ) 2 [ 1] ( ) 1
s x s xv v
x s x s x s xv v
s x Z s s x e Z s eA s
s x Z s s x e e Z s e
(3)
We can write ( )
( )( ) ( )
iLL
iL inpl
U sI s
Z s Z s
(4)
where ( )inplZ s is an input impedance of the TL of the length l, terminated by a loading
impedance ( )iRZ s :
Fig.1 Uniform transmission line of length l
Fig. 2 Two-port model of uniform transmission line
2 ( )
11 12
2 ( )
21 22
( ) ( ) ( )( ) 1 ( )( ) ( )
( ) ( ) ( ) ( ) 1 ( )
s l
l iR lL Rinpl v s l
L l iR l R
a s Z s a sU s s eZ s Z s
I s a s Z s a s s e
, (5)
and ( ) ( )
( )( ) ( )
iR vR
iR v
Z s Z ss
Z s Z s
(6)
is a reflection coefficient on the TL right side. Based on a cascade matrix xA ,
21 ( ) 22
( )( )
( ) ( ) ( )
Lx
x inp l x x
I sI s
a s Z s a s
, (7)
where 2 ( )[ ]
( ) 2 ( )[ ]
( ) 1 ( )( ) ( )
( ) 1 ( )
s l x
x Rinp l x v s l x
x R
U s s eZ s Z s
I s s e
(8)
is an input impedance of the two-port with l xA matrix, i.e. a loading impedance of the two-
port with the matrix xA . After designation
( ) ( )
( )( ) ( )
iL vL
iL v
Z s Z ss
Z s Z s
(9)
as a reflection coefficient on the TL left side, we can write
( ) ( )[2 ]
2 ( )
( )1( ) ( )
( ) ( ) 1 ( ) ( )
s x s l x
Rx iL s l
iL v L R
e s eI s U s
Z s Z s s s e
(10)
and ( ) ( )[2 ]
2 ( )
( ) ( )( ) ( )
( ) ( ) 1 ( ) ( )
s x s l x
v Rx iL s l
iL v L R
Z s e s eU s U s
Z s Z s s s e
. (11)
To determine the current ( )xI s and the voltage ( )xU s when only the right source ( )iRU s is in
action, we can use backward cascade matrices (which are, however, equal to the forward ones
due to reciprocity and longitudinal symmetry of the uniform TL). The resultant current
( ) ( ) ( )x x xI s I s I s and voltage ( ) ( ) ( )x x xU s U s U s are
( ) ( )[2 ] ( )[ ] ( )[ ]
2 ( )
( ) ( )1( ) ( ) ( )
1 ( ) ( ) ( ) ( ) ( ) ( )
s x s l x s l x s l x
R Lx iL iRs l
L R iL v iR v
e s e e s eI s U s U s
s s e Z s Z s Z s Z s
(12)
( ) ( )[2 ] ( )[ ] ( )[ ]
2 ( )
( ) ( ) ( )( ) ( ) ( )
1 ( ) ( ) ( ) ( ) ( ) ( )
s x s l x s l x s l x
v R Lx iL iRs l
L R iL v iR v
Z s e s e e s eU s U s U s
s s e Z s Z s Z s Z s
(13)
2. Transmission line modeled by means of П- or T-networks
A TL is modeled as a cascade connection of m - or T-networks, see Fig. 3.
Individual elements are defined as:
dR Rl m , dL Ll m ,
dG Gl m , dC Cl m (14)
A cascade matrix of a longitudinal symmetric two-port can be expressed in a wave form
0 0 0
0 0 0
cosh ( ) ( )sinh ( )( )
sinh ( ) ( ) cosh ( )d
g s Z s g sA s
g s Z s g s
, (15)
where a characteristic impedance
0 ( )Z s and an image transfer constant 0 ( )g s are
120
21
( )( )
( )
d
d
a sZ s
a s (16) and 2
0 11 11( ) ln ( ) ( ) 1d dg s a s a s (17)
We look for the voltage ( )kU s and current ( )kI s in the output of the k-th cell of the TL
model. Under action of the left source iLU the two-port model in Fig. 4 can be considered.
A cascade matrix of a cascade connection of k cells ( ) ( )k
k dA s A s . In a wave form
2 2
0 0 0 0 0 0
2 20 0 0 0 0 0 0
cosh ( ) ( )sinh ( ) ( ) 1 ( )[ ( ) 1]1( )
sinh ( ) ( ) cosh ( ) 2 ( ) [ ( ) 1] ( ) ( ) 1
k k
k k k k
kg s Z s kg s G s Z s G sA s
kg s Z s kg s G s G s Z s G s
(18)
a) -network b) T-network
Fig. 3 Elements of sectional model of uniform transmission line
Fig.4 Two-port sectional model of uniform TL
where 0 ( )G s is an image transmission
0 ( ) 2
0 11 11( ) ( ) ( ) 1g s
d dG s e a s a s . (19)
Formally, the cascade matrix (18) corresponds to the one of a uniform TL, xA according to
(3). The solution is thus possible to formulate based on (12) and (13), replacing ( )vZ s for
0 ( )Z s , ( )se for 0 ( )G s , and a length l and x for numbers of cells m and k:
(2 ) ( ) ( )
0 0 0 0
2
0 0 0
( ) ( ) ( ) ( ) ( ) ( )1( ) ( ) ( )
1 ( ) ( ) ( ) ( ) ( ) ( ) ( )
k m k m k m k
R Lk iL iRm
L R iL iR
G s s G s G s s G sI s U s U s
s s G s Z s Z s Z s Z s
(20)
(2 ) ( ) ( )
0 0 0 0 0
2
0 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
1 ( ) ( ) ( ) ( ) ( ) ( ) ( )
k m k m k m k
R Lk iL iRm
L R iL iR
Z s G s s G s G s s G sU s U s U s
s s G s Z s Z s Z s Z s
(21)
Here ( )L s and ( )R s denote reflection coefficients at left and right ends of the TL model,
which can be stated from (6) and (9) after replacement ( )vZ s for 0 ( )Z s .
When denoting ( )d d dZ s R sL (22) ( )d d dY s G sC , (23) the cascade matrices of -network or T-network are equal
( ) ( )1 ( )
2( )
( ) ( ) ( ) ( )( ) 1 1
4 2
d dd
d
d d d dd
Z s Y sZ s
A sZ s Y s Z s Y s
Y s
(24)
( ) ( ) ( ) ( )1 1
2 4( )
( ) ( )( ) 1
2
d d d dd
d
d dd
Z s Y s Z s Y sZ
A sZ s Y s
Y s
(25)
A characteristic impedance (16) cant hen be expressed through TL’s secondary parameters
02
( )( )
1 ( )1
4
vZ sZ s
s l
m
(26), or
2
0
1 ( )( ) ( ) 1
4v
s lZ s Z s
m
(27)
for -network, or T-network, respectively. The image transfer (19) is the same for both
22 2
0
1 ( ) 1 ( )( ) 1 1 1
2 2
s l s lG s
m m
(28)
In case of an infinite number of cells, m , the equations (20) and (21) become (12) and
(13), for lim ( ) ( )m
vZ s Z s
0 and ( )
0lim ( )m s l
mG s e
, respectively.
SIMULATION OF SIMPLE CIRCUITS WITH
MULTICONDUCTOR TRANSMISSION LINES (MTL)
1. Formation of the MTL model
a transmission line with n active conductors,
existence of impedance or admittance matrices of linear (n+1)-poles is
supposed (generalized Thévenin or Norton equivalents),
the MTL is considered as a linear 2n-port.
We denote U = [U1, U2,..., Un]T and I = [I1, I2,..., In]
T as vectors of voltages and
currents, respectively.
a) 2n-port model with voltage sources
Ui = [U10, U20, ..., Un0]T are vectors of open voltages of the linear circuits, Zi are
respective internal impedance matrices (Thévenin models).
ZiL ZiR
UiL UiR
IL -IR
UL UR
n
2
1 1
2
n
IL1
ILn
IL2
-IR1
-IRn
-IR2
Iz Iz
n-conductor
transmission line
LINEÁR (n+1)-pole (L) (operational
scheme)
LINEÁR (n+1)-pole (R) (operational
scheme)
. .
.
. .
.
The solution follows cascade equations in the matrix form:
UL = A11UR + A12(–IR) , IL = A21UR + A22(–IR)
The vectors of currents can be derived as
IR = [ZiL(A22 + A21ZiR) + A12 + A11ZiR]–1
[(A11 + ZiLA21)UiR – UiL]
IL = A21UiR – (A22 + A21ZiR)IR ,
and the vectors of voltages are given by boundary conditions
UR = UiR – ZiRIR , UL = UiL – ZiLIL
b) 2n-port model with current sources
Ii = [I1k, I2k, ..., Ink]T are vectors of short-circuited currents of the linear circuits,
Yi are respective internal admittance matrices (Norton models).
The vectors of voltages can be derived as
UR = [YiL(A11 + A12YiR) + A21 + A22YiR]–1
[(A22 + YiLA12)IiR + IiL] ,
UL = (A11 + A12YiR)UR – A12IiR
and the vectors of currents are given by boundary conditions
IR = IiR – YiRUR , IL = IiL – YiLUL .
c) combination of the models a) and b)
voltage model left – current model right:
UR = [ZiL(A21 + A22YiR) + A11 + A12YiR]–1
[(A12 + ZiLA22)IiR + UiL]
UL = (A11 + A12YiR)UR – A12IiR
YiL YiR IiL IiR
IL -IR
UL UR
current model left – voltage model right:
IR = [YiL(A12 + A11ZiR) + A22 + A21ZiR]–1
[(A21 + YiLA11)UiR – IiL] ,
IL = A21UiR – (A22 + A21ZiR)IR .
Remaining quantities are again given by respective boundary conditions.
d) determination of a cascade matrix A
A discrete model of the MTL based on a cascade connection of 2n-port T- or -
networks
Matrices of primary parameters are R0, L0, G0, C0, and a length is l.
Zp and Yp denote partial longitudinal impedance and shunt admittance matrices
Zp(s) = l(R0 + sL0)/m and Yp(s) = l(G0 + sC0)/m ,
where m is a number of sections of the MTL model.
Partial cascade matrices of the T- or -networks are
p p p p p
p
p p p
/ 2 [ / 4]
/ 2
E Z Y Z E Y ZA
Y E Y Z ,
or
p p p
p
p p p p p
/ 2
[ / 4] / 2
E Z Y ZA
Y E Z Y E Y Z ,
wher E is a unit matrix of the order n.
Example of derivation of a submatrix A11p for the T-network: the output port
open, i.e. -I2 = 0:
U1 = A11pU2 = A11pYp–1
I1 = A11pYp–1
(Zp/2 + Yp–1
)–1
U1 =>
E = A11pYp–1
(Zp/2 + Yp–1
)–1
=> A11p = [Yp–1
(Zp/2 + Yp–1
)–1
]–1
= E + ZpYp/2
Zp/2 Zp/2
Yp U1 U2
I1 -I2 Zp
Yp/2 Yp/2 U1 U2
I1 -I2
The resultant cascade matrix: A = Apm
Determination of voltage and current distributions along the MTL conductors:
it is necessary to determine vectors of voltages Uk and currents Ik on the
output of the k-th section of the MTL model, 0 < k < m:
Uk = Ak11UR + Ak12(–IR) , Ik = Ak21UR + Ak22(–IR) ,
where Ak is a cascade matrix corresponding to m-k MTL sections: Ak = Apm–k
.
the vectors Uk and Ik correspond to vectors Ux and Ix of the real MTL in a
length mklx from its left end.
2. Application of numerical inverse Laplace transformation
Designating f(t) as an n-dimensional time vector of voltages u(t) or currents i(t),
i.e. the original in the Laplace transformation, then
1 2
1FIm1,f )(f
n
na
tnj
t
a
t
eatt ,
where F(s) is the image of the vector of voltages U(s) or currents I(s) calculated
for complex frequencies tnjtas 21 .
the error is approximately ae 2 , in practice, a = 6 is often chosen,
necessary number of terms in the basic sum depends on values of time delays
of the lines, usually nsum = 100 to 200,
another roughly ndif = 6 terms is undergone to so-called Euler transformation,
being weighted by factors
difkn
nknnkV
dif
difsum ,2,1,2)1( 1
,
where
r
nVV
dif
rr 1 and 11 V .
Example 1: An MTL with 2 bounded conductors, with resistive terminations
A voltage ui(t) is a trapezoidal pulse, with rise/fall times 1.5ns and top duration
4.5ns. The line length l = 0.3048 m, matrices of primary parameters:
0
0.1 0.02 Ω
0.02 0.1 m
R , 0
494.6 63.3 nH
63.3 494.6 m
L ,
0
0.1 0.01 S
0.01 0.1 m
G , 0
62.8 4.9 pF
4.9 62.8 m
C
A discrete MTL model had m = 104 sections [in case of c), d), R0 = 0 , G0 = 0]
Ri1= 50 R1= 100
R2= 100 Ri2= 100 ui(t)
Example 2: An MTL with 2 bounded conductors, with reactive terminations
All the remaining parameters correspond to Example 1.
The vectors of short-circuited currents and internal admittance matrices:
0
))106exp(1())105.1exp(1(
3
104)(I 2
997
iL s
sss ,
0
0)(IiR s
ss
9iL1001.00
002.0)(Y ,
ss
ssss
99
99
iR1002.010
10)01.05(110)(Y
Ri1=50
R1=5 R2=50 Ri2=100
ui(t)
Ci=1 nF
C=1 nF
L=10 mH
Voltage and current waves distributions along the conductors:
A result of the animation of pulse propagation along a simple TL (normalized
values R0 = 1, G0 = 0.2, L0 = 8, C0 = 2, l = 1)
Resistive load
Capacitive load
FORMULATION OF MTL MATRIX EQUATIONS AND THEIR
SOLUTION
- simple multiconductor transmission line system
- basic MTL equations
),(I
),(V
0)(C
)(L0
),(I
),(V
0)(G-
)(R-0
),(I
),(V
0
0
0
0
tx
tx
tx
x
tx
tx
x
x
tx
tx
x
- after Laplace transformation w.r. to t
)0,(I
)0,(V
0)(C
)(L0
),(I
),(V
0),(Y-
,Z-0
),(I
),(V
0
0
x
x
x
x
sx
sx
sx
s)(x
sx
sx
dx
d
where )(L)(R),(Z 00 xsxsx , )(C)(G),(Y 00 xsxsx
- the compact matrix form
)0,(W),(N),(W),(M),(W xsxsxsxsxdx
d
- the solution of the last equation
dsssxssx
x
x
xξ
xx )0,(W),(N)(Φ),(W)(Φ),(W
0
0 0
where )(Φ0x sx is so-called integral matrix.
- under zero initial conditions it follows
),(W)(Φ),(W 00
sxssx xx
LINEAR NETWORK
(L)
LINEAR NETWORK
(R)
(n+1)-conductor transmission
line
IL IR
VL VR
0 l x
- for homogenous line
][)(M)(M),(Mx
0
0)(Φ xxs
ssxx es −⋅
==
- multiport model of the simple MTL system - cascade matrix of the part of the line
a) homogenous line
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅⎥
⎦
⎤⎢⎣
⎡==− )(
0)(Y)(Z0
exp)(Φ),(A startendxxstartend xx
ss
ssxx start
end
b) non-homogenous line
∏=
− ⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅⎥
⎦
⎤⎢⎣
⎡==
m
kkk
k
kxxendstart xx
sξsξ
ssxx start
end
11 )(
0),(Y),(Z0
exp)(Φ~),,(A~
where kkk xxξ ,1−∈ , k=1, 2, ..., m and startxx =0 , endm xx = . - the solution depends on the models of terminating linear networks
Norton left – Norton right
]I)AYA(I[]Y)AYA(AYA[V iR12iL22iL1
iR12iL2211iL21R ++⋅+++= −
iRRiRR IVYI −=−
LINEAR NETWORK
(L)
LINEAR NETWORK
(R)
I(0) = IL I(l) = -IR
V(0) =VL V(l) =VR AL(x) AR(x)
I(x)
V(x)
A(l)=AL(x).AR(x)
Thévenin left – Thévenin right
]V)AZA(V[]Z)AZA(AZA[I- iR21iL11iL1
iR21iL1122iL12R +−⋅+++= −
)I(ZVV RiRiRR −+=
Thévenin left - Norton right
]I)AZA(V[]Y)AZA(AZA[V iR22iL12iL1
iR22iL1221iL11R ++⋅+++= −
iRRiRR IVYI −=−
Norton left – Thévenin right
]V)AYA(I[]Z)AYA(AYA[I- iR11iL21iL1
iR11iL2112iL22R +−⋅+++= −
)I(ZVV RiRiRR −+= - voltage V(x,s) and current I(x,s) vectors
⎥⎦
⎤⎢⎣
⎡−
⋅⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡)(I)(V
),(A),(A),(A),(A
),(I),(V
R
R
22R21R
12R11R
ss
sxsxsxsx
sxsx
.
- computation of the cascade matrix AR(x,s)
),(A)(Φ),(A Rx1R1 sxssx k
xk
k
k
−=− , where 1,2,...,1, −= mmk and lxm = , 00 =x . - numerical inversion of Laplace transforms F(s)
( ) ( )∑∞
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+−==
1 21FIm1,f )(f
n
na
tnj
ta
teatt π
Euler transformation of difn terms helps to speed up the convergence
∑∑−
=+−
−−
=
+1
0
1
0
2dif
sumdif
dif
sumn
kknkn
nn
nn FWF where ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+ m
nWW dif
mm 1 and 11 =W
1. MNA Matrix Equation Formulation – a linear network containing initially excited MTLs
– network’s modified nodal admittance (MNA) matrix equation in
the time domain
)()()()(1
tttdt
tdM
P
kkkMM
MM iiDvGvC ∑
==++ , (1)
where
MC , MG – NN × constant matrices with entries determined by the lumped memory and memoryless components, respectively,
)(tMv – 1×N vector of node voltages appended by currents of independent voltage sources and inductors,
)(tMi – 1×N vector of source waveforms,
)(tki – 1×kn vector of currents entering the k-th MTL,
kD – knN × selector matrix with entries 1,0, ∈jid mapping the vector )(tki into the node space of the network.
)1(1i
)2(1i
)1(2i )2(
2i )1(Pi
MTL1
a section with lumped-parameter components
)2(Pi
MTL2 MTLP
– a frequency–domain representation of the MNA matrix equation
)0()()()(][1
MMM
P
kkkMMM sss vCIIDVCsG +=++ ∑
= . (2)
– MTLs consist of 2kk nN = active conductors and they are regarded
as kN2 -ports – the )(skI in (2) is formed to contain vectors of currents entering the
input and output ports as Tkkk sss )](),([)( )2()1( III =
Description of multiconductor transmission lines – a length of the MTL is l – per-unit-length matrices )(xR , )(xL , )(xG , )(xC
– frequency–domain MTL’s equations
⎥⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡)0,()0,(
)()(
),(),(
),(,
),(),(
xx
xx
sxsx
sxsx
sxsx
dxd
iv
0CL0
IV
0Y-)(Z-0
IV
, (3)
where – [ ]),(),( txsx vV L= , [ ]),(),( txsx iI L= are column vectors of
Laplace transforms of instantaneous voltages and currents, respectively,
– )0,(xv , )0,(xi are column vectors of initial voltage and current distributions, respectively,
– )()(),( xsxsx LRZ += , )()(),( xsxsx CGY += are series impedance and shunting admittance matrices, respectively.
– more formally written
)0,()(),(),(),( xxsxsxsxdxd wNWMW += . (4)
– the solution of (4)
ξξξ dssssll
lξ
l )0,()()(),0()(),(0
0 wNWW ∫+= ΦΦ , (5)
where )(0 slΦ is an integral matrix (matrizant), defined generally by so–called Volterra product integral:
[ ]∫ += ll dxsxs 00 ),()( MEΦ . (6)
– a practical evaluation of )(0 slΦ : – a uniform MTL => the exact solution
lsssx
l es ⋅
== )(
)(),()( M
MM0Φ . (7)
– a nonuniform MTL => an approximate solution e.g. as
)(~)(~ 1
0),(
0 ses jjjj xxsx −⋅= Δ ΦΦ ζM with E=)(~ 00 sΦ . (8)
where
1−−=Δ jjj xxx , jjj xx ,1−∈ζ , mj ,,2,1 …= ,
and 00 =x , lxm = ,
m is a number of the MTL’s sections.
– in terms of the multiport theory the integral matrix acts as the chain matrix )(sΦ
– denoting Tsss )](),([),0( )1()1( IVW = , (9)
Tsssl )](),([),( )2()2( IVW −= , (10)
Tl
ll sssds )](),([)()0,()()( )()(
00
ΓΓ∫ == IVwN ΓΦ ξξξξ , (11)
then for the k–th MTL is valid
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡Γ
Γ
)()(
)()(
)()()()(
)()(
)(
)(
)1(
)1(
2221
1211)2(
)2(
ss
ss
ssss
ss
k
k
k
k
kk
k
IV
IV
I-V
ΦΦΦΦ
(12)
admittance equations taking MTL’s nonzero initial conditions
into account
⎥⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡Γ
Γ
)()(
)()(
)()(
)()()()(
)()(
)(
)(
22
12)2(
)1(
2221
1211)2(
)1(
ss
ss
ss
ssss
ss
k
k
kk
k
kk
k
IV
EY0Y
VV
YYYY
II
(13)
where )()()( 11
11211 sss ΦΦ−−=Y ,
)()()( 1122222 sss −−= ΦΦY ,
)()( 11212 ss −= ΦY ,
)()( 1221 ss TYY =
– in the compact matrix form
)()()()()( sssss kkkkk ΓXVYI −= (14)
– substituting (14) into (2) =>
Resultant MNA matrix equation
⎥⎦⎤
⎢⎣⎡ ++⎥⎦
⎤⎢⎣⎡ ++= ∑∑
=
−
=
P
kkkkMMM
Tk
P
kkkMMM sssss
1
1
1)()()0()()()( ΓXDvCIDYDCsGV . (15)
– to solve the voltage and current at a coordinate x from a beginning
(1) of the k-th MTL the (12) can be rewritten as
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡Γ
Γ
),(),(
)()(
),(),(),(),(
),(),(
)(
)(
2221
1211
sxsx
ss
sxsxsxsx
sxsx
k
k
k
k
kk
k
IV
IV
IV
(1)
(1)
ΦΦΦΦ
, (16)
where – ),( sxkΦ is a partial chain matrix,
– the column vector )()],(),,([ 0)()( ssxsx xT
kk Γ=ΓΓ IV is expressed by the matrix convolution integral as
∫=x
xx dss0
0 )0,()()()( ξξξξ wNΦΓ . (17)
– the voltage and current vectors )()1( skV and )()1( skI are extracted
from equations
)()( ss MTkk VDV = (18)
and (14), respectively. – a numerical calculation of (17) can be made using FFT method as
follows:
Matrix convolution integral calculation 1. a three-dimensional array of a cumulative product of matrices
according to (8) is computed, with m = 2N , N integer, as
m
kx sk
103
)(~=
><Φ = ΦΩ , (19)
where – xkxk Δ= , and xΔ is taken equidistantly,
– the superscript <3> means the array is formed along to 3rd dimension
– this array is also used to determine ),( sxkΦ in (16). 2. after designation )0,()()( xxx wNΨ = , see in eq. (17), another three-
dimensional array is created as
mk
Tnk
T x 123
)( =
><Ψ ⊗= 1ΨΩ , (20)
where – ]1111[2 =n1 is n21× row vector with all elements equal to 1,
– the symbol ⊗ means Kronecker tensor product of matrices.
3. the values of )(0 sxΓ can be determined from the array
( ) ( )( )⎟⎠⎞
⎜⎝⎛= ∑
><Ψ><Φ><><Γ
2 333ΩFFTΩFFTIFFTΩ
ml , (21)
where – the FFT and IFFT denotes 2m-point fast Fourier transformation
operation and its inversion, respectively, – subscripts <2> and <3> determine dimensions along which
necessary operations are performed, – the symbol designates so-called element-by-element product (as
is in the Matlab language defined)
2. Enhanced FFT-based NILT Method
Theoretical base – to get the original )(tf to a Laplace transform )(sF the Bromwich
formula is used
∫∞+
∞−=
jc
jc
stdsesFj
tf )(21)(π
, (1)
under tKetf α≤)( , K real positive, α as exponential order of )(tf ,
0≥t , and )(sF defined for α>]Re[s . – the rectangular rule of integration leads to an approximate formula
in the discrete form )(~~ kTffk =
]Re[2~0
0FzFCf
n
nknkk −= ∑
∞
= , 1,,0 −= Nk , (2)
with Ω−= jkTk ez , ckT
k eCπ2
Ω= , )( Ω−= jncFFn , (3)
where T and )(2 NTπ=Ω are sampling periods in the original and transform domain, respectively. – the maximum time is supposed to be TMtm )1( −= , with 2NM =
as the number of resultant computed points – the coefficient c in (3) can approximately be determined as
rEc ln2
⋅Ω
−≈π
α , (4)
where rE is the desired relative error.
FFT and quotient-difference algorithm application – the formula (2) can be rewritten into the form
]Re[2~0
0
1
0FzFzFCf
n
nNknN
N
n
nknkk −+= ∑∑
∞
=
++
−
= , 1,,0 −= Nk . (5)
– the finite sum is evaluated by the FFT supposing mN 2= , m
integer, when only M first points is further considered – the infinite sum can be arranged into the form
nk
nn
nk
nnN
n
nNknN zGzFzF ∑∑∑
∞
=
∞
=+
∞
=
++ ==
000 , 1,,0 −= Mk , (6)
where the equality 12 == − kjN
k ez π , k∀ , was considered. – the convergence of the infinite sum (6) can be accelerated using the
quotient-difference algorithm of Rutishauser – just for a power series this algorithm corresponds to rational Padé
approximation of the series, but expressed as a continued fraction
)))1(1(1()( 210 +++= kkk zdzddzv , k∀ . (7)
– taking only 12 +P terms into account, i. e. considering the power
series
∑=
=P
n
nknk zGPzu
2
0),( , k∀ , (8)
the continued fraction is constructed as
))1(1(),( 210 kPkk zdzddPzv +++= , k∀ . (9)
The quotient-difference algorithm diagram
)4(0
)3(1
)2(1
)3(0
)1(2
)2(1
)0(2
)1(1
)2(0
)0(2
)1(1
)0(1
)1(0
)0(1
)0(0
eq
eeqq
eeeqq
eeq
e
– the terms nd , Pn 2,,0= , are calculated using the q-d algorithm:
– the first two columns are formed as
0)(0 =ie , Pi 2,,0= , (10)
ii
i GGq 1)(
1 += , 12,,0 −= Pi , (11) and then successive columns are given by the rules – for Pr ,,1= ,
)1(1
)()1()( +−
+ +−= ir
ir
ir
ir eqqe , rPi 22,,0 −= , (12)
– for Pr ,,2= ,
)(1
)1(1
)1(1
)( ir
ir
ir
ir eeqq −
+−
+−= , 122,,0 −−= rPi . (13)
– the coefficients nd , Pn 2,,0= , are given by
00 Gd = , )0(12 mm qd −=− , )0(
2 mm ed −= , Pm ,,1= . (14)
– for any kz the recurrence formulae are valid
)()()( 21 knknknkn zAzdzAzA −− += (15)
)()()( 21 knknknkn zBzdzBzB −− += ,
Pn 2,,1= , k∀ , with 01 =−A , 11 =−B , 00 dA = , and 10 =B .
– finally the continued fraction (9) can also be expressed in the form
)()(),( 22 kPkPk zBzAPzv = , k∀ , (16)
– the result of (16) is used instead of the infinite sum in (5). Brief comparison with the NILT method based on the FFT
and the ε–algorithm of Wynn – unlike the ε–algorithm, the quotient-difference algorithm does not
require recalculating the coefficients nd , Pn 2,,0= , for each new variable kz => the NILT method can be faster
– the NILT method under consideration is more numerically stable,
while the accuracy is approximatelly the same – a numerical instability of the ε–algorithm results from its basic
computational formula
)()1()1(
1)(1
1s
rs
r
sr
sr εε
εε−
+= ++
−+ , 2,1,0, =sr (17)
namely, due to the occurrence of difference in the denominator.
Experimental error analysis
Laplace transforms and their originals
1 2 3 4 5 6 )1(1 +s 2)1(1 +s )4(2 22 ππ +s 11 2 +s se s− se s−
te− tte− )2sin( tπ )(0 tJ ))t(21erfc( )1( −t1
Computed originals and their errors
0 2 40
0 . 5
1
f1
O R I G I N A L S
0 2 4
1 0- 1 0
1 00
E R R O R S
0 5 1 00
0 . 2
0 . 4
f2
0 2 4 6 8
1 0- 1 0
1 00
0 2 4- 1
0
1
f3
t0 2 4
1 0- 1 0
1 00
t
0 1 0 2 0 3 0- 0 . 5
0
0 . 5
1
f4
O R I G I N A L S
0 1 0 2 0 3 0
1 0- 1 0
1 00
E R R O R S
0 5 1 0 1 50
0 . 5
1
f5
0 5 1 0 1 5
1 0- 1 0
1 00
0 1 2 30
1
2
f6
t0 1 2 3
1 0- 1 0
1 00
t
Scalar, vector and matrix NILT versions A. Laplace transform has a scalar form )(sF :
])(Re[2ˆ0MM
PNMM FFT FFCf −+= VRM . (18)
B. Laplace transform has a vector form T
JJ sFsFsFs )](),(),([)( 21=F :
])(Re[2~
02
MJMJP
NJMJMJMJ FFT ×××
><
××× −+= FFCf VR . (19) C. Laplace transform has a matrix form
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=×
)()()(
)()()()()()(
)(
21
22221
11211
sFsFsF
sFsFsFsFsFsF
s
JLJJ
L
L
LJF
])(Re[2~02
LMJLMJP
LNJLMJLMJLMJ FFT ××××××
><
×××××× −+= FFCf VR . (20) where – all the terms are upper indexed vectors (A.), matrices (B.) or 3D
arrays (C.) – the R. are operators of MN → arrays length reduction – >< 2 designates the FFT operation runs along the 2nd dimension,
but in parallel on all the remaining ones – designates so-called Hadamard product of matrices (element-by-
element product in terms of Matlab language)
Matlab function definition – matrix NILT version %******************************* NILTM function **********************************% function [ft,t,x]=niltm(F,tm,pl); global ft t x; alfa=0; M=256; P=3; Er=1e-10; % adjustable N=2*M; qd=2*P+1; t=linspace(0,tm,M); NT=2*tm*N/(N-2); omega=2*pi/NT; c=alfa-log(Er)/NT; s=c-i*omega*(0:N+qd-1); Fsc=feval(F,s); ft=fft(Fsc,N,2); ft=ft(:,1:M,:); dim1=size(Fsc,1); dim3=size(Fsc,3); d=zeros(dim1,qd,dim3); q=Fsc(:,N+2:N+qd,:)./Fsc(:,N+1:N+qd-1,:); e=d; d(:,1,:)=Fsc(:,N+1,:); d(:,2,:)=-q(:,1,:); for r=2:2:qd-1 w=qd-r; e(:,1:w,:)=q(:,2:w+1,:)-q(:,1:w,:)+e(:,2:w+1,:); d(:,r+1,:)=-e(:,1,:); if r>2 q(:,1:w-1,:)=q(:,2:w,:).*e(:,2:w,:)./e(:,1:w-1,:); d(:,r,:)=-q(:,1,:); end end A2=zeros(dim1,M,dim3); B2=ones(dim1,M,dim3); A1=repmat(d(:,1,:),[1,M]); B1=B2; z=repmat(exp(-i*omega*t),[dim1,1,dim3]); for n=2:qd Dn=repmat(d(:,n,:),[1,M]); A=A1+Dn.*z.*A2; B=B1+Dn.*z.*B2; A2=A1; B2=B1; A1=A; B1=B; end ft=ft+A./B; ft=2*real(ft)-repmat(real(Fsc(:,1,:)),[1,M]); ft=repmat(exp(c*t)/NT,[dim1,1,dim3]).*ft; ft(:,1,:)=2*ft(:,1,:); feval(pl); %************************************************************************************% %******************************** PLOT function *********************************% function pl3 global ft t x; m=length(t); tgr=[1:m/64:m,m]; % 65 time points for k=1:size(ft,3) figure; mesh(t(tgr),x,ft(:,tgr,k)); xlabel('t'); ylabel('x'); zlabel(strcat('f_',num2str(k))); end %************************************************************************************%
3. Examples Example 1: MTL network No.1 driven with external source
MTLs description: – lenghts: ml 05.01 = , ml 04.02 = , ml 03.03 =
– per-unit-length matrices
mnH
⎥⎦
⎤⎢⎣
⎡=
6.4943.633.636.494
L , mpF
⎥⎦
⎤⎢⎣
⎡−
−=
8.629.49.48.62
C , mΩ
⎥⎦
⎤⎢⎣
⎡=
75151575
R , mS
⎥⎦
⎤⎢⎣
⎡−
−=
1.001.001.01.0
G
Application of the vector NILT version:
0 0.2 0.4 0.6 0.8 1x 10-8
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (Seconds)
Vol
tage
(Vol
ts)
vin
vout
0 0.2 0.4 0.6 0.8 1x 10-8
-5
0
5x 10-4
Time (Seconds)
Cur
rent
(Am
pere
s)
i2
MTL1
MTL2
MTL3
1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
I1
I2
vin
50Ω
50Ω 75Ω
25Ω
25Ω
100Ω
100Ω 100Ω
100Ω 50Ω
1pF
2pF
10nH
1pF
vout
Application of the matrix NILT version:
Example 2: MTL network No.2 driven with external source
MTLs description: – lenghts: mll 1.021 ==
– per-unit-length matrices: – two-conductor MTL1: the same as in the Example 1 – four-conductor MTL2:
mnH
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
6.4943.638.703.636.4943.638.7
8.73.636.4943.6308.73.636.494
L , mpF
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−−−−−
−−
=
8.629.43.009.48.629.43.03.09.48.629.4
03.09.48.62
C ,
mΩ
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
50101010501011105010011050
R , mS
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−−−−−
−−
=
1.001.0001.0001.01.001.0001.0
001.001.01.001.00001.001.01.0
G .
50Ω
75Ω 100Ω
25Ω
50Ω
25Ω
100Ω 100Ω
50Ω
100Ω
2pF 1pF
1pF
vout
+
–
vin
10nH
MTL1
MTL2
– the 1V pulse with 0.4 ns rise/fall time and 5 ns duration is applied at the input
– overall the 15 nodal voltages and 2 currents are the variables to be solved by the MNA method (the system of 17 equations is solved)
0 0 .5 1 1 .5 2
x 1 0-8
-0 .2
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
T ime (S econds)
Inpu
t (V
olts
)
0 0 .5 1 1 .5 2
x 1 0-8
-0 .05
0
0 .05
0 .1
0 .15
0 .2
0 .25
0 .3
0 .35
T ime (S econds)
Out
put (
Vol
ts)
– using a PC Pentium IV 2GHz/256MB the CPU time was under one second
– the vector NILT version has been used to get the time–domain solutions
Example 3: MTL network with initially excited MTL1
MTLs description: – lenghts: mll 2.021 ==
– per-unit-length matrices:
mnH
⎥⎦
⎤⎢⎣
⎡=
6.4943.633.636.494
L , mpF
⎥⎦
⎤⎢⎣
⎡−
−=
8.629.49.48.62
C , mΩ
⎥⎦
⎤⎢⎣
⎡=
1.002.002.01.0
R , mS
⎥⎦
⎤⎢⎣
⎡−
−=
1.001.001.01.0
G
– a nonzero initial voltage distribution of the 1st wire of the MTL1
⎟⎠
⎞⎜⎝
⎛⎥⎦⎤
⎢⎣⎡ −=
234sin)0,( 2
1 lxxv π if lxl
85
83
≤≤ , 0)0,(1 =xv otherwise,
– to get the nodal voltages and/or branch currents the vector NILT
version has again been used 0 0.2 0.4 0.6 0.8 1 1.2
x 10-9
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
t [s]
v 1 [V]
Node voltage v1 waveform
0 0.2 0.4 0.6 0.8 1 1.2x 10-9
-5
-4
-3
-2
-1
0
1
2
3
4
5x 10-3
t [s]
i 3 [A]
Current i3 waveform
MTL1
1
2 4
I1
10Ω
5
MTL2
10Ω 10Ω
10pF 10pF
1nH 1nH
1nH
3
6
7
8
I7
I3
– to get the wave propagations along the MTLs‘ wires the matrix
NILT version has been used
– using the same PC as in the first two examples the CPU time was
under 10 seconds
0 0.2 0.4 0.6 0.8 1 1.2x 10-9
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
t [s]
v 7 [V]
Node voltage v7 waveform
0 0.2 0.4 0.6 0.8 1 1.2x 10-9
-5
-4
-3
-2
-1
0
1
2
3
4
5x 10-3
t [s]
v 6 [V]
Node voltage v6 waveform
Comparative Comparative SStudytudy of of MMethodsethods for for SSensitivityensitivity DDeterminationetermination in in MTLMTL SSystemsystems
LectureLecture OutlineOutline
IntroductionIntroduction && ProblemProblem FFormulationormulationMNAMNA FrequencyFrequency--Domain Domain SSensitivityensitivity FormulaeFormulae
LumpedLumped--ParameterParameter SensitivitySensitivityDistributedDistributed--ParameterParameter SensitivitySensitivity
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproachChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion ApproachApproachGettingGetting TimeTime--Domain SensitivityDomain SensitivityExamplesExamples && ConclusionsConclusions
linear hybrid multiconductor-transmission-line circuit
the circuit will be described by modified nodal analysis(MNA) method
IntroductionIntroduction & P& Problemroblem FFormulationormulation
i1(1) i1(2)
section with lumped-parameter components
MTL1 MTL2 MTLP
i2(1) i2(2) iP(1) iP(2)
v1(1) v1
(2) v2(1) v2
(2) vP(1) vP
(2)
CircuitCircuit MNA Matrix EquationMNA Matrix Equation
The admittance equation of k–th MTL
( ) ( ) ( )k k ks s s=I Y V (3)
where (1) (2)( ) [ ( ), ( )]Tk k ks s s=I I I (1) (2)( ) [ ( ), ( )]T
k k ks s s=V V V
Description in the time domain
1
( ) ( ) ( ) ( )P
MM M M k k M
k
d t t t tdt =
+ + =∑vC G v D i i (1)
1
[ s ] ( ) ( ) ( ) (0)P
M M M k k M M Mk
s s s=
+ + = +∑G C V D I I C v (2)
Description in the frequency domain
CircuitCircuit MNA Matrix EquationMNA Matrix Equation
Substituting (3) into (2) we have
[ ]-1( ) ( ) ( ) (0)M M M M Ms s s= +V Y I C v (4)
where
1( ) s ( )
PT
M M M k k kk
s s=
= + + ∑Y G C D Y D (5)
Determined via- modal analysis technique- chain/admittance matrix conversion
The s–domain solution is prepared serving for thederivation of sensitivities in the frequency domain
Consider γ as some lumped or distributed parameter
MNA equation differentiation
Then going out from eq. (4) in the form
we have( ) ( ) ( ) (0)M M M M Ms s s= +Y V I C v (6)
( ) ( )( ) ( ) (0)M M MM M M
s ss sγ γ γ
∂ ∂ ∂+ =
∂ ∂ ∂Y V CV Y v (7)
where there were considered
( )M s γ∂ ∂ =I 0 (0)M γ∂ ∂ =v 0
From (7) we can write
-1( ) ( )( ) (0) ( )M M MM M M
s ss sγ γ γ
⎛ ⎞∂ ∂ ∂= −⎜ ⎟∂ ∂ ∂⎝ ⎠
V C YY v V (8)
1( ) s ( )
PT
M M M k k kk
s s=
= + + ∑Y G C D Y D
FrequencyFrequency--Domain SensitivityDomain Sensitivity
LumpedLumped--Parameter SensitivityParameter Sensitivity
The parameter γ is certainly included in orMC MG
( )-1( ) ( )( ) (0) ( ) ( )M M MM M M M
s ss s s sγ γ γ
⎛ ⎞∂ ∂ ∂= − −⎜ ⎟∂ ∂ ∂⎝ ⎠
V C GY v V V (9)⇒
is a memory-element parameterMcγ ≡
( )-1( ) ( ) (0) ( )M MM M M
M M
s s s sc c
∂ ∂= −
∂ ∂V CY v V (10)
is a memoryless-element parameterMgγ ≡
(11)-1( ) ( )( ) ( )M MM M
M M
s ss sg g
∂ ∂= −
∂ ∂V GY V
DistributedDistributed--Parameter SensitivityParameter Sensitivity
• MTL’s length l• a component of per-unit-length matrices R0, L0, G0, C0
• a general physical parameter affecting p.-u.-l. matrices
The γ is a parameter of the k-th MTL:
-1( ) ( )( ) (0) ( )M M MM M M
s ss sγ γ γ
⎛ ⎞∂ ∂ ∂= −⎜ ⎟∂ ∂ ∂⎝ ⎠
V C YY v V1
( ) s ( )P
TM M M k k k
ks s
=
= + + ∑Y G C D Y D
Remember about eq. (8)
-1 ( )( ) ( ) ( )TkMM k k M
ss s sγ γ
∂∂= −
∂ ∂YV Y D D V (12)
MTLk admittance matrix derivative
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
Frequency-Domain MTL Equations Formulation
0 0 0( )s s= +Z R L 0 0 0( )s s= +Y G Cwhere
2
0 02
( , ) ( ) ( ) ( , )d x s s s x sdx
− =V Z Y V
2
0 02
( , ) ( ) ( ) ( , )d x s s s x sdx
− =I Y Z I (14)
0( , ) ( ) ( , )d x s s x sdx
− =V Z I 0
( , ) ( ) ( , )d x s s x sdx
− =I Y V (13)
Elimination of variables in (13) leads to
Decoupling is done via treating Z0Y0 matrix. There are found:- eigenvalues λi
2
- associated eigenvectors xi , i=1,2,....,nk
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
There are created matrices:
1kn
i idiag λ
==Λ 1 2, , ,
kv n⎡ ⎤= ⎣ ⎦S x x x (15)
and another matrices computed as
10i v−=S Z S Λ
1 1coth( ) kn
i idiag lλ
==E 2 1
1 sinh( ) kni i
diag lλ=
= −E
(16)
1 111 12 1 2
1 112 11 2 1
i v i vk
i v i v
− −
− −
⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦
Y Y S E S S E SY
Y Y S E S S E S
(17)
=> MTL admittance matrix
(18)
1 11 21 11 2 12
1 12 12 12 1 11
i v i vi v i v
k
i v i vi v i v
γ γ γ γ γ γγ
γ γ γ γ γ γ
− −
− −
⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂∂ ∂+ − + −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂∂ ⎝ ⎠ ⎝ ⎠⎢ ⎥=
⎢ ⎥∂ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂∂ ∂+ − + −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
S S S SE EE S Y S E S Y SY
S S S SE EE S Y S E S Y S
MTL admittance matrix derivative
(19)
where (16) is used to get
1 00
i vv iγ γ γ γ
−∂ ⎛ ∂ ∂ ⎞∂= −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
S S ZZ S SΛΛ +
1 2, ,γ γ γ∂ ∂ ∂ ∂ ∂ ∂E EΛ depend on the eigenvalues sensitivities
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
depends on the eigenvectors sensitivities
(20)
v γ∂ ∂S
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
( )20 0i iλ − =I Z Y x 0
Consider basic equation
(21)
Differentiating (21) with respect to γ
( ) ( )20 02
0 0i i
i i iλλ
γ γ γ∂∂ ∂
− + =∂ ∂ ∂
Z YxI Z Y x x (22)
and appending equation 1Ti i =x x 0T i
i γ∂
=∂xx⇒
( )1 0 020 0
2 00
i
ii iTii
γ λγ
λγ
−∂⎡ ⎤
⎡ ⎤∂⎢ ⎥∂ ⎡ ⎤− ⎢ ⎥⎢ ⎥ = ∂⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ⎣ ⎦ ⎢ ⎥⎣ ⎦⎢ ⎥∂⎣ ⎦
xZ Y
xI Z Y xx
Resultant systemnk+1 equations:
(23)
(24)
212
i i
i
λ λγ λ γ
∂ ∂=
∂ ∂
The system (24) is solved repeatedly for 1,2, , ki n=
1
kni
i
diag λγ γ =
⎧∂ ⎫∂= ⎨ ⎬∂ ∂⎩ ⎭
Λ
12
1
1sinh ( )
kn
ii
i i
ldiag ll
λ λγ λ γ γ
=
⎧ ⎫⎛ ∂ ⎞∂ − ∂= +⎨ ⎬⎜ ⎟∂ ∂ ∂⎝ ⎠⎩ ⎭
E
22
1
cosh( )sinh ( )
kn
i ii
i i
l ldiag ll
λ λ λγ λ γ γ
=
⎧ ⎫⎛ ∂ ⎞∂ ∂= +⎨ ⎬⎜ ⎟∂ ∂ ∂⎝ ⎠⎩ ⎭
E
⇒ 1 2, , , knv
γ γ γ γ∂⎡ ⎤∂ ∂ ∂
= ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦
xS x x
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
⇒
(25)
(26)
(27)
(28)
where (29)
ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion
MTL equations (13) in the matrix form
0
0
- ( )( , ) ( , )- ( )( , ) ( , )
sx s x sdsx s x sdx
⎡ ⎤⎡ ⎤ ⎡ ⎤= ⋅⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦
0 ZV VY 0I I
(30)
with the solution(2) (1)
11 12(2) (1)
21 11
( ) ( )( ) ( )( ) ( )( ) ( )T
s ss ss ss s
⎡ ⎤ ⎡ ⎤⎡ ⎤= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦
V VI I
Φ ΦΦ Φ
(31)
The chain matrix ( )( ) s ls e= MΦ (32)0
0
- ( )( )
- ( )s
ss
⎡ ⎤= ⎢ ⎥
⎣ ⎦
0 ZM
Y 0
Rearranging eq. (31)(1) (1)
11 12(2) (2)
12 11
( ) ( )( ) ( )( ) ( )( ) ( )s ss ss ss s
⎡ ⎤ ⎡ ⎤⎡ ⎤= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ ⎦⎣ ⎦ ⎣ ⎦
Y YI VY YI V
(33)
ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion
Conversion Φ(s) into Y(s) yields
1 112 11 12
1 112 12 11
( ) ( ) ( )( )
( ) ( ) ( )ks s s
ss s s
− −
− −
⎡ ⎤−= ⎢ ⎥−⎣ ⎦
YΦ Φ Φ
Φ Φ Φ(34)
and derivative is
Thus we have to determine the chain matrix derivative
(35)
( )( ) s ls eγ γ
∂ ∂=
∂ ∂MΦ
(36)
1 1 1 112 11 1212 12 11 12 12
1 1 1 112 12 1112 12 12 12 11
kγ γ γ
γγ γ γ
− − − −
− − − −
⎡ ⎤⎛ ⎞∂ ∂ ∂− −⎢ ⎥⎜ ⎟∂ ∂ ∂∂ ⎝ ⎠⎢ ⎥=
⎢ ⎥∂ ⎛ ⎞∂ ∂ ∂− −⎢ ⎥⎜ ⎟∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦
YΦ Φ Φ
Φ Φ Φ Φ Φ
Φ Φ ΦΦ Φ Φ Φ Φ
ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion
⇒
Performing necessary multiplications
12 0 11 00 21 0 11
11 0 21 00 11 0 12
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) T
s s s ss s s sss s s sl s s s s
Τ⎡ ⎤ ⎡ ⎤∂= − = −⎢ ⎥ ⎢ ⎥∂ ⎣ ⎦⎣ ⎦
Y ZΖ ZY ZY Y
Φ ΦΦ ΦΦΦ ΦΦ Φ
(38)
( ) ( ) ( ) ( ) ( )s s s s sl
∂= =
∂M MΦ
Φ Φ (37)
Comparing (38) and (36) we have needed matrix derivative
12 0 12 11 0 12
11 0 12 12 0 12
( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )
k s s s s s sss s s s s sl
⎡ ⎤∂= − ⎢ ⎥∂ ⎣ ⎦
Y Z Y Y Z YYY Z Y Y Z Y (39)
lγ ≡MTL-Length Sensitivity:
ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion
Parameter γ is contained in M(s) matrix, while l is constant
( )( ) s ls eγ γ
∂ ∂=
∂ ∂MΦ
(40)
First, we need to find the derivative with respect to M(s)
( )
0( )
!
ks l k
k
le sk
∞
=
= ∑M M0
(s) ( )!
k k
k
l skγ γ
∞
=
∂ ∂=
∂ ∂∑ MΦ⇒
To evaluate (41) effectively the recurrence formula can be used
(41)
(s) (s)(s) (s) (s)(s)+ (s)k -1k k-1
k -1
γ γ γ γ
⎡ ⎤∂∂ ∂ ∂⎣ ⎦= =∂ ∂ ∂ ∂
M MM M MM M (42)
starting with k = 2
MTL-Primary-Parameter Sensitivity
ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion
For a computation the derivative is prepared as ( )s γ∂ ∂M
0 :ijRγ ≡ ∈R
0( )ij
ij
s RR
∂⎡ ⎤−∂ ⎢ ⎥∂= ⎢ ⎥∂⎢ ⎥⎣ ⎦
R0M
0 0
0 :ijLγ ≡ ∈L
0( )ij
ij
ss LL
∂⎡ ⎤−∂ ⎢ ⎥∂= ⎢ ⎥∂⎢ ⎥⎣ ⎦
L0M
0 0
0 :ijGγ ≡ ∈G
0( )
ijij
sG G
⎡ ⎤∂ ⎢ ⎥∂= ⎢ ⎥−∂
∂⎢ ⎥⎣ ⎦
0 0M G 0
0 :ijCγ ≡ ∈C
0( )
ijij
ssC C
⎡ ⎤∂ ⎢ ⎥∂= ⎢ ⎥−∂
∂⎢ ⎥⎣ ⎦
0 0M C 0
(43) (44)
(45) (46)
MTLMTL--PhysicalPhysical--Parameter SensitivityParameter Sensitivity
• width of the line wires• spacing between the wires• material properties• etc.
Suppose γ as a general MTL’s physical parameter:
The MTL’s admittance matrix derivative is gained via chain rule
1
m mij ij ij ijk k k k k
i j i ij ij ij ij
R L G CR L G Cγ γ γ γ γ= =
⎛ ⎞∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂= + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
∑∑Y Y Y Y Y(47)
where m denotes the order of the per-unit-length matrix.
GettingGetting TimeTime--Domain SensitivityDomain Sensitivity
Application of NILT method
-1( ) ( )M Mt sγ γ
⎡ ⎤∂ ∂= ⎢ ⎥∂ ∂⎣ ⎦
v VL (48)
Semirelative sensitivity
( ) ( )( ), = MM
ttγ γ γγ
∂∂
vS v (49)
• The used NILT method is based on FFT and quotient-differencealgorithm and is running on all the vector elements in parallel
• The NILT procedure has been created in Matlab language environment
vout
vin
MTL1
MTL2
MTL3
i1
i2
R1=50Ω
50Ω75Ω
25Ω
25Ω
100Ω
100Ω 100Ω
100Ω50Ω
1pF
C2=2pF
10nH
1pF
ExampleExample 1 1 –– circuit in circuit in viewview
Hybrid linear network with three MTLs
ExampleExample 1 1 –– MTL MTL parametersparameters
0
75 1515 75 m
⎡ ⎤ Ω= ⎢ ⎥
⎣ ⎦R 0
494.6 63.363.3 494.6
nHm
⎡ ⎤= ⎢ ⎥
⎣ ⎦L
0
0.1 0.010.01 0.1
Sm
−⎡ ⎤= ⎢ ⎥−⎣ ⎦
G 0
62.8 4.94.9 62.8
pFm
−⎡ ⎤= ⎢ ⎥−⎣ ⎦
C
The MTLs’ per-unit-length matrices
1V pulse 1.5 ns rise/fall times and 7.5 ns width acts on the input
1 0.05l m= 2 0.04l m= 3 0.03l m=
The MTLs differ only in their lengths
9 9 91.5 10 6 10 7.5 10
9 2
1( )1.5
s s
ine e eV s
e s
− − −− ⋅ − ⋅ − ⋅
−
− − +=
ExampleExample 1 1 -- resultsresults
Input and output voltages
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Inpu
t/out
put v
olta
ges
(vol
ts)
Time (seconds)
vinvout
Semirel. sensitivity vout w. r. to R1
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Time (seconds)Se
mire
lativ
e se
nsiti
vity
SR
1 (vol
ts)
ExampleExample 1 1 -- resultsresults
Sem. sensitivity vout w. r. to C2
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (seconds)
Sem
irela
tive
sens
itivi
ty S
C2 (v
olts
)
Sem. sens. vout w. r. to l of MTL2
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Time (seconds)S
emire
lativ
e se
nsiti
vity
SM
TL2
l (v
olts
)
ExampleExample 1 1 -- resultsresults
Sem. sens. vout w. r. to R11 of MTL2 Sem. sens. vout w. r. to L11 of MTL2
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-6
-5
-4
-3
-2
-1
0
1
2
3x 10-3
Time (seconds)
Sem
irela
tive
sens
itivi
ty S
MTL
2R
11 (v
olts
)
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (seconds)S
emire
lativ
e se
nsiti
vity
SM
TL2
L11
(vol
ts)
ExampleExample 2 2 –– circuit in circuit in viewview
vin
MTL1
MTL2
MTL3
i1
i2
R1=50Ω
50Ω75Ω
25Ω
25Ω
100Ω
100Ω 100Ω
100Ω50Ω
1pF
C2=2pF
10nH
1pF
vcross
Hybrid linear network with three lossless MTLs
ExampleExample 2 2 –– MTLs MTLs structurestructure
0
493.11 63.0463.04 493.11
nHm
⎡ ⎤= ⎢ ⎥
⎣ ⎦L 0
69.62 7.097.09 69.62
pFm
−⎡ ⎤= ⎢ ⎥−⎣ ⎦
C
The MTLs structure
Ground plane
Laminate, εr
w
h
l
d
Flat conductors
2 20
12 21 1 12ln 1
4r
L Cw hC C K Kh d
ε επ
⎡ ⎤⎛ ⎞ ⎛ ⎞= ≈ − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
11 22 0 1 12r CwC C K Ch
ε ε ⎛ ⎞= ≈ −⎜ ⎟⎝ ⎠
20
12 212ln 1
4r hL L
dµ µ
π⎡ ⎤⎛ ⎞= ≈ +⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦
011 22 12
1
r
L
hL L LK w
µ µ ⎛ ⎞= ≈ −⎜ ⎟⎝ ⎠
10( 1)
120 ,r
LhK
Z wε
π
=
⎛ ⎞= ⎜ ⎟⎝ ⎠
( )1 1
r effC L
r
K Kε
ε=
0( 1)860ln
4r
h wZw hε =
⎛ ⎞≈ +⎜ ⎟⎝ ⎠
0.58w mm=1.17h mm= 2.49d mm=
ExampleExample 2 2 -- resultsresults
Semirelative sensitivity vcross w.r. to w/d of MTL2 (keeping h/w constant)
0 0.2 0.4 0.6 0.8 1 1.2x 10-8
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Time (seconds)
Sem
irela
tive
sens
itivi
ty S
MTL
2w
/d (v
olts
)
1
m mij ij ij ijk k k k k
i j i ij ij ij ij
R L G CR L G Cγ γ γ γ γ= =
⎛ ⎞∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂= + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
∑∑Y Y Y Y Y
-1 ( )( ) ( ) ( )TkMM k k M
ss s sγ γ
∂∂= −
∂ ∂YV Y D D V
Reminder of computationmethod:
ConclusionsConclusions
The results were verified by perturbing γ as follows
[ ] [ ]-1 -12 1
2 1
( , ) ( , )( ) ( ) M MM M s st t γ γγ γ γ γ
−∂ ∆≈
∂ ∆ −V Vv v L L
= (34)
where a central difference was always chosen as 0.1% ofthe nominal value 1 2( ) 2γ γ γ= +
γ∆
The obtained RMS errors about 10-8 - 10-9
The modal analysis technique is slightly faster
The proposed method is more general:- nonuniform MTL structures- voltage/current wave sensitivities along the MTLs wires
Comparison of some Mathematical Models for Comparison of some Mathematical Models for MTL Transient and Sensitivity AnalysisMTL Transient and Sensitivity Analysis
LECTURE LECTURE OOUTLINEUTLINE
Introduction & Problem motivationIntroduction & Problem motivationMTL MTL CContinuousontinuous MModelsodels
((ss,x,x))--domaindomain solution + 1D NILTsolution + 1D NILT((ss,q,q))--domaindomain solution + 2D NILTsolution + 2D NILT
MTL MTL SSemiemiddiscreteiscrete MModelsodels((ss,,xxkk))--domain solution + 1D NILTdomain solution + 1D NILT
Chain matrix approachChain matrix approachStateState--variable methodvariable method
Errors comparison & ExamplesErrors comparison & ExamplesConclusionConclusion
Simple linear system consisting of uniform (n+1)-conductor TL
The aimvoltage/current distributions along the MTL wiressensitivities with respect to MTL primary parameters, MTL length or lumped parameters of terminating networks
MTL telegrapher equations in (t,x)-domain
LINEARNETWORK
(L)
LINEARNETWORK
(R)
(n+1) - conductortransmission line
iL iR
vL vR
0 lx
PROBLEM MOTIVATIONPROBLEM MOTIVATION
0 0
0 0
( , ) ( , ) ( , )( , ) ( , ) ( , )t x t x t xt x t x t xx t
∂ ∂⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤− = +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦0 R 0 Lv v v
G 0 C 0i i i
MTL chain matrix
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((s,xs,x))--domain domain voltagevoltage/current distribution/current distribution
After Laplace transform (for zero MTL initial conditions)
where 0 0 0( )s s= +Z R L 0 0 0( )s s= +Y G C
0
0
( )( , ) ( , ) ( , )( )
( )( , ) ( , ) ( , )ss x s x s xd s
ss x s x s xdx⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= − =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
0 ZV V VM
Y 0I I I
( )( , ) ( ,0)
,( , ) ( ,0)
s x ss x
s x s⎡ ⎤ ⎡ ⎤
=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
V VΦ
I IThe solution
Given by boundary conditions
( ) 11 12( )T
21 11
( , ) ( , ),
( , ) ( , )s x s x s x
s x es x s x
⎡ ⎤= = ⎢ ⎥
⎣ ⎦M Φ Φ
ΦΦ Φ
Boundary conditions by generalized Thévenin equivalents
11 12( )
21 11
( ) ( )( )
( ) ( )s l s s
s es sΤ
⎡ ⎤= = ⎢ ⎥
⎣ ⎦M Φ Φ
ΦΦ Φ
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((s,xs,x))--domain domain voltagevoltage/current distribution/current distribution
R iR iR R( ) ( ) ( ) ( )s s s s= −V V Z I
LINEARNETWORK
(L)
LINEARNETWORK
(R)
(n+1) - conductortransmission line
IL IR
VL VR
L iL iL L( ) ( ) ( ) ( )s s s s= −V V Z I
( )( )
1TL 11 iR 21 iL iR 11 12
11 iR 21 iL iR
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
s s s s s s s s
s s s s s
−⎡ ⎤= − + −⎣ ⎦⎡ ⎤× − −⎣ ⎦
I Φ Z Φ Z Z Φ Φ
Φ Z Φ V V
MTL full chainmatrix
MTL voltage/current sensitivity w. r. to a parameter γ
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((ss,x,x))--domain voltage/current sensitivitydomain voltage/current sensitivity
( ) ( )L L
L L
( ) ( )( , ) ,,
( ) ( )( , )s ss x s x
s xs ss xγ γ γ
∂ ⎡ ⎤ ⎡ ⎤⎡ ⎤∂ ∂= +⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦
V VV ΦΦ
I II
( )
( )
1TL11 iR 21 iL iR 11 12
iL11 iR 21
11 iR 2121 iR L T
TiR 11 1211 iR
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( )
s s s s s s s s
ss s ss s ss s s
s s ss s
γ
γγ γ γ
γ γ γ
−∂ ⎡ ⎤= − + −⎣ ⎦∂
∂⎛ ⎞− +⎜ ⎟∂⎛ ⎞∂ ∂ ∂ ⎜ ⎟× − − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ + −⎜ ⎟∂ ∂ ∂⎝ ⎠
I Φ Z Φ Z Z Φ Φ
ZΦ Z ΦΦ Z ΦΦ Z V I
Z Φ ΦΦ ZL( )s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
L iL LL iL
( ) ( ) ( )( ) ( )s s ss sγ γ γ
∂ ∂ ∂= − −
∂ ∂ ∂V Z II ZMethod of computation
depends on γ
Method for practical computation – recursive formulae
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((ss,x,x))--domain distribution and sensitivitydomain distribution and sensitivity
( ) 1p
1
( ) ( )( ) ( )
k k
k k
s ss
s s−
−
⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
V VΦ
I I
( ) ( )p 1 1p
1 1
( ) ( ) ( )( ) ( ) ( )
k k k
k k k
ss s ss
s s sγ γ γ− −
− −
∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦
ΦV V VΦ
I I I
( ) ( )p ,s s x= ΔΦ ΦPartial chain matrix
1 , 1,2, ,k kx x x l m k m−Δ = − = = L
Advantagein case of uniform MTL the Φp(s) is evaluated only oncean easy generalization for nonuniform MTLs
Taylor series expansion with scaling & squaringAugmented matrix utilizationEigenvalues decompositionLaplace transform approach with scaling & squaringConvolution integral evaluationPadé approximation with scaling & squaring
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((s,xs,x))--domain voltage/current sensitivitydomain voltage/current sensitivity
( )( ) s ls eγ γ
∂ ∂=
∂ ∂MΦ
MTL-primary-parameter sensitivity: ( )sγ ∈M
is used to evaluate the boundary conditions, or
12 0 11 00 21 0 11T
11 0 21 00 11 0 12
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )s s s ss s s sss s s sl s s s s
Τ⎡ ⎤ ⎡ ⎤∂= − = −⎢ ⎥ ⎢ ⎥∂ ⎣ ⎦⎣ ⎦
Y ZΖ ZY ZY Y
Φ ΦΦ ΦΦΦ ΦΦ Φ
( )( ) ( ) ( ) ( ) ( )s ls e s s s sl l
∂ ∂= = =
∂ ∂M M MΦ
Φ Φ
γ l≡MTL-length sensitivity:
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((s,xs,x))--domain voltage/current sensitivitydomain voltage/current sensitivity
( )p
p p
( ) ( ) ( )( ) ( )s l ms e s ss s
l l m m∂ ∂
= = =∂ ∂
MΦ M MΦ Φ
To evaluate a recursive formula we use
Substitution x = lz into the telegr. eq. and doing Łtz. operation
L
L
( )( , )( , )
( )( , )ss q
s qss q
⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦
VVΨ
II
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((ss,,qq))--domain distribution and sensitivitydomain distribution and sensitivity
[ ] 1( )( , ) ( , ) ( )s lzz zs q s z e q s l −= = =MΨ Φ I - ML Lwhere
L L
L L
( ) ( )( , ) ( , ) ( , )( ) ( )( , )s ss q s q s qs ss qγ γ γ
⎡ ⎤ ⎡ ⎤⎡ ⎤∂ ∂ ∂= +⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦
V VV Ψ ΨI II
An absolute sensitivity
( , ) ( )( , ) ( , )s q ss q s q lγ γ
∂ ∂=
∂ ∂Ψ MΨ Ψ ( , ) ( , ) ( ) ( , )s q s q s s q
l∂
=∂
Ψ Ψ M Ψ
( )sγ ∈M lγ ≡
Łtz-1. always on z∈<0,1>
Boundary conditions
Generalized Π network
0
0
0
0
k
k
k
k
l ml ml ml m
==
=
=
Model paraL LR RC CG
m rs
G
ete
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, chain matrixchain matrix approachapproach
Generalized T network
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, chain matrixchain matrix approachapproach
i-thwire
j-thwire
couplings
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, chain matrixchain matrix approachapproach
( ) ( ) ( )2( )
( ) ( ) ( ) ( )( )4 2
k kn k
kk k k k
n k n
s s ss
s s s ss
⎡ ⎤+ −⎢ ⎥⎢ ⎥=⎛ ⎞⎢ ⎥− + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Z YI ZΦ
Y Z Y ZI Y I%
Generalized Π network: partial chain matrix
11( ) ( ) ( )( ) ( )
γ γ γ
k kkd d dd d
s s ss s−
−∂ ∂ ∂= +
∂ ∂ ∂
% % %% %Φ Φ ΦΦ Φ
Uniform MTL, Φk(s) = Φd(s) , for all k1( ) ( ) ( )k k
d d ds s s−=% % %Φ Φ Φ
k = 2,3,...,m.
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, chain matrixchain matrix approachapproach
Φd(s) submatrices derivatives
11( ) ( ) ( )1 ( ) ( )γ 2 γ γ
d d dd d
s s ss s⎛ ⎞∂ ∂ ∂
+⎜ ⎟∂ ∂ ∂⎝ ⎠
Φ Z Y= Y Z%
12 ( ) ( )γ γ
d ds s∂ ∂∂ ∂
Φ Z= -%
21( ) ( ) ( ) ( ) ( ) ( )1 ( ) ( ) ( )γ 4 γ γ 4 γ
d d d d d dd d d n
s s s s s ss s s⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞+ − +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
Φ Y Z Y Z Y= - Z Y Y I%
22 ( ) ( ) ( )1 ( ) ( )γ 2 γ γ
d d dd d
s s ss s⎛ ⎞∂ ∂ ∂
+⎜ ⎟∂ ∂ ∂⎝ ⎠
Φ Y Z= Z Y%
STL model reduced to 2 Π sections in cascade
1 1 1
2 2 2
3 3 3
12 12
23 23
2 0 0 0 0 ( ) 2 0 0 1 0 ( )0 0 0 0 ( ) 0 0 1 1 ( )0 0 2 0 0 ( ) 0 0 2 0 1 ( )0 0 0 0 ( ) 1 1 0 0 ( )0 0 0 0 ( ) 0 1 1 0 ( )
d d S S
d d S
d d S
d d
d d
C v t G G v t GC v t G G v tdC v t G G v t
dtL i t R i tL i t R i t
+⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ = − + − ⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1 1
2 2
3 3
( )( )( )
00
S
S S
S S
v tG v tG v t
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
vS3Cd2
Cd Gd
Rd RdLd Ld
Gd2
RS1
vS1 v1v2 v3
Cd2
Gd2
1 2 3i12iS1
iS3 RS3
RS2
vS2
iS2
i23
0
5 state variables:3 capacitor voltages2 inductor currents ( )( ) ( ) ( )d t t t
dt= − +
Formal matrix description :xM H + P x Pu
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
STL model composed of m Π sections in cascade
2m+1 state variables:m+1 capacitor voltagesm inductor currents
( )( ) ( ) ( )d t t tdt
= − +xM H + P x Pu
0 0
0 0
Model parameters:,,
d d
d d
L L l m R R l mC C l m G G l m
= == =
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
i-thwire
j-thwire
couplings
0 0
0 0
,,
d d
d d
l m l ml m l m
= =
= =
Matrix model parameters :L L R RC C G G
Generally, n(2m+1)state variables
( )( ) ( ) ( )d t t tdt
= − +xM H + P x Pu
( )( ) ( )C
L
tt t⎡ ⎤= ⎢ ⎥⎣ ⎦vx i
⎡ ⎤= ⎢ ⎥⎣ ⎦C 0M 0 L
m d= ⊗+1C I C
m d= ⊗L I L T⎡ ⎤⎢ ⎥⎣ ⎦
G EH =
-E Rm d= ⊗+1G I G
m d= ⊗R I R
State equations
where n(m+1) capacitor voltages
nm inductor currents
S⎡ ⎤⎢ ⎥⎣ ⎦Y 0P = 0 0
( )( ) S tt ⎡ ⎤⎢ ⎥⎣ ⎦vu = 0
Formed by internalmatrices of Thévenin
equivalents Rsk-1
Formed by internalvoltage vectors of
Thévenin equivalents
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
( )( ) ( ) ( )d t t tdt
= − +xM H + P x Pu
State equations
After Laplace transform
( ) ( )1( ) ( )s s s−= + + +0x H P M Mx Pu
where x(s) = Łx(t) , u(s) = Łu(t) and x0 = x(t)|t = 0
Considering only zero initial conditions, x0 = 0, we have
( ) 1( ) ( )s s s−= + +x H P M Pu
P ≡ P(s) for external networks, or M ≡ M(s), H ≡ H(s) for p.-u.-l. matrices
Generally, if necessary, frequency dependences can be incorporated by
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
Solution in the s-domain
Absolute sensitivity w. r. to γ
( ) 1( ) ( )s s s−= + +x H P M Pu
( ) ( )1( ) ( ) ( ) ( )s s s s s sγ γ γ γ
− ⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂= − + + + − −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦
x H M PH P M x u x
Let us consider a parameter γ as
• a distributed parametera component of any p.-u.-l. matrix C0, L0, G0, R0 (M or H influenced) the length l of the MTL (both M and H influenced)
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
( ) 1( ) ( )s s s sγ γ γ
− ⎛ ⎞∂ ∂ ∂= − + + +⎜ ⎟∂ ∂ ∂⎝ ⎠
x H MH P M x
MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, statestate--variablevariable approachapproach
EXPERIMENTAL ERROR ANALYSISEXPERIMENTAL ERROR ANALYSISSensitivity inSensitivity in perfectly matchedperfectly matched Thomson cableThomson cable
RS1=50Ω RL1=100Ω
RL2=50ΩRS2=100ΩvS1(t)
00.1 0.02
0.02 0.1 mΩ⎡ ⎤= ⎢ ⎥⎣ ⎦
R0494.6 63.363.3 494.6
nHm
⎡ ⎤= ⎢ ⎥⎣ ⎦L
062.8 4.9
4.9 62.8pFm
−⎡ ⎤= ⎢ ⎥−⎣ ⎦C 0
0.1 0.010.01 0.1
Sm
−⎡ ⎤= ⎢ ⎥−⎣ ⎦G
50 00 100S
⎡ ⎤= Ω⎢ ⎥⎣ ⎦R
100 00 50L
⎡ ⎤= Ω⎢ ⎥⎣ ⎦R
1( )( ) 0S
Sv tt ⎡ ⎤= ⎢ ⎥⎣ ⎦
v
0( ) 0L t ⎡ ⎤= ⎢ ⎥⎣ ⎦v
Théveninequivalents
MTL p.-u.-l. matrices
EXAMPLESEXAMPLESSensitivity in (2+1)Sensitivity in (2+1)--conductor TL systemconductor TL system
Linear hybrid multiconductor-transmission-line circuit
The modified nodal analysis can successfully be applied
SOLUTION OF COMPLEX SYSTEMSSOLUTION OF COMPLEX SYSTEMS
i1(1) i1(2)
section with lumped-parameter components
MTL1 MTL2 MTLP
i2(1) i2(2) iP(1) iP(2)
v1(1) v1
(2) v2(1) v2
(2) vP(1) vP
(2)
Fully TimeFully Time--Domain Simulation of Domain Simulation of Multiconductor Transmission Line SystemsMulticonductor Transmission Line Systems
page 2
Presentation schedule
IntroductionImplicit Wendroff formulaMTL boundary conditions incorporation
Simply terminated MTL (Thévenin equivalents)MTL within a lumped circuit (MNA formulation)General MTL systems (MNA, Euler method)
Experimental error analysisExamples of MTL simulationCPU time evaluationConclusion
page 3
Introduction
LUMPEDCIRCUIT
(L)
LUMPEDCIRCUIT
(R)
(n+1) - conductortransmission line
iL iR
vL vR
0 lx
0 0 0 0( , ) ( , ) ( , ) ( , )( ) ( , ) ( ) , ( ) ( , ) ( )t x t x t x t xx t x x x t x xx t x t
∂ ∂ ∂ ∂− = + − = +
∂ ∂ ∂ ∂v i i vR i L G v C
R0(x), L0(x), G0(x), C0(x) – nonuniform MTL’s n × n per-unit-length matrices v(t,x), i(t,x) – n × 1 column vectors of voltages and currents of n active wires
MTL telegraphic equations
Simply terminatedMTL
MTL within a lumped circuit
General MTL system
MTL1 MTLPMTL2
( )RPv(2)
Rv(1)Rv
(1)Li
(1)Ri
(2)Li
(2)Ri ( )
LPi
SECTION WITH LUMPED-PARAMETER ELEMENTS
( )RPi
(2)Lv ( )
LPv(1)
Lv
MTL
SECTION WITH LUMPED-PARAMETER ELEMENTS
vL vR
iL iR
page 4
Implicit Wendroff formula
Voltage and current vectors and their derivatives are replaced by
Equations expressed for (k+1)-th section and j-th time instance
with
where R0k = R0(ξk), L0k = L0(ξk), G0k = G0(ξk), C0k = C0(ξk), with ξk ∈ (xk, xk+1)
( )
1 1 1 11 1 1 1
, ,
1 11 1,
( , ) 1 ( , ) 1,2 2
( , ) 4
j j j j j j j jk k k k k k k k
j k j k
j j j jk k k kj k
t x t xt t t x x x
t x
− − − −+ + + +
− −+ +
⎛ ⎞ ⎛ ⎞− − − −∂ ∂= + = +⎜ ⎟ ⎜ ⎟∂ Δ Δ ∂ Δ Δ⎝ ⎠ ⎝ ⎠
=
u u u u u u u uu u
u u + u + u + u
1 1 1 11 1 1 1
j j j j j j j jk k vk k vk k k k vk k vk k
− − − −+ + + +− + + = − + + +v v A i A i v v B i B i
1 1 1 11 1 1 1
j j j j j j j jk k ik k ik k k k ik k ik k
− − − −+ + + +− + + = − + + +i i A v A v i i B v B v
( ) ( )( ) ( )
v 0 0 v 0 0
i 0 0 i 0 0
2 , 2
2 , 2k k k k k k
k k k k k k
t x t x
t x t x
= − + Δ Δ = − Δ Δ
= − + Δ Δ = − Δ Δ
A R L B R L
A G C B G C
page 5
Simply terminated MTL
Matrix recursive formulation
TT T T TT T T T T T1 2 1 1 2 1with , , , , , ,, ,+ +⎡ ⎤ ⎡ ⎤= =⎣ ⎦= ⎣⎤⎣ ⎦⎡ ⎦ … …j j j j j j
Kj jj j
Kj vx v v v i i i iv i
1−= +j j jAx Bx D
Equation internal structure (MTL divided on K = 3 sections)
Boundary conditions via generalized Thévenin equivalents
v1 v1 1 v1 v1
v2 v2 2 v2 v2
v3 v3 3
i1 i1 4
i2 i2 1
i3 i3 2
iL 3
iR 4
--
--
--
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
jI I 0 0 A A 0 0 v -I I 0 0 B B 0 00 I I 0 0 A A 0 v 0 -I I 0 0 B B 00 0 I I 0 0 A A v 0 0 -I
A A 0 0 I I 0 0 v0 A A 0 0 I I 0 i0 0 A A 0 0 I I iI 0 0 0 R 0 0 0 i0 0 0 I 0 0 0 -R i
11
2
3v3 v3
4i1 i1
1i2 i2
2i3 i3
3 iL
4 iR
−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⋅ +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
j jv 0v 0v 0I 0 0 B Bv 0B B 0 0 -I I 0 0i 00 B B 0 0 -I I 0i 00 0 B B 0 0 -I Ii v0 0 0 0 0 0 0 0i v0 0 0 0 0 0 0 0
L iL L iL+ =v R i v
LUMPEDCIRCUIT
(L)
LUMPEDCIRCUIT
(R)
(n+1) - conductortransmission line
iL iR
vL vR
0 lxR iR R iR- =v R i vBoundary conditions
page 6
Modified nodal analysis descriptionMTL
SECTION WITH LUMPED-PARAMETER ELEMENTS
vL vR
iL iR
NN L L R R N
( ) ( ) ( ) ( ) ( )+ + + =d t t t t t
dtvC Gv S i S i i
MTL boundary conditionsT T
L L N R R N( ) ( ) , ( ) ( )= =t t t tv S v v S v
Matrix recursive formulationTT T T
1 2 1
TT T T1 2 1
1L L R R N N N
TL
T
L N
T
TN
T TN
R R
with , , ,
, , ,
, ,
-
-
j j j jK
j j j jK
j j j j j
j j
j j j
j j
j+
+
−
⎡⎡ ⎤= ⎣ ⎤= ⎣ ⎦
⎡ ⎤= ⎣ ⎦+ + =
⎦
+
=
=
v v v v
i i i i
S i S i Hv Fv i
v S v 0
v S
v i v
v 0
x …
…
1−= +j j jAx Bx D
Wendroff method
MNA equations via implicit Euler method
= +Δ
=Δ
t
t
CH G
CF
MTL within a lumped circuit (1)
page 7
Equation internal structure (MTL divided on K = 3 sections)
v1 v1 L
v2 v2 2
v3 v3 3
i1 i1 R
i2 i2 L
i3 i3 2
3
R
L R N
- --
--
--
-- -
-
j⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
TLTR
I I 0 0 A A 0 0 0 v I0 I I 0 0 A A 0 0 v0 0 I I 0 0 A A 0 v
A A 0 0 I I 0 0 0 v0 A A 0 0 I I 0 0 i0 0 A A 0 0 I I 0 iI 0 0 0 0 0 0 0 S i0 0 0 I 0 0 0 0 S i0 0 0 0 S 0 0 S H v
1v1 v1 L
v2 v2 2
v3 v3 3
i1 i1 R
i2 i2 L
i3 i3 2
3
R
N N
--
--
-
-
j−⎡ ⎤ ⎡ ⎤ ⎡⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
I 0 0 B B 0 0 0 v 00 I I 0 0 B B 0 0 v 00 0 I I 0 0 B B 0 v 0
B B 0 0 I I 0 0 0 v 00 B B 0 0 I I 0 0 i 00 0 B B 0 0 I I 0 i 00 0 0 0 0 0 0 0 0 i 00 0 0 0 0 0 0 0 0 i 00 0 0 0 0 0 0 0 F v i
j⎤
⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Boundary conditions via MNA and implicit Euler methods
v v
i i
N2 c
r
, ,
±
±
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦
I A 0 I B 00A I 0 B I 0
A B DiI 0 S 0 0 0
0 S H 0 0 F
∓
∓1−= +j j jAx Bx D
MTL within a lumped circuit (2)
page 8
Sensitivity with respect to a parameter γ (j = 1,2,…)
( )
11 1
1 1with
j j jj j
j j j
γ γ γ γ γ
−− −
− −
⎛ ⎞∂ ∂ ∂ ∂ ∂= − + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
= +
x x A B DA B x x
x A Bx D
v v
i i
N2 c
r
, ,
±
±
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦
I A 0 I B 00A I 0 B I 0
A B DiI 0 S 0 0 0
0 S H 0 0 F
∓
∓
Sensitivity determination
page 9
MNA description
MTL boundary conditions( ) ( )T ( ) ( )TL L N R R N( ) ( ) , ( ) ( )k k k kt t t t= =v S v v S v
Matrix recursive formulation
( )
T( ) ( ) T ( ) T
( ) ( ) ( ) ( ) 1L L R R N N N
1
( ) ( )T ( )
T(1) T (2) T ( ) T T
( )TL L
N
N R R N
, with ,
- , -
, , k j k j k j
Pk k j k k j j j j
kk j k j k
j j j P j j
j k j
−
=
⎡ ⎤= ⎣ ⎦
+ + = +
= =
⎡ ⎤= ⎣ ⎦
∑
w v i
S i S i Hv Fv i
v S v 0 v S v
x w w w , v
0
…
1−= +j j jAx Bx D Wendroff method
MNA equations via implicit Euler method
MTL1 MTLPMTL2
( )RPv(2)
Rv(1)Rv
(1)Li
(1)Ri
(2)Li
(2)Ri ( )
LPi
SECTION WITH LUMPED-PARAMETER ELEMENTS
( )RPi
(2)Lv ( )
LPv(1)
Lv
General MTL systems (1)
( )( ) ( ) ( ) ( )NN L L R R N
1
( ) ( ) ( ) ( ) ( )P
k k k k
k
d t t t t tdt =
+ + + =∑vC Gv S i S i i
page 10
Matrix recursive formulation 1−= +j j jAx Bx D
General MTL systems (2)
( )
( ) ( ) ( )20 c( )0r
( )
( )
k
k k k
k
k
k
±⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
A 0A I S
S H
B 0B 0 0
0 F
∓
±
±
±
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
ww
wv
∓
∓
∓
(1) (1)
(2) (2)
(1)
( ) (2) ( )
(1) (1)20 c
(2) (2) ( )20 c
N( ) ( )20 c
(1) (2) ( )0r 0r 0r
j
P P
P
P P
P
A 0 0 0 B 0 0 00 A 0 0 0 B 0 0
0 0 A 0 0 0 B 0I 0 0 S 0 0 0 00 I 0 S
0 0 I SS S S H
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
0w0w
0wiv
1(1)
(2)
( )
NN
j j
P0 0 0 0
0 0 0 00 0 0 F
( )( )( )
(1) (2) ( )
(1) (2) ( )
(1) (2) ( )20 20 20 20
T(1)T (2)T ( )Tc c c c
(1) (2) ( )0r 0r 0r 0r
diag , , ,
diag , , ,
diag , , ,
, , ,
, , ,
P
P
P
P
P
± ± ± ±=
=
=
⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦
A A A A
B B B B
I I I I
S
S
∓ ∓ ∓ ∓
…
…
…
…
…
S S S
S S S
1
20 cN
0r
j j j−±⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⋅ = ⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
N N
A 0 B 0w w 0
I S 0 0v v i
S H 0 F
∓
Thomson transmission line (uniform)
( ) ( ) 2 ( , )
iL
iL
0 0 iL 0 0
( , ) 1( ) erfc ( ) ( , )
( , ) 1( ) erfc ( , ) ( , ) , where
( ) , ( , ) 2
a t a t b t xi t x t R e a t b t x
v t x t b t x R i t x
a t R t C R b t x x R C t
Known analytical solutions
page 11
Experimental error analysis
page 12
Uniform/Nonuniform MTLs: responses to external driving
Examples: Thévenin equivalents
iL1iL
iL2
iL1iL
00
0
RR
v
⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤= ⎢ ⎥⎣ ⎦
R
v
iR1iR
iR2
iR
00
00
RR
⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤= ⎢ ⎥⎣ ⎦
R
v
11 120 0
12 22
0 0 0 0 0
( )
, , ,
px pxP Px e e
P P⎡ ⎤
= = ⎢ ⎥⎣ ⎦
∈
P P
P R L G C
2 -9iL1v ( ) = sin ( /2 10 )t tπ ⋅
Voltage distributions and their sensitivities:
uniform vs. nonuniform MTL
page 13
Uniform/Nonuniform MTLs:
( )21
1
( ,0) = sin (4 / 3/2) , if 3 /8 < < 5 /8( ,0) = 0, otherwise
v x x l l x lv x
π −
nonzero initial condition external driving & nonlinear MTL
2 -9 -9iL1
iL1
( ) = sin ( /2 10 ) , if 0 2 10( ) = 0 , otherwise
v t t tv t
π ⋅ ≤ ≤ ⋅
2( ) = /(1+ / )i ii i pC v C v V
Examples: Thévenin equivalents
page 14
Nonuniform MTL, reactive terminations
Example: MNA + Euler method (1)
L1 L1
L1 L1
L2L R
R1 R1
R2
0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0, , , ,
0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
G GG G
CG C
G
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
= = = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
G S S C L1 R1N L R
L2 R2
L1
000
, , .00
i ii i
v
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎡ ⎤
= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥
⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
i i i
NN L L R R N
( ) ( ) ( ) ( ) ( )+ + + =d t t t t t
dtvC Gv S i S i i
0 lx
GL1
GR1
GR2CL2 vL1
iL1
iL2
iR1
iR2
1 2
3
4
5
iv
CR1
page 15
Voltage and current distributions
Example: MNA + Euler method (2)
Nodal voltage waveforms
page 16
Voltage distributions sensitivities
Example: MNA + Euler method (3)
Nodal voltage sensitivities
page 17
Example: General MTL system
MTL1
MTL2
MTL3
i1
i2
vin
R1
R9R2
R7
R4
R3
R8 R10
R6 R5
C1
C2
L
C3
vout
1 2
3
4
5
6
7
8
9
10 12
11 13
14
15
page 18
CPU times for PC 2GHz/2GB, sparse matrix notations
CPU times evaluation
RiR1
RiR10
viL1(t)
RiL2 RiR2
RiL10
RiL1