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TRANSMISSION LINE SYSTEMS (Soustavy s přenosovými vedeními) Lecture in terms of Ph.D. Study L. Brančík Dept. of Radio Electronics FEEC BUT in Brno

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Microsoft PowerPoint - Obvody s penosovými vedenímiLecture in terms of Ph.D. Study
L. Braník
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

i x t
u x 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 s s l
Z s I s s l
v
v
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
Y Z l Z l
Z l Z l

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
L Ll
m R
m d 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
2 4

.
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 pizpsobeného
bezeztrátového vedení s 50 lánky a zpodním 5
SPECTRAL FUNCTIONS MODULI
a) rectangular b) sin 2 ⋅ 3 ⋅ c) sin2 2 ⋅
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 a e
t F
t F
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.
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
s l
s l
( ) [ ( ) ( )]
( ) [ ( ) ( )]
( )
( )

The transmission line can therefore be replaced by a circuit, in which voltage sources 1 and 2 are dependent on voltages of opposed ends of the TL at the time instance − .
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
( , ) ( , )( , )
( , ) ( , )( , )
∂ ∂ − = +
∂ ∂ ∂ ∂
− = + ∂ ∂
− =
− =
0 0 0 0 0 0( ) , ( )Z s R sL Y s G sC= + = +
2 2
γ
γ
Characteristic equation:
2 2 1,2( ) 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 s dx
− =
γ γ−= − , 0
=
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 1 1(0, ) ( ) ( )v
I s I s K K Z s
= = − ( )( ) ( ) 2 2 1
v
I l s I s K e K e Z 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 l vU s I s Z s K eγ+ =
L.T.
[ ] [ ]
v
U s I s Z s I s Z s K
I s Z s U s I s Z
E s
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
γ
γ
γ

= +
= +
= +

+
= +
Practical realization of the method requires: a) replacement of the wave impedance
Z s R sL
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 s R
G H yv v k
k
L
k
e e e s
k
k
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
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 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
(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 ( )
s x s x v v
x s x s x s x v v
s x Z s s x e Z s e A s

(4)
where ( )inplZ s is an input impedance of the TL of the length l, terminated by a loading
impedance ( )iRZ s :
Fig. 2 Two-port model of uniform transmission line
2 ( )
L l iR l R
a s Z s a sU s s e Z s Z s

Z s Z s
(6)
is a reflection coefficient on the TL right side. Based on a cascade matrix xA ,
21 ( ) 22
I s I s

x R
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
( ) ( )
( ) ( ) ( )
Z s Z s
(9)
as a reflection coefficient on the TL left side, we can write
( ) ( )[2 ]
2 ( )
( )1 ( ) ( )
( ) ( ) 1 ( ) ( )
iL v L R

v R x iL s l
iL v L R
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 L x iL iRs l
L R iL v iR v
e s e e s e I s U s U s

1 ( ) ( ) ( ) ( ) ( ) ( )
s x s l x s l x s l x
v R L x iL iRs l
L R iL v iR v
Z s e s e e s e U s U s U s

2. Transmission line modeled by means of - 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
g s Z s g s

where a characteristic impedance
0 ( )Z s and an image transfer constant 0 ( )g s are
12 0
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
2 2 0 0 0 0 0 0 0
cosh ( ) ( )sinh ( ) ( ) 1 ( )[ ( ) 1]1 ( )
sinh ( ) ( ) cosh ( ) 2 ( ) [ ( ) 1] ( ) ( ) 1
k k
k k k k

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
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 ) ( ) ( )
R L k iL iRm
L R iL iR
G s s G s G s s G s I s U s U s

2
R L k iL iRm
L R iL iR
Z s G s s G s G s s G s U s U s U 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 ( )
Z s Y s Z s
A s Z s Y s Z s Y s
Y s
d
A s Z s Y s
Y s
(25)
A characteristic impedance (16) cant hen be expressed through TL’s secondary parameters
0 2
m
(27)
for -network, or T-network, respectively. The image transfer (19) is the same for both
2 2 2
2 2
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
MULTICONDUCTOR TRANSMISSION LINES (MTL)
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
scheme)
scheme)
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]
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] ,
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]
[(A21 + YiLA11)UiR – IiL] ,
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
Y E Y Z ,
/ 2
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
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) ,
.
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

e att ,
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
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
C
A discrete MTL model had m = 10 4 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:

ui(t)
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)
SOLUTION
- basic MTL equations

- the compact matrix form
dsssxssx
x
x
- under zero initial conditions it follows
),(W)(Φ),(W 00
= =
- multiport model of the simple MTL system - cascade matrix of the part of the line
a) homogenous line
s s
ssxx start
1 1 )(
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
iR12iL2211iL21R ++⋅+++= −
I(x)
V(x)
iR21iL1122iL12R +−⋅+++= −
iR22iL1221iL11R ++⋅+++= −
iR11iL2112iL22R +−⋅+++= −

=

),(A)(Φ),(A Rx1R 1 sxssx k
x k
k
k
( ) ( )∑ ∞
=

−+−==
t eatt π
∑∑ −
= +−
− −
=
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)
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
)2( Pi
MTL2 MTLP
)0()()()(][ 1
= . (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 T kkk 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

+

=

, (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
– the solution of (4)
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
ls ssx
l es ⋅
)(~)(~ 1
0 ),(
0 ses jjjj xxsx −⋅= Δ ΦΦ ζM with E=)(~ 0 0 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)
then for the k–th MTL is valid

+

=

into account
)()()( 1 122222 sss −−= ΦΦY ,
)()( 1 1212 ss −= ΦY ,
)()()()()( sssss kkkkk ΓXVYI −= (14)
– substituting (14) into (2) =>
Resultant MNA matrix equation
1 )()()0()()()( ΓXDvCIDYDCsGV . (15)
– to solve the voltage and current at a coordinate x from a beginning

+

=

– the column vector )()],(),,([ 0 )()( ssxsx xT
kk Γ=ΓΓ IV is expressed by the matrix convolution integral as
∫= x
0 )0,()()()( ξξξξ wNΦΓ . (17)
– the voltage and current vectors )()1( skV and )()1( skI are extracted
from equations
)()( ss M T kk 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
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
{ }m k
T nk
>< Ψ ⊗= 1Ψ , (20)
where – ]1111[2 =n1 is n21× row vector with all elements equal to 1,
– the symbol ⊗ means Kronecker tensor product of matrices.
( ) ( )( )
= ∑
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
, (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
k eC π2
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 ln 2
FFT and quotient-difference algorithm application – the formula (2) can be rewritten into the form
}]Re[2{~ 0
= , 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
n k
n n
n k
n nN
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
The quotient-difference algorithm diagram
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
– for Pr ,,2= ,
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
1 s
r s
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 4 0
0 2 4
0 5 1 0 0
0 . 2
0 . 4
1 0 - 1 0
0
0 1 0 2 0 3 0
1 0 - 1 0
0 5 1 0 1 5 0
0 . 5
1 0 - 1 0
1
2
f6
1 0 - 1 0
t
Scalar, vector and matrix NILT versions A. Laplace transform has a scalar form )(sF :
}])}({Re[2{ˆ 0 MM
P NMM FFT FFCf −+= VRM . (18)
B. Laplace transform has a vector form T
J J sFsFsFs )](),(),([)( 21=F :
}])}({Re[2{~

)()()(
)()()( )()()(
)(
><
×××××× −+= 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
m nH
0 0.2 0.4 0.6 0.8 1 x 10-8
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-5
0
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:
m nH
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
x 1 0 -8
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:
m nH
G

−=
8 5
8 3
≤≤ , 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
-5
-4
-3
-2
-1
0
1
2
3
4
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
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-5
-4
-3
-2
-1
0
1
2
3
4
LectureLecture OutlineOutline
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion ApproachApproach GettingGetting TimeTime--Domain SensitivityDomain Sensitivity ExamplesExamples && ConclusionsConclusions
linear hybrid multiconductor-transmission-line circuit
the circuit will be described by modified nodal analysis (MNA) method
IntroductionIntroduction & P& Problemroblem FFormulationormulation
The admittance equation of k–th MTL
( ) ( ) ( )k k ks s s=I Y V (3)
where (1) (2)( ) [ ( ), ( )]T k 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
k
+ + =∑vC G v D i i (1)
1
s s s =
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
s s =
Determined via - modal analysis technique - chain/admittance matrix conversion
The s–domain solution is prepared serving for the derivation 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 M M M M
s ss s γ γ γ
∂ ∂ ∂ + =
where there were considered
From (7) we can write
-1( ) ( )( ) (0) ( )M M M M M M
s ss s γ γ γ
∂ ∂ ∂ = − ∂ ∂ ∂
1 ( ) s ( )
P T
s s =
FrequencyFrequency--Domain SensitivityDomain Sensitivity
LumpedLumped--Parameter SensitivityParameter Sensitivity
s ss s s s γ γ γ
∂ ∂ ∂ = − − ∂ ∂ ∂
is a memory-element parameterMcγ ≡
M M
∂ ∂ = −
is a memoryless-element parameterMgγ ≡
M M
∂ ∂ = −
∂ ∂ 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 M M M M
s ss s γ γ γ
∂ ∂ ∂ = − ∂ ∂ ∂
( ) s ( ) P
k s s
ss s s γ γ
∂∂ = −
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
− = V Z Y V
− = I Y Z I (14)
0 ( , ) ( ) ( , )d x s s x s dx
− = V Z I 0
− = I Y V (13)
Elimination of variables in (13) leads to
Decoupling is done via treating Z0Y0 matrix. There are found: - eigenvalues λi
2
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
There are created matrices:
and another matrices computed as
1 0i v −=S Z S Λ
{ }1 1 coth( ) kn
i i diag lλ
diag lλ =
i v i v k
i v i v
Y Y S E S S E S
(17)
i v i v i v i v
k
γ γ γ γ γ γ γ
γ γ γ γ γ γ
− −
− −
∂ ∂ ∂ ∂∂ ∂ + − + − ∂ ∂ ∂ ∂ ∂ ∂∂ =
∂ ∂ ∂ ∂ ∂∂ ∂ + − + − ∂ ∂ ∂ ∂ ∂ ∂
S S S SE EE S Y S E S Y S Y
S S S SE EE S Y S E S Y S
1 0 0
−∂ ∂ ∂ ∂ = − ∂ ∂ ∂ ∂
S S ZZ S SΛ Λ +
1 2, ,γ γ γ∂ ∂ ∂ ∂ ∂ ∂E EΛ depend on the eigenvalues sensitivities
ModalModal AnalysisAnalysis TechniqueTechnique ApproachApproach
(20)
Consider basic equation
( ) ( )2 0 02
and appending equation 1T i i =x x 0T i
i γ ∂
2 0 0
(23)
(24)
∂ ∂ =
∂ ∂
The system (24) is solved repeatedly for 1,2, , ki n=
1
=
∂ ∂ − ∂ = + ∂ ∂ ∂
=
∂ ∂ ∂ = + ∂ ∂ ∂
0
0
= ⋅

(30)
11 12 (2) (1)
= ⋅ −
0
11 12 (2) (2)
= ⋅

(33)
( ) ( ) ( ) ( )
(35)
∂ ∂ =
∂ ∂ MΦ
(36)
1 1 1 112 11 12 12 12 11 12 12
1 1 1 112 12 11 12 12 12 12 11
k γ γ γ
γ γ γ γ
Φ Φ Φ Φ Φ Φ Φ Φ
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) T
s s s ss s s ss s s s sl s s s s
Τ ∂ = − = − ∂
(38)
∂ = =
12 0 12 11 0 12
11 0 12 12 0 12
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )
∂ = − ∂
Y Z Y Y Z YY Y Z Y Y Z Y (39)
lγ ≡MTL-Length Sensitivity:
Parameter γ is contained in M(s) matrix, while l is constant
( )( ) s ls e γ γ
∂ ∂ =
∂ ∂ MΦ
(40)
( )
To evaluate (41) effectively the recurrence formula can be used
(41)
k -1
starting with k = 2
For a computation the derivative is prepared as ( )s γ∂ ∂M
0 :ijRγ ≡ ∈R
0 :ijCγ ≡ ∈C
(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 m ij ij ij ijk k k k k
i j i ij ij ij ij
∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
where m denotes the order of the per-unit-length matrix.
GettingGetting TimeTime--Domain SensitivityDomain Sensitivity
Application of NILT method
∂ ∂ = ∂ ∂
vS v (49)
• The used NILT method is based on FFT and quotient-difference algorithm 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
Hybrid linear network with three MTLs
ExampleExample 1 1 –– MTL MTL parametersparameters
0
=
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
e s
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 x 10-8
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0 0.2 0.4 0.6 0.8 1 1.2 x 10-8
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
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 1.2 x 10-8
-6
-5
-4
-3
-2
-1
0
1
2
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
vin
MTL1
MTL2
MTL3
i1
i2
ExampleExample 2 2 –– MTLs MTLs structurestructure
0
4 r
ε ε π
= ≈ − +
11 22 0 1 12r C wC C K C h
ε ε = ≈ −
2 0
µ µ = ≈ −
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 1.2 x 10-8
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
m m ij ij ij ijk k k k k
i j i ij ij ij ij
∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
-1 ( )( ) ( ) ( )TkM M k k M
ss s s γ γ
∂∂ = −
Reminder of computation method:
[ ] [ ]-1 -1 2 1
2 1
−∂ ≈
= (34)
where a central difference was always chosen as 0.1% of the nominal value 1 2( ) 2γ γ γ= +
γ
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
((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 approach StateState--variable methodvariable method
Errors comparison & ExamplesErrors comparison & Examples ConclusionConclusion
Simple linear system consisting of uniform (n+1)-conductor TL
The aim voltage/current distributions along the MTL wires sensitivities with respect to MTL primary parameters, MTL length or lumped parameters of terminating networks
MTL telegrapher equations in (t,x)-domain
LINEAR NETWORK
iL iR
vL vR
0 0
0 0
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) t x t x t x t x t x t xx t
∂ ∂ − = + ∂ ∂ 0 R 0 Lv v v
G 0 C 0i i i
MTL chain matrix
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 xdx
= − =
Y 0I I I
s x s
s x e s x s x
= =
11 12( )
21 11
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
LINEAR NETWORK
IL IR
VL VR
( ) ( )
11 iR 21 iL iR
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
s s s s s
− = − + − × − −
Φ Z Φ V V
MTL full chain matrix
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS ((ss,x,x))--domain voltage/current sensitivitydomain voltage/current sensitivity
( ) ( )L L
L L
s x s ss xγ γ γ
∂ ∂ ∂ = + ∂ ∂ ∂
iL 11 iR 21
TiR 11 12 11 iR
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( )
s s ss s
ZΦ Z Φ Φ Z ΦΦ Z V I
Z Φ ΦΦ Z L( )s

( ) ( ) ( )( ) ( )s s ss s γ γ γ
∂ ∂ ∂ = − −
depends on γ
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS ((ss,x,x))--domain distribution and sensitivitydomain distribution and sensitivity
( ) 1 p
− −
∂ ∂ ∂ = + ∂ ∂ ∂
( ) ( )p ,s s x= ΔΦ ΦPartial chain matrix
1 , 1,2, ,k kx x x l m k m−Δ = − = = L
Advantage in case of uniform MTL the Φp(s) is evaluated only once an easy generalization for nonuniform MTLs
Taylor series expansion with scaling & squaring Augmented matrix utilization Eigenvalues decomposition Laplace transform approach with scaling & squaring Convolution integral evaluation Padé approximation with scaling & squaring
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS ((s,xs,x))--domain voltage/current sensitivitydomain voltage/current sensitivity
( )( ) s ls e γ γ
∂ ∂ =
11 0 21 00 11 0 12
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) s s s ss s s ss s s s sl s s s s
Τ ∂ = − = − ∂
( )( ) ( ) ( ) ( ) ( )s ls e s s s s l l
∂ ∂ = = =
( ) p
l l m m ∂ ∂
To evaluate a recursive formula we use
Substitution x = lz into the telegr. eq. and doing tz{.} operation
L
L
MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS ((ss,,qq))--domain distribution and sensitivitydomain distribution and sensitivity
{ } { } [ ] 1( )( , ) ( , ) ( )s lz z zs q s z e q s l −= = =MΨ Φ I - ML Lwhere
L L
L L
∂ ∂ ∂ = + ∂ ∂ ∂
An absolute sensitivity
∂ ∂ =
∂ ∂ Ψ MΨ Ψ ( , ) ( , ) ( ) ( , )s q s q s s q
l ∂
Boundary conditions
= =
=
=
m rs
Generalized T network
i-th wire
j-th wire
( ) ( ) ( ) 2( )
n k n
1 1( ) ( ) ( )( ) ( )
s s ss s −
Uniform MTL, Φk(s) = Φd(s) , for all k 1( ) ( ) ( )k k
d d ds s s−=% % %Φ Φ Φ
k = 2,3,...,m.
Φd(s) submatrices derivatives
d d d d d
s s ss s ∂ ∂ ∂
12 ( ) ( ) γ γ
d ds s∂ ∂ ∂ ∂
d d d d d d d 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 d d d
s s ss s ∂ ∂ ∂
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
d d
C v t G G v t G C v t G G v tdC v t G G v t
+ + − ⋅ = − + − ⋅ +
− −

RS2
vS2
iS2
i23
0
5 state variables: 3 capacitor voltages 2 inductor currents ( )( ) ( ) ( )d t t t
dt = − +
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 voltages m inductor currents
( )( ) ( ) ( )d t t t dt
= − + xM H + P x Pu
0 0
0 0
Model parameters: , ,
d d
d d
= = = =
i-th wire
j-th wire
= =
= =
Matrix model parameters : L L R R C C G G
Generally, n(2m+1) state variables
( )( ) ( ) ( )d t t t dt
= − + xM H + P x Pu
( )( ) ( ) C
L
m d= ⊗L I L T
G E H =
m d= ⊗R I R
State equations
nm inductor currents
( )( ) S tt vu = 0
equivalents Rsk -1
Thévenin equivalents
( )( ) ( ) ( )d t t t dt
= − + xM H + P x Pu
State equations
( ) ( )1( ) ( )s s s−= + + +0x H P M Mx Pu
where x(s) = {x(t)} , u(s) = {u(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
( ) 1( ) ( )s s s−= + +x H P M Pu
− ∂ ∂ ∂ ∂ = − + + + − − ∂ ∂ ∂ ∂
Let us consider a parameter γ as
• a distributed parameter a 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 s γ γ γ
− ∂ ∂ ∂ = − + + + ∂ ∂ ∂
EXPERIMENTAL ERROR ANALYSISEXPERIMENTAL ERROR ANALYSIS Sensitivity inSensitivity in perfectly matchedperfectly matched Thomson cableThomson cable
RS1=50 RL1=100
RL2=50RS2=100 vS1(t)
nH m
Linear hybrid multiconductor-transmission-line circuit
SOLUTION OF COMPLEX SYSTEMSSOLUTION OF COMPLEX SYSTEMS
i1(1) i1(2)
page 2
Presentation schedule
Introduction Implicit Wendroff formula MTL boundary conditions incorporation
Simply terminated MTL (Thévenin equivalents) MTL within a lumped circuit (MNA formulation) General MTL systems (MNA, Euler method)
Experimental error analysis Examples of MTL simulation CPU time evaluation Conclusion
page 3
iL iR
vL vR
0 lx
∂ ∂ ∂ ∂ − = + − = +
∂ ∂ ∂ ∂ 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 terminated MTL
General MTL system
vL vR
iL iR
page 4
Voltage and current vectors and their derivatives are replaced by
Equations expressed for (k+1)-th section and j-th time instance
with
( )
, ,
( , ) 4
j j j j j j j j k k k k k k k k
j k j k
t x t x t t t x x x
t x
u u + u + u + u
1 1 1 1 1 1 1 1
j j j j j j j j k 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 1 1 1 1 1
j j j j j j j j k k ik k ik k k k ik k ik k
( ) ( ) ( ) ( )
2 , 2
k k k k k k
t x t x
t x t x
= − + Δ Δ = − Δ Δ
= − + Δ Δ = − Δ Δ
page 5
Simply terminated MTL
Matrix recursive formulation
TT T T TT T T T T T 1 2 1 1 2 1with , , , , , ,, ,+ + = = = … …j j j j j j
K j jj j
K j 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
- -
- -
- -
⋅ =
jI I 0 0 A A 0 0 v -I I 0 0 B B 0 0 0 I I 0 0 A A 0 v 0 -I I 0 0 B B 0 0 0 I I 0 0 A A v 0 0 -I
A A 0 0 I I 0 0 v 0 A A 0 0 I I 0 i 0 0 A A 0 0 I I i I 0 0 0 R 0 0 0 i 0 0 0 I 0 0 0 -R i
1 1
− ⋅ +

j jv 0 v 0 v 0I 0 0 B B v 0B B 0 0 -I I 0 0 i 00 B B 0 0 -I I 0 i 00 0 B B 0 0 -I I i v0 0 0 0 0 0 0 0 i v0 0 0 0 0 0 0 0
L iL L iL+ =v R i v
LUMPED CIRCUIT
iL iR
vL vR
0 lx R iR R iR- =v R i vBoundary conditions
page 6
vL vR
iL iR
( ) ( ) ( ) ( ) ( )+ + + = d t t t t t
dt vC Gv S i S i i
MTL boundary conditions T T
L L N R R N( ) ( ) , ( ) ( )= =t t t tv S v v S v
Matrix recursive formulation TT T T
1 2 1
1 L L R R N N N
T L
j j
v S v 0
Wendroff method
= + Δ
= Δ
t
t
page 7
v1 v1 L
v2 v2 2
v3 v3 3
i1 i1 R
i2 i2 L
i3 i3 2
T L T R
I I 0 0 A A 0 0 0 v I 0 I I 0 0 A A 0 0 v 0 0 I I 0 0 A A 0 v
A A 0 0 I I 0 0 0 v 0 A A 0 0 I I 0 0 i 0 0 A A 0 0 I I 0 i I 0 0 0 0 0 0 0 S i 0 0 0 I 0 0 0 0 S i 0 0 0 0 S 0 0 S H v
1 v1 v1 L
j− ⋅ +
I 0 0 B B 0 0 0 v 0 0 I I 0 0 B B 0 0 v 0 0 0 I I 0 0 B B 0 v 0
B B 0 0 I I 0 0 0 v 0 0 B B 0 0 I I 0 0 i 0 0 0 B B 0 0 I I 0 i 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 F v i
j
v v
i i
N2 c
, ,
±
±
= = =

I A 0 I B 0 0A I 0 B I 0
A B D iI 0 S 0 0 0
0 S H 0 0 F

page 8
( )
j j j
− − −
− −
∂ ∂ ∂ ∂ ∂ = − + + ∂ ∂ ∂ ∂ ∂
= +
x A Bx D
, ,
±
±
= = =

I A 0 I B 0 0A I 0 B I 0
A B D iI 0 S 0 0 0
0 S H 0 0 F

Sensitivity determination
page 9
MNA description
MTL boundary conditions ( ) ( )T ( ) ( )T L L N R R N( ) ( ) , ( ) ( )k k k kt t t t= =v S v v S v
Matrix recursive formulation
1
( ) ( )T ( )
( )T L L
P k k j k k j j j j
k k j k j k
j j j P j j
j k j
x w w w , v
0
MNA equations via implicit Euler method
MTL1 MTLPMTL2
( ) R Pv(2)
Rv(1) Rv
(1) Li
(1) Ri
(2) Li
(2) Ri ( )
L Pi
1
( ) ( ) ( ) ( ) ( ) P
page 10
General MTL systems (2)
S H
0 F
j
P
A 0 0 0 B 0 0 0 0 A 0 0 0 B 0 0
0 0 A 0 0 0 B 0 I 0 0 S 0 0 0 0 0 I 0 S
0 0 I S S S S H

⋅ +

( ) ( ) ( )
T(1)T (2)T ( )T c c c c
(1) (2) ( ) 0r 0r 0r 0r
diag , , ,
diag , , ,
diag , , ,
S H 0 F
( , ) 1( ) erfc ( ) ( , )
( ) , ( , ) 2
a t a t b t x i t x t R e a t b t x
v t x t b t x R i t x

Examples: Thévenin equivalents
( )
, , ,
P P
2 -9 iL1v ( ) = sin ( /2 10 )t tπ ⋅
Voltage distributions and their sensitivities:
uniform vs. nonuniform MTL
1
( ,0) = sin (4 / 3/2) , if 3 /8 < < 5 /8 ( ,0) = 0, otherwise
v x x l l x l v x
π −
2 -9 -9 iL1
v t t t v t
π ⋅ ≤ ≤ ⋅
2( ) = /(1+ / )i ii i pC v C v V
Examples: Thévenin equivalents
L1 L1
L1 L1
R2
0 0 0 1 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 0 0 1 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 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
G G G G
L2 R2
( ) ( ) ( ) ( ) ( )+ + + = d t t t t t
dt vC Gv S i S i i
0 lx
Nodal voltage waveforms
Nodal voltage sensitivities
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