TRANSMISSION LINE SYSTEMS - vutbr.cz

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

xRi x t L

xGu x t C

( , )( , )

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

( ) ( ) cosh ( ) [ ( )] ( ) sinh ( )

( ) ( )sinh ( )

( )[ ( )]cosh ( )

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

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

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

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

( ) ( , ) ( ) Re

f t f t ae

( ) ( , ) ( ) Im

In case of

f t f t a f t a f t aa s c( ) ( , ) ( , ) ( , ) 1

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

( ) ( ) ( ) ( )

E s U s E s e

( ) [ ( ) ( )]

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

( , ) ( , )( , )

u x t i x tR i x t Lx t

i x t u x tG u x t Cx t

∂ ∂− = +

∂ ∂∂ ∂

− = +∂ ∂

( , ) ( ) ( , )

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= + = +

( , ) ( ) ( , )

dU x s s U x sdx

dI x s s I x sdx

= , 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

I x s K e K eZ s

γ γ−= − , 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

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+ =

R E M A R K S

[ ][ ]

1 1( )

( ) ( ) ( )( ) ( ) 2

( ) ( ) ( ) ( ) ( )

( ) ( )

2 ( ) ( ) s l

U s I s Z sI s Z s K

I s Z s U s I s Z

U s E s

s s eI Z

= += +

[ ][ ]

( )2 2

( )2 1 1

( ) ( ) ( )

( ) ( ) 2

( ) ( ) ( ) ( ) ( )

2 ( )( ) ( )

s lv v

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

U s E s e

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

kn( ) ( ) exp ( )

Z s Z ss

CH zv v k

kn( ) ( ) exp ( )

and an exponential propagation constant into the form

e e es

e e H zs l sk

( ) exp ( )0 01

where the exponential functions have been approximately replaced by rational functions Hn (in a sence of so-called Padé approximation).

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:

R sLZ s

(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

We can write ( )

( )( ) ( )

iL inpl

U sI s

Z s Z s

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

( ) ( ) ( )( ) 1 ( )( ) ( )

( ) ( ) ( ) ( ) 1 ( )

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

and ( ) ( )

( )( ) ( )

Z s Z ss

Z s Z s

is a reflection coefficient on the TL right side. Based on a cascade matrix xA ,

21 ( ) 22

( )( )

( ) ( ) ( )

x inp l x x

I sI s

a s Z s a s

where 2 ( )[ ]

( ) 2 ( )[ ]

( ) 1 ( )( ) ( )

( ) 1 ( )

x Rinp l x v s l x

U s s eZ s Z s

I s s e

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 ss

Z s Z s

as a reflection coefficient on the TL left side, we can write

( ) ( )[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

and ( ) ( )[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 ] ( )[ ] ( )[ ]

( ) ( )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

( ) ( )[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

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

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

( )( )

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

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

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

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

( ) ( ) ( ) ( ) ( ) ( )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

(2 ) ( ) ( )

0 0 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

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 ( )

( ) ( ) ( ) ( )( ) 1 1

d d d dd

Z s Y sZ s

A sZ s Y s Z s Y s

( ) ( ) ( ) ( )1 1

2 4( )

( ) ( )( ) 1

d d d dd

Z s Y s Z s Y sZ

A sZ s Y s

A characteristic impedance (16) cant hen be expressed through TL’s secondary parameters

( )( )

1 ( )1

vZ sZ s

(26), or

1 ( )( ) ( ) 1

s lZ s Z s

for -network, or T-network, respectively. The image transfer (19) is the same for both

1 ( ) 1 ( )( ) 1 1 1

s l s lG s

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

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

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

/ 2 [ / 4]

E Z Y Z E Y ZA

Y E Y Z ,

p p p p p

[ / 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

E = A11pYp–1

(Zp/2 + Yp–1

=> A11p = [Yp–1

(Zp/2 + Yp–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

1FIm1,f )(f

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

nknnkV

difsum ,2,1,2)1( 1

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:

0.1 0.02 Ω

0.02 0.1 m

494.6 63.3 nH

63.3 494.6 m

0.1 0.01 S

0.01 0.1 m

62.8 4.9 pF

4.9 62.8 m

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:

))106exp(1())105.1exp(1(

104)(I 2

0)(IiR s

9iL1001.00

002.0)(Y ,

iR1002.010

10)01.05(110)(Y

Ri1=50

R1=5 R2=50 Ri2=100

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

- after Laplace transformation w.r. to t

0),(Y-

where )(L)(R),(Z 00 xsxsx , )(C)(G),(Y 00 xsxsx

- the compact matrix form

)0,(W),(N),(W),(M),(W xsxsxsxsxdx

- the solution of the last equation

dsssxssx

xx )0,(W),(N)(Φ),(W)(Φ),(W

where )(Φ0x sx is so-called integral matrix.

- under zero initial conditions it follows

),(W)(Φ),(W 00

sxssx xx

LINEAR NETWORK

(n+1)-conductor transmission

- for homogenous line

][)(M)(M),(Mx

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

ssxx start

b) non-homogenous line

− ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⎥

⎤⎢⎣

kxxendstart xx

sξsξ

ssxx start

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

I(0) = IL I(l) = -IR

V(0) =VL V(l) =VR AL(x) AR(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

22R21R

12R11R

sxsxsxsx

- computation of the cascade matrix AR(x,s)

),(A)(Φ),(A Rx1R1 sxssx k

−=− , where 1,2,...,1, −= mmk and lxm = , 00 =x . - numerical inversion of Laplace transforms F(s)

( ) ( )∑∞

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+−==

1 21FIm1,f )(f

teatt π

Euler transformation of difn terms helps to speed up the convergence

∑∑−

−−

sumdif

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

MM iiDvGvC ∑

==++ , (1)

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(2i )2(

2i )1(Pi

a section with lumped-parameter components

MTL2 MTLP

– a frequency–domain representation of the MNA matrix equation

)0()()()(][1

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,(

),(),(

0Y-)(Z-0

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

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

l es ⋅

)(),()( M

MM0Φ . (7)

– a nonuniform MTL => an approximate solution e.g. as

)(~)(~ 1

0 ses jjjj xxsx −⋅= Δ ΦΦ ζM with E=)(~ 00 sΦ . (8)

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)

ll sssds )](),([)()0,()()( )()(

ΓΓ∫ == IVwN ΓΦ ξξξξ , (11)

then for the k–th MTL is valid

⎥⎦

⎤⎢⎣

⎡+⎥

⎤⎢⎣

⎡⋅⎥

⎤⎢⎣

⎡=⎥

⎤⎢⎣

)()()()(

1211)2(

ΦΦΦΦ

admittance equations taking MTL’s nonzero initial conditions

into account

⎥⎦

⎤⎢⎣

⎡⋅⎥

⎤⎢⎣

⎡−⎥

⎤⎢⎣

⎡⋅⎥

⎤⎢⎣

⎡=⎥

⎤⎢⎣

)()()()(

1211)2(

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

⎥⎦⎤

⎢⎣⎡ ++⎥⎦

⎤⎢⎣⎡ ++= ∑∑

kkkkMMM

kkkMMM sssss

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

⎥⎦

⎤⎢⎣

⎡+⎥

⎤⎢⎣

⎡⋅⎥

⎤⎢⎣

⎡=⎥

⎤⎢⎣

),(),(

),(),(),(),(

),(),(

sxsxsxsx

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

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

><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

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

∫∞+

∞−=

stdsesFj

tf )(21)(π

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

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

−≈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

0FzFzFCf

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

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

nknk zGPzu

0),( , k∀ , (8)

the continued fraction is constructed as

))1(1(),( 210 kPkk zdzddPzv +++= , k∀ . (9)

The quotient-difference algorithm diagram

– 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)

i GGq 1)(

1 += , 12,,0 −= Pi , (11) and then successive columns are given by the rules – for Pr ,,1= ,

)()1()( +−

+ +−= ir

ir eqqe , rPi 22,,0 −= , (12)

– for Pr ,,2= ,

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(

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

O R I G I N A L S

1 0- 1 0

E R R O R S

0 5 1 00

0 2 4 6 8

1 0- 1 0

0 2 4- 1

t0 2 4

1 0- 1 0

0 1 0 2 0 3 0- 0 . 5

O R I G I N A L S

0 1 0 2 0 3 0

1 0- 1 0

E R R O R S

0 5 1 0 1 50

0 5 1 0 1 5

1 0- 1 0

0 1 2 30

t0 1 2 3

1 0- 1 0

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~

NJMJMJMJ FFT ×××

××× −+= FFCf VR . (19) C. Laplace transform has a matrix form

⎥⎥⎥⎥

⎢⎢⎢⎢

)()()(

)()()()()()(

sFsFsF

sFsFsFsFsFsF

])(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

⎥⎦

⎤⎢⎣

6.4943.633.636.494

L , mpF

⎥⎦

⎤⎢⎣

⎡−

8.629.49.48.62

C , mΩ

⎥⎦

⎤⎢⎣

75151575

R , mS

⎥⎦

⎤⎢⎣

⎡−

1.001.001.01.0

Application of the vector NILT version:

0 0.2 0.4 0.6 0.8 1x 10-8

Time (Seconds)

0 0.2 0.4 0.6 0.8 1x 10-8

5x 10-4

Time (Seconds)

50Ω 75Ω

100Ω 100Ω

100Ω 50Ω

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:

⎥⎥⎥⎥

⎢⎢⎢⎢

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

⎥⎥⎥⎥

⎢⎢⎢⎢

50101010501011105010011050

R , mS

⎥⎥⎥⎥

⎢⎢⎢⎢

−−−−−−−−

−−

1.001.0001.0001.01.001.0001.0

001.001.01.001.00001.001.01.0

75Ω 100Ω

100Ω 100Ω

2pF 1pF

– 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

T ime (S econds)

0 0 .5 1 1 .5 2

x 1 0-8

-0 .05

T ime (S econds)

– 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:

⎥⎦

⎤⎢⎣

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

– a nonzero initial voltage distribution of the 1st wire of the MTL1

⎟⎠

⎞⎜⎝

⎛⎥⎦⎤

⎢⎣⎡ −=

234sin)0,( 2

1 lxxv π if lxl

≤≤ , 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

v 1 [V]

Node voltage v1 waveform

0 0.2 0.4 0.6 0.8 1 1.2x 10-9

5x 10-3

i 3 [A]

Current i3 waveform

10Ω 10Ω

10pF 10pF

1nH 1nH

– 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

v 7 [V]

0 0.2 0.4 0.6 0.8 1 1.2x 10-9

5x 10-3

v 6 [V]

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

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

( ) ( ) ( ) ( )P

MM M M k k M

d t t t tdt =

+ + =∑vC G v D i i (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)

1( ) s ( )

M M M k k kk

= + + ∑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 ( )

M M M k k kk

= + + ∑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

s s s sc c

∂ ∂= −

∂ ∂V CY v V (10)

is a memoryless-element parameterMgγ ≡

(11)-1( ) ( )( ) ( )M MM 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

= + + ∑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

( , ) ( ) ( ) ( , )d x s s s x sdx

− =V Z Y V

( , ) ( ) ( ) ( , )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

- associated eigenvectors xi , i=1,2,....,nk

There are created matrices:

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

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

=> MTL admittance matrix

1 11 21 11 2 12

1 12 12 12 1 11

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

where (16) is used to get

i vv iγ γ γ γ

−∂ ⎛ ∂ ∂ ⎞∂= −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

S S ZZ S SΛΛ +

1 2, ,γ γ γ∂ ∂ ∂ ∂ ∂ ∂E EΛ depend on the eigenvalues sensitivities

depends on the eigenvectors sensitivities

v γ∂ ∂S

( )20 0i iλ − =I Z Y x 0

Consider basic equation

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

ii iTii

γ λγ

−∂⎡ ⎤

⎡ ⎤∂⎢ ⎥∂ ⎡ ⎤− ⎢ ⎥⎢ ⎥ = ∂⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ⎣ ⎦ ⎢ ⎥⎣ ⎦⎢ ⎥∂⎣ ⎦

xI Z Y xx

Resultant systemnk+1 equations:

λ λγ λ γ

∂ ∂=

∂ ∂

The system (24) is solved repeatedly for 1,2, , ki n=

diag λγ γ =

⎧∂ ⎫∂= ⎨ ⎬∂ ∂⎩ ⎭

1sinh ( )

ldiag ll

λ λγ λ γ γ

⎧ ⎫⎛ ∂ ⎞∂ − ∂= +⎨ ⎬⎜ ⎟∂ ∂ ∂⎝ ⎠⎩ ⎭

cosh( )sinh ( )

l ldiag ll

λ λ λγ λ γ γ

⎧ ⎫⎛ ∂ ⎞∂ ∂= +⎨ ⎬⎜ ⎟∂ ∂ ∂⎝ ⎠⎩ ⎭

⇒ 1 2, , , knv

γ γ γ γ∂⎡ ⎤∂ ∂ ∂

= ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

xS x x

where (29)

ChainChain//AdmittanceAdmittance MatrixMatrix ConversionConversion

MTL equations (13) in the matrix form

- ( )( , ) ( , )- ( )( , ) ( , )

sx s x sdsx s x sdx

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⋅⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦

0 ZV VY 0I I

with the solution(2) (1)

11 12(2) (1)

( ) ( )( ) ( )( ) ( )( ) ( )T

s ss ss ss s

⎡ ⎤ ⎡ ⎤⎡ ⎤= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦

V VI I

Φ ΦΦ Φ

The chain matrix ( )( ) s ls e= MΦ (32)0

- ( )( )

- ( )s

⎡ ⎤= ⎢ ⎥

⎣ ⎦

Rearranging eq. (31)(1) (1)

11 12(2) (2)

( ) ( )( ) ( )( ) ( )( ) ( )s ss ss ss s

⎡ ⎤ ⎡ ⎤⎡ ⎤= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦

Y YI VY YI V

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

( )( ) s ls eγ γ

∂ ∂=

∂ ∂MΦ

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

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

( ) ( ) ( ) ( ) ( )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:

Parameter γ is contained in M(s) matrix, while l is constant

( )( ) s ls eγ γ

∂ ∂=

∂ ∂MΦ

First, we need to find the derivative with respect to M(s)

ks l k

= ∑M M0

(s) ( )!

l skγ γ

∂ ∂=

∂ ∂∑ MΦ⇒

To evaluate (41) effectively the recurrence formula can be used

(s) (s)(s) (s) (s)(s)+ (s)k -1k k-1

γ γ γ γ

⎡ ⎤∂∂ ∂ ∂⎣ ⎦= =∂ ∂ ∂ ∂

M MM M MM M (42)

starting with k = 2

MTL-Primary-Parameter Sensitivity

For a computation the derivative is prepared as ( )s γ∂ ∂M

0 :ijRγ ≡ ∈R

0( )ij

∂⎡ ⎤−∂ ⎢ ⎥∂= ⎢ ⎥∂⎢ ⎥⎣ ⎦

0 :ijLγ ≡ ∈L

0( )ij

∂⎡ ⎤−∂ ⎢ ⎥∂= ⎢ ⎥∂⎢ ⎥⎣ ⎦

0 :ijGγ ≡ ∈G

⎡ ⎤∂ ⎢ ⎥∂= ⎢ ⎥−∂

∂⎢ ⎥⎣ ⎦

0 0M G 0

0 :ijCγ ≡ ∈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

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

R1=50Ω

50Ω75Ω

100Ω 100Ω

100Ω50Ω

C2=2pF

ExampleExample 1 1 –– circuit in circuit in viewview

Hybrid linear network with three MTLs

ExampleExample 1 1 –– MTL MTL parametersparameters

75 1515 75 m

⎡ ⎤ Ω= ⎢ ⎥

⎣ ⎦R 0

494.6 63.363.3 494.6

⎡ ⎤= ⎢ ⎥

⎣ ⎦L

0.1 0.010.01 0.1

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

62.8 4.94.9 62.8

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

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

1( )1.5

ine e eV s

− − −− ⋅ − ⋅ − ⋅

− − +=

ExampleExample 1 1 -- resultsresults

Input and output voltages

0 0.2 0.4 0.6 0.8 1 1.2x 10-8

Time (seconds)

vinvout

Semirel. sensitivity vout w. r. to R1

0 0.2 0.4 0.6 0.8 1 1.2x 10-8

Time (seconds)Se

1 (vol

Sem. sensitivity vout w. r. to C2

0 0.2 0.4 0.6 0.8 1 1.2x 10-8

Time (seconds)

Sem. sens. vout w. r. to l of MTL2

0 0.2 0.4 0.6 0.8 1 1.2x 10-8

Time (seconds)S

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

3x 10-3

Time (seconds)

0 0.2 0.4 0.6 0.8 1 1.2x 10-8

Time (seconds)S

ExampleExample 2 2 –– circuit in circuit in viewview

R1=50Ω

50Ω75Ω

100Ω 100Ω

100Ω50Ω

C2=2pF

vcross

Hybrid linear network with three lossless MTLs

ExampleExample 2 2 –– MTLs MTLs structurestructure

493.11 63.0463.04 493.11

⎡ ⎤= ⎢ ⎥

⎣ ⎦L 0

69.62 7.097.09 69.62

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

The MTLs structure

Ground plane

Laminate, εr

Flat conductors

12 21 1 12ln 1

L Cw hC C K Kh d

ε επ

⎡ ⎤⎛ ⎞ ⎛ ⎞= ≈ − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

11 22 0 1 12r CwC C K Ch

ε ε ⎛ ⎞= ≈ −⎜ ⎟⎝ ⎠

12 212ln 1

4r hL L

dµ µ

π⎡ ⎤⎛ ⎞= ≈ +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

011 22 12

hL L LK w

µ µ ⎛ ⎞= ≈ −⎜ ⎟⎝ ⎠

10( 1)

120 ,r

⎛ ⎞= ⎜ ⎟⎝ ⎠

( )1 1

r effC L

0( 1)860ln

h wZw hε =

⎛ ⎞≈ +⎜ ⎟⎝ ⎠

0.58w mm=1.17h mm= 2.49d mm=

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

-0.005

Time (seconds)

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

( , ) ( , )( ) ( ) 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= +

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

(n+1) - conductortransmission line

PROBLEM MOTIVATIONPROBLEM MOTIVATION

( , ) ( , ) ( , )( , ) ( , ) ( , )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

( )( , ) ( , ) ( , )( )

( )( , ) ( , ) ( , )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⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

I IThe solution

Given by boundary conditions

( ) 11 12( )T

( , ) ( , ),

( , ) ( , )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( )

( ) ( )( )

( ) ( )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 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

( ) ( )( , ) ,,

( ) ( )( , )s ss x s x

s xs ss xγ γ γ

∂ ⎡ ⎤ ⎡ ⎤⎡ ⎤∂ ∂= +⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

V VV ΦΦ

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

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

( ) ( )( ) ( )

s s−

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

( ) ( )p 1 1p

( ) ( ) ( )( ) ( ) ( )

ss s ss

s s sγ γ γ− −

− −

∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

ΦV V VΦ

( ) ( )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

γ l≡MTL-length sensitivity:

MTL CONTINUOUS MODELSMTL CONTINUOUS MODELS((s,xs,x))--domain voltage/current sensitivitydomain voltage/current sensitivity

( ) ( ) ( )( ) ( )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

( )( , )( , )

( )( , )ss q

s qss q

⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

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

( ) ( )( , ) ( , ) ( , )( ) ( )( , )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

Ψ Ψ M Ψ

( )sγ ∈M lγ ≡

Łtz-1. always on z∈<0,1>

Boundary conditions

Generalized Π network

l ml ml ml m

Model paraL LR RC CG

MTL MTL SEMIDISCRETESEMIDISCRETE MODELSMODELS((ss,,xxkk))--domain solution, domain solution, chain matrixchain matrix approachapproach

Generalized T network

i-thwire

j-thwire

couplings

( ) ( ) ( )2( )

( ) ( ) ( ) ( )( )4 2

k kn k

kk k k k

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.

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

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

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

+⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ = − + − ⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

( )( )( )

v tG v tG v t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

vS3Cd2

Rd RdLd Ld

vS1 v1v2 v3

1 2 3i12iS1

iS3 RS3

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

Model parameters:,,

L L l m R R l mC C l m G G l m

= == =

i-thwire

j-thwire

couplings

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

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

( )( ) ( ) ( )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

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)

( ) 1( ) ( )s s s sγ γ γ

− ⎛ ⎞∂ ∂ ∂= − + + +⎜ ⎟∂ ∂ ∂⎝ ⎠

x H MH P M x

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

⎡ ⎤= ⎢ ⎥⎣ ⎦L

062.8 4.9

4.9 62.8pFm

−⎡ ⎤= ⎢ ⎥−⎣ ⎦C 0

0.1 0.010.01 0.1

−⎡ ⎤= ⎢ ⎥−⎣ ⎦G

50 00 100S

⎡ ⎤= Ω⎢ ⎥⎣ ⎦R

100 00 50L

⎡ ⎤= Ω⎢ ⎥⎣ ⎦R

1( )( ) 0S

Sv tt ⎡ ⎤= ⎢ ⎥⎣ ⎦

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

Fully TimeFully Time--Domain Simulation of Domain Simulation of Multiconductor Transmission Line SystemsMulticonductor Transmission Line Systems

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

Introduction

LUMPEDCIRCUIT

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

(2)Ri ( )

SECTION WITH LUMPED-PARAMETER ELEMENTS

( )RPi

(2)Lv ( )

LPv(1)

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

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

− − − −+ + + +

− −+ +

⎛ ⎞ ⎛ ⎞− − − −∂ ∂= + = +⎜ ⎟ ⎜ ⎟∂ 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 , 2k k k k k k

k k k k k k

t x t x

= − + Δ Δ = − Δ Δ

A R L B R L

A G C B G C

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

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

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

3v3 v3

4i1 i1

1i2 i2

2i3 i3

−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⋅ +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

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

0 lxR iR R iR- =v R i vBoundary conditions

Modified nodal analysis descriptionMTL

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

TT T T1 2 1

1L L R R N N N

with , , ,

j j j jK

j j j j j

⎡⎡ ⎤= ⎣ ⎤= ⎣ ⎦

⎡ ⎤= ⎣ ⎦+ + =

v v v v

i i i i

S i S i Hv Fv i

v S v 0

1−= +j j jAx Bx D

Wendroff method

MNA equations via implicit Euler method

MTL within a lumped circuit (1)

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

j⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

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

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

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Boundary conditions via MNA and implicit Euler methods

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

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)

Sensitivity with respect to a parameter γ (j = 1,2,…)

1 1with

j j jj j

γ γ γ γ γ

−− −

− −

⎛ ⎞∂ ∂ ∂ ∂ ∂= − + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

x x A B DA B x x

x A Bx D

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

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

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

( ) ( )T ( )

T(1) T (2) T ( ) T T

( )TL L

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

⎡ ⎤= ⎣ ⎦

+ + = +

⎡ ⎤= ⎣ ⎦

S i S i Hv Fv i

v S v 0 v S v

x w w w , v

1−= +j j jAx Bx D Wendroff method

MNA equations via implicit Euler method

MTL1 MTLPMTL2

( )RPv(2)

Rv(1)Rv

(2)Ri ( )

( )RPi

(2)Lv ( )

LPv(1)

General MTL systems (1)

( )( ) ( ) ( ) ( )NN L L R R N

( ) ( ) ( ) ( ) ( )P

k k k k

d t t t t tdt =

+ + + =∑vC Gv S i S i i

Matrix recursive formulation 1−= +j j jAx Bx D

General MTL systems (2)

( ) ( ) ( )20 c( )0r

±⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

A 0A I S

B 0B 0 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(1) (1)

(2) (2)

( ) (2) ( )

(1) (1)20 c

(2) (2) ( )20 c

N( ) ( )20 c

(1) (2) ( )0r 0r 0r

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

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

P0 0 0 0

0 0 0 00 0 0 F

( )( )( )

(1) (2) ( )

(1) (2) ( )20 20 20 20

T(1)T (2)T ( )Tc c c c

(1) (2) ( )0r 0r 0r 0r

diag , , ,

± ± ± ±=

⎡ ⎤= ⎣ ⎦⎡ ⎤= ⎣ ⎦

A A A A

B B B B

I I I I

∓ ∓ ∓ ∓

j j j−±⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⋅ = ⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

A 0 B 0w w 0

I S 0 0v v i

S H 0 F

Thomson transmission line (uniform)

( ) ( ) 2 ( , )

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

Experimental error analysis

Uniform/Nonuniform MTLs: responses to external driving

Examples: Thévenin equivalents

⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤= ⎢ ⎥⎣ ⎦

11 120 0

0 0 0 0 0

px pxP Px e e

P P⎡ ⎤

= = ⎢ ⎥⎣ ⎦

P R L G C

2 -9iL1v ( ) = sin ( /2 10 )t tπ ⋅

Voltage distributions and their sensitivities:

uniform vs. nonuniform MTL

Uniform/Nonuniform MTLs:

( ,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

( ) = 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

Nonuniform MTL, reactive terminations

Example: MNA + Euler method (1)

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

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= = = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

G S S C L1 R1N L R

, , .00

i ii i

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤ ⎡ ⎤

= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

NN L L R R N

( ) ( ) ( ) ( ) ( )+ + + =d t t t t t

dtvC Gv S i S i i

GR2CL2 vL1

Voltage and current distributions

Nodal voltage waveforms

Voltage distributions sensitivities

Nodal voltage sensitivities

Example: General MTL system

R8 R10

CPU times for PC 2GHz/2GB, sparse matrix notations

CPU times evaluation

viL1(t)

RiL2 RiR2

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