DIRECTRIZ PARA EL USO DE LA PROTESIS PARCIAL REMOVIBLE

where x-, is the zth column o f matrix X defined as

* = [*, * 2 jc J = B Q . (4.18)

Since £ and g are diagonal, the admittance matrix YB defined in (4.13) may be decomposed as:

r . ^ + C + E i , (4. 19)

i = l

The matrices Yb and YBB are defined as

Yb — A] and YBB — A2 (4.20)

where A i and A 2 are the component matrices o f A defined in Appendix B.

The block-diagram o f this new resonant model may be shown as in Fig. 4.3. For an m-line interconnect network, each component matrix in (4.19) has the dimension

2m x2m and the number o f terms in the summation is n=m x _{(K -l) . The new model thus}

represents the interconnect network, relative to its boundary terminals, by a set o f

2+ m x(K -l) multiterminal admittances, connected in parallel.

* 0

front-end transformation core distribution o f model matrices o f model matrix

Fig. 4.3. Resonant model o f a transmission line

The main advantage o f this new structure is that it has potential for straightforward conversion into the time domain.

4.1.4. The time-domain conversion

To convert the frequency-domain resonant model described by (4.13) and (4.14) into a time-domain counterpart, the next step is to obtain ^-dom ain representations for

each o f the constituent elements o f the model. Usually the transformation and distribution matrices P and Q are real and time and frequency independent and there is no need to approximate them. The coefficients o f the approximating functions are obtained using auto-regressive moving average (ARMA) modelling [NNA96], [NNA97], where frequency-dependant elements are approximated with ^-dom ain rational functions as described in the following section.

4.1.4.1. A R M A m odelling

In order to perform the conversion to the time domain, the various frequency- dependent elements o f the frequency-domain resonant model are approximated with Z - domain transfer functions defined as:

¿nr, , \ an + a .z'1 + ... + a z~m anz ” + a lzn'1 + ... + a z n'm

= — --- -— = —--- --- ;--- --- > (4 -21)

m,nK l + b,z■I +... + bnz-" z n + b ,zn-‘ +... + bn

Typically APPm/n is a low-order ^-tran sfer function and choosing m and n less than 3 will suffice in most cases.

In general, the frequency-dependent element f(c o ) to be approximated is a complex num ber in which case it may be written as x + j y and equated with an approximating function (4.21)

^ . . a0z n + aIzn-1 +... + a z n-m

f(<o ) = x + j y = —--- L--:--- --- (4.22)

z" +b1z n +... + bn

Cross m ultiplying and substituting for z, where z is:

z = eja>Al = Re+ j Im (4.23) in (4.22) yields

(X + jy) [(Re + j l mr +b1(Re + j l mr 1 + ...+ 6 J =

°o(R. + j I m )n + a i(R e + J I mr I + - . + am(R, + j l mr

Here A t is the time step o f the model.

Equating the real and imaginary parts on both sides, the following matrix equation is obtained:

/^Alx(m+n)AB(m+„)x\ = (4.25)

where A A 2x(m+n) = A A ( x ,y ,R e, I m) and B B 2xl = B B ( x ,y ,R e, I m) and the ARMA coefficients are collected in:

CHAPTER 4__________________Development of interconnect models from the Telegrapher’s Equations

CHAPTER 4 Development of interconnect models from the Telegrapher’s Equations

) x \ — Q m (4.26)

b.n

The matrix equation (4.25) is then solved in a least square sense to obtain the ARMA

coefficients a0,...,a m,bx,...b no f the approximation APPm/n(z) that is a ^-dom ain

representation for each o f the constituent frequency-dependant elements o f the resonant model. The approximants are then checked for stability. Any poles or zeros that are outside unit circle are reflected back inside thus guaranteeing the stability o f all approximations. The significance o f obtaining these coefficients is that it is now straightforward to calculate the time-domain response.

4.I.4 .2 . The choice o f approxim ating functions

The individual elements o f the matrices g, Yb, Ç and Ybb that need to be

approximated only require low-order 2-transfer functions. Typically, the maximum order is three. This is shown in [C98] where exact Z-domain expressions are derived for the elements o f the matrices g, Yb, Ç and ŸBB for the case o f a lossless line:

I f losses and frequency dependence are to be taken into account, it is recommended [C98] that the order o f both the numerator and denominator in (4.27) and (4.28) is increased for approximation o f the elements o f g and Yb matrices where the

(4.27)

i = J

(4.28)

(4.29)

losses are such that this is necessary, For the lossy lines encountered in this thesis it is

CHAPTER 4 Development of interconnect models from the Telegrapher’s Equations

recommended to model the individual elements o f the frequency-dependant matrices g,

Yb, ¿'and Ym as:

as z 1 + a gz 2 _1______2 l + bsz + bs z + b fz_{I 2 J} -3 (4.30) YbO , j ) = < a bz 1 + abz 2 I____________ -1 l + bbz - ‘ + bbz~2 + bDzb_-3 a bz 1 + a bz 2 _I______2 l + bbz -' + bbz-2 + bbzb —3 l = J i * j (4.31)

Ç (i,i) = 2YBB( i ,i ) = ^a t -va1’ z 1

1 + ^ z - 1 a f = - a t (4.32)

4.I.4 .3 . T im e dom ain m odel

After calculating the ARMA coefficients, the line model may be written in the following form: I s 0 ) - w

= [Y

The elements o f the matrix YB(z) are calculated from (4.19) after each frequency dependant element is replaced with a suitably chosen approximation o f the form (4.21).

Equation (4.33) translates directly to the time domain yielding:

(4.34)

where the superscript V ’ denotes the value at the time tr. The elements o f the y s matrix

are determined from the coefficients o f the ARMA models collected in Yb(z) . It is

important to note that the history currents ihisi and ihi.,2 are dependent only on past values

o f the terminal voltages and currents. Contrary to the convolution approach that requires all past values to calculate the value at the next time point, only a few last values are

necessary to obtain ihisi and ihiS2. The required number o f past points is determined by

the chosen order o f denominator, n, in the approximation (4.21). As shown in Section 4.1.4.2, this order is usually very low, typically up to three. Thus, calculation o f the history currents is not computationally expensive. A detailed derivation for the

expressions for ihisi and iuS2 can be found in Appendix C. W hile the model is derived

with a given time-step A t, this is not a limitation as time-domain interpolation is possible with minimal loss o f accuracy. Furthermore, this modelling procedure avoids

CHAPTER 4 Development of interconnect models from the Telegrapher’s Equations

the many numerical difficulties and stability issues involved in direct approximation o f the Y parameters. The whole modelling procedure is illustrated in the following section where the resonant model for a single lossy interconnect with frequency-dependant parameters is obtained.