Patent Application: US-98266804-A

Abstract:
the adjoint network reduction technique has been shown to reduce 50 % of the computational complexity of constructing the congruence transformation matrix . the method was suitable for analyzing the special multi - port driving - point impedance of rlc interconnect circuits . this paper extends this technique for the general circumstances of rlc interconnects . comparative studies among the conventional methods and the proposed methods are also investigated . experimental results will demonstrate the accuracy and the efficiency of the proposal method .

Description:
the dynamics of rlc interconnect networks can be represented by the following modified nodal analysis ( mna ) formulae [ reference 8 and 11 ]: m ⁢ ⅆ x ⁡ ( t ) ⅆ t = - nx ⁡ ( t ) + bu ⁡ ( t ) , y ⁡ ( t ) = d t ⁢ x ⁡ ( t ) ⁢ ⁢ m = [ c 0 0 l ] , n = [ g e - e t r ] , x ⁡ ( t ) = [ v ⁡ ( t ) i ⁡ ( t ) ] , ( 1 ) x ( t ) εr n is the state vector , u ( t ) εr m is the input vector , y ( t ) εr p is the output vector , and m , nεr nxn , bεr nxm , and dεr nxp are so - called the mna matrices . m and n containing capacitances in c , inductances in l , conductances in g and resistances in r are positive definite , m is symmetric and n is non - symmetric . e presents the incident matrix for satifying kirchhoff &# 39 ; s current law . x ( t ) contains node voltages v ( t ) εr nv and branch currents of inductors i ( t ) εr ni , where n = n v + n i . if the m - port driving - point impedance is concerned , then p = m and d = b . let the signature matrix be defined as s = diag ( i nv ,− i ni ) so that the symmetric properties of the mna matrices are as follows [ reference 11 ]: under this situation , if port driving - point impedance is concerned , that is , each port is connected with a current source , then b t =└ b v t 0 ┘, where b v εr nvxm , and sb = b . the transfer functions of the state variables and of the outputs are defined as x ( s )=( n + sm ) − 1 and y ( s )= d t x ( s ). given a frequency s 0 εc , let a =−( n + s 0 m ) − 1 m and r =( n + s 0 m ) − 1 b , where n + s 0 m is assumed nonsingular . the taylor series expansion of x ( s ) about s 0 is given by x ⁡ ( s ) = ∑ i = 0 ∞ ⁢ x ( i ) ⁡ ( s 0 ) ⁢ ( s - s 0 ) , x ( i ) ⁡ ( s 0 ) = a i ⁢ r is the ith - order system moment about s 0 . similarly , the ith - order output moment about s 0 is calculated asy ( i ) ( s 0 )= d t x ( i ) ( s 0 ). suppose that the above system is large - scale and sparse . the model - order reduction problem is to seek a q - order system , where q & lt ;& lt ; n , such that m ^ ⁢ ⅆ x ^ ⁡ ( t ) ⅆ t = - n ^ ⁢ x ^ ⁡ ( t ) + b ^ ⁢ u ⁡ ( t ) , y ^ ⁡ ( t ) = d ^ t ⁢ x ^ ⁡ ( t ) ( 3 ) where { circumflex over ( x )}( t ) εr q ,{ circumflex over ( m )},{ circumflex over ( n )} εr qxm ,{ circumflex over ( b )} εr qxm , and { circumflex over ( d )} εr qxp . the corresponding ith - order system moment and output moment about s 0 is { circumflex over ( x )} ( i ) ( s 0 )=(−({ circumflex over ( n )}+ s 0 { circumflex over ( m )}) − 1 { circumflex over ( m )}) i ({ circumflex over ( n )}+ s 0 { circumflex over ( m )}) − 1 { circumflex over ( b )} and ŷ ( i ) ( s 0 )={ circumflex over ( d )} t { circumflex over ( x )} ( i ) ( s 0 ). the purpose of the moment matching is to construct a reduced - order system such that ŷ ( i ) ( s 0 )= y ( i ) ( s 0 ) for 0 ≦ i ≦ k − 1 , where k is the order of moment matching . one conventional solution for moment matching is using the one - sided krylov subspace projection method [ references 3 , 8 and 11 ]. the kth - order block krylov subspace is defined as k ( a , r , k )= colsp ([ rar . . . a k − 1 r ]). k ( a , r , k ) is indeed the subspace spanned by x ( i ) ( s 0 ) for 0 ≦ i ≦ k − 1 . the projection can be achieved by constructing v q εr nxq , q ≦ km , from the krylov subspace k ( a , r , k ). under this framework , we have x ( t )= v q { circumflex over ( x )} q ( t ) and the reduced - order model can be expressed as { circumflex over ( m )}= v q t mv q ,{ circumflex over ( n )}= v q t nv q ,{ circumflex over ( b )}= b q t b ,{ circumflex over ( d )}= v q t d ( 4 ) it has been shown that x ( i ) ( s 0 )= v q { circumflex over ( x )} ( i ) ( s 0 ) and ŷ ( i ) ( s 0 )= y ( i ) ( s 0 ) for 0 ≦ i ≦ k − 1 . the reduced - order model is guaranteed stable { circumflex over ( m )} and { circumflex over ( n )} are positive definite . furthermore , it will be passive if the multi - port driving - point impedance is concerned . two types of algorithms can be employed to generate v q from the krylov subspace : the arnoldi type [ references 8 and 11 ] and the lanczos type [ references 4 and 5 ]. we use the notation v q ( a ) to denote the orthonormal basis generated from the block arnoldi algorithm from the krylov subspace k ( a , r , k ). similarly , we use the notation v q ( l ) and w q ( l ) to represent the biorthogonal bases yielded from the block lanczos algorithm from the krylov subspaces k ( a , r , k ) and k ( a t , d , k ), respectively . in this case , w q ( l ) t v q ( l ) = δ q , where δ q is a full rank diagonal matrix . in the past , either v q ( a ) or v q ( l ) has been used to generate the reduced - order model ( 4 ). traditionally , w q ( l ) and v q ( l ) are used to perform oblique projection { circumflex over ( m )}=− w q ( l ) t av q ( l ) ,{ circumflex over ( n )}= w q ( l ) t ( i + s 0 a ) v q ( l ) ,{ circumflex over ( b )}= w qt ( l ) t r ,{ circumflex over ( d )}= v q ( l ) t d although the reduced - order system can match up to 2k - order moments , this model can not be guaranteed to be stable and passive . variations of the lanczos - type algorithms have also been proposed . for example , if the m - port driving - point impedance is concerned ( d = b ), the symmetric block lanczos algorithm has been investigated . in this case , w q = d q v q ( l ) , where d q is a diagonal matrix , only a half of the computational cost and storage are required [ references 3 and 5 ]. using both w q ( l ) and v q ( l ) in the one - sided projection method is still possible . it can be achieved by the adjoint network reduction technique . the details can be developed in the following section . one technique for model - order reductions is tot apply the corresponding adjoint mna formulae . the adjoint network ( or the dual system [ reference 11 ]) of the system ( 1 ) is represented as [ reference 10 ] m ⁢ ⅆ x a ⁡ ( t ) ⅆ t = - n t ⁢ x a ⁡ ( t ) + du ⁡ ( t ) , y a ⁡ ( t ) = b t ⁢ x a ⁡ ( t ) ( 5 ) the system transfer function and its ith - order system moment about s 0 are defined as x a ( s )=( n t + sm ) − 1 d and x a ( i ) ( s 0 ), respectively . by using the information of x a ( i ) ( s 0 ), the following theorem has been shown in [ reference 11 ]. theorem 1 : if a matrix u is chosen as the congruence transformation matrix such that x ( i ) ( s 0 ), x a ( j ) ( s 0 )} ε colsp ( u ) for 0 ≦ i ≦ k − 1 and 0 ≦ j ≦ l − 1 , ( 6 ) the technique can overcome the numerical instability problem when generating the basis matrix u if order k + l is extremely high . in this section , we summarize some properties of the adjoint network reduction technique . relationships between x a ( i ) ( s 0 ) and x ( i ) ( s 0 ) are derived explicitly . it can be contributed to reduce the computational cost of constructing u . in this subsection , we will show that the congruence transformation matrix u can be constructed with the resultant biorthogonal bases v q ( l ) and w q ( l ) from the lanczos - type algorithms . the theorems will be disclosed as below . by changing the state variables : x a ( t )=( n t + s 0 m ) − 1 z a ( t ). equation ( 5 ) can be rewritten as m ⁡ ( n t + s 0 ⁢ m ) - 1 ⁢ ⅆ z a ⁡ ( t ) ⅆ t = - n t ⁡ ( n t + s 0 ⁢ m ) - 1 ⁢ z a ⁡ ( t ) + du , ⁢ y a ⁡ ( t ) = b t ⁡ ( n t + s 0 ⁢ m ) - 1 ⁢ z a ⁡ ( t ) . z a ( s )=( n t + s 0 m )( n t + sm ) − 1 d =( n t + s 0 m ) x a ( s ) . thus the ith - order system moments of z a ( s ) and x a ( s ) about s 0 can be derived as follows : z a ( i ) ( s 0 )=( n t + s 0 m ) x a ( i ) ( s 0 ) ( 7 ) through the introduction of z a ( i ) ( s 0 ), the relationships between x a ( i ) ( s 0 ) and w q ( l ) are observed in the following lemma . lemma 1 : suppose that k ( a t , d , k )= colsp ( w q ( l ) ). we have x a ( i ) ( s 0 ) εcolsp (( n t + s 0 m ) − 1 w q ( l ) , ( 9 ) z a ( i ) ⁡ ( s 0 ) = ( n t + s 0 ⁢ m ) ⁡ [ - ( n t + s 0 ⁢ m ) - 1 ⁢ m ] i ⁢ ( n t + s 0 ⁢ m ) ⁢ d = [ - m ⁡ ( n t + s 0 ⁢ m ) - 1 ] i ⁢ d = ( a t ) i ⁢ d . colsp ⁡ ( [ z a ( 0 ) ⁡ ( s 0 ) ⁢ z a ( 1 ) ⁡ ( s 0 ) ⁢ ⁢ … ⁢ ⁢ z a ( k - 1 ) ⁡ ( s 0 ) ] ) = k ⁡ ( a t , d , k ) = colsp ⁡ ( w q ⁡ ( l ) ) . second , the above result implies that z a ( i ) ( s 0 )= w q ( l ) { circumflex over ( z )} a ( i ) ( s 0 ) for 0 ≦ i ≦ k − 1 . thus ( 9 ) can be proven as below : x a ( i ) ( s 0 )=( n t + s 0 m ) − 1 w q ( l ) { circumflex over ( z )} a ( i ) ( s 0 ), for 0 ≦ i ≦ k − 1 . each x a ( i ) ( s 0 ) exists in the subspace spanned by columns of ( n t + s 0 m ) − 1 w q ( l ) . this completes the proof . theorem 2 : suppose that x ( i ) ( s 0 ) εcolsp ( v q ( l ) ), for 0 ≦ i ≦ k − 1 , is a set of moments of x ( s ) about s 0 . x a ( i ) ( s 0 ) εcolsp (( n t + s 0 m ) − 1 w q ( l ) ) for 0 ≦ i ≦ k − 1 . suppose that v q ( l ) and w q ( l ) are biorthogonal matrices generated by the block lanczos algorithm . let u =[ v q ( n t + s 0 m ) − 1 w q ] be the congruence transformation matrix for model - order reductions in ( 4 ), then ŷ ( s 0 )= y ( i ) ( s 0 ), for 0 ≦ i ≦ 2 k − 1 , ( 10 ) proof : from the projection theory in section 2 and lemma 1 , we have x ( i ) ( s 0 )= u { circumflex over ( x )} ( i ) ( s 0 ), for 0 i ≦ k − 1 , x a ( i ) ( s 0 )= u { circumflex over ( x )} a ( i ) ( s 0 ), for 0 ≦ i ≦ k − 1 . theorem 2 demonstrates that a stable reduced - order model for general rlc circuits can be generated by the lanczos - type algorithms . moreover , 2k - order output moments are matched by performing k iterations of the algorithm . theorem 3 : suppose that x ( i ) ( s 0 ) εcolsp ( v q ( a ) ) for 0 ≦ i — k − 1 is a set of moments of x ( s ) about s 0 . v q ( a ) is theorthonormal matrix generated by the block arnoldi algorithm . let u =[ v q ( a ) sv q ( a ) ] be the congruence transformation matrix for model - order reductions in ( 4 ), then ŷ ( i ) ( s 0 )= y ( i ) ( s 0 ), for 0 ≦ i ≦ 2 k − 1 ( 11 ) proof : let p be a q × n diagonal matrix . since w q ( l ) = p ( n t + s 0 m ) v q ( l ) and v q ( l ) = v q ( a ) [ references 3 and 5 ], it can be shown that the subspace spanned by u =[ v q ( a ) sv q ( a ) ] and u =[ v q ( l ) ( n t + s 0 m ) − 1 w q ( l ) ] are the same . a mesh twelve - line circuit , presented in fig1 , is studied to show the efficiency of the proposed method . the line parameters are resistance : 1 . 0 o / cm , capacitance : 5 . 0 pf / cm , inductance : 1 . 5 nh / cm , driver resistance 3 o , and load capacitance : 1 . 0 pf . each line is divided into 50 sections . therefore , the dimension of the mna matrices is n = 1198 , m = 1 , and p = 1 , and d ≠ b . we set the expansion frequency s 0 = 1 ghz , the iteration number k = 15 , and use a total of 1001 frequency points distributed uniformly between the frequency range { 0 , 15 ghz } for simulations . the frequency responses of the original model and the reduced - order model generated by the following projections : ( 1 ) w q ( l ) and v q ( l ) ; ( 2 ) v q ( a ) ; ( 3 ) u = v 2q ( a ) ; ( 4 ) u =[ v q ( l ) ( n t + s 0 m ) − 1 w q ( l ) )] i are compared in fig2 . their corresponding relative error , | y ( s )− ŷ ( s )|/| y ( s )|, are illustrated in fig3 . the program was implemented in matlab 6 . 1 with pentium iv 2 . 8 ghz cpu and 1024 mb dram . the time to generate each reduced - order model and the corresponding average 1 - norm relative error are summarized . the average 1 - norm relative error is approximated by ( ∑ i = 1 1001 ⁢  y ⁡ ( s i ) - y ^ ⁡ ( s i )  /  y ⁡ ( s i )  ) / 1001 . note that only 60 % work is needed to generate the similar frequency response by using the proposed method . although the invention has been explained in relation to its preferred embodiment , it is to be understood that many other possible modifications and variations can be made without departing from the spirit and scope of the invention as hereinafter claimed . 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