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A Comparison of Two Latency Insertion Methods in ... - IEEE Xplore

cannot implement the closed loop **in** Eq(15). The solution is tonreplace I byn 1vccs I − **in** the n 1/2vccs V + update equations. The above2equations then becomen−1 n+1/2 n−1/2Ivccs= g⋅( V −V)(16)1 2−1n+ 1/2 ⎛C1 ⎞ ⎡C1n−1/2n1= +1 1−⎤V ⎜ G ⎟ ⎢ V I ⎥⎝Δt⎠ ⎣Δt⎦−1n+ 1/2 ⎛C⎞ ⎡2C2n−1/2n n−1⎤V2 = ⎜ + G2⎟ ⎢ V2−( − I + Ivccs) ⎥⎝ Δt⎠ ⎣Δt⎦Eq.(16) **in**troduces an approximation which leads to a decrease**in** accuracy.The problem **in** Eq.(15) can be easily solved by theamplification matrix LIM without any approximation. Eq. (15)can be rewritten **in**to a block matrix form as follows:⎛1⎞⎜0 ⎟⎛V⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝V⎠ −1 −1 −1⎝ ⎠⎜( 1 tC2 ( G2 g ))( 1 tC1 G1) ( 1 tC2 ( G2g ))⎟⎝+Δ − +Δ +Δ −⎠−1⎛−ΔtC⎞1⎜−1⎟1+ΔtC1 G1n+⎜⎟I⎜ −1 −1ΔtC ⎡2g ⋅ΔtC⎤ ⎟1⎜⎢1+ ⎥⎟1 1n+ −1n−2 1+ΔtC1 G121V1=⎜⎟−1⎜−ΔtC2 2g1 ⎟ V2(17)⎜−1 −11 tC2 ( G2 g ) 1 tC1 G ⎟⎝ +Δ − ⎣ +Δ1⎦⎠Eqs (15) and (17) are entirely equivalent. However the scalarLIM cannot implement Eq. (15) without sacrific**in**g theaccuracy due to the closed loop. The amplification matrix LIMcan implement Eq.(17) accurately.For generality, the amplification matrix LIM formulationsfor general node and branch circuits **in**clud**in**g dependentsources are presented.Figure 3. General node topology with dependent sourcesFig. 3 shows a node i’s topology **in**clud**in**g• voltage controlled current source(VCCS) :- B ik V k : V k is the controll**in**g voltage at node k.• current controlled current source(CCCS) :-S ip I p : I p is the controll**in**g current **of** branch p;-U ij H j : the controll**in**g current H j is an **in**dependentcurrent source at a node.• Node **in**ductance L i ; conductance G i ; capacitance C i• Independent current source H i• Independent voltage source E iFig. 4 shows the topology **of** a branch between node i and j**in**clud**in**g• voltage controlled voltage source(VCVS) :-T ijk V k : V k is the controll**in**g voltage **of** node k.-W ijmn E mn :the controll**in**g voltage E mn is an **in**dependentvoltage source **in** a branch.• current controlled voltage source(CCVS) :-Z ijpq I pq : I pq is the controll**in**g current **of** branch pq.• Branch capacitance C ij ; **in**ductance L ij ; resistance R ij• Independent voltage source E ij• Independent current source H mnFigure 4.General branch topology with dependent sourcesThe block matrix formulation **in** semi-implicit LIM is1 1 11⎛n+ ⎞ ⎛n− ⎞ nn−2 2⎛ 0 PU' ⎞⎛+ ⎞⎛2⎛ 0 P ⎞ ⎞V2V V + E+⎜ ⎟ CN= A'⎜ ⎟+ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎜ ⎟⎜ n+ 1 ⎟ ⎜ n ⎟ ⎜ Tn Tn−1I I QW+' QM+' PU + '⎟⎜ ⎟ ⎜H Q+ QM+' P⎟ ⎝ ⎠ ⎝ +⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎠ ⎜ IL⎟⎝ N ⎠where⎛ P+ P− −P M ' ⎞+A'= ⎜⎟⎜ TTQ+ M ' P+ P− Q+ Q− − Q+M ' P+M ' ⎟⎝⎠⎛ C G'P−= − −Δt⋅LN⎜⎝Δt2−1⎞⎟⎠⎛ '+2 ⎟ ⎟ ⎞⎜ L RQ = +⎜ Δt⎝ ⎠T TR' = R − Z M ' = M + T W ' = E + W−1P+⎛ ' ⎞⎜ C G= + ⎟⎜ 2 ⎟⎝Δt⎠−1⎛ L R'Q−= − −Δt⋅CN⎜⎝Δt2(18)G'= G − B M ' = M − S U ' = E + USome restrictions apply **in** the use **of** dependent sources forthe amplification matrix LIM, namely:• The controll**in**g sources and the controlled sourcesmust be **in** the same block;• The controll**in**g current cannot be the current throughan element(G i /C i /L i /E i ) at a node;• If the controlled voltage source is at a node, it shouldbe transformed **in**to a controlled current source with avery small resistance **in**serted by a Nortontransformation.• If the controlled current source is **in** a branch, itshould be changed to a controlled voltage source witha very large resistance us**in**g a Théven**in**transformation.We can use the new amplification matrix A ' to predict thestability **of** a time step Δt**in** the presence **of** dependent sources.V. EXAMPLEIn this section we use a real case to compare theperformance between the two LIMs **in** handl**in**g circuits withdependent sources. Fig. 5 shows a circuit **in**clud**in**g a voltagecontrolled voltage source (VCVS). The controll**in**g coefficiente is 1.1. S**in**ce the controlled voltage source is at a node, itshould be transformed **in**to a controlled current source by the−1⎞⎟⎠297