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

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

cannot implement the

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) introduces an approximation which leads to a decreasein accuracy.The problem in Eq.(15) can be easily solved by theamplification matrix LIM without any approximation. Eq. (15)can be rewritten into 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 sacrificing 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 including dependentsources are presented.Figure 3. General node topology with dependent sourcesFig. 3 shows a node i’s topology including• voltage controlled current source(VCCS) :- B ik V k : V k is the controlling voltage at node k.• current controlled current source(CCCS) :-S ip I p : I p is the controlling current of branch p;-U ij H j : the controlling current H j is an independentcurrent source at a node.• Node inductance 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 jincluding• voltage controlled voltage source(VCVS) :-T ijk V k : V k is the controlling voltage of node k.-W ijmn E mn :the controlling voltage E mn is an independentvoltage source in a branch.• current controlled voltage source(CCVS) :-Z ijpq I pq : I pq is the controlling current of branch pq.• Branch capacitance C ij ; inductance 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 controlling sources and the controlled sourcesmust be in the same block;• The controlling 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 into a controlled current source with avery small resistance inserted by a Nortontransformation.• If the controlled current source is in a branch, itshould be changed to a controlled voltage source witha very large resistance using a Thévenintransformation.We can use the new amplification matrix A ' to predict thestability of a time step Δtin the presence of dependent sources.V. EXAMPLEIn this section we use a real case to compare theperformance between the two LIMs in handling circuits withdependent sources. Fig. 5 shows a circuit including a voltagecontrolled voltage source (VCVS). The controlling coefficiente is 1.1. Since the controlled voltage source is at a node, itshould be transformed into a controlled current source by the−1⎞⎟⎠297

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