The Indefinite Admittance Matrix

Transcription

1 Subject: ndefinite Adittance Matrices Date: June 6, 998 The ndefinite Adittance Matrix The indefinite adittance atrix, designated F for short, is a circuit analsis technique i,ii,iii which lends itself well to an topolog. Once the nodal equations of the circuit are written, basic inherent properties of the F allow an N N adittance atrix to be collapsed to a twoport adittance atrix. Fro this point, R standard two-port relationships in either - or S- paraeters can be utilized to calculate input, output ipedance, voltage or power gain, R stabilit, etc. R A fundaental propert of the F technique is that an individual row or colun of the F atrix sus to zero. This is due to Kirchoff s current law at each node. One inor exception to this rule, however, is in the presence of independent current sources connecting specific nodes. n those cases the suation of currents will not equal zero. Figure best illustrates the application of the Figure F ethod. Standard nodal equations at each of the individual nodes are written; these are given in []. Rewriting [] in standard atrix for gives [] which, adopting standard adittance notation, can be equivalentl expressed as in []. REF g ( - ),, g, g,, g, g,, [] g, g, g g,,,,,, [] where is adittance and,,,,,, [], = +, = -, = -, = -( + g ) = +, + g, = -,, = -( g, = -(, + g ), =, +, n actual use the F atrix is converted first, through a series of atrix reduction ethods, fro an N N atrix to a atrix coposed of a) input, b) output, and c) rerence nodes. At this point in tie an one of the three nodes can be considered the rerence. Once the rerence node is identified and the reduction fro the to a final atrix copleted, the indefinite atrix becoes definite. When the rerence is identified, all voltages for the reaining two ports are rerenced to this rerence node with none of the reaining nodes floating. ( J.E. Crawford All Rights Reserved )

2 Subject: ndefinite Adittance Matrices Date: June 6, 998 f onl of the 9 eleents in a F atrix are known, it is possible to coplete all entries of the atrix using the zero-su propert that each colun and row ust obe. The reverse operation on the definite atrix in this anner allows the rerence to be shifted to a difrent node, i.e. coon-base versus a coon-eitter configuration. Once again if independent sources are present, the zero-row, zero-colun properties do not hold. f the circuit network is passive, the F atrix is also setrical. The F atrix for the transistor is shown in [] and Figure. t is a siple algebraic atter to absorb the rerence terinal ( base, eitter, or collector ) as the rerence culinating in two-port -paraeters for the respective transistor topolog. [] b c e bb cb eb bc cc ec be ce ee F cc ce [] ec ee B C E [] b B Transistor e E c C REF Figure oltages & Currents for Transistor When E is set to zero, for exaple, the indefinite atrix in [] and Figure describes the coon-eitter configuration. The F in [] describes a coon-base configuration. To coplete the atrix the zero-su propert of the atrix can be used, giving [6]. ob fb rb ib ob rb fb ib F ob fb ob fb [6] rb ib rb ib The steps to deterine the indefinite atrix for the overall network are the following:. abel each node. Break the circuit up into coponent networks one network for the passive eleents and separate networks for each active eleent. Deterine the F for each coponent network. Add the individual Fs to give the coplete indefinite atrix. Each of the atrices for the coponent networks, as well as the overall atrix, have diensions of N N, where N is the nuber of nodes in the circuit. The row and colun that correspond to an unconnected node of the coponent network are set to zero. For a circuit fragent that includes a transistor, the letters B, E, and C are placed against the respective nodes to which the base, eitter, and collector are connected. These three nodes are treated as if the were the indefinite atrix that represents the transistor. Specificall, for a coon-eitter transistor connected with ( B, E, C ) to nodes (,, ) respectivel, the following sub-atrix, and, B, E, C associated row-colun nubers, applies., B F, E, C bb cb ie bc Application of the zero-su propert for rows and coluns ( J.E. Crawford All Rights Reserved ) cc re oe [7]

3 Subject: ndefinite Adittance Matrices Date: June 6, 998 copletes the atrix in [7] as shown in [8]. The final result is obtained b transrring the eleents to their proper position in the N N atrix for the coplete network. The F atrix for the passive portion of the circuit can be written b inspection, following the following rules:. Each diagonal eleent rr equals the su of all adittances connected to node r.. An off-diagonal eleent rs equals inus the adittance connected between node r and node s.. Eleents in rows and coluns that correspond to unconnected nodes are zero. F, B, E, C, B ie ie ie re oe re oe, E ie oe re, C oe re [8] n the process of node reduction to a final adittance atrix, the nodes which are suppressed are no longer available for connection to external coponents or other sources. The corresponding current at each of these nodes, j, ust be zero. For exaple, if node is being suppressed for a circuit with a total of nodes, the current entering node is identicall zero, giving [9]. 0,,, [9] The expression in [9] is then used to solve for, and this expression added back to the original atrix for all entries of, thus suppressing an rerence to.,,, For an exaple circuit in which node is reoved, the expression in [0] is substituted for all appearances of in the nodal equation, giving for the first row of the new, reduced atrix that in []. After consolidating and siplifing ters, [] takes on the for of []. [0],,,,,,, [],,,,,, [] The copleted F after the suppression of node is shown in []. A further suppression of nodes is then perfored, driving toward having a defined rerence and input and output nodes. At this point, coon two-port relationships are then applied. ( J.E. Crawford All Rights Reserved )

4 Subject: ndefinite Adittance Matrices Date: June 6, 998,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Reduced Adittance Matrix After Suppression of Node,,,,,,, [] Using the copleted two-port adittance paraeters, the following calculations iv of interest a be perfored. Alternativel, a transforation fro to S paraeters a be ade and a siilar set of calculations perfored in the S-doain. oltage Gain A A Current Gain,, nput Adittance Output Adittance Stern K-Factor in out, K,,, where S g, GS g G Re n the actual reduction forulation, the first process is to order the coluns and rows to reflect the nodes desired for input, output, and rerence, respectivel, in order,, and. Then the reduction coences eliinating row/colun N, followed b N-, etc. until onl a F atrix reains. Before each successive reduction, additional reordering aong the rows/coluns reaining to be suppressed should be done to iniize round-off errors, etc. The actual procedure suitable for this atrix reduction is shown in the Appendix. ( J.E. Crawford All Rights Reserved )

5 Subject: ndefinite Adittance Matrices Date: June 6, 998 Appendix oid Reduce_Mat( int n_node, int Out_node, N int) Den double; // agnitude of rr r int; // row and colun being used i,j int; int; // argest diagonal eleent row XX double; // real part of coon ter double; // iaginar part of coon ter S, S double; // tests for largest diagonal eleent oid Swap( double x, double x) XX double; XX = x; x = ; = XX; f( n_node <> ) or ( Out_node <> ) then for j = to N do Swap( [,j].r, [n_node,j].r; Swap( [,j].i, [n_node,j].i; Swap( [,j].r, [Out_node,j].r; Swap( [,j].i, [Out_node,j].i; for j = to N do Swap( [j,].r, [j, n_node].r; Swap( [j,].i, [j, n_node].i; Swap( [j,].r, [j, Out_node].r; Swap( [j,].i, [j, Out_node].i; while N > do = ; S = abs([,].r) + abs([,].i); f N > then for i = to N do S = abs([,].r) + abs([,].i); f S > S then Begin S = S; = ; if <> N then // swap n and Out ( J.E. Crawford All Rights Reserved )

6 Subject: ndefinite Adittance Matrices Date: June 6, 998 for j = to N do Swap( [,j].r, [N,j].r); Swap( [,j].i, [N,j].i); For j = to N do Swap( [j,].r, [j,n].r); Swap( [j,].i, [j,n].i); Den = sqr([n,n].r) + sqr([n,n].i); R = N; N = N-; for i = to N do if([i,r].r <> 0) or ([i,r].i <> 0 ) then XX = ([,r].r * [r,r].r + [r,r].i*[,r].i) / Den; = ([r,r].r * [,r].i [,r].r * [r,r].i ) / Den; For j = to N do Begin if([r,j].r <> 0) or ([r,j].i <> 0 ) then [i,j].r = [i,j].r [r,j].r*xx + [r,j].i*; [,j].i = [,j].i [r,j].r * [r,j].i * XX; end. i Unif Two-Port Calculations, Daruvala, D. J., Electronic Design, Januar, 97, pp. -6 ii Consider the ndefinite Matrix, Daruvala, D. J., Electronic Design, Januar 8, 97, pp iii Coputer Methods for Circuit Analsis and Design, lach, J., Singhal, K., an Nostrand Reinhold, 98 iv Solid State Radio Engineering, Krauss, H.., Bostian, C.W., Raab, F. H., John Wile & Sons, 980 ( J.E. Crawford All Rights Reserved ) 6