Classify by number of ports and examine the possible structures that result. Using only one-port elements, no more than two elements can be assembled.

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1 Jnction elements in network models. Classify by nmber of ports and examine the possible strctres that reslt. Using only one-port elements, no more than two elements can be assembled. Combining two two-ports yields another two-port. At most two one-port elements cold be connected. Ths three-port jnction elements are necessary. They are also sfficient insofar as jnctions with more than three ports may be constrcted from assemblages of three-ports. An assembly of two three-port elements yields a for-port element. Using two- and three-port jnctions, an arbitrary nmber of one-port (or n-port) elements may be assembled into a system. Three-port jnction elements A jnction strctre or jnction element is treated as a system with certain special properties which are derived from energetic considerations. December 6, 1994 page 1 Neville Hogan

2 An ideal jnction strctre is power-continos: power is transmitted instantaneosly withot storing or dissipating energy. instantaneos power ot eqals instantaneos power in. Power flow in and ot of a three-port may be characterized by six real-valed wave-scattering variables 1, 2, 3, define the inpt power flow v 1, v 2, v 3, define the otpt power flow Using vector notation: = v = v 1 v 2 v 3 (A.1) (A.2) The inpt and otpt power flows are the sqare of the length of these vectors, their inner prodcts. 3 Pin = i 2 = t i=1 3 Pot = v i 2 = v t v i=1 (A.3) (A.4) December 6, 1994 page 2 Neville Hogan

3 Constittive eqations of the jnction strctre may be written in vector form as an algebraic relation between inpt and otpt scattering variables. v = f() (A.5) This algebraic relation is constrained by the reqirement that power in eqal power ot. Geometrically, the length of the vector v mst eqal the length of the vector. The inpt and otpt vectors are both confined to the srface of a sphere. 3 v 3 2 v 2 v 1 v 1 Conseqently, for any two particlar vales of and v, the algebraic relation f(. ) is eqivalent to a rotation operator and eqation A.5 may be re-written as follows. v = S() (A.6) where the sqare matrix S is known as a scattering matrix. December 6, 1994 page 3 Neville Hogan

4 S need not be a constant matrix, bt may in general depend on the power flx throgh the jnction, hence the notation S(). However, S is sbject to important restrictions. In particlar, vtv = tsts = t (A.7) S is ortho-normal: the vectors formed by each of its rows (or colmns) are (i) orthogonal and (ii) have nit magnitde; its transpose is its inverse. StS = 1 (A.8) (This is simply a mathematical statement of the conclsion that S is a rotation operator.) December 6, 1994 page 4 Neville Hogan

5 Symmetric three-port jnction elements To evalate S, we assme an additional symmetry property in the mathematical sense of invariance nder the action of a grop operator. Specifically, assme the constittive eqation is invariant nder the operation of permting or exchanging the ports of the jnction strctre. In terms of the coefficients of the scattering matrix S: S = abb bab bba (A.9) The power otpt along any port depends pon the power inpt along all three ports, itself and its two neighbors. Each of the neighboring ports contribte the same fraction, b, of their inpt power That fraction may be different from the portion, a, of the power inpt which is "reflected" back ot the same port. Each of the three ports is the same in this regard. Using eqation A.9 in eqation A.8 yields three eqations for the coefficients a and b, bt only two are distinct. a 2 + 2b 2 = 1 b 2 + 2ab = 0 (A.10) (A.11) These eqations admit only two soltions, a = 1/3; b = -2/3 (A.12) or a = -1/3; b = 2/3 (A.13) so the symmetry condition constrains the scattering matrix to be a constant. There are only two possible symmetric, power-continos, three-port jnctions. They are both linear. December 6, 1994 page 5 Neville Hogan

6 Transform back to effort and flow variables. e = e 1 e 2 e 3 f = f 1 f 2 f3 (A.14) (A.15) The relation between efforts and wave-scattering variables is as follows. e = ( - v) c = c (1 - S) (A.16) where c is a scaling constant. Using eqation A.12 for the vale of S S = 1/3-2/3-2/3-2/3 1/3-2/3-2/3-2/3 1/3 (A.17) it can be seen that e = c (A.18) The efforts on each of the ports are eqal. This is a common effort or type zero jnction. e 1 = e 2 = e 3 (A.19) December 6, 1994 page 6 Neville Hogan

7 The relation between flows and wave-scattering variables is as follows. f = ( + v)/c = 1/c (1 + S) (A.20) Using the same vale for S f = 1 c 4/3-2/3-2/3-2/3 4/3-2/3-2/3-2/3 4/3 (A.21) Smming down the colmns, it can be seen that the relation between flows on the ports is f 1 + f 2 + f 3 = 0 (A.22) This is a continity eqation 1. It is a derived property of the zero jnction. 1 Note that power on all ports of the jnction has been defined to be positive inwards. December 6, 1994 page 7 Neville Hogan

8 Using the other of the two vales of S (eqation A.13). S = -1/3 2/3 2/3 2/3-1/3 2/3 2/3 2/3-1/3 (A.23) From eqation A.20 we find f = 1 c (A.24) The flows on each of the ports of this jnction are eqal. This is a common flow or type one jnction. f 1 = f 2 = f 3 (A.25) Using eqation A.16 e = c 4/3-2/3-2/3-2/3 4/3-2/3-2/3-2/3 4/3 (A.26) Smming down the colmns, it can be seen that the relation between efforts is e 1 + e 2 + e 3 = 0 (A.27) This is a generalized compatibility eqation. It is a derived property of the zero jnction. Eqating effort and voltage, flow with crrent, we have derived Kirchhoff's crrent and voltage laws from energetic "first principles". We have also shown that an exactly analogos pair of eqations can be derived for any of the energetic media and that they mst always be linear. December 6, 1994 page 8 Neville Hogan

9 Conservation principles and symmetric jnctions In terms of efforts and flows, power continity constrains the constittive eqations for a jnction as follows. e i f i = 0 (A.28) The bond graph zero-jnction is associated with a generalization of Kirchhoff's node (crrent) law, a statement of flow (crrent) continity that constrains the constittive eqations for a zero jnction as follows. f i = 0 (A.29) Bt these two constraints are not sfficient to define the zero jnction. A jnction element known as a circlator (sed in microwave circit theory) satisfies the reqirements of power continity (eqation A.28) and crrent continity (eqation A.29), bt it is qite distinct from a zero jnction. The defining property of a zero jnction is its symmetry the effort is invariant nder the operation of permting or exchanging the ports of the jnction strctre. e i = e j i, j (A.30) Combined with this condition, either of the previos two constraints implies the other. Ths, sppressing the sbscript on the common effort, the power continity condition may be written as follows. e f i = 0 (A.31) If this relation is to be identically tre for all vales of the (common) effort, the flow continity condition mst be satisfied. A similar argment holds for the one-jnction. Ths symmetry is the fndamental property, continity or compatibility are not. December 6, 1994 page 9 Neville Hogan