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MIT 2 141 - Junction elements in network models

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Junction elements in network models.Three-port junction elementsSymmetric three-port junction elementsConservation principles and symmetric junctionsJunction elements in network models. 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. Combining two two-ports yields another two-port. At most two one-port elements could be connected. Thus three-port junction elements are necessary. They are also sufficient insofar as junctions with more than three ports may be constructed from assemblages of three-ports. An assembly of two three-port elements yields a four-port element. Using two- and three-port junctions, an arbitrary number of one-port (or n-port) elements may be assembled into a system. Three-port junction elements A junction structure or junction element is treated as a system with certain special properties which are derived from energetic considerations. December 6, 1994 page 1 Neville HoganAn ideal junction structure is power-continuous: power is transmitted instantaneously without storing or dissipating energy. instantaneous power out equals instantaneous power in. Power flow in and out of a three-port may be characterized by six real-valued wave-scattering variables u1, u2, u3, define the input power flow v1, v2, v3, define the output power flow Using vector notation: u = ⎣⎢⎡⎦⎥⎤u1u2u3 (A.1) v = ⎣⎢⎡⎦⎥⎤v1v2v3 (A.2) The input and output power flows are the square of the length of these vectors, their inner products. Pin = ∑i=13ui2 = utu (A.3) Pout = ∑i=13vi2 = vtv (A.4) December 6, 1994 page 2 Neville HoganConstitutive equations of the junction structure may be written in vector form as an algebraic relation between input and output scattering variables. v = f(u) (A.5) This algebraic relation is constrained by the requirement that power in equal power out. Geometrically, the length of the vector v must equal the length of the vector u. The input and output vectors are both confined to the surface of a sphere. v13v2vu1u2u3uv Consequently, for any two particular values of u and v, the algebraic relation f(.) is equivalent to a rotation operator and equation A.5 may be re-written as follows. v = S(u) u (A.6) where the square matrix S is known as a scattering matrix. December 6, 1994 page 3 Neville HoganS need not be a constant matrix, but may in general depend on the power flux through the junction, hence the notation S(u). However, S is subject to important restrictions. In particular, vtv = utStSu = utu (A.7) S is ortho-normal: the vectors formed by each of its rows (or columns) are (i) orthogonal and (ii) have unit magnitude; its transpose is its inverse. StS = 1 (A.8) (This is simply a mathematical statement of the conclusion that S is a rotation operator.) December 6, 1994 page 4 Neville HoganSymmetric three-port junction elements To evaluate S, we assume an additional symmetry property in the mathematical sense of invariance under the action of a group operator. Specifically, assume the constitutive equation is invariant under the operation of permuting or exchanging the ports of the junction structure. In terms of the coefficients of the scattering matrix S: S = ⎣⎢⎢⎡⎦⎥⎥⎤abbbabbba (A.9) The power output along any port depends upon the power input along all three ports, itself and its two neighbors. Each of the neighboring ports contribute the same fraction, b, of their input power That fraction may be different from the portion, a, of the power input which is "reflected" back out the same port. Each of the three ports is the same in this regard. Using equation A.9 in equation A.8 yields three equations for the coefficients a and b, but only two are distinct. a2 + 2b2 = 1 (A.10) b2 + 2ab = 0 (A.11) These equations admit only two solutions, 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-continuous, three-port junctions. They are both linear. December 6, 1994 page 5 Neville HoganTransform back to effort and flow variables. e = ⎣⎢⎡⎦⎥⎤e1e2e3 (A.14) f = ⎣⎢⎡⎦⎥⎤f1f2f3 (A.15) The relation between efforts and wave-scattering variables is as follows. e = (u - v) c = c (1 - S) u (A.16) where c is a scaling constant. Using equation A.12 for the value 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 ⎣⎢⎢⎡⎦⎥⎥⎤2/3 2/3 2/32/3 2/3 2/32/3 2/3 2/3 u (A.18) The efforts on each of the ports are equal. This is a common effort or type zero junction. e1 = e2 = e3 (A.19) December 6, 1994 page 6 Neville HoganThe relation between flows and wave-scattering variables is as follows. f = (u + v)/c = 1/c (1 + S) u (A.20) Using the same value for S f = 1c ⎣⎢⎢⎡⎦⎥⎥⎤ 4/3 -2/3 -2/3-2/3 4/3 -2/3-2/3 -2/3 4/3 u (A.21) Summing down the columns, it can be seen that the relation between flows on the ports is f1 + f2 + f3 = 0 (A.22) This is a continuity equation1. It is a derived property of the zero junction. 1 Note that power on all ports of the junction has been defined to be positive inwards. December 6, 1994 page 7 Neville HoganUsing the other of the two values of S (equation A.13). S = ⎣⎢⎢⎡⎦⎥⎥⎤-1/3 2/3 2/3 2/3 -1/3 2/3 2/3 2/3 -1/3 (A.23) From equation A.20 we find f = 1c ⎣⎢⎢⎡⎦⎥⎥⎤2/3 2/3 2/32/3 2/3 2/32/3 2/3 2/3 u (A.24) The flows on each of the ports of this junction are equal. This is a common flow or type one junction. f1 = f2 = f3 (A.25) Using equation A.16 e = c ⎣⎢⎢⎡⎦⎥⎥⎤ 4/3 -2/3 -2/3-2/3 4/3 -2/3-2/3 -2/3 4/3 u (A.26) Summing down the columns, it can be seen that the relation between efforts is e1 + e2 + e3 = 0 (A.27) This is a generalized compatibility equation. It is a derived property of the zero junction. Equating effort and voltage, flow with current, we have derived Kirchhoff's current and voltage laws from energetic "first principles". We have also shown that an exactly analogous pair of equations can be derived for any of the energetic media and that they must always be linear. December 6, 1994 page 8 Neville HoganConservation principles and symmetric junctions In terms of efforts and flows, power continuity constrains the constitutive equations for a junction as follows.


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MIT 2 141 - Junction elements in network models

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