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Network Graphs and Tellegen’s Theorem The concepts of a graph Cut sets and Kirchhoff’s current laws Loops and Kirchhoff’s voltage laws Tellegen’s Theorem The concepts of a graph The analysis of a complex circuit can be perform systematically Using graph theories. Graph consists of nodes and branches connected to form a circuit. Fig. 1 Network P M Network P Graph Graph The concepts of a graph Special graphs 1 4 3 2 Isolate node Self loop Non plannar Fig. 2 The concepts of a graph Subgraph G1 is a subgraph of G if every node of G1 is the node of G and every branch of G1 is the branch of G 1 4 3 2 G 1 Fig. 3 1 3 2 3 2 4 G2 G1 1 1 4 3 3 2 2 G3 G4 G5 The concepts of a graph Associated reference directions The kth branch voltage and kth branch current is assigned as reference directions as shown in fig. 4 + jk + vk vk jk - Fig. 4 Graphs with assigned reference direction to all branches are called oriented graphs. The concepts of a graph 1 1 2 3 2 5 Fig. 5 Oriented graph Branch 4 is incident with node 2 and node 3 Branch 4 leaves node 3 and enter node 2 3 4 6 4 The concepts of a graph Incident matrix The node-to-branch incident matrix Aa is a rectangular matrix of nt rows and b columns whose element aik defined by 1 aik 1 0 If branch k leaves node i If branch k enters node i If branch k is not incident with node i The concepts of a graph For the graph of Fig.5 the incident matrix Aa is 1 1 Aa 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 Cutset and Kirchhoff ’s current law If a connected graph were to partition the nodes into two set by a closed gussian surface , those branches are cut set and KCL applied to the cutset Cutset branches Guassian surface Fig. 6 Cutset Cutset and Kirchhoff ’s current law A cutset is a set of branches that the removal of these branches causes two separated parts but any one of these branches makes the graph connected. An unconnected graph must have at least two separate part. Fig. 7 Connected Graph Unconnected Graph Cutset and Kirchhoff ’s current law Connected Graph removal removal Unconnected Graph Fig. 8 Cutset and Kirchhoff ’s current law 1 1 2 6 5 2 7 1 4 1 2 8 3 3 3 (a) Fig. 9 Cutset 1,2,3 Cutset 1,2,3 (b) 1 3 2 4 19 5 6 7 11 9 15 8 16 10 14 18 21 22 24 23 25 26 29 27 17 28 13 12 Cut set (c) Fig. 9 20 Cutset and Kirchhoff ’s current law For any lumped network , for any of its cut sets, and at any time, the algebraic sum of all branch currents traversing the cut-set branches is zero. From Fig. 9 (a) j1 (t ) j2 (t ) j3 (t ) 0 for all t for all t And from Fig. 9 (b) j1 (t ) j2 (t ) j3 (t ) 0 Cutset and Kirchhoff ’s current law Cut sets should be selected such that they are linearly independent. III 10 8 9 2 1 3 4 6 I 7 5 II Fig. 10 Cut sets I,II and III are linearly dependent Cutset and Kirchhoff ’s current law Cut set I j1 (t ) j2 (t ) j3 (t ) j4 (t ) j5 (t ) 0 Cut set II j4 (t ) j5 (t ) j8 (t ) j10 (t ) 0 Cut set III j1 (t ) j2 (t ) j3 (t ) j8 (t ) j10 (t ) 0 KCLcut set III = KCLcut set I + KCLcut set II Loops and Kirchhoff’s voltage laws A Loop L is a subgraph having closed path that posses the following properties: The subgraph is connected Precisely two branches of L are incident with each node Not a loop a loop Fig. 11 Not a loop Loops and Kirchhoff’s voltage laws 2 1 2 3 4 2 3 41 4 1 3 5 I II III 7 8 6 4 5 9 3 IV 2 1 Cases I,II,III and IV violate the loop V 10 12 11 Case V is a loop Fig. 12 Loops and Kirchhoff’s voltage laws For any lumped network , for any of its loop, and at any time, the algebraic sum of all branch voltages around the loop is zero. Example 1 Write the KVL for the loop shown in Fig 13 1 2 3 4 5 8 7 6 KVL v2 (t ) v5 (t ) v7 (t ) v8 (t ) v4 (t ) 0 9 for all 8 10 Fig. 13 t Tellegen’s Theorem Tellegen’s Theorem is a general network theorem It is valid for any lump network For a lumped network whose element assigned by associate reference direction for branch voltage v k and branch current j k The product element k vk jk is the power delivered at time t by the network to the If all branch voltages and branch currents satisfy KVL and KCL then b v k 1 k jk 0 b = number of branch Tellegen’s Theorem Suppose that vˆ1 , vˆ2 ,...... vˆb and ˆj1 , ˆj2 ,...... ˆjb ˆjk voltages and branch currents and if vˆ k and Then b k 1 b k 1 b v vˆk ˆjk 0 vk ˆjk 0 is another sets of branch satisfy KVL and KCL k jk 0 k 1 b and vˆ k jk k 1 0 Tellegen’s Theorem Applications Tellegen’s Theorem implies the law of energy conservation. b Since v k jk 0 k 1 “The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”. Applications Conservation of energy Conservation of complex power The real part and phase of driving point impedance Driving point impedance Conservation of Energy b v (t ) j (t ) 0 k 1 k k For all t “The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”. Conservation of Energy Resistor 2 k k R j For kth resistor Capacitor 1 2 C k vk 2 For kth capacitor Inductor 1 2 Lk ik 2 For kth inductor Conservation of Complex Power b 1 Vk J k 0 k 1 2 Vk = Branch Voltage Phasor J k = Branch Current Phasor J k = Branch Current Phasor Conjugate J 2 V2 J1 V1 J4 V4 J 3 V3 b 1 1 V1J1 Vk J k 2 k 2 2 V1 V2 J1 J2 N Linear time-invariant RLC Network Jk Vk Conservation of Complex Power The real part and phase of driving point impedance J1 V1 Vk Jk Z in Linear timeinvariant RLC one-port V1 J1Zin ( j) From Tellegen’s theorem, and let P = complex power delivered to the one-port by the source 1 1 2 P V1 J 1 Z in ( j ) J 1 2 2 b 1 1 2 Vk J k Z k ( j ) J k 2 2 k 2 Taking the real part 1 2 Pav Re[Z in ( j )] J 1 2 b 1 Re[Z k ( j )] J k 2 k 2 2 All impedances are calculated at the same angular frequency i.e. the source angular frequency Driving Point Impedance 1 2 P Z in ( j ) J 1 2 1 b 2 Z m ( j ) J m 2 k 2 1 1 1 1 2 2 Ri J i j Lk J k Jl 2 i 2 k 2 l jCl R L C 2 Exhibiting the real and imaginary part of P 1 1 1 1 2 2 2 P Ri J i 2 j Lk J k 2 J l 2 i 4 l Cl 4 k Average power dissipated Pav Average Magnetic Energy Stored M Average Electric Energy Stored P Pav 2 j M E E From 1 2 P Z in ( j ) J 1 2 Zin ( j) 2P J1 2 P Pav 2 j M E Driving Point Impedance Given a linear time-invariant RLC network driven by a sinusoidal current source of 1 A peak amplitude and given that the network is in SS, The driven point impedance seen by the source has a real part = twice the average power Pav and an imaginary part that is 4 times the difference of EM and EE