Networ 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. M Networ Graph
The concepts of a graph Special graphs Fig.
The concepts of a graph Subgraph G is a subgraph of G if every node of G is the node of G and every branch of G is the branch of G 4 G 3 4 Fig. 3 G 3 G 3 4 3 3 G 3 G 4 G 5
The concepts of a graph Associated reference directions The th branch voltage and th branch current is assigned as reference directions as shown in fig. 4 v j v j Fig. 4 Graphs with assigned reference direction to all branches are called oriented graphs.
The concepts of a graph 4 3 3 5 6 4 Fig. 5 Oriented graph Branch 4 is incident with node and node 3 Branch 4 leaves node 3 and enter node
The concepts of a graph Incident matrix The node-to-branch incident matrix A a is a rectangular matrix of n t rows and b columns whose element a i defined by a i = If branch leaves node i If branch enters node i If branch is not incident with node i
The concepts of a graph For the graph of Fig.5 the incident matrix A a is = A a
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 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 maes 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 removal Connected Graph removal Unconnected Graph Fig. 8
Cutset and Kirchhoff s current law Fig. 9
3 4 6 5 7 8 9 5 6 8 7 9 3 4 4 9 5 6 7 8 3 Cut set (c) Fig. 9
Cutset and Kirchhoff s current law For any lumped networ, 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) j ( t) j ( t) + j3( t) = for all t And from Fig. 9 (b) j ( t) + j ( t) j ( t) = for all t 3
Cutset and Kirchhoff s current law Cut sets should be selected such that they are linearly independent. Cut sets I,II and III are linearly dependent Fig.
Cutset and Kirchhoff s current law j ( t) + j ( t) + j ( t) + j ( t) + j ( t) = Cut set I 3 4 5 Cut set II Cut set III j ( t) j ( t) j ( t) j ( t) = 4 5 8 j ( t) + j ( t) + j ( t) j ( t) j ( t) = 3 8 KCL cut set III = KCL cut set I + KCL cut 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 Fig.
Loops and Kirchhoff s voltage laws I II III V IV Fig. Cases I,II,III and IV violate the loop Case V is a loop
Loops and Kirchhoff s voltage laws For any lumped networ, for any of its loop, and at any time, the algebraic sum of all branch voltages around the loop is zero. Example Write the KVL for the loop shown in Fig 3 KVL v ( t) v5( t) v7 ( t) + v8 ( t) + v4( t) = for all t Fig. 3
Tellegen s Theorem Tellegen s Theorem is a general networ theorem It is valid for any lump networ For a lumped networ whose element assigned by associate reference direction for branch voltage v and branch current j v j The product is the power delivered at time by the networ to the element If all branch voltages and branch currents satisfy KVL and KCL then b = v j = b = number of branch t
Tellegen s Theorem ˆ ˆ ˆ Suppose that vˆ, vˆ,... vˆ b and j, j,... j b is another sets of branch voltages and branch currents and if vˆ and ˆ j satisfy KVL and KCL Then b = v ˆj = b = vˆ ˆj = and b = vˆ j =
Tellegen s Theorem Applications Tellegen s Theorem implies the law of energy conservation. Since b v j = = The sum of power delivered by the independent sources to the networ is equal to the sum of the power absorbed by all branches of the networ.
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) = For all t The sum of power delivered by the independent sources to the networ is equal to the sum of the power absorbed by all branches of the networ.
Conservation of Energy Resistor R j For th resistor Capacitor C v For th capacitor Inductor L i For th inductor
Conservation of Complex Power b = V J = V = Branch Voltage Phasor J = Branch Current Phasor J = Branch Current Phasor Conjugate
J V J J 4 V V 4 J 3 V 3 V J = V J b =
V J V J N Linear time-invariant RLC Networ J V Conservation of Complex Power
The real part and phase of driving point impedance J J V V Z in
V = J Z ( jω) in From Tellegen s theorem, and let P = complex power delivered to the one-port by the source P = V J = Z in( jω ) J = V J = Z ( jω) J b =
Taing the real part P = Re[ Z ( jω )] J av in b = Re[ Z ( jω)] J = All impedances are calculated at the same angular frequency i.e. the source angular frequency
Driving Point Impedance P = Z in( jω ) J b = Zm( jω) J = m = R J + jωl J + J jωc i i l i l l R L C
Exhibiting the real and imaginary part of P P = R J + jω L J J 4 4 i i l i l ω Cl Average power dissipated P av Average Magnetic Energy Stored Ε M ( ) P = P + jω Ε Ε av M E Average Electric Energy Stored Ε E
From P = Z in( jω ) J Z ( ) in jω = P J ( ) P = P + jω Ε Ε av M E
Driving Point Impedance Given a linear time-invariant RLC networ driven by a sinusoidal current source of A pea amplitude and given that the networ is in SS, The driven point impedance seen by the source has a real part = twice the average power P av and an imaginary part that is 4ω times the difference of E M and E E