Chapter 3. Loop and Cut-set Analysis

Chapter 3. Loop and Cut-set Analysis By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical Engineering, K.N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/electriccircuits2.htm References: Basic Circuit Theory, by Ch. A. Desoer and E. S. Kuh, 1969

Chapter Contents 0. Introduction 1. Fundamental theorem of graph theory 2. Loop analysis 3. Cut-set analysis 4. Comments on loop and cut-set analysis 5. Relation between B and Q 6. Summary Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 2

0. Introduction o We have learned to perform systematically: node analysis of any LTI network, mesh analysis for any planar LTI network. o Here, we briefly, discuss 2 generalizations (variations) of these methods: cut-set analysis, loop analysis. o There are 2 reasons for studying loop and cut-set analysis: 1. These methods are much more flexible than mesh and node analysis, 2. They use concepts that are useful for writing state equations. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 3

1. Fundamental theorem of graph theory o Let G be a connected graph and T a subgraph of G. o T is a tree of connected graph G if: 1. subgraph T is connected, 2. it contains all nodes of G, and 3. it contains no loops. o Given a connected graph G and a tree T: branches of T are called tree branches, branches of G not in T are called links (cotree branches, or chords). o If a graph has n t nodes and has a single branch connecting every pair of nodes, then it has trees: when n t = 5, there are 125 trees, when n t = 10, there are 10 8 trees. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 5

1. Fundamental theorem of graph theory Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 6

1. Fundamental theorem of graph theory THEOREM o Given a connected graph G of n t nodes and b branches, and a tree T of G: 1. There is a unique path along tree between any pair of nodes. 2. There are n t -1 tree branches and b-n t +1 links. 3. Every link of G and unique tree path between its nodes constitute a unique loop (fundamental loop associated with link). 4. Every tree branch of T together with some links defines a unique cut set of G (fundamental cut set associated with tree branch). o For proof, please refer to pages 478 and 479 of book. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 7

1. Fundamental theorem of graph theory COROLLARY o Suppose: G has n t nodes, b branches, and s separate parts, T 1, T 2,..., T s are trees of each separate part: Set {T 1, T 2,..., T s } is called a forest of G. Forest has n t -s branches, G has b-n t +s links. Remaining statements of theorem are true. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 8

2. Loop analysis 2.1. Two basic facts of loop analysis o Consider a connected graph with b branches and n t nodes. o Pick an arbitrary tree T. o There are n = n t -1 tree branches and l = b-n links. o Number branches as follows: links first from 1 to l, tree branches next from l+1 to b. o Every link defines a fundamental loop: loop formed by link and unique tree path between nodes of that link. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 10

2. Loop analysis 2.1. Two basic facts of loop analysis o b = 8, n t = 5, n = 4, and l = 4. o To apply KVL to each fundamental loop, we adopt a reference direction for loop which agrees with reference direction of link which defines that fundamental loop: fundamental loop 1 has same orientation as link 1. o KVL equations can be written for 4 fundamental loops in terms of branch voltage. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 11

2. Loop analysis 2.1. Two basic facts of loop analysis Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 12

2. Loop analysis 2.1. Two basic facts of loop analysis o l linear homogeneous algebraic equations in v 1, v 2,, v b obtained by applying KVL to each fundamental loop constitute a set of l linearly independent equations. o B is an l b matrix called fundamental loop matrix. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 13

2. Loop analysis 2.1. Two basic facts of loop analysis o Since: each fundamental loop includes one link only, orientations of loop and link are picked to be same, number of links are 1, 2,..., l, tree branches are l +1, l +2,..., b, matrix B has form: o 1 l designates a unit matrix of order l. o F designates a rectangular matrix of l rows and n columns. o Rank of B is l. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 14

2. Loop analysis 2.1. Two basic facts of loop analysis o KCL implies that any current that comes to a node must leave this node. o Call i 1, i 2,..., i l currents in l links of tree T. o Each of these currents flowing in its respective fundamental loop. o Each tree branch current is superposition of one or more loop currents. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 15

2. Loop analysis 2.1. Two basic facts of loop analysis Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 16

2. Loop analysis 2.2. Loop analysis for LTI networks o For simplicity, consider a linear time-invariant resistive network with b branches, n t nodes, and one separate part. o Extension to general case is exactly the same as generalization discussed in previous chapter. o A typical branch is shown. o It includes independent sources. o Branch equations are of form: Electric Circuits 2 Chapter 2. Node and Mesh Analyses 17

2. Loop analysis 2.2. Loop analysis for LTI networks o Branch equations are written in matrix form as: o R is a diagonal branch resistance matrix of dimension b. o v s and j s are voltage source and current source vectors. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 18

2. Loop analysis 2.2. Loop analysis for LTI networks o Z l is loop impedance matrix of order l. o e s is loop voltage source vector. o Loop impedance matrix has properties similar to those of mesh impedance matrix discussed in previous chapter. o R is symmetric. o Z l is symmetric. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 19

2. Loop analysis 2.2. Loop analysis for LTI networks Example: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 20

2. Loop analysis 2.2. Loop analysis for LTI networks Example: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 21

2. Loop analysis 2.3. Properties of the loop impedance matrix o Analysis of a resistive network and sinusoidal steady-state analysis of a similar network are very closely related. o Main difference is in the appearance of phasors and impedances. 1. If network has no coupling elements: branch impedance matrix Z b (jω) is diagonal, loop impedance matrix Z l (jω) is symmetric. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 22

2. Loop analysis 2.3. Properties of the loop impedance matrix 2. If network has no coupling elements, Z l (jω) can be written by inspection: a. z ii is equal to sum of impedances in loop i. z ii is called self-impedance of loop i. b. z ik is equal to plus or minus sum of impedances of branches common to loop i and to loop k. Plus sign applies if, in branches common to loop i and loop k, loop reference directions agree. Minus sign applies when they are opposite. 3. If all current sources are converted, by Thevenin's theorem, into voltage sources, forcing term e si is algebraic sum of all source voltages in loop i. Voltage sources whose reference direction pushes current in i th loop reference direction are assigned a positive sign, others a negative sign. 4. If network is resistive and if all its resistances are positive, det(z l ) > 0. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 23

3. Cut-set analysis 3.1. Two basic facts of cut-set analysis o Cut-set analysis is dual of loop analysis. o First, we pick a tree (T). o Next we number branches; as before, links range from 1 to l, tree-branches range from l+1 to b. o We know that every tree branch defines (for given tree) a unique fundamental cut set. o That cut set is made up: of links and of one tree branch (which defines cut set). Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 25

3. Cut-set analysis 3.1. Two basic facts of cut-set analysis o Let us number cut sets as follows: cut set 1 is associated with tree branch 5, cut set 2 with tree branch 6, etc. o For each fundamental cut set, we adopt a reference direction for cut set which agrees with that of tree branch defining cut set. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 26

3. Cut-set analysis 3.1. Two basic facts of cut-set analysis o Under these conditions, if we apply KCL to 4 cut sets: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 27

3. Cut-set analysis 3.1. Two basic facts of cut-set analysis o n linear homogeneous algebraic equations in j 1, j 2,..., j b obtained by applying KCL to each fundamental cut set constitute a set of n linearly independent equations. o KCL equations are of form: o Q = [q ik ] is an n b matrix and called fundamental cut-set matrix: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 28

3. Cut-set analysis 3.1. Two basic facts of cut-set analysis E is an appropriate n l matrix with elements -1, +1, 0. 1 n is n n unit matrix. Q has a rank n. n fundamental cut-set equations in terms of branch currents are linearly independent. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 29

3. Cut-set analysis 3.1. Two basic facts of cut-set analysis o From KVL, each branch voltage can be expressed as a linear combination of tree-branch voltages. o For convenience, let us label tree-branch voltages by e 1, e 2,..., e n : Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 30

3. Cut-set analysis 3.2. Cut-set analysis for LTI networks o For case of LTI resistive networks, branch equations are easily written in matrix form: o G is diagonal branch conductance matrix of dimension b. o j s and v s are source vectors. o KCL and KVL are: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 31

3. Cut-set analysis 3.2. Cut-set analysis for LTI networks o Y q is cut-set admittance matrix. o i s is cut-set current source vector. o In scalar form, cut-set equations are: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 32

3. Cut-set analysis 3.3. Properties of cut-set admittance matrix For sinusoidal steady-state analysis, cut-set admittance matrix Y q is: 1. If network has no coupling elements: branch admittance matrix Y b (jω) is diagonal, cut-set admittance matrix Y q (jω) is symmetric. 2. If network has no coupling elements, Y q (jω) can be written by inspection: a. y ii is equal to sum of admittances of branches of the i th cut set. b. y ik is equal to sum of all admittances of branches common to cut set i and cut set k when, in branches common to their 2 cut sets, their reference directions agree; otherwise, y ik is negative of that sum. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 33

3. Cut-set analysis 3.3. Properties of cut-set admittance matrix 3. If all voltage sources are transformed to current sources, i sk is algebraic sum of all current sources in cut set k. Current sources whose reference direction is opposite to that of k th cut set reference direction are assigned a positive sign. All others are assigned a negative sign. 4. If network is resistive and if all its resistances are positive, det(y q ) > 0. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 34

3. Cut-set analysis 3.3. Properties of cut-set admittance matrix Example: Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 35

4. Comments on loop and cut-set analysis o Both loop analysis and cut-set analysis start with choosing a tree for given graph. o Since number of possible trees for a graph is usually large, 2 methods are extremely flexible. o They are more general than mesh analysis and node analysis. o Fundamental loops for particular tree coincide with 4 meshes of graph. o Thus, mesh currents are identical with fundamental loop currents. o Mesh analysis for this particular example is a special case of loop analysis. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 37

4. Comments on loop and cut-set analysis o Fundamental cut sets for particular tree coincide with sets of branches connected to nodes 1, 2, 3, and 4. o If node 5 is picked as datum (reference) node, tree-branch voltages are identical with node-to-datum voltages. o Node analysis for this particular example is a special case of cut-set analysis. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 38

4. Comments on loop and cut-set analysis o It should be pointed out that for this graph, meshes are not special cases of fundamental loops. o There exists no tree such that 5 meshes are fundamental loops. o Similarly, in this graph, if node 4 is picked as datum node, there exists no tree which gives tree-branch voltages identical to node-to-datum voltages. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 39

4. Comments on loop and cut-set analysis o As far as relative advantages of cut-set analysis and loop analysis, conclusion is same as that between mesh analysis and node analysis. o It depends: on graph as well as on kind and number of sources in network. o For example, if number of tree branches (n) is much smaller than number of links (l), cut-set method is usually more efficient. o In previous chapter, duality applied only to planar graphs and planar networks (node and mesh analysis). o Now, duality extends to non planar graphs and networks: For example, cut sets and loops are dual concepts. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 40

5. Relation between B and Q o If we: start with an oriented graph G, pick any one of its trees, say T, write fundamental loop matrix B and fundamental cut-set matrix Q, we should expect to find a very close connection between B and Q. o B tells us which branch is in which fundamental loop. o Q tells us which branch is in which fundamental cut set. o Precise relation between B and Q is stated in following theorem. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 42

5. Relation between B and Q THEOREM Call B fundamental loop matrix and Q fundamental cut-set matrix of same oriented graph G, and let both matrices pertain to same tree T; then: If we number links from 1 to l and number tree branches from l+1 to b, then: Product of l b matrix B and b n matrix Q T is l n zero matrix. Product of every row of B and every column of Q T is 0. Product of every row of Q by every column of B T is 0. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 43

5. Relation between B and Q o Relation between B and Q is extremely useful. o Whenever we know one of these matrices, we can write other one by inspection. o Both matrices B and Q are uniquely specified by l n matrix F. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 44

6. Summary o Analogies between 4 methods of analysis deserve to be emphasized: o Each one of connection matrices A, M, Q, and B is of full rank. Electric Circuits 2 Chapter 3. Loop and Cut-set Analysis 46