What is a super node

Basics of electrical engineering 3

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1 Lorenz-Peter Schmidt Gerd Schaller Siegfried Martius Basics of Electrical Engineering 3 Networks an imprint from Pearson Education Munich Boston San Francisco Harlow, England Don Mills, Ontario Sydney Mexico City Madrid Amsterdam

2 Analysis of networks 3.1 Introduction of the mesh flow method Node potential method OVERVIEW

3 3 CHAPTER 3: ANALYSIS OF NETWORKS 3.1 Introduction The analysis of small networks with a few nodes and branches can usually be carried out with little effort using Kirchhoff's rules. In this chapter we want to address networks with greater complexity and introduce network analysis methods that keep the computational effort manageable and, through systematization, can potentially also be used in network analysis programs. Here, too, we want to limit ourselves to linear networks and use the complex vector representation for the operational variables. The (still quite manageable) circuit in Figure 3.1 with K = 5 nodes and Z = 9 branches serves as an example. Figure 3.1: Network with K = 5 nodes and Z = 9 branches Since transmitters and controlled sources should initially be excluded, each branch n between two nodes A and B consists of an impedance Z n or admittance Y n and a constant voltage source U qn or a constant current source I qn, as shown in Figure 3.2. The voltage U n along the branch n, which connects the nodes A and B, is referred to as the branch voltage, the current I n, which flows in and out of the branch, is referred to as the branch current. Convention: We agree that branch voltage and branch current in a branch always have the same direction, as can be seen in Figure 3.2. There may be branches in which the source voltage or the source current disappears to zero, so that only the branch impedance or admittance remains. In a complete analysis of the circuit according to Figure 3.1, Z branch voltages and Z branch currents are searched for, so there are formally 2 Z unknowns. Through A U n B A U n B I n Z n U qn I n Iqn Y n Figure 3.2: Branch n between nodes A and B with constant voltage source or constant current source 56

Using the current / voltage relationships for each branch (equation (3.1)), we can reduce the number of unknowns to Z. U n = Z n I n U qn or I n = Y n U n I qn (3.1) With a conventional approach, we can use Kirchhoff's laws for circuit analysis as follows: Nodal equations: There are K 1 linearly independent nodal equations; the K-th node can be represented by a super node of the other nodes and would therefore only result in a linearly dependent equation. Mesh equations: Z (K 1) independent mesh equations are still required to arrive at the total number of Z equations for our Z unknowns. When setting up the linearly independent mesh equations, it must be ensured that each new mesh contains at least one branch that has not been recorded before and that each branch is run through at least once. In our example, four nodal equations and five mesh equations are required, which together result in an equation system of nine equations with nine unknowns. Although this network still has a very low level of complexity, a rather extensive system of equations has to be solved using the conventional approach in order to calculate the branch voltages and branch currents that are initially unknown. In the further course of this chapter, two methods are described in which the number of variables in the system of equations to be solved is significantly lower and the setting up of the equations can also be formalized so that a computer-aided network analysis is possible. 3.2 Mesh flow method The network in Figure 3.3 is used as an example to explain the mesh flow method. For the elements occurring in the network, we will initially limit ourselves to R, L, C elements and independent voltage sources. We are looking for all branch currents and branch voltages in the network. With a conventional solution with direct listing of Kirchhoff's equations for the six branch voltages and six branch currents, we need K 1 = 3 node equations and Z (K 1) = 3 mesh equations and thus get six equations with six unknowns, with which we can calculate the branch voltages, for example . The branch currents result from this with the help of the current / voltage relationships for each branch (equation (3.1)) Mesh currents With the aim of reducing the number of variables and thus the size of the system of equations to be solved, we have so-called mesh currents 57 in the network

5 3 CHAPTER 3: ANALYSIS OF NETWORKS I 3 U q3 DZ 3 I M2 I 2 AZ 1 Z 2 I M1 I 5 I 6 Z 5 CZ 4 I 4 I M3 Z 6 I 1 U q1 B Figure 3.3: Network with mesh flows I. M1, I M2, I M3, which represent self-contained current paths. In our example in Figure 3.3, these are the mesh currents I M1, I M2, I M3 that have already been drawn in. By means of these mesh currents, Kirchhoff's node equations are fulfilled from the start, since each mesh current flows into and out of a node under consideration. In each branch, the introduced mesh flows are superimposed (with the correct sign) to form the entire branch flow. This allows a link between the six branch currents and the three mesh currents in the following form: I 1 = I 2 = I 3 = I M1 I M1 I M2 I M2 I 4 = I M1 I M3 I 5 = I M2 I M3 I 6 = I M3. (3.2) These six equations can also be represented more compactly in matrix notation. To do this, we introduce the column vector I of the branch currents and the column vector IM of the mesh currents, which are linked to one another by the so-called incidence matrix A according to equation (3.2): I = IM or I = AIM (3.3) Obviously, the elements a ij of the incidence matrix have the value 0 or +1 or 1, depending on whether the relevant branch i is not traversed by the mesh flow j or not or in the same direction or in the opposite direction to the branch flow. 58

6 3.2 Mesh flow method In the next step, mesh circulations are now carried out with the help of the branch voltages along the individual mesh flows (in the mesh flow direction). This results in the equations U 1 + U 2 U 4 = 0 U 2 + U 3 U 5 = 0 U 4 U 5 + U 6 = 0. (3.4) The matrix representation of these equations shows that the branch voltages over the transpose of the incidence matrix are linked with each other: U 1. U 6 = 0 or AT U = 0. (3.5) The same mesh equations should now not be set up with the branch voltages, but with the help of the mesh currents, in order to reduce the number of variables from the outset. When performing the mesh revolutions, we must of course always take into account all mesh currents (with the correct sign) that flow through this impedance for the voltage drops at the individual impedances. If we write the constant voltages occurring in the revolutions on the right-hand side of the equation, the following three equations result: Z 1 I M1 + Z 2 (I M1 I M2) + Z 4 (I M1 + I M3) = U q1 Z 2 ( I M2 I M1) + Z 3 I M2 + Z 5 (I M2 + I M3) = U q3. Z 4 (I M3 + I M1) + Z 5 (I M3 + I M2) + Z 6 I M3 = 0 (3.6) We can still sort the terms on the left-hand side of the equation according to the mesh flows and get three equations with the three mesh flows as unknowns: (Z 1 + Z 2 + Z 4) I M1 Z 2 I M2 + Z 4 I M3 = U q1 Z 2 I M1 + (Z 2 + Z 3 + Z 5) I M2 + Z 5 I M3 = U q3 Z 4 I M1 + Z 5 I M2 + (Z 4 + Z 5 + Z 6) I M3 = 0. (3.7) In the matrix representation we finally get the following equation: Z 1 + Z 2 + Z 4 Z 2 Z 4 Z 2 Z 2 + Z 3 + Z 5 Z 5 Z 4 Z 5 Z 4 + Z 5 + Z 6 I M1 I M2 I M3 U q1 = U q3. (3.8) 0 The system of equations (3.7) or the matrix equation (3.8) can be solved for the mesh flows using common methods of linear algebra. Once the mesh currents have been determined, all branch currents can be calculated using equation (3.3) and the individual branch voltages can be calculated using the current / voltage relationships in equation (3.1). The network is now in the figure

7 3 CHAPTER 3: ANALYSIS OF NETWORKS fully analyzed. By introducing the mesh flows, we have achieved that only an equation system of three equations with three unknowns has to be set up or solved. Let us now turn to the obvious laws according to which the matrix equation (3.8) is built. For this purpose, let us realize that the three equations and thus the three lines of the impedance matrix in equations (3.8) are the result of the three mesh circulations along the three mesh flows: The main diagonal elements Z ii of the matrix contain the sums of all network impedances that occur in the respective mesh circulations be touched. The impedance matrix is ​​symmetrical to the main diagonal as long as there are no controlled sources in the network. The elements Z ij outside the main diagonal are formed by the sum of those network elements through which the mesh currents I Mi and I Mj flow together. The column vector on the right-hand side of the matrix equation contains (with the correct sign) the voltages of the constant voltage sources that are detected by the relevant mesh circulation. If only individual currents or voltages are searched for in the network, it is advisable to select the mesh currents in such a way that the branches with the desired sizes are traversed by mesh currents only once. Procedure for the mesh flow method: Select suitable meshes and mesh flows Establish relationships between mesh and branch flows Apply Kirchhoff's rule of meshes to all meshes with mesh flows as variables Solve a linear system of equations Calculate branch currents with the help of the incidence matrix Calculate branch voltages using current / voltage relationships for the individual branches Networks it becomes increasingly difficult to ensure the linear independence and completeness of the mesh equations. For the automation of the process it is also necessary to be able to use a simple and operationally reliable process with which linearly independent mesh flows can be established. For this purpose we use the so-called graph representation for a network, which has the following properties: 60

8 3.2 Mesh flow method Graph display: The node structure of the network is retained Branches are replaced by connecting lines between nodes Voltage sources are set to zero, i.e. replaced by short circuits in the graph Current sources are also set to zero, i.e. replaced by idle circuits in the graph As an example, we convert using these Convert the network in Figure 3.1 into a graph: Figure 3.4: Graph (or complex) of the network from Figure 3.1 The network and thus also the graph consists (as before) of K = 5 nodes and Z = 9 branches. In the next step we convert the graph into a complete tree by removing just enough branches that there are no more meshes in the entire network. However, all nodes must remain connected to one another. We then get, as can easily be seen, a tree with exactly K 1 tree branches. So Z (K 1) independent branches that do not belong to the tree have been removed. It turns out that several different trees can be found that meet all of the requirements mentioned. Figure 3.5 shows two trees (from a large number of possible trees) for the network under consideration. or Figure 3.5: Two complete trees for the network under consideration In both cases, K 1 = 4 tree branches and Z (K 1) = 5 independent branches. We find that every node of the network can be reached via tree branches. With each independent branch of the network, exactly one mesh can be formed which, besides this independent branch, only contains tree branches. In this way, exactly Z (K 1) mesh flows can be determined which are linearly independent with certainty and which completely describe the network. 61

9 3 CHAPTER 3: ANALYSIS OF NETWORKS Figure 3.6: Representation of the considered network with tree branches (thick lines) and independent branches (thin lines). Also shown are the Z (K 1) = 5 mesh circulations, which each contain only one independent branch and are defined by the mesh flows that are independent of one another. As an example, for the first of the two trees from Figure 3.5, exactly the required number of meshes Z (K 1) = 5 is shown in Figure 3.6. With this method it can therefore be ensured in a simple manner that the required number of linearly independent meshes are precisely defined even in very complex networks. The mesh current method can then be used in the manner described. Sources and transmitters in the network So far, for the sake of simplicity, we have assumed that our network only contains R, L, C elements and constant voltage sources, but no constant current sources, no controlled sources and no transformers. In the following, we want to remove this restriction and deal with sources and transmitters in the network in general. Constant voltage sources: It is useful to design the tree for a network in such a way that the voltage sources are in independent branches. Then each source appears only in one mesh and is only present in one equation in the system of equations to be set up. Constant current sources: 1. Constant current sources are converted into constant voltage sources before the network analysis (see chapter 1.1). 2. With Y i = 0, a direct conversion of the current source into a voltage source is not possible. However, pre-split and offsetting can correct this problem. Controlled sources: 1. All controlled sources are first converted into mesh current controlled voltage sources. 2. When setting up the system of equations, they are initially treated like constant voltage sources and become elements of the voltage vector on the right-hand side of the matrix equation (compare: Equation (3.7) and (3.8)). 62

10 3.2 Mesh flow method 3. In a subsequent calculation step, the dependency of the controlled sources on the mesh flows is used and taken into account on the left-hand side of the equation for the mesh flow coefficients. Transmitter: 1. If a transmitter is included in the network, the meshes are first determined in the usual way. Each transmitter side is treated like a branch of the network. 2. If there is a transformer winding in the mesh circulation, then in addition to the current through this winding, the mesh current through the coupled winding must be taken into account (via the mutual inductance M) when setting up the mesh equations completed in matrix representation. Using the equations (3.3) and (3.5) and combining all the links between branch voltages and branch currents to form a matrix equation, we can determine the inhomogeneous system of equations for calculating the mesh currents, as an alternative to the calculation in Chapter, also very easily in matrix notation in a generally valid form: Combination of Branch currents and mesh currents: I = AIM with incidence matrix A [ZZ (K 1)] (3.9) Establishment of Kirchhoff's mesh equations: ATU = 0 [Z (K 1) equations] (3.10) Linking branch voltages and branch currents: U = ZIU q [ Z is the diagonal matrix of the branch impedances] (3.11) Inserting equation (3.9) into (3.11) and inserting the resulting equation into equation (3.10) results in the system of equations for the mesh currents: ATZAIM = ATU q [Z (K 1) equations] (3.12 ) The matrix ATZA = ZM is called the mesh impedance matrix. The system of equations can thus be set up directly from the easy-to-create sub-matrices according to equation (3.12) (after preparing the network according to chapter and and with consistent consideration of all signs!), Which is very useful for an automated circuit analysis. Literature: [5], [11], [20] 63

11 3 CHAPTER 3: ANALYSIS OF NETWORKS 3.3 Knot potential method The knot potential method can be viewed as a dual method for mesh flow analysis. Its use has advantages in networks with many branches and few nodes and in cases where there are numerous power sources in the network. In addition, the node potential method has advantages when used in circuit simulation programs. The circuit in Figure 3.7 is used as an exemplary network, which is the result of converting the voltage sources into current sources from Figure 3.3. For further considerations, we want to restrict ourselves to networks with R, L, C elements and constant current sources. Node potentials and node voltages We first assign a potential to each node in the network, so we start from the potentials ϕ in our exemplary circuit in Figure 3.7 A, ϕ B, ϕ C, ϕ D at the K = 4 nodes, which are marked with the letters A, B, C, D. In the next step we select one of the nodes as a reference or reference node and define so-called node voltages as potential differences between the relevant node and the reference node for the remaining K 1 nodes. In the present case we declare node D to be the reference node and obtain the three node voltages: U K1 = ϕ A ϕ D; U K2 = ϕ B ϕ D; U K3 = ϕ C ϕ D (3.13) From Figure 3.7 it is easy to see that each branch voltage can be expressed by one or (over a loop of the mesh) by several node voltages.This combination of branch voltages and node voltages is shown by the following system of equations. It should be noted that U K1 Y 3 DI 3 AI q3 Y 2 I 2 U K3 CI 5 I6 Y 5 Y 6 U K2 I 1 I q1 Y 4 Y 1 I 4 B Figure 3.7: Network example with reference node D and node voltages U K1, U K2, U K3 64

12 3.3 Node potential method according to Figure 3.2 Branch voltages and branch currents in a branch always have the same direction. U 1 = U K1 + U K2 U 2 = U K1 U K3 U 3 = U 4 = U 5 = U 6 = U K1 U K2 U K3 U K3 U K2 (3.14) We can again represent and maintain the system of equations in matrix notation the incidence matrix B as a link between the column vector U of the branch voltages and the column vector UK of the node voltages: U = UK or U = BU K. (3.15) Here, too, only the values ​​0, +1, 1 appear in the incidence matrix for the matrix elements b ij depending on whether the respective knot tension is used for the individual loop loops and whether the directions of loop loop and the respective knot tension are the same or opposite. Just as in the mesh flow method by introducing the mesh currents, Kirchhoff's nodal equations are fulfilled from the start, in the nodal potential method by introducing the nodal voltages, Kirchhoff's mesh equations are fulfilled from the outset. With the K 1 nodal voltages we have again introduced a compact set of unknowns for which we can find a solution by setting up the K 1 linearly independent Kirchhoff equations. First we set up the K 1 = 3 node equations for the nodes A, B, C of our example network with the help of the branch currents, whereby outflowing currents are given a positive sign: I 1 + I 2 + I 3 = I 1 + I 4 + I 6 = 0 or I = 0. (3.16) I 2 I 4 + I 5 = The matrix representation of these equations shows that the branch currents are linked to one another via the transpose of the incidence matrix B: BTI = 0 (3.17) 65

13 3 CHAPTER 3: ANALYSIS OF NETWORKS We can set up the same nodal equations as a function of the branch voltages by using the links between the branch currents and the branch voltages according to equation (3.1). (This gives, for example, I 1 = Y 1 U 1 + I q1 etc.) The node equations are thus: Y 1 U 1 + Y 2 U 2 + Y 3 U 3 = I q1 + I q3 Y 1 U 1 + Y 4 U 4 + Y 6 U 6 = I q1 Y 2 U 2 + Y 4 U 4 + Y 5 U 5 = 0. (3.18) Now, with the help of equations (3.14), the branch voltages in the equation system (3.18) are converted into node voltages replaced, the result is the inhomogeneous system of equations sought to calculate the initially unknown node voltages (in matrix notation): Y 1 + Y 2 + Y 3 Y 1 Y 2 Y 1 Y 1 + Y 4 + Y 6 Y 4 Y 2 Y 4 Y 2 + Y 4 + Y 5 U K1 U K2 U K3 I q1 + I q3 = I q1. (3.19) 0 This inhomogeneous system of equations can be solved for the nodal stresses using methods of linear algebra. If the node voltages are known, we can use equations (3.14) to calculate all branch voltages from them, and equation (3.1) to calculate all branch currents in the network. By introducing the nodal voltages, we have achieved that only an equation system of three equations with three unknowns has to be set up or solved. The somewhat cumbersome way to set up the system of equations, which we have chosen here for the purpose of a clear derivation, can be shortened if we have looked at the formation laws of the matrix equation. It should be noted that each line of the system of equations or the matrix equation is assigned to a node in the network. The main diagonal elements Y ii of the admittance matrix contain the sums of all network admittances that are connected to the relevant node. The admittance matrix is ​​symmetrical to the main diagonal as long as there are no controlled sources in the network. The elements Y ij outside the main diagonal are formed by the negative sums of those admittances that lie on the path between the node i and the node j. The column vector on the right-hand side of the matrix equation contains (with the correct sign) the sum of the source currents of all branches connected to the respective node. Currents flowing to the node have a positive sign. 66

14 3.3 Node potential method Basic procedure for the node potential method: Define reference nodes and K 1 node voltages Establish relationships between node and branch voltages Apply Kirchhoff's knot rule to K 1 nodes with node voltages as variables Solve a linear system of equations Calculate branch voltages using an incidence matrix from the node voltages Determine branch currents (using current / voltage relationships for the individual branches) Determination of the node voltages In principle, the reference node can be freely selected for the application of the node potential method. Usually a central node is chosen that has as many direct connections as possible to other network nodes. This is the ground node in many networks. The completeness and linear independence of the equation system can be ensured very easily. A reference node is selected in the network with K nodes, and K 1 node voltages are necessary and sufficient for a complete network description. Since the complete tree of a network contains exactly K 1 tree branches, it is a safe practice to select the voltages on the K 1 tree branches as nodal voltages. Each node voltage then extends over exactly one branch of the tree. If possible, the tree is created in such a way that the tree branches run from a central reference node directly to the other nodes RK 1 (a) (b) Figure 3.8: (a) Graph of a network with 7 nodes and 14 branches, (b) Tree for this network with a central reference node Figure 3.8 shows a network example with K = 7 nodes and Z = 14 branches and a tree is drawn in the manner described. 67

15 3 CHAPTER 3: ANALYSIS OF NETWORKS By the way: For the mesh flow method, Z (K 1) = 8 mesh flows would be required for this network, with the node potential method K 1 = 6 node voltages are sufficient! Each branch voltage on an independent branch can be determined with a clear mesh circulation exclusively via tree branches (node ​​voltages). For each tree branch (and the associated node voltage), a cut can be made through the network in an unambiguous way, in which only this one tree branch is cut. This intersection defines a knot or super knot and the associated Kirchhoff knot equation. As mentioned, with the node potential method it is usually advantageous to choose a star-shaped tree with a central reference node, which is usually identical to the ground node. However, there are also cases in which no node can be found from which all other nodes can be reached directly. Such a case is sketched as an example in Figure 3.9. In these cases, we continue to proceed in such a way that node voltages are chosen along tree branches, even if they do not all start from the reference node, as is the case in the example for tree branch 5. (This generalization of the node potential method is also referred to as intersection analysis in the literature.) Since the tree branch 4 does not have an open end, a super node is defined in this case, which in turn only intersects the associated tree branch and otherwise only independent branches. For this super node, the Kirchhoff node equation belonging to tree branch 4 and the assigned node voltage can now be set up RK (a) (b) Figure 3.9: (a) Graph of a network with 6 nodes and 11 branches, (b) Tree for this network with reference nodes which is not a central node Sources and transmitters in the network Here, too, we want to remove the restriction to networks with R, L, C elements and constant current sources and extend our consideration to circuits with controlled and constant sources and transmitters. 68

16 3.3 Node potential method, constant current sources: It is advisable to select the tree for a network with the node potential method in such a way that the constant current sources are only located in tree branches. Then each source occurs only in the nodal equation belonging to this branch of the tree. Constant voltage sources: 1. Constant voltage sources are converted into constant current sources before the network analysis (see Section 1.1). 2. With Z i = 0, a direct conversion of the voltage source into a current source is not possible. However, source sharing and relocation can correct this problem. Controlled sources: 1. All controlled sources are first converted into node voltage controlled current sources. 2. When setting up the system of equations, they are initially treated like constant current sources and become elements of the current vector on the right-hand side of the matrix equation (compare: Equation (3.19)). 3. In a subsequent calculation step, the dependency of the controlled sources on the node voltages is used and taken into account on the left side of the equation in the admittance matrix. Transformer: 1. An equivalent circuit with two voltage-controlled current sources can be used for transmitters in the network (see Figure 3.10). The admittances in the equivalent circuit can easily be obtained from the usual transformer equations in impedance form by inverting the system of equations. I 1 I 2 Y 12 á U 2 U 1 U 2 Y 11 Y 21 á U 1 Y 22 Figure 3.10: Transformer equivalent circuit with voltage-controlled current sources 69

17 3 CHAPTER 3: ANALYSIS OF NETWORKS Matrix representation Even with the node potential method, when using the matrix representation, we can come to the system of equations in a very simple and clear manner, which enables a calculation of the selected node voltages and thus a complete analysis of the network: Linking branch voltages and node voltages: U = BUK with incidence matrix B [K 1 Z] (3.20) Establishment of Kirchhoff's nodal equations: BTI = 0 [K 1 equations] (3.21) Linking branch currents and voltages: I = YUI q [Y is diagonal matrix of branch admittances] (3.22 ) Inserting equation (3.20) into (3.22) and inserting the resulting equation into equation (3.21) results in the equation system for the node voltages: BTYBUK = BTI q [K 1 equations] (3.23) The matrix BTYB = YK is called the node admittance matrix. The system of equations can in turn, after suitable preparation of the network according to Sect. This means that there are good prerequisites for automated circuit analysis. Literature: [5], [11], [20] 70