Reference node

Notice that we have added a bubble at the voltage reference node so that we can easily plot the voltage. The name for this part in Capture is Vcc. Select Place and then Power from the menus to place it. We will run the same temperature analysis as... 

Fig. 26 Graph Theoretical Distances (GTD) From Reference Node (0) in Ceo...

In nodal analysis, the voltages between adjacent nodes of the network are chosen as the unknowns. This can commonly be achieved by selecting a reference node from the graph of the network. Equations are then formed if KCL is employed. By equating the sum of the currents flowing through admittances associated with one node to the sum of the currents flowing out of the current sources associated with the same node, a set of equations can be established with the form of [F][V] = [/] ... 

Tn, T22, T33,. .., T v ijA- i, known as the self node admittances, are given by the sum of all admittances directed to a given node with all other nodes shorted to the reference node. 

Each node mutual admittance 7 (i k) is the sum of the admittances between two given nodes i and k. The current Y kV in the mutual admittances between nodes i and k is negative if the voltages of nodes i and k have the same assumed polarity relative to the reference node. The current Y kV is positive if the voltages of nodes i and k have the opposite assumed polarity relative to the reference node. In a linear network, we have... 

Every worksheet should be provided with an identification and a means to correlate it to the node and design conditions it was evaluated against. Locations for date, location, drawing reference, node identification or description, and design parameters should be noted on each worksheet. 

For instance, in an electrical circuit diagram the unwritten, heuristic rule is that the reference node is at the bottom, nodes related to sources at the left, and nodes related to loads at the right, while the nodes are kept as much as possible at a grid and the symbols of the labeled edges are connected to the nodes by straight lines (Fig. 1.9). 

This makes clear that any two electrical terminals, also if one is grounded and not represented as such, form a port. However, in many treatments of electrical circuits the logical order is reversed First, a distinction is made between potential and potential differences. Next a relation between the potential differences and the potentials is derived via the incidence matrix and only then it is recognized that one of the rows (balance equations) refers to a reference node (ground) which should be omitted from the incidence matrix to obtain the so-called reduced incidence matrix. This culture may have led Willems to drawing this conclusion. 

As each conserved state determines a domain, additional connection constraints can be found for various port types. For instance, a bond connected to one side of a 0-junction may be connected to a C-type storage port or a source port, as these ports do not violate the balance equation. However, in principle, one should be more careful when connecting an I-type, R-type, TF-type, or GY-typeport, because these ports cannot absorb the conserved state related to the flow. However, all domains with relative equilibrium-determining variables have a non-displayed balance for the reference node (this balance equation is dependent on the balance equations for the rest of the network and corresponds to the row that is omitted in an incidence matrix to turn it into a reduced incidence matfix of an electrical circuit, for example). This additional balance compensates for this flow, such that it is still possible to connect these ports without violating the balance equation. Note that the I-type port in principle is a connection to a GY-type port that connects to the storage in another domain. Some domains have absolute equilibrium-determining variables, like temperature and pressure, but since in most cases it is not practical to choose the absolute zero point as a reference, usually another reference state is chosen, such that these variables are treated as differences with respect to an arbitrary reference and an additional balance too. 

Let us arrange the subgraphs G° in the manner that G , , G - K < K) are those subgraphs which are not isolated nodes observe that if G is an isolated node, the corresponding (scalar) equation (node balance) in (3.2.2) becomes automatically one of the node balances of the reduced graph G. Having selected a reference node in each G for k = I, , K , let B be the reduced incidence matrix of G, thus B is of full row rank let further A, be the corresponding... 

Having completed the classification according to steps (a) and (b) above, we can make use of the additional information obtained in the described manner. First, the reduced graph G determines, having selected a reference node, the reduced incidence matrix A. It is the matrix occurring in (3.2.4)2, thus in the constraint equation for the measured vector m (in fact, only for the subvector m of redundant variables). The equation is employed for adjusting the given values if the components of m"" have been actually measured then for reconciliation by statistical methods. 

The inverse R2 of the reduced incidence matrix is found as in the example at the end of Section A.3 for each node n e N2, the reference node excluded, we go backwards through the sequence of predecessors. We have... 

In addition, the incidence matrix of G determines the conditions the measured variables have to obey in order to have the system solvable. Taking node , as reference node, the reduced incidence matrix equals (cf. Fig. 3-9c)... 

Irrespective of how we have obtained the spanning tree G°, we choose a reference node in the mass balance problem, it will be clearly the environment node, thus C is the matrix in (3.1.6). We now can classify the nodes (units) n N = N - o according to their distance from Mq. Thus N is partitioned... 

The matrix C of elements C j (n e N , 7 e J) is the reduced incidence matrix of graph G, with the environment node as reference node see Chapter 3. Let us now admit the presence of energy distributors. Then the matrix D of... 

As shown in Section A.2, G being a tree, having fixed the reference node o > for any node n ttg there exists a unique sequence of arcs from n to n. Let us call j(n) the last arc of this sequence, thus n is endpoint of j(n) let p(n) be the other endpoint that precedes n in the corresponding sequence of endpoints. Let us call predecessors of n the members of the nodes sequence, with the exception of n itself Ietp( ) be (called) the last predecessor. Clearly, if m is a predecessor of n and m n then p(m) is also a predecessor of n by the uniqueness of the sequence, and j m) is a member of the sequence of arcs from /Iq to n. In Section A.4, we will show an effective way of finding the predecessors. Observe also that any arc y e J equals j(n) for a unique n ), the other endpoint of j being the last predecessor of n. 

The algorithm is basic and makes different other operations possible. Observe that the subsets N above represent the sets of nodes of distance p from node n, this is a meaningful classification for a connected component, if node n, is regarded as reference node. Having a connected graph G [N, J] (or a connected component as found above), we can find a spanning tree T [N, J ] from node n,. We suppose again J 0. Then... 

The node nj stands for the reference node n introduced in Section A.3. If in particular G is a tree then jin) is that introduced before the formula (A.5), and pin) is the last predecessor of node n. More generally, T and J are those introduced later in Section A.3 and in particular, the knowledge of jin) enables one to compute the inverse R of A (reduced incidence matrix of T). 

The case (a) need not be considered in detail. If the deletion of any arc of G disconnects G then G is a tree and the formula (A. 18) holds. Else we can find, in G, an arc j with endpoints denoted as , and Bq, which obeys hypothesis (b). We can take as the reference node and n, as above. [Generally, if n is reference node and n another node, by addition of columns n n ) to column n and then of rows n ) to row n we obtain new matrix Z that corresponds to the choice of as reference node, while the determinant remains unaltered. Thus the determinant detZ is, indeed, independent of the choice of the reference node.]... 

Moreover, we use an improved version of the adaptive mechanism proposed in [4] in which just an automatically elected reference node, namely the one with the lowest ID among the active ones, senses the delays in the team transmissions and adjusts the round phase. All other nodes transmit after receiving from the reference node in each round and with an offset equal to that of the respective slot (Fig. 3). The reference node is automatically reassigned in case the current one leaves the team. 

The reference node (node 0) determines its transmission instant in the following round (I +l) as follows, considering the previously computed transmission instant for the current round (i) and the delays 5 suffered by nodes 1 to k in this round (if lower than the maximum allowed A adjustment) ... 

If the reference node does not receive from any of the other nodes during one round (or receives with a delay larger than A) then it transmits Ttup later. Similarly, if the other nodes do not receive from the reference node in one round, they tfansmit T p after their previous transmission. 

Fig. 3. The improved Adaptive TDMA protocol with a single adapting reference node...

This mechanism is particularly better than the originally proposed one when in the presence of an AP with the power management feature active and a fairly long DTIM period. In such case the transmissions instants in the former approach would be altered and the team would not synchronize. With the new mechanism, the reference node transmission instants may still be altered (implying a potential large error in % i) but this will impact the start of the round, only. The remaining transmissions in the team are stili separated in time according to the respective slots in the round. 

Fig. 5. Transmission offsets in the round with respect to the reference node in both clock synchronized and Adaptive TDMA with 1KB ping every 20ms of external traffic...

As a result of this systematic process four points or reference nodes were identified and considered relevant, shown in Figure 4, and the parameters and deviations associated with the guide words are described in Table 4. For each deviation identified, in each node, the causes for the events were investigated using the technique of tree failures that allows us to identify faults in simultaneously. Through the FMEA technique allowed us to identify the medium to detect these causes and present the possible maintenance policies. 

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