In this paper we show how to extend the known algorithm of nodal analysis in such a way that, in the case of circuits without nullors and controlled sources (but allowing for both, independent current and voltage sources), the system of nodal equations describing the circuit is partitioned into one part, where the nodal variables are explicitly given as linear combinations of the voltage sources and the voltages of certain reference nodes, and another, which contains the node variables of these reference nodes only and which moreover can be read off directly from the given circuit. Neither do we need preparational graph transformations, nor do we need to introduce additional current variables (as in MNA). Thus this algorithm is more accessible to students, and consequently more suitable for classroom presentations.
Deep Dive into Supernodal Analysis Revisited.
In this paper we show how to extend the known algorithm of nodal analysis in such a way that, in the case of circuits without nullors and controlled sources (but allowing for both, independent current and voltage sources), the system of nodal equations describing the circuit is partitioned into one part, where the nodal variables are explicitly given as linear combinations of the voltage sources and the voltages of certain reference nodes, and another, which contains the node variables of these reference nodes only and which moreover can be read off directly from the given circuit. Neither do we need preparational graph transformations, nor do we need to introduce additional current variables (as in MNA). Thus this algorithm is more accessible to students, and consequently more suitable for classroom presentations.
It is a well known fact -already taught in most undergraduate courses on circuit theory [1] -that, when the node equations for a standard Nodal Analysis (NA) of a linear circuit containing only admittances and independent current sources are set up in matrix form
where v n denotes the vector of node voltages v 1 , . . . , v n ‘! &" %# $ of those nodes different from some fixed reference node 0 '! &" %# $ , the entries of the node-admittance matrix Y n and the node current-source vector J n can be directly read off from the circuit itself. I.e., each diagonal term of Y n in position (i, i) is given by the sum of admittances incident with the node i , each off-diagonal term in position (i, j), i = j, is described by the negative of the sum of admittances connecting the nodes i and j ‘! &" %# $ ; the i-th entry of the vector J n is the sum of all independent currents leaving or entering the node i with a plus sign attached only to those currents directed toward the node and a minus sign to all the others.
As is equally well known, while it is easy to extend nodal analysis to deal with circuits, which furthermore contain voltage controlled current sources or nullors, massive problems arise, when voltage sources of any kind have to be taken into consideration, as well. Although this obstacle has been basically overcome by the invention of the Modified Nodal Analysis (MNA), which gives a universal method for any kind of linear circuit, with good reasons most teachers of Electrical Engineering seem to be very reluctant to confront their students with the MNA-algorithm, especially in undergraduate courses.
Accordingly, several authors ( [2,3]) have proposed another alternative, the so-called Supernodal Analysis (SNA), which seems to be more accessible to students and thus has been incorporated into existing undergraduate and graduate courses1 ([4, 5, 7]): Starting with a linear circuit with admittances and all kinds of independent sources, the initial set of equations can be reduced to a smaller set; these resulting SNA-equations again can be described in matrix form as
where v N is a vector of selected node voltages, Y N is a matrix of admittances and J N consists of suitably chosen linear combinations of the independent sources. Although in the literature ( [3,4]) instructions are given how to calculate the entries of Y N and J N , as far as we know, no algorithm has been developed, yet, which in analogy to standard nodal analysis allows one to directly read them off from the circuit. This paper was written to remedy this situation.
Remark: Throughout this paper, to keep notation as simple as possible, while we freely talk about circuits with admittances, all the examples will be linear circuits considered in the time domain, which besides independent sources consist of resistors with positive conductances (symbolized by capital letters), controlled sources and/or nullors. The experienced reader will know how to generalize the results, which will be presented, to other linear circuits containing inductors and capacitors.
The only notational convention we will strictly adhere to is using boldface letters for vectors and matrices.
Without loss of generality (cp. [8], 1.5.3) we demand that all circuits under consideration are connected.
INDEPENDENT SOURCES To keep matters simple at the beginning, in this section we will only consider circuits without controlled sources and nullors.
The basis of our discussion is the concept of a supernode. The definition, we will give, slightly differs both from the one presented in [2], as well as from the one in [4], chapter 4.2. This was done to streamline the formulation of the “traditional” algorithm and to prepare for our improvements.
Definition 1: A subcircuit of a given circuit which is connected, consists only of nodes and (independent) voltage sources, and which is maximal with these two properties 2 is called a supernode.
Let us remark, that by this definition an ordinary node which is not incident with any voltage source is regarded as a supernode, as well. Furthermore, any supernode defines a cut of the circuit. The contraction3 Γ of a circuit Γ, obtained by contracting all the branches of all supernodes and removal of all resulting loops, is a circuit, which by the initial prerequisite of this section only contains resistors and independent current sources. During the course of this paper we will call Γ the contraction along the supernodes.
If, moreover, one removes all branches associated to current sources from Γ, the result is the deactivated circuit in the sense of [3,4].
We are now able to adapt the general algorithm of Supernodal Analysis as presented in [2,3,4], and formulate it in our terminology.
/) .* -+ , , consisting only of independent sources and resistors. ( * Node 0 ‘! &" %# $ will be our global reference node (ground/datum), and we set v 0 = 0. * ) Output:
- a set of equations for the node voltages v 0 , v 1 , . . . , v n ‘! &" %# $ ,
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
