An Introduction to the Mofied Nodal AnalysisMichael HankeMay 30, 20061 IntroductionGilbert Strang provides an introduction to the analysis of electrical circuits in hisbook Introduction to Applied Mathematics. His presentation aims at having thetheory as close as possible to his general construction principles for mathematicalmodels which occur throughout the above mentioned book. This notation mightbe rather unusual for an electrical engineer working in electric circuit analysis.The followingexplanations are intended to put the derivations into the enginneringcontext.2 The PrinciplesThe numerical simulation of electric networks is closely related to the networkmodeling. A well established approach is the description of the network by agraph with branches and nodes. Each branch represents an electric element whoseterminals are coupled together at the nodes. (See the simple example in Figure 1.)The simplest network elements are fully described by a relation between a singlebranch current and the respective branch voltage (e.g., resistors, capacitors, induc-tors, (independent) voltage and current sources). The describing current-voltagerelations are called characteristic equations.The state of the network at any given time is now completely described byall branch voltages, branch currents, and node potentials. The node potentialsare only defined up to a constant. Therefore, one node is assigned the voltage(potential) 0V. This node is called the mass (or, ground) node. The other nodepotentials are given with respect to this reference node (this way being unique).In order to complete the network model, the topology of the elements (i.e.,their mutual interconnections) has to be taken into account. Assuming the electri-cal connections between circuit elements to be ideally conducting and the nodesto be ideal, the topology can be described by Kirchhoff’s laws.1Figure 1: A sample circuit: Schematic of a low-pass filter2Thus, the network modeling consists of two steps:1. Describe the network elements.2. Apply Kirchhoff’s laws.3 Network elementsIn order to make the presentation as simple as possible, I will only consider un-controlled, two-terminal elements. Moreover, the elements will be assumed to belinear. The respective branch voltage will be denoted by v while the branch cur-rent is i. Note that both can be positive or negative depending on the orientationof the branch.3.1 Passive ElementsIn network modeling one distinguishes three different types of passive elements:resistors, capacitors, and inductors. Their characteristic equations can be de-scribed as follows:Resistors limit, or resist, the flow of electrical current, following the law v = R i.R is the resistance value. Sometimes this relation is written as i = Gvwhere G = 1/R is called conductivity.Capacitors store energy in an electrostatic field following the law q = Cv whereq is the electric charge. C is the capacitance value. The voltage-currentcharacteristics is given by i =ddtq = Cdvdt(if C is constant).Inductors store energy in an electromagnetic field following the law Φ = Liwhere Φ is the magnetic flux. L is the self-inductance value. The voltage-current characteristic is given by v =ddtΦ = Ldidt(if L is constant).3.2 Independent SourcesAgain for the sake of simplicity, independent sources will be the only active ele-ments we consider. The generalization to controlled elements is straightforward.Voltage source The current-voltage characteristic is given by v = E with E be-ing the strength of the source. Note that v does not depend on the branchcurrent i.Current source The current-voltage characteristic is given by i = I with I beingthe strength of the source. Note that i does not depend on the branch voltagev.34 Kirchhoff’s LawsThe electrical behavior of the network is completely described by Kirchhoff’slaws.Considering one node with branch currents i1, . . . , ilentering this node wemay describe Kirchhoff’s current law (KCL) as i1+ · ·· + il= 0 , that means, thesum of all branch currents entering a node equals zero.If we consider a loop with the branch voltages v1, . . . , vm, then we may for-mulate Kirchhoff’s voltage law (KVL) as v1+ · · ·+ vm= 0 , that means, the sumof all branch voltages in a loop equals zero.In a practical network there are very many nodes and loops. In order to de-scribe the topology of the circuit, one must write down KCL and KVL for eachof them. It becomes obvious that ones needs a systematic way to derive all theseequations for a given network. Fortunately, there is a very elegant descriptionof all these individual equations. The magic happens if one uses the (reduced)incidence matrix A of the circuit.Assume that we have a circuit with n + 1 nodes and b branches. Number thenodes and branches accordingly. The ground node is omitted here. For everybranch, define an orientation, that is, one node of the branch is assumed to be the“starting” node, the other one is the “end” node. The reduced1incidence matrixdescribes which nodes belong to which branch:akl=1, if branch l has start node k−1, if branch l has end node k0, else.It is nowconvenientto collect all branch currents into one vector2i = [i1, i2, . . . , ib]′.Then, the compact notation of KCL isAi = 0.The incidence matrix allows, additionally, a simple description of the relation be-tween node potentials and branch voltages of the network. If v = [v1, v2, . . . , vb]′is the vector of all branch voltages and e = [e1, e2, . . . , en]′denotes the vector ofall node potentials (excluding the reference (mass) node), then the relationv = A′eis satisfied.1The notion “reduced” comes from the fact that the ground node is left out.2By convention, all our vectors are assumed to be column vectors. In order to come as close aspossible to the MATLAB notation, I will use the apostrophe for denoting the transpose of a matrixor vector: AT= A′. This is valid as long as A is a real matrix.45 Network AnalysisAs introduced previously, let i be the vector of all branch currents, v be the vectorof all branch voltages, and e be the vector of all node potentials. The first stepconsists of writing down the network equations and the characteristic equations ofall network elements as described in the previous subsections. This gives rise to asystem of dimension 2b + n for the unknowns i, v, e. The approach leading to thissystem is called sparse tableau analysis.The so-called modified nodal analysis (MNA) requires a