Nodal Analysis: A Method for Solving Circuits with Resistors and Current Sources, Study notes of Law

Download Nodal Analysis: A Method for Solving Circuits with Resistors and Current Sources and more Study notes Law in PDF only on Docsity! 3.1 PMcL Contents Index 2019 3 - Nodal Analysis 3 Nodal Analysis Contents Introduction ..................................................................................................... 3.2 3.1 Nodal Analysis .......................................................................................... 3.3 3.1.1 Circuits with Resistors and Independent Current Sources Only .. 3.6 3.1.2 Nodal Analysis Using Branch Element Stamps ........................... 3.9 3.1.3 Circuits with Voltage Sources .................................................... 3.12 3.1.4 Summary of Nodal Analysis ....................................................... 3.14 3.2 Summary .................................................................................................. 3.15 3.3 References ............................................................................................... 3.15 Exercises ........................................................................................................ 3.16 3.2 Index Introduction PMcL 3 - Nodal Analysis 2019 Introduction After becoming familiar with Ohm’s Law and Kirchhoff’s Laws and their application in the analysis of simple series and parallel resistive circuits, we must begin to analyse more complicated and practical circuits. Physical systems that we want to analyse and design include electronic control circuits, communication systems, energy converters such as motors and generators, power distribution systems, mobile devices and embedded systems. We will also be confronted with allied problems involving heat flow, fluid flow, and the behaviour of various mechanical systems. To cope with large and complex circuits, we need powerful and general methods of circuit analysis. Nodal analysis is a method which can be applied to any circuit. This method is widely used in hand design and computer simulation. 3.5 PMcL Nodal Analysis Index 2019 3 - Nodal Analysis Rewriting in matrix notation, we have:                     2 3 2.12.0 2.07.0 2 1 v v These equations may be solved by a simple process of elimination of variables, or by Cramer’s rule and determinants. Using the latter method we have: V 5.2 8.0 2 8.0 6.04.1 8.0 22.0 37.0 V 5 8.0 4 04.084.0 4.06.3 2.12.0 2.07.0 2.12 2.03 2 1              v v Everything is now known about the circuit – any voltage, current or power in the circuit may be found in one step. For example, the voltage at node 1 with respect to node 2 is   V 5.221  vv , and the current directed downward through the  2 resistor is A 5.221 v . 3.6 Index Nodal Analysis PMcL 3 - Nodal Analysis 2019 3.1.1 Circuits with Resistors and Independent Current Sources Only A further example will reveal some interesting mathematical features of nodal analysis, at least for the case of circuits containing only resistors and independent current sources. We will find it is much easier to consider conductance, G, rather than resistance, R, in the formulation of the equations. EXAMPLE 3.2 Nodal Analysis with Independent Sources Only A circuit is shown below with a convenient reference node and nodal voltages specified. -8 A 25 A 1/3  1 1v v2 -3 A 1/2  1/5  v3 1/4  We sum the currents leaving node 1:         11437 03843 321 3121   vvv vvvv At node 2:     3263 03213 321 32212   vvv vvvvv At node 3:     251124 025524 321 32313   vvv vvvvv 3.7 PMcL Nodal Analysis Index 2019 3 - Nodal Analysis Rewriting in matrix notation, we have:                                   25 3 11 1124 263 437 3 2 1 v v v For circuits that contain only resistors and independent current sources, we define the conductance matrix of the circuit as:               1124 263 437 G It should be noted that the nine elements of the matrix are the ordered array of the coefficients of the KCL equations, each of which is a conductance value. Thr first row is composed of the coefficients of the Kirchhoff current law equation at the first node, the coefficients being given in the order of 1v , 2v and 3v . The second row applies to the second node, and so on. The major diagonal (upper left to lower right) has elements that are positive. The conductance matrix is symmetrical about the major diagonal, and all elements not on this diagonal are negative. This is a general consequence of the systematic way in which we ordered the equations, and in circuits consisting of only resistors and independent current sources it provides a check against errors committed in writing the circuit equations. We also define the voltage and current source vectors as:                       25 3 11 3 2 1 iv v v v The conductance matrix defined 3.10 Index Nodal Analysis PMcL 3 - Nodal Analysis 2019 Thus, the branch between nodes i and j contributes the following element stamp to the conductance matrix, G :         GG GG j i ji (3.3) If node i or node j is the reference node, then the corresponding row and column are eliminated from the element stamp shown above. For any circuit containing only resistors and independent current sources, the conductance matrix can now be built up by inspection. The result will be a G matrix where each diagonal element iig is the sum of conductances connected to node i, and each off-diagonal element ijg is the total conductance between nodes i and j but with a negative sign. Now consider a current source connected between nodes i and j: vv I i j Figure 3.2 In writing out the ith KCL equation we would introduce the term: 0  I (3.4) In writing out the jth KCL equation we would introduce the term: 0  I (3.5) The element stamp for a conductance 3.11 PMcL Nodal Analysis Index 2019 3 - Nodal Analysis Thus, a current source contributes to the right-hand side (rhs) of the matrix equation the terms:       I I j i (3.6) Thus, the i vector can also be built up by inspection – each row is the addition of all current sources entering a particular node. This makes sense since iGv  is the mathematical expression for KCL in the form of “current leaving a node = current entering a node”. EXAMPLE 3.3 Nodal Analysis Using the “Formal” Approach We will analyse the previous circuit but use the “formal” approach to nodal analysis. -8 A 25 A 1/3  1 1v v2 -3 A 1/2  1/5  v3 1/4  By inspection of each branch, we build the matrix equation:                                           25 3 38 24524 22313 4343 3 2 1 3 2 1 321 v v v This is the same equation as derived previously. The element stamp for an independent current source 3.12 Index Nodal Analysis PMcL 3 - Nodal Analysis 2019 3.1.3 Circuits with Voltage Sources Voltage sources present a problem in undertaking nodal analysis, since by definition the voltage across a voltage source is independent of the current through it. Thus, when we consider a branch with a voltage source when writing a nodal equation, there is no way by which we can express the current through the branch as a function of the nodal voltages across the branch. There are two ways around this problem. The more difficult is to assign an unknown current to each branch with a voltage source, proceed to apply KCL at each node, and then apply KVL across each branch with a voltage source. The result is a set of equations with an increased number of unknown variables. The easier method is to introduce the concept of a supernode. A supernode encapsulates the voltage source, and we apply KCL to both end nodes at the same time. The result is that the number of nodes at which we must apply KCL is reduced by the number of voltage sources in the circuit. EXAMPLE 3.4 Nodal Analysis with Voltage Sources Consider the circuit shown below, which is the same as the previous circuit except the  21 resistor between nodes 2 and 3 has been replaced by a 22 V voltage source: -8 A 25 A 1/3  1 1v v2 -3 A 22 V 1/5  v3 1/4  supernode The concept of a supernode 3.15 PMcL Summary Index 2019 3 - Nodal Analysis 3.2 Summary  Nodal analysis can be applied to any circuit. Apart from relating source voltages to nodal voltages, the equations of nodal analysis are formed from application of Kirchhoff’s Current Law.  In nodal analysis, a supernode is formed by short-circuiting a voltage source and treating the two ends as a single node. 3.3 References Hayt, W. & Kemmerly, J.: Engineering Circuit Analysis, 3rd Ed., McGraw- Hill, 1984. 3.16 Index Exercises PMcL 3 - Nodal Analysis 2019 Exercises 1. (a) Find the value of the determinant: 1003 2304 1011 3012      (b) Use Cramer’s rule to find 1v , 2v and 3v if: 0283 56432 03352 312 123 321    vvv vvv vvv