Nodal Analysis

10 Key excerpts on "Nodal Analysis"

• 

  eBook - ePub

  The Fundamentals of Electrical Engineering

  for Mechatronics

  • Felix Hüning(Author)
  • 2014(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  5 Circuit analysis

  In the previous chapter I presented basic electric circuit concepts like KCL, KVL or Wye-Delta transformation. Now I want to introduce some more sophisticated circuit analysis techniques for practically and efficiently solving problems associated with circuit operations. We will start with two basic analysis techniques, nodal and mesh analysis. These two techniques are based on KCL and KVL and make use of two fundamental facts about electric circuits:

  1. In any electric network with n nodes (n–1) independent equations for the nodes can be found
  2. In any electric network with m meshes m independent equations can be found

  5.1 Nodal Analysis

  Nodal Analysis can be used for any electric circuit and in particular for circuits with few nodes (but rather many loops) as the number of equations will be small. It is based on the definition of voltage: Voltage is the difference between two electrical potentials.

  Fig. 5.1: Voltages (including a reference potential U0 ) in an electric circuit.

  It is common to define a reference potential and to refer the other voltages to this reference potential. The reference potential, or ground potential is marked with a special sign and is defined to have a voltage of U0 = 0 V. For a circuit with n nodes there are (n–1) nodal voltages referring to ground potential. With Nodal Analysis the voltages of all nodes referring to the reference potential can be calculated. The procedure of Nodal Analysis will be introduced by an example before the general approach is presented.

    An example of a Nodal Analysis

  Refer to Fig. 5.2 for the first example of Nodal Analysis. The circuit contains 4 nodes, 0–3, and node 0 is defined as ground potential. The direction of the current vectors can be chosen arbitrarily (at the end of the calculation the sign of the current will show whether the current flows in the chosen, or in opposite direction). By using KCL and Ohm’s law the three nodal voltages U10 , U20 and U30 can be calculated. For simplification of the calculation each resistance Ri is substituted by corresponding conductance Gi , i.e. Gi = 1 / Ri

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Electric Circuit Theory

  Applied Electricity and Electronics

  • R. Yorke, P. Hammond(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 4 CIRCUIT ANALYSIS 4 . 0 I N T R O D U C T I O N The process of circuit analysis consists of determining the currents in, and potential differences across, all branches of a network or circuit. The data normally comprise all source voltages and/or currents and all branch resistances/impedances. Ohm's and Kirchhoff's Laws form the basis of all methods of analysis, either directly or in modified form, according to circumstances. Thus, in circuits containing only voltage sources, mesh or loop analysis is generally to be preferred to Kirchhoff's Laws themselves since fewer simultaneous equations result and, when only current sources are pres-ent, Nodal Analysis is generally best. In addition, many network theorems exist which, although not strictly falling within the above definition of circuit analysis, are nevertheless useful aids in the process. A few of these will be dealt with in this chapter. 4 . 1 N E T W O R K T E R M I N O L O G Y First a few definitions. (a) A branch of a circuit is that part of the circuit containing a single active or passive element. (b) A node is the point of interconnection of two or more branches. The junction of three or more branches is referred to as a principal node; a secondary node is the junction of two branches. 200 CIRCUIT ANALYSIS 201 (c) A planar network is one whose schematic may be drawn on a plane surface without branch crossings. (d) A non-planar network is one whose schematic may not be so drawn. (e) A loop is any closed path in a network which does not pass through any branch or node more than once. (f) A mesh is a loop which contains no other loops. It is defined only for planar networks. 4.2 KIRCHHOFF'S LAWS The reader is referred to section 1.3 for a statement of the laws, which are valid for both planar and non-planar networks. An unknown current is assigned to every branch and the first law applied at as many nodes as will provide independent equations.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms, Amalendu Patnaik(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  EXPERIMENTS THAT HELP STUDENTS DEVELOP AN UNDERSTANDING OF LOOP AND NODAL TECHNIQUES FOR CIRCUIT ANALYSIS ARE: ■ Kirchhoff’s Laws: Prove Kirchhoff’s laws experimentally and the conservation of power from voltage and current measurements in simple circuits. ■ Mesh Current and Node Voltage Analyses: Build and ana- lyze a circuit using mesh current or node voltage analysis and examine the effect of multiple sources on a circuit parameter by accumulating the results of the individual sources using superposition. ■ Wheatstone Bridge: Predict the temperature using a Wheatstone bridge with a thermistor as the element in one leg of the bridge. Nodal and Loop Analysis Techniques CHAPTER 3 THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO: ■ Calculate the branch currents and node voltages in circuits containing multiple nodes using KCL and Ohm’s law in Nodal Analysis. ■ Calculate the mesh currents and voltage drops and rises in circuits containing multiple loops using KVL and Ohm’s law in loop analysis. ■ Identify the most appropriate analysis technique that should be utilized to solve a particular problem. 89 3.1 Nodal Analysis In a Nodal Analysis, the variables in the circuit are selected to be the node voltages. The node voltages are defined with respect to a common point in the circuit. One node is selected as the reference node, and all other node voltages are defined with respect to that node. Quite often, this node is the one to which the largest number of branches are connected. It is commonly called ground because it is said to be at ground-zero potential, and it sometimes represents the chassis or ground line in a practical circuit. 90 CHAPTER 3 NODAL AND LOOP ANALYSIS TECHNIQUES + – 1 2 3 4 5 V a = 3 V I 1 I 3 I 5 I 2 I 4 12 V 6 kΩ 4 kΩ 3 kΩ 9 kΩ 9 kΩ 3 kΩ + − + − + − V S V 1 + − V 3 + − V 5 + − V b = — V 3 2 V c = — V 3 8 Figure 3.1 Circuit with known node voltages.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Basic Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  79 3.1 Nodal Analysis In a Nodal Analysis, the variables in the circuit are selected to be the node voltages. The node voltages are defined with respect to a common point in the circuit. One node is selected as the reference node, and all other node voltages are defined with respect to that node. Quite often, this node is the one to which the largest number of branches are connected. It is commonly called ground because it is said to be at ground-zero potential, and it sometimes represents the chassis or ground line in a practical circuit. We will select our variables as being positive with respect to the reference node. If one or more of the node voltages are actually negative with respect to the reference node, the analysis will indicate it. In order to understand the value of knowing all the node voltages in a network, we con- sider once again the network in Fig. 2.32, which is redrawn in Fig. 3.1. The voltages, V S , V a , V b , and V c , are all measured with respect to the bottom node, which is selected as the reference and labeled with the ground symbol . Therefore, the voltage at node 1 is V S = 12 V with respect to the reference node 5, the voltage at node 2 is V a = 3 V with respect to the reference node 5, and so on. Now note carefully that once these node voltages are known, we can immediately calculate any branch current or the power supplied or absorbed by any element, since we know the voltage across every element in the network.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Essentials of Electrical and Computer Engineering

  • J. David Irwin, David V. Kerns, Jr.(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 3 Circuit Analysis Techniques LEARNING OBJECTIVES • To understand how to apply Ohm’s law and Kirchhoff’s laws to analyze circuits • To be able to determine voltages and currents anywhere in the network using both nodal and loop analysis • To be able to effectively apply both superposition and Thevenin’s theorems in network analysis • To understand and apply maximum power transfer In this chapter, we present several techniques that find wide application in the analysis of electric circuits. These basic techniques will include those that represent, in essence, a sledgehammer approach to solving electric circuits as well as others that provide some actual insight into net- work performance. Nodal Analysis Our first technique is called nodal analyses. In order to understand the power of this approach to circuit analysis, consider the network shown in Figure 3.1. Suppose for a moment that we somehow know that the voltages V 1 = 6 V and V 2 = 4 V, i.e. we know the voltage across the 3k Ω resistor and the voltage across the (4/3)k Ω resistor. We can now show that this knowledge is sufficient to determine the current in each element and, in turn, the voltage across each element. For example, since V 1 = 6 V, applying Ohm’s law yields I B = 6∕3k = 2 mA Similarly, since V 2 = 4 V I D = 4∕(4∕3)k = 3 mA Now, the systematic application of KVL to the loops in the network yields the following equations: −12 + (2k)I A + V 1 = 0 −V 1 + (2k)I C + V 2 = 0 and −12 + (4k)I E + V 2 = 0 which produces the unknown currents, I A = 3 mA, I C = 1 mA, and I E = 2 mA. Note that KCL is satisfied at every node in the network, and KVL is satisfied around every loop. In addition, this information permits us to calculate the power absorbed or supplied by every element. So in essence, a knowledge of all the node voltages provides us with a complete analysis of the circuit. As we indicated earlier in the book, the voltage must be referenced to some specific point.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 4 Methods of Analysis of Resistive Circuits I N T H I S C H A P T E R 4.1 Introduction 4.2 Node Voltage Analysis of Circuits with Current Sources 4.3 Node Voltage Analysis of Circuits with Current and Voltage Sources 4.4 Node Voltage Analysis with Dependent Sources 4.5 Mesh Current Analysis with Independent Voltage Sources 4.6 Mesh Current Analysis with Current and Voltage Sources 4.7 Mesh Current Analysis with Dependent Sources 4.8 The Node Voltage Method and Mesh Current Method Compared 4.9 Circuit Analysis Using MATLAB 4.10 Using PSpice to Determine Node Voltages and Mesh Currents 4.11 How Can We Check . . . ? 4.12 DESIGN EXAMPLE— Potentiometer Angle Display 4.13 Summary Problems PSpice Problems Design Problems 4.1 I n t r o d u c t i o n To analyze an electric circuit, we write and solve a set of equations. We apply Kirchhoff’s current and voltage laws to get some of the equations. The constitutive equations of the circuit elements, such as Ohm’s law, provide the remaining equations. The unknown variables are element currents and voltages. Solving the equations provides the values of the element current and voltages. This method works well for small circuits, but the set of equations can get quite large for even moderate-sized circuits. A circuit with only 6 elements has 6 element currents and 6 element voltages. We could have 12 equations in 12 unknowns. In this chapter, we consider two methods for writing a smaller set of simultaneous equations:  The node voltage method.  The mesh current method. The node voltage method uses a new type of variable called the node voltage. The “node voltage equations” or, more simply, the “node equations,” are a set of simultaneous equations that represent a given electric circuit. The unknown variables of the node voltage equations are the node voltages. After solving the node voltage equations, we determine the values of the element currents and voltages from the values of the node voltages.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Electric Circuits

  • James A. Svoboda, Richard C. Dorf(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  The voltage at any node of the circuit, relative to the reference node, is called a node voltage. In Figure 4.2-1b, there are two node voltages: the voltage at node a with respect to the reference node, node c, and the voltage at node b, again with respect to the reference node, node c. In Figure 4.2-1c, voltmeters are added to measure the node voltages. To measure node voltage at node a, connect the red (a) (b) R 1 R 1 R 2 R 2 R 3 i s i s R 3 v a – + v b – + Voltmeter Voltmeter a b c (c) R 1 R 2 i s R 3 a b c FIGURE 4.2-1 (a) A circuit with three nodes. (b) The circuit after the nodes have been labeled and a reference node has been selected and marked. (c) Using voltmeters to measure the node voltages. Node Voltage Analysis of Circuits with Current Sources 115 probe of the voltmeter at node a and connect the black probe at the reference node, node c. To measure node voltage at node b, connect the red probe of the voltmeter at node b and connect the black probe at the reference node, node c. The node voltages in Figure 4.2-1c can be represented as v ac and v bc , but it is conventional to drop the subscript c and refer to these as v a and v b . Notice that the node voltage at the reference node is v cc ¼ v c ¼ 0 V because a voltmeter measuring the node voltage at the reference node would have both probes connected to the same point. One of the standard methods for analyzing an electric circuit is to write and solve a set of simultaneous equations called the node equations. The unknown variables in the node equations are the node voltages of the circuit. We determine the values of the node voltages by solving the node equations. To write a set of node equations, we do two things: 1. Express element currents as functions of the node voltages. 2. Apply Kirchhoff’s current law (KCL) at each of the nodes of the circuit except for the reference node. Consider the problem of expressing element currents as functions of the node voltages.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Electrical Engineer's Reference Book

  • G R Jones(Author)
  • 2013(Publication Date)
  • Newnes

    (Publisher)

  4.2.3.2 Node-voltage equations Of the network nodes, one is chosen as a reference node to which all other node voltages are related. The sources are represented by current generators feeding specified currents into their respective nodes and the branches are in terms of admittance Y. Then for the n independent nodes 'a = 'a*aa · ^M ab ' ··· ' *n*an I b = V a Yba+V b Y bb +...+ V n Y bn 'ln=V a Yna+V b Y nb +...+ V„Y nn Here Y aa , Y bb ,..., Y m are the self-admittances of nodes 0, b,..., n, i.e. the sum of the admittances terminating on nodes a, b,..., n; and Y ab , Y pq ,..., are the mutual admittances , those that link nodes a and , ρ and q,..., respectively, and which are usually negative. The mesh-current and node-voltage methods are general and basic; they are applicable to all network conditions. Simplified and auxiliary techniques are applied in special cases. 4.2.3.3 Techniques Steady-state conditions Transient phenomena are absent. For d.c. networks the constant current implies absence of induct-ive effects, and capacitors (having a constant charge) are equivalent to an open circuit. Only resistance is taken into account, using the Ohm law. For a.c. networks with sinusoidal current and voltage, complexor algebra, phasor diagrams, locus diagrams and symmetrical components are used, while for a.c. networks with periodic but non-sinusoidal waveforms harmonic analysis with superposition of harmonic components is employed. Transient conditions Operational forms of stimuli and para-meters are used and the solutions are found using Laplace transforms. 4.2.4 Network theorems Network theorems can simplify complicated networks, facil-itate the solution of specific network branches and deal with particular network configurations (such as two-ports). They are applicable to linear networks for which superposition is valid, and to any form (scalar, complexor, or operational) of voltage, current, impedance and admittance.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Graph Theory with Applications to Engineering and Computer Science

  • Narsingh Deo(Author)
  • 2017(Publication Date)
  • Dover Publications

    (Publisher)

  Fig. 13-2 Electrical network and its graph.

  That is,

  Thus each of e edge voltages can be expressed as a linear combination of n − 1 quantities v

  N1

  (t), v

  N2

  (t) . . ., vN(n-1 (t). These are called node voltages, and they represent the voltage at each of n − 1 independent vertices with respect to the reference vertex.

  Substituting Eq. (13-7 ) into Eq. (13-4 ), we get

  Let us now illustrate with an example the loop currents and node voltages and how they are obtained from the edge currents and edge voltages, respectively. Figure 13-2(a) shows an electrical network with five vertices and seven edges. The corresponding directed graph is shown in Fig. 13-2(b) . For this graph the reduced incidence matrix A

  f

  with respect to vertex N5 and the fundamental circuit matrix B

  f

  , with respect to the spanning tree {1, 4, 5, 7} (shown in heavy lines), are

  The edge-current vector expressed in terms of loop-current vector is

  The edge voltages in terms of the node voltages (with respect to N5 ) are

  13-4. RLC NETWORKS WITH INDEPENDENT SOURCES: Nodal Analysis

  In this section we shall restrict ourselves to electrical networks containing resistors, inductors, and capacitors (RLC) with independent voltage and current sources. In spite of its inherent simplicity, the RLC network covers a very large class of electrical networks in practice. In fact, it has been shown by Brune and Bott and Duffin that any time-invariant, two-terminal, linear, passive electrical element can be formed by a combination of R, L, and C (with real positive values of R, L, and C). A further stipulation may be made, without any loss of generality, that the voltage sources may only be connected in series with RLC elements and that current sources may only be connected in parallel with these elements. This stipulation allows us to convert all the energy sources either into a set of voltage sources or into a set of current sources.

  Noda. Analysis:

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Analysis and Design of Linear Circuits

  • Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  3. Node equations are required at supernodes and all other nonreference nodes except those that are directly connected to the reference by voltage sources. 4. Use KCL to write node equations at the nodes identified in step 3. Express element currents in terms of node voltages or the currents produced by inde- pendent current sources. 5. Write expressions relating the node voltages to the voltages produced by independent voltage sources. 6. Substitute the expressions from step 5 into the node equations from step 4 and arrange the resulting equations in standard form. 7. Solve the equations from step 6 for the node voltages of interest. Manual techniques may be efficient for lower-order problems. Computer tools, such as MATLAB or Multisim, are usually more practical and faster for higher- order problems. 3–2 M E S H - C U R R E N T A N A L Y S I S Mesh currents are analysis variables that are useful in circuits containing many ele- ments connected in series. To review terminology, a loop is a closed path formed by (b) + – XMM1 R 2 R 1 V 1 5 V R 3 4 kΩ R 4 10 kΩ 10 kΩ 1 kΩ + – XMM1 R 2 R 1 V 1 V 2 5 V 10 V R 3 4 kΩ R 4 10 kΩ 10 kΩ 1 kΩ FIGURE 3–18 76 C H A P T E R 3 C IRCUIT A NALYSIS T ECHNIQUES passing through an ordered sequence of nodes without passing through any node more than once. A mesh is a special type of loop that does not enclose any elements. For example, loops A and B in Figure 3–19 are meshes, while the loop Q is not a mesh because it encloses an element, X. Mesh-current analysis is restricted to planar circuits. A planar circuit can be drawn on a flat surface without crossovers in the “window pane” fashion shown in Figure 3–19. To define a set of variables, we associate a mesh current (i A , i B , i C , etc.) with each window pane and assign a reference direction. The reference directions for all mesh currents are customarily taken in a clockwise sense. There is no momentous reason for this, except perhaps tradition.

  Sign up to read

  Learn more about book