Kirchhoff's Junction Rule

8 Key excerpts on "Kirchhoff's Junction Rule"

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  eBook - ePub

  Fundamental Electrical and Electronic Principles

  • Jo Verhaevert, Christopher R. Robertson(Authors)
  • 2024(Publication Date)
  • Routledge

    (Publisher)

  Applying Kirchhoff’s current law yields: Figure 2.17 Kirchhoff’s current law I 1 − I 2 + I 3 + I 4 − I 5 = 0 where ‘+’ signs have been used to denote currents arriving and ‘–’ signs for currents leaving the junction. This equation can be transposed to comply with the alternative statement for the law, thus: I 1 + I 3 + I 4 = I 2 + I 5 Gustav Robert Kirchhoff (1824–1887) was a German physicist. Besides his laws in the field of electrical engineering, he also performed research in the domain of spectroscopy and radiation of black bodies under heating. Together with the German chemist Robert Bunsen (1811–1899), he co-discovered the chemical elements cesium and rubidium. Worked Example 2.8 Q For the network shown in Figure 2.18 calculate the values of the marked currents. Figure 2.18 The circuit diagram for Worked Example 2.8 Junction A: I 2 = 40 + 10 = 50 A Junction C: I 1 + I 2 = 80 A I 1 + 50 A = 80 A I 1 = 30 A Junction D: I 3 = 80 + 30 = 110. A Junction E: I 4 + 25 = I 3 I 4 = 110 − 25 = 85 A Junction F: I 5 + I 4 = 30 A I 5 + 85 A = 30 A I 5 = 30 − 85 = − 55 A Note : The minus sign in the last answer tells us that the current I 5 is actually flowing away from the junction rather than towards it as shown. 2.5 Kirchhoff’s Voltage Law This law also has already been used – in the explanation of p.d. and in the series and series/parallel circuits. This law states that in any closed network the algebraic sum of the emfs is equal to the algebraic sum of the p.d.s taken in order about the network. Once again, the law sounds very complicated, but it is really only common sense, and is simple to apply. So far, it has been applied only to very simple circuits, such as resistors connected in series across a source of emf. In this case we have said that the sum of the p.d.s is equal to the applied emf (e.g. V 1 + V 2 = E). However, these simple circuits have had only one source of emf, and could be solved using simple Ohm’s law techniques

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  Laws and Theories of Physics

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  or The algebraic sum of currents in a network of conductors meeting at a point is zero. (Assuming that current entering the junction is taken as positive and current leaving the junction is taken as negative). ________________________ WORLD TECHNOLOGIES ________________________ Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as: n is the total number of branches with currents flowing towards or away from the node. This formula is also valid for complex currents: The law is based on the conservation of charge whereby the charge (measured in coulombs) is the product of the current (in amperes) and the time (which is measured in seconds). Changing charge density Physically speaking, the restriction regarding the capacitor plate means that Kirchhoff's current law is only valid if the charge density remains constant in the point that it is applied to. This is normally not a problem because of the strength of electrostatic forces: the charge buildup would cause repulsive forces to disperse the charges. However, a charge build-up can occur in a capacitor, where the charge is typically spread over wide parallel plates, with a physical break in the circuit that prevents the positive and negative charge accumulations over the two plates from coming together and cancelling. In this case, the sum of the currents flowing into one plate of the capacitor is not zero, but rather is equal to the rate of charge accumulation. However, if the displacement current d D /dt is included, Kirchhoff's current law once again holds.

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  eBook - ePub

  Basic Electric Circuit Theory

  A One-Semester Text

  • Isaak D. Mayergoyz, W. Lawson(Authors)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  4 = 0 as claimed. Second, note that the negative sum of the two cut set equations is equal to the KCL equation for node D (the only node which is not contained within a cut set). This example clearly demonstrates that the cut set equations follow from the node equations (and vice versa) and leads us to the notion of independent equations, which will be discussed later in this chapter.

  2.3.2 Kirchhoff’s Voltage Law

  Kirchhoff’s voltage law is the second fundamental axiom of circuit theory and is often abbreviated as KVL. KVL is applied to voltages in loops and states that the algebraic sum of branch voltages around any loop of an electric circuit is equal to zero at every instant of time. Mathematically, it can be written as follows:

  (2.2)

  To actually write KVL equations, we shall need the following rule for voltage polarities. We start by introducing reference voltage polarities for each branch. (Remember that these reference voltages are coordinated with the reference currents for the passive elements.) Then, we trace each loop in an arbitrary direction (it is helpful to trace every loop in the same direction—so let us always agree to go clockwise in this text). If, while tracing through the loop, we enter the “plus” terminal of an element and exit its “minus” terminal, then we take the voltage across this element with a positive sign. If, on the other hand, we enter the negative terminal while tracing the loop, we take the corresponding voltage with a negative sign.

  To demonstrate the above rule, consider the example circuit shown in Figure 2.6 with the loops already labeled (keep in mind that these are not the only possible loops for this circuit). KVL for loop I in Figure 2.6 yields − v 1 + v 3 + v 2 = 0. Likewise, for loop III we will get − v 5 − v 6 + v 9 = 0 from KVL.

  Figure 2.6 Example circuit with four loops shown as I, II, III, IV.

  Since for passive elements, reference polarities for voltage and reference directions for current are coordinated, we can formulate the rule for determining the signs of branch voltages in a KVL equation in terms of the reference current directions. Namely, if the tracing direction coincides with the reference current direction, the branch voltage is taken with a positive sign in the KVL equation. Otherwise, the branch voltage requires a minus sign in the KVL equation. In loop I, the tracing direction coincides with reference directions of i 2 and i 3 but is opposite to i 1 , so we would get − v 1 + v 3 + v 2 = 0 as we must.

  Because the reference current directions and voltage polarities are not necessarily coordinated for sources, one should always use the previous rule for voltage polarities to evaluate the signs of the voltages across the sources in the KVL equations.

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  eBook - ePub

  Fundamentals Of Electric Power Engineering

  • Isaak D Mayergoyz, Patrick McAvoy(Authors)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  Previously we discussed the terminal relations, which are determined by the physical nature of the circuit elements. Now, we proceed to the brief discussion of the relations which are due to the connectivity of elements in an electric circuit. There are two types of such relations. We begin with the Kirchhoff Current Law (KCL). KCL equations are written for nodes of electric circuits. A node of an electric circuit is a “point” where three or more elements are connected together. KCL states that the algebraic sum of electric currents at any node of an electric circuit is equal to zero at every instant of time. This is mathematically expressed as follows:

  The term “algebraic sum” implies that some currents are taken with positive signs while others are taken with negative signs. Two equivalent rules can be used for sign assignments. One rule is that positive signs are assigned to currents with reference directions toward the node, while negative signs are assigned to currents with reference directions from the node. KCL equations can be written for any node. However, only (n − 1) equations will be linearly independent, where n is the number of nodes in a given circuit. The (n − 1) nodes for which KCL equations are written can be chosen arbitrarily. The KCL equation for the last (n-th) node can be obtained by summing up the previously written KCL equations. This clearly suggests that the equation for the last (n-th) node is not linearly independent. A “point” in an electric circuit where only two elements are connected is not qualified as a node because of the triviality of the KCL equation in this case, which simply suggests that the same current flows through both circuit elements, i.e., these two circuit elements are connected in series.

  Next, we discuss equations written by using the Kirchhoff Voltage Law (KVL). These equations are written for loops. A loop is defined as a set of branches that form a closed path with the property that each node is encountered only once as the loop is traced. A branch is defined as a single two-terminal element or several two-terminal elements connected in series. KVL states that the algebraic sum of branch voltages around any loop of an electric circuit is equal to zero at every instant of time. This is mathematically expressed as follows:

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  eBook - ePub

  Electric Circuits and Signals

  • Nassir H. Sabah(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  2

  Basic Circuit Connections and Laws

  Overview

  There are two basic circuit laws, known as Kirchhoff’s laws, which are expressions of conservation of charge and conservation of energy. They must be obeyed, therefore, by electric circuits under all conditions, independently of the circuit configuration and the circuit elements involved.

  Kirchhoff’s laws are first applied in this chapter to analyze some simple circuits, namely, resistive voltage dividers and current dividers. These simple circuits feature the two basic circuit connections: the series and parallel connections. Circuit elements connected in series or in parallel can be combined with other circuit elements in series or in parallel combinations to obtain circuits of any desired complexity. When resistors are connected in series or in parallel, the individual resistances or conductances combine according to some simple rules to give an equivalent resistance or conductance. The concept of equivalence is fundamental to circuit analysis and is applied to the Δ-Y transformation and transformation between voltage and current sources.

  The voltage divider supplying a load at a reduced voltage is considered at the end of the chapter to illustrate a common and important feature of engineering design in general, namely, the trade-off between conflicting performance requirements. The example also illustrates the practical effects of tolerance of resistance values due to inevitable variations in component values during manufacturing.

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  eBook - PDF

  Electrical Principles and Technology for Engineering

  • John Bird(Author)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Part 2 Electrical Principles This page intentionally left blank 11 D.c. circuit theorems At the end of this chapter you should be able to: • state and use Kirchhoff s laws to determine unknown currents and voltages in d.c. circuits • understand the superposition theorem and apply it to find currents in d.c. circuits • understand general d.c. circuit theory • understand Thevenin's theorem and apply a procedure to determine unknown currents in d.c. circuits • recognize the circuit diagram symbols for ideal voltage and current sources • understand Norton's theorem and apply a procedure to determine unknown currents in d.c. circuits • appreciate and use the equivalence of the Thevenin and Norton equivalent networks • state the maximum power transfer theorem and use it to determine maximum power in a d.c. circuit 11.1 Introduction The laws which determine the currents and voltage drops in d.c. networks are: (a) Ohm's law (see Chapter 2); (b) the laws for resistors in series and in parallel (see Chapter 5); and (c) Kirchhoff s laws (see section 11.2). In addition, there are a number of circuit theorems which have been developed for solving problems in electrical networks. These include: (i) the superposition theorem (see section 11.3); (ii) Thevenin's theorem (see section 11.5); (iii) Norton's theorem (see section 11.7); and (iv) the maximum power transfer theorem (see section 11.8). 11. 2 Kirchhofes laws Kirchhoff s laws state: (a) Current law. At any junction in an electric circuit the total current flowing towards that junction is equal to the total current flowing away from the junction, i.e. / = 0. Thus, referring to Fig. 11.1: h + h = h + h + h or h + h -h -h ~ h = ° (b) Voltage law. In any closed loop in a network, the algebraic sum of the voltage drops (i.e. products of current and resistance) taken around the loop is equal to the resultant e.m.f acting in that loop. Thus, referring to Fig. 11.2:

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  eBook - ePub

  Circuit Analysis and Feedback Amplifier Theory

  • Wai-Kai Chen(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  2 Network Laws and Theorems 2.1    Kirchhoff’s Voltage and Current Laws

  Nodal Analysis • Mesh Analysis • Fundamental Cutset-Loop Circuit Analysis • References

  2.2    Network Theorems

  The Superposition Theorem • The Thévenin Theorem • The Norton Theorem • The Maximum Power Transfer Theorem • The Reciprocity Theorem

  Ray R. Chen San Jose State University Artice M. Davis San Jose State University Marwan A. Simaan University of Pittsburgh 2.1    Kirchhoff’s Voltage and Current Laws Ray R. Chen and Artice M. Davis

  Circuit analysis, like Euclidean geometry, can be treated as a mathematical system; that is, the entire theory can be constructed upon a foundation consisting of a few fundamental concepts and several axioms relating these concepts. As it happens, important advantages accrue from this approach — it is not simply a desire for mathematical rigor, but a pragmatic need for simplification that prompts us to adopt such a mathematical attitude.

  The basic concepts are conductor, element, time, voltage, and current. Conductor and element are axiomatic; thus, they cannot be defined, only explained. A conductor is the idealization of a piece of copper wire; an element is a region of space penetrated by two conductors of finite length termed leads and pronounced “leeds”. The ends of these leads are called terminals and are often drawn with small circles as in Figure 2.1 .

  Conductors and elements are the basic objects of circuit theory; we will take time, voltage, and current as the basic variables. The time variable is measured with a clock (or, in more picturesque language, a chronometer). Its unit is the second, s. Thus, we will say that time, like voltage and current, is defined operationally, that is, by means of a measuring instrument and a procedure for measurement. Our view of reality in this context is consonant with that branch of philosophy termed operationalism [1 ].

  Voltage is measured with an instrument called a voltmeter, as illustrated in Figure 2.2 . In Figure 2.2 , a voltmeter consists of a readout device and two long, flexible conductors terminated in points called probes that can be held against other conductors, thereby making electrical contact with them. These conductors are usually covered with an insulating material. One is often colored red and the other black. The one colored red defines the positive polarity of voltage, and the other the negative polarity. Thus, voltage is always measured between two conductors. If these two conductors are element leads, the voltage is that across the corresponding element. Figure 2.3 is the symbolic description of such a measurement; the variable v, along with the corresponding plus and minus signs, means exactly the experimental procedure depicted in Figure 2.2 , neither more nor less. The outcome of the measurement, incidentally, can be either positive or negative. Thus, a reading of v = −12 V, for example, has meaning only when viewed within the context of the measurement. If the meter leads are simply reversed after the measurement just described, a reading of vʹ = +12 V will result. The latter, however, is a different variable; hence, we have changed the symbol to v

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  eBook - ePub

  Introduction to Nonlinear Oscillations

  • Vladimir I. Nekorkin(Author)
  • 2015(Publication Date)
  • Wiley-VCH

    (Publisher)

  11 Hz has been reached.

  11.2 Equivalent Circuit of the Junction

  Apart from the supercurrent given by expression (11.9 ), a regular current also flows through a Josephson junction due to the tunneling of separate electrons. The junction also has capacitive properties due to the specific junction configuration. Taking all these properties into account, we obtain the electric diagram of the junction presented in Figure 11.1 .

  Figure 11.1

  Equivalent circuit diagram of the Josephson junction.

  Let us write down the Kirchhoff law for the total current through the junction

  (11.11)

  By eliminating the variable V in (11.11 ) by means of the relationship (11.10 ), we obtain the following equation:

  (11.12)

  We now introduce in (11.12 ) the new time and parameters:

  As a result, (11.11 ) takes the following equivalent form:

  (11.13)

  where the dot denotes differentiation with respect to the time τ.

  Note that (11.13 ) also describes the dynamics of the physical systems that are completely different from the superconductivity theory, namely, the mechanical pendulum in a viscous (parameter λ) medium under the action of a constant external torque γ (see Chapter 8 ) and the phase-locked loop (PLL) system, which contains in the control circuit a linear filter whose time constant is characterized by the parameter λ (see Chapter 4 ). In the case of the pendulum, the variable φ is the angle of deviation from the equilibrium, and in the case of the PLL system, it is the phase difference of two generators having the initial frequency detuning γ. Therefore, the results of studying the dynamics of (11.13 ) obtained next can also be used to understand the behavior of these physical objects.

  11.3 Dynamics of the Model

  Let us rewrite (11.13 ) as the following equivalent system:

  (11.14)

  Consider the system (11.14 ) in the following parameter domain: D = {λ, γ | γ ≥ 0, λ ≥ 0}. System (11.14 ) has a cylindrical phase space, G = S1 × R, as its right-hand side is 2π-periodic with respect to the variable φ

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