Kirchhoff's Laws

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  Laws and Theories of Physics

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

    (Publisher)

  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. (This is really only required if one wants to apply the current law to a point on a capacitor plate . In circuit analyses, however, the capacitor as a whole is typically treated as a unit, in which case the ordinary current law holds since exactly the current that enters the capacitor on the one side leaves it on the other side.) More technically, Kirchhoff's current law can be found by taking the divergence of Ampère's law with Maxwell's correction and combining with Gauss's law, yielding: This is simply the charge conservation equation (in integral form, it says that the current flowing out of a closed surface is equal to the rate of loss of charge within the enclosed volume (Divergence theorem)). Kirchhoff's current law is equivalent to the statement that ________________________ WORLD TECHNOLOGIES ________________________ the divergence of the current is zero, true for time-invariant ρ, or always true if the displacement current is included with J . Uses A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. Kirchhoff's voltage law (KVL) The sum of all the voltages around the loop is equal to zero. v 1 + v 2 + v 3 - v 4 = 0 This law is also called Kirchhoff's second law , Kirchhoff's loop (or mesh) rule , and Kirchhoff's second rule . The principle of conservation of energy implies that ________________________ WORLD TECHNOLOGIES ________________________ The directed sum of the electrical potential differences (voltage) around any closed circuit is zero. or More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop.

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  Fundamental Theorems, Laws and Theories of Physics

  • (Author)
  • 2014(Publication Date)
  • Academic Studio

    (Publisher)

  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. (This is really only required if one wants to apply the current law to a point on a capacitor plate . In circuit analyses, however, the capacitor as a whole is typically treated as a unit, in which case the ordinary current law holds since exactly the current that enters the capacitor on the one side leaves it on the other side.) More technically, Kirchhoff's current law can be found by taking the divergence of Ampère's law with Maxwell's correction and combining with Gauss's law, yielding: This is simply the charge conservation equation (in integral form, it says that the current flowing out of a closed surface is equal to the rate of loss of charge within the enclosed volume (Divergence theorem)). Kirchhoff's current law is equivalent to the statement that ________________________ WORLD TECHNOLOGIES ________________________ the divergence of the current is zero, true for time-invariant ρ, or always true if the displacement current is included with J . Uses A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. Kirchhoff's voltage law (KVL) The sum of all the voltages around the loop is equal to zero. v 1 + v 2 + v 3 - v 4 = 0 This law is also called Kirchhoff's second law , Kirchhoff's loop (or mesh) rule , and Kirchhoff's second rule . The principle of conservation of energy implies that The directed sum of the electrical potential differences (voltage) around any closed circuit is zero. ________________________ WORLD TECHNOLOGIES ________________________ or More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop.

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  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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  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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  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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  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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  Electric Circuit Theory

  Applied Electricity and Electronics

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

    (Publisher)

  (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. The second law is then applied to as many independent loops as are necessary to form, in total, the same number of simultaneous equations as there are unknowns. They are solved for the branch currents by substitution, determinants or matrix methods. All branch voltages are then easily found by applying Ohm's Law. With simple circuits it is relatively easy to choose nodes and loops which yield independent equations. A rule of thumb, satisfactory in simple cases, is to choose each new node or loop so as to involve at least one branch current not already involved, either explicitly or implicitly. For more complicated circuits some care is needed and for guaranteed independence of the equations, the methods of section 4.5 must be used. The method of application of the laws is most readily understood by dealing initially with steady (direct current) conditions. PROBLEM 4.1 Calculate, using Kirchhoff's Laws, all the branch currents and voltages in the network represented by Fig. 4.1. Solution The circuit has six branches and four nodes and we begin by assigning six unknown currents, one to each branch (Fig. 4.1(a)). 202 ELECTRIC CIRCUIT THEORY 2 5 0 a FIG. 4.1. (a) Diagram of the network for Problem 4.1. 250 a 95 a 250 & FIG. 4.1. (b) The network for Fig. 4.1(a) with the number of unknowns reduced to three. FIG. 4.1. (c) Analysing the circuit of Fig. 4.1(a) using the loop-current method. Applying the first Law at node A: /i = / 2 + h-At node B: At node D: h - U = / 6 . h + h = / 5 -(4.1) (4.2) (4.3)

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  Grounds for Grounding

  A Handbook from Circuits to Systems

  • Elya B. Joffe, Kai-Sang Lock(Authors)
  • 2022(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Faraday’s Law is by far one of the two most important aspects of Maxwell’s equations associated with grounding theory (the other being Ampere’s Law), forming the basis for the definition of inductance, discussed in Section 2.3. 2.1.2.4 Ampere’s Law All the rivers flow to the sea, but the sea is not full. — Ecclesiastes 1:7 Faraday’s Law, discussed earlier, demonstrated that a time-varying magnetic field can produce an electric field. Similarly, Maxwell’s fourth equation, also known as Maxwell’s Electrical Current Law or Ampere’s Law (Equations (2.14) and (2.19)) acknowledges that, in addition to net movement of electric charge, currents as well as time-varying electric flux crossing through an area, S C , enclosed by a contour, C, can produce a magnetic field: C H •d l = SC J V + ∂D ∂t •d s ∇ × H = J + ∂D ∂t 2 33 In other words, Ampere’s Law states that the line integral of the magnetic field intensity, H, (A/m), around any closed con- tour, C, equals the total current crossing the surface area, S C , enclosed by C; thus, where J V is related to motion of charges and ∂D/∂t represents the displacement current density. U Z 1 c U in Contour –C E•dl = U Z i –U in = 0 Σ i → → U Z 2 Figure 2.11 Equivalence between Faraday’s Law and Kirchhoff’s Voltage Law in the absence of magnetic flux. 5 Inducrance is discussed later in this chapter. 6 Note that Kirchhoff’s voltage and current laws (KVL and KCL, respectively) apply in lumped-circuit models only. Voltages and currents obtained from them are valid only so long as the largest physical dimension of the circuit is electrically small; that is, much less than a wavelength at the frequency of excitation, f, of that circuit. 18 2 Fundamental Concepts Ampere’s Law reveals, therefore, that magnetic fields emerge from two sources.

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