Kirchhoff's Loop Rule

10 Key excerpts on "Kirchhoff's Loop Rule"

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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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  Laws and Theories of Physics

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

    (Publisher)

  In this potential field, regardless of what electronic components are present, the gain or loss in energy given by the potential field must be zero when a charge completes a closed loop. Electric field and electric potential Kirchhoff's voltage law could be viewed as a consequence of the principle of con-servation of energy. Otherwise, it would be possible to build a perpetual motion machine that passed a current in a circle around the circuit. Considering that electric potential is defined as a line integral over an electric field, Kirchhoff's voltage law can be expressed equivalently as ________________________ WORLD TECHNOLOGIES ________________________ which states that the line integral of the electric field around closed loop C is zero. In order to return to the more special form, this integral can be cut in pieces in order to get the voltage at specific components. Limitations This is a simplification of Faraday's law of induction for the special case where there is no fluctuating magnetic field linking the closed loop. Therefore, it practically suffices for explaining circuits containing only resistors and capacitors. In the presence of a changing magnetic field the electric field is not conservative and it cannot therefore define a pure scalar potential—the line integral of the electric field around the circuit is not zero. This is because energy is being transferred from the magnetic field to the current (or vice versa). In order to fix Kirchhoff's voltage law for circuits containing inductors, an effective potential drop, or electromotive force (emf), is associated with each inductance of the circuit, exactly equal to the amount by which the line integral of the electric field is not zero by Faraday's law of induction.

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  Circuit Analysis and Feedback Amplifier Theory

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

    (Publisher)

  voltage drop. If the signs of a voltage are reversed and a negative sign is affixed to the voltage variable, the value of that variable remains unchanged; thus, note that a negative rise is a drop, and vice versa.

  We are now in a position to state our fourth and final axiom:

  Axiom 4 (Kirchhoff’s Voltage Law): The sum of the voltage rises around any loop is identically zero at all instants of time.

  We refer to this law as KVL for the sake of economy of space. Just as KCL was phrased in terms of charge, KVL could just as well be phrased in terms of flux linkage. Flux linkage is the time integral of voltage, so it can be said that the sum of the flux linkages around a loop is zero. In voltage form, we write

  ∑ loop v rises = 0

  (2.8)

  We observed that a negative rise is a drop, so

  ∑ loop v drops = 0

  (2.9)

  or

  ∑ loop v rises = ∑ loop v drops

  (2.10)

  Thus, in Figure 2.13 , we could write [should we choose to use the form of (2.8) ]

  FIGURE 2.13 Illustration of Kirchhoff’s voltage law.

  FIGURE 2.14 Path form of KVL.

  v x − v y − v a + v b = 0

  (2.11)

  Clearly, one can rearrange KVL into many different algebraic forms that are equivalent to those just stated; one form, however, is more useful in circuit computations than many others. It is known as the path form of KVL. To better appreciate this form, review Figure 2.13 . This time, however, the paths are defined a bit differently. As illustrated in Figure 2.14 , we consider two paths, P1 and P2 , having the same initial and terminal nodes, 1 and 4, respectively.2 We can rearrange (2.11)

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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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  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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  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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  Engineering Circuit Analysis

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

    (Publisher)

  STEP 2. Using Ohm’s law, define a voltage across each resistor in terms of the defined current. STEP 3. Apply KVL to the single-loop circuit. STEP 4. Solve the single KVL equation for the current i(t). If i(t) is positive, the current is flowing in the direction assumed; if not, then the current is actually flowing in the opposite direction. SINGLE-LOOP CIRCUITS SECTION 2.4 SINGLE-NODE-PAIR CIRCUITS 45 R 2 R 1 υ(t) i(t) i 1 (t) i 2 (t) + − Figure 2.22 Simple parallel circuit. CURRENT DIVISION An important circuit is the single-node-pair circuit. If we apply KVL to every loop in a single-node-pair circuit, we discover that all of the elements have the same volt- age across them and, therefore, are said to be connected in parallel. We will, however, apply Kirchhoff’s current law and Ohm’s law to determine various unknown quantities in the circuit. Following our approach with the single-loop circuit, we will begin with the simplest case and then generalize our analysis. Consider the circuit shown in Fig. 2.22. Here we have an independent current source in parallel with two resistors. Since all of the circuit elements are in parallel, the voltage υ (t) appears across each of them. Furthermore, an examination of the circuit indicates that the current i(t) is into the upper node of the circuit and the currents i 1 (t) and i 2 (t) are out of the node. Since KCL essen- tially states that what goes in must come out, the question we must answer is how i 1 (t) and i 2 (t) divide the input current i(t). Applying Kirchhoff’s current law to the upper node, we obtain i(t) = i 1 (t) + i 2 (t) and, employing Ohm’s law, we have i(t) = υ (t) — R 1 + υ (t) — R 2 = ( 1 — R 1 + 1 — R 2 ) υ (t) = υ (t) — R p where 1 — R p = 1 — R 1 + 1 — R 2 2.16 R p = R 1 R 2 _______ R 1 + R 2 2.17 Therefore, the equivalent resistance of two resistors connected in parallel is equal to the product of their resistances divided by their sum.

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  Basic Engineering Circuit Analysis

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

    (Publisher)

  We know from KCL that there is only one current for a single-loop circuit. This current is assumed to be flowing either clockwise or counterclockwise around the loop. STEP 2 Using Ohm’s law, define a voltage across each resistor in terms of the defined current. STEP 3 Apply KVL to the single-loop circuit. STEP 4 Solve the single KVL equation for the current i(t). If i(t) is positive, the current is flowing in the direction assumed; if not, then the current is actually flowing in the opposite direction. 40 CHAPTER 2 Resistive Circuits E2.10 Find I and V bd in the circuit in Fig. E2.10. I 6 V 12 V 40 kΩ 80 kΩ a b c d − + + − FIGURE E2.10 E2.11 In the network in Fig. E2.11, if V ad is 3 V, find V S . − + V S 20 kΩ 25 kΩ 15 kΩ a b c d FIGURE E2.11 Answer: I = −0.05 mA; V bd = 10 V. Answer: V S = 9 V. Learning Assessments 2.4 Single-Node-Pair Circuits Current Division An important circuit is the single-node-pair circuit. If we apply KVL to every loop in a single- node-pair circuit, we discover that all of the elements have the same voltage across them and, therefore, are said to be connected in parallel. We will, however, apply Kirchhoff’s current law and Ohm’s law to determine various unknown quantities in the circuit. Following our approach with the single-loop circuit, we will begin with the simplest case and then generalize our analysis. Consider the circuit shown in Fig. 2.22. Here we have an independent current source in parallel with two resistors. Since all of the circuit elements are in parallel, the voltage υ (t) appears across each of them. Furthermore, an examination of the circuit indicates that the current i(t) is into the upper node of the circuit and the currents i 1 (t) and i 2 (t) are out of the node. Since KCL essen- tially states that what goes in must come out, the question we must answer is how i 1 (t) and i 2 (t) divide the input current i(t).

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  Electrical and Electronic Principles 3 Checkbook

  The Checkbook Series

  • J O Bird, A J C May(Authors)
  • 2016(Publication Date)
  • Newnes

    (Publisher)

  1 Circuit theorems A. MAIN POINTS CONCERNED WITH D.C. CIRCUIT ANALYSIS 1 The laws which determine the currents and voltage drops in d.c. networks are: (a) Ohm's law, (b) the laws for resistors in series and in parallel, and (c) Kirchhoffs laws. In addition, there are a number of circuit theorems which have been devel-oped for solving problems in electrical networks. These include: (i) the superposition theorem, (ii) Thévenin's theorem, (iii) Norton's theorem, and (iv) the maximum power transfer theorem. 2 The superposition theorem states: 7/1 any network made up of linear resistances and containing more than one source ofemf, the resultant current flowing in any branch is the algebraic sum of the currents that would flow in that branch if each source was considered separately y all other sources being replaced at that time by their respective internal resistances. * (See Problems 1 and 2) 3 The following points involving d.c. circuit analysis need to be appreciated before proceeding with problems using Thevenin's and Norton's theorems: (i) The open-circuit voltage, E, across terminals AB in Fig 1 is equal to 10 V, since no current flows through the 2 Ω resistor and hence no voltage drop occurs. (ii) The open-circuit voltage, E, across terminals AB in Fig 2(a) is the same as the voltage across the 6 Ω resistor. The ciruit may be redrawn as shown in Fig 2(b) 'tèi)™ by voltage division in a series circuit, i.e. E = 30 V 1 / (iii) (iv) (v) (vi) For the circuit shown in Fig 3(a) represent-ing a practical source supplying energy, V = E—Ir, where E is the battery emf, V is the battery terminal voltage and r is the internal resistance of the battery. For the circuit shown in Fig 3(b), V V = E-{-Dr, i.e. V = E+Ir The resistance 'looking-in' at terminals AB in Fig 4(a) is obtained by reducing the circuit in stages as shown in Figs 4(b) to (d). Hence the equivalent resistance across AB is 7 Ω. For the circuit shown in Fig 5(a), the 3 Ω resistor carries no current and the p.d.

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  Principles of Physics: Extended, International Adaptation

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  LEARNING OBJECTIVES Multiloop Circuits Figure 27.2.1 shows a circuit containing more than one loop. For simplicity, we assume the batteries are ideal. There are two junctions in this circuit, at b and d, and there are three branches connecting these junctions. The branches are the left branch (bad), the right branch (bcd), and the central branch (bd). What are the currents in the three branches? We arbitrarily label the currents, using a different subscript for each branch. Current i 1 has the same value everywhere in branch bad, i 2 has the same value everywhere in branch bcd, and i 3 is the current through branch bd. The directions of the currents are assumed arbitrarily. Consider junction d for a moment: Charge comes into that junction via incoming currents i 1 and i 3 , and it leaves via outgoing current i 2 . Because there is no variation in the charge at the junction, the total incoming current must equal the total outgoing current: i 1 + i 3 = i 2 . (27.2.1) You can easily check that applying this condition to junction b leads to exactly the same equation. Equation 27.2.1 thus suggests a general principle: This rule is often called Kirchhoff’s junction rule (or Kirchhoff’s current law). It is simply a statement of the conservation of charge for a steady flow of charge—there is neither a buildup nor a depletion of charge at a junction. Thus, our basic tools for solving complex circuits are the loop rule (based on the conservation of energy) and the junction rule (based on the conservation of charge). Equation 27.2.1 is a single equation involving three unknowns. To solve the circuit completely (that is, to find all three currents), we need two more equations involving those same unknowns. We obtain them by applying the loop rule twice. In the circuit of Fig. 27.2.1, we have three loops from which to choose: the left- hand loop (badb), the right-hand loop (bcdb), and the big loop (badcb).

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