Thevenin Theorem

12 Key excerpts on "Thevenin Theorem"

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  eBook - ePub

  A Practical Introduction to Electrical Circuits

  • John E. Ayers(Author)
  • 2024(Publication Date)
  • CRC Press

    (Publisher)

  5 Thevenin’s and Norton’s Theorems

  DOI: 10.1201/9781003408529-5

  5.1 Thevenin’s Theorem

  Thevenin’s theorem is a useful tool for solving certain types of problems. It states that for any two-terminal network involving resistors and sources, there is a Thevenin equivalent network involving a single voltage source

  V Th

  and a single resistance

  R Th

  , as shown in Figure 5.1 . The Thevenin circuit is externally equivalent to the original circuit, in the sense that if we connect additional circuitry to the two terminals a and b, the voltages and currents in this external circuitry will be the same for the Thevenin equivalent as the original circuit, regardless of the complexity of the original circuit. They are not internally equivalent, however. This means that we cannot use the Thevenin equivalent circuit to directly find voltages and currents inside the original circuit.

  Long Description for Figure 5.1

  On the left is a two-terminal network comprising resistors and sources; it is represented by a rectangular box with two terminals on the right hand side, and the top and bottom terminal are labeled a and b, respectively. On the right hand side is the Thevenin equivalent for the original network, also with two terminals labeled a and b. Connected to terminal b is the negative terminal of a voltage source of value VTh. The positive terminal of this voltage source is connected to a resistor with value RTh, and the other side of the resistor is connected to terminal a. In between the original network and the Thevenin equivalent is an arrow, pointing from left to right, which indicates the transformation from the original circuit to the Thevenin equivalent.

  FIGURE 5.1 A network containing resistors and sources and its Thevenin equivalent.

  Because the Thevenin circuit is externally equivalent to the original circuit, we can analyze the behavior with simple external connections to find

  V Th

  and

  R Th

  . One such connection involves a short circuit as shown in Figure 5.2 . If we place a wire directly between the terminals a and b to create a short circuit, the same current should flow in either case. The short-circuit current for the Thevenin equivalent is

  V Th

  /

  R Th

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  Basic Electric Circuit Theory

  A One-Semester Text

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

    (Publisher)

  Figure 5.3 . In the following section, the proof of Thevenin’s theorem is given.

  Figure 5.3 Graphical illustration of Thevenin’s theorem.

  5.4.1 Proof of Thevenin’s Theorem

  The proof consists of the following three steps: Step I:

  First, we introduce two voltage sources into the branch with impedance Z (see Figure 5.4 ). Both have the same peak value of voltage but opposite polarities. We can see that doing this will not affect the current through Z at all, because the two voltage sources will cancel out in any KVL equations due to their opposite polarities.

  Figure 5.4 Introduce two voltage sources.

  Step II: Next, we shall use the superposition principle. To do this, we divide all the sources in the network into two groups:

  1. The first group consists of all the sources in the active network and the left voltage source introduced into the branch with impedance Z .

  2. The second group consists of only the right voltage source in the same branch.

  Now, we can consider two separate regimes, shown in Figure 5.5 . The first regime contains the sources of the first group, and the second regime contains the only source of the second group. Note that, in the second regime, the active network has been replaced by a passive network. This passive network is formed when the sources in the active network are set to zero. This is accomplished by replacing voltage sources with short-circuit branches and current sources with open branches.

  Figure 5.5 The two regimes.

  By using the superposition principle we find that the total current through Z is equal to the sum of the branch currents in the above two regimes:

  (5.8)

  Step III:

  Consider the first regime. We choose in a way that causes to be zero. This is equivalent to having the branch with Z open. Consequently, the voltage across the terminals A and B will be equal to the open-circuit voltage . This situation is shown in Figure 5.6 . To find such a which guarantees that will be equal to zero, we write the KVL for the loop shown in Figure 5.6

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  Feedback Networks: Theory and Circuit Applications

  • J Choma, W K Chen;;;(Authors)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  of initial capacitive and inductive energies have dissipated and no longer possess engineering significance. A third possibility is that initial capaci-tor voltages and inductor currents are treated as additional independently applied input excitations, similar to the signal sources, V s , and I s . In the present circumstance, it is tacitly assumed that analytical attention focuses exclusively on steady state electrical characteristics. Thévenin’s theorem states that the electrical characteristics at any port (or terminal pair) of a linear electrical network can be modeled by a volt-age source in series with an impedance, as suggested by Fig. 1.1(b). The indicated voltage source, V th , is termed the Thévenin voltage of the port undergoing scrutiny, while the subject series impedance, Z th , is known as the Thévenin impedance of said port. If the port at which Thévenin’s the-orem is applied happens to be the output port of the network where signal responses to applied input excitations are to be delivered, the Thévenin impedance is also known as the network output impedance . When V th and Z th are correctly measured or calculated, the Thévenin equivalent circuit, or Thévenin model, “seen” by the load establishes a load voltage, V , and a load current, I , that are respectively identical to the load voltage and current supported by the original system in Fig. 1.1(a). It is important to underscore the fact that the foregoing assertions are independent of the nature of the load connected to the network port undergoing a Thévenin investigation. This is to say that the load at hand can be a passive, an active, a linear, or even a nonlinear electrical branch. An alternative to Thévenin’s theorem is Norton’s theorem, which stip-ulates that any port of a linear electrical network can be represented as a Circuit and System Fundamentals 5 Z th I n Load I + − V Figure 1.2. The Norton equivalent circuit for the system given in Fig.

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  Basic Electronics for Scientists and Engineers

  • Dennis L. Eggleston(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  This is represented by the example in Fig. 1.10 . The sources can include both voltage and current sources (the current source is described below). A more general version of the theorem replaces the word resistor with impedance , a concept we will develop in Chapter 2 . The point of Thevenin’s theorem is that when we connect a component to the terminals, it is much easier to analyze the circuit on the right than the circuit on the left. But there is no free lunch – we must first determine the values of V th and R th . V th is the voltage across the circuit terminals when nothing is connected to the terminals. This is clear from the equivalent circuit: if nothing is connected to the terminals, then no current flows in the circuit and there is no voltage drop across R th . The voltage across the terminals is thus the same as V th . In practice, the voltage across the terminals must be calculated by analyzing the original circuit. There are two methods for calculating R th ; you can use whichever is easiest. In the first method, you start by short circuiting all the voltage sources and open circuiting all the current sources in the original circuit. This means that you replace the voltage sources by a wire and disconnect the current sources. Now only resistors are left in the circuit. These are then combined into one resistor using the resistor equivalent circuit laws. This one resistor then gives the value of R th . In the second method, we calculate the current that would flow in the circuit if we shorted (placed a wire across) the terminals. Call this the short circuit current I sc . Then from the Thevenin equivalent circuit it is clear that R th = V th I sc . There is also a similar result known as Norton’s theorem. This theorem states that any two-terminal network of sources and resistors can be replaced by a parallel 1.2 Resistors 11 I nor R nor Figure 1.11 Equivalent circuit of Norton’s theorem. combination of a single resistor R nor and current source I nor .

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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)

  Fig 7 (viii) In Fig 7(a), to find the equivalent resistance across AB the circuit may be redrawn as in Figs 7(b) and (c). From Fig 7(c), the equivalent resistance across AB _ 5 X 15^ 12X3 5 + 1 5 12 + 3 = 3.75+2.4 = 6.15 Ω (ix) In the worked problems in section B it may be considered that Thévenin's and Norton's theorems have no obvious advantages compared with, say, Kirchhoff's laws. However, these theorems can be used to analyse part of a circuit and in much more complicated networks the principle of replacing the supply by a constant voltage source in series with a resistance (or impedance) is very useful. Thévenin's theorem states: 'The current in any branch of a network is that which would result if an emf equal to the p.d. across a break made in the branch, were introduced into the branch, all other emf s being removed and represented by the internal resistances of the sources.' The procedure adopted when using Thévenin's theorem is summarized below. To determine the current in any branch of an active network (i.e. one containing a source of emf): I ->~ (i) remove the resistance R from that branch, (ii) determine the open-circuit voltage, E, across the break, (iii) remove each source of emf and replace them by their internal resistances and then determine the resistance, r, 'looking-in' at the break, (iv) determine the value of the current from the equivalent circuit shown in Fig 8, i.e. T _ E Fig 8 (See Problems 3 to 8). ~R + r A source of electrical energy can be represented by a source of emf in series with a resistance. In para. 5, the Thévenin constant-voltage source consisted of a constant emf E in series with an internal resistance r. However this is not the only form of representation. A source of electrical energy can also be represented by a constant-current source in parallel with a resistance. It may be 4

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  Basic Engineering Circuit Analysis

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

    (Publisher)

  The theorems help us to understand the effect of using a transducer at the input of an amplifier with a given input resistance. They help us explain the effect of a load, such as a speaker, at the output of an amplifier. We derive none of this information from a node or loop analysis. In fact, as a simple example, suppose that a network at a specific pair of terminals has a Thévenin equivalent circuit consisting of a voltage source in series with a 2-kΩ resistor. If we connect a 2-Ω resistor to the network at these terminals, the voltage across the 2-Ω resistor will be essentially nothing. This result is fairly obvious using the Thévenin theorem approach; however, a node or loop analysis gives us no clue as to why we have obtained this result. We have studied networks containing only dependent sources. This is a very important topic because all electronic devices, such as transistors, are modeled in this fashion. Motors in power systems are also modeled in this way. We use these amplification devices for many different purposes, such as speed control for automobiles. In addition, it is interesting to note that when we employ source transformation as we did in Example 5.13, we are simply converting back and forth between a Thévenin equivalent circuit and a Norton equivalent circuit. Finally, we have a powerful tool at our disposal that can be used to provide additional insight and understanding for both circuit analysis and design. That tool is Microsoft Excel, and it permits us to study the effects on a network of varying specific parameters. The follow- ing example will illustrate the simplicity of this approach. We wish to use Microsoft Excel to plot the Thévenin equivalent parameters V oc and R Th for the circuit in Fig. 5.16 over the R x range 0 to 10 kΩ. EXAMPLE 5.14 FIGURE 5.16 Circuit used in Example 5.14. − + −+ V oc + − 12 V 6 V 4 kΩ R x R Th Solution The Thévenin resistance is easily found by replacing the voltage sources with short circuits.

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  Mechatronic Systems, Sensors, and Actuators

  Fundamentals and Modeling

  • Robert H. Bishop(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  In studying node voltage and mesh current analysis, you may have observed that there is a certain correspondence (called duality) between current sources and voltage sources, on the one hand, and parallel and series circuits, on the other. This duality appears again very clearly in the analysis of equivalent circuits: it will shortly be shown that equivalent circuits fall into one of two classes, involving either voltage or current sources and (respectively) either series or parallel resistors, reflecting this same principle of duality. The discussion of equivalent circuits begins with the statement of two very important theorems, summarized in Figures 11.27 and 11.28. FIGURE 11.27 Illustration of Thévenin theorem. FIGURE 11.28 Illustration of Norton theorem. FIGURE 11.29 Computation of Thévenin resistance. The Thévenin Theorem As far as a load is concerned, any network composed of ideal voltage and current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal voltage source, v T, in series with an equivalent resistance, R T The Norton Theorem As far as a load is concerned, any network composed of ideal voltage and current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal current source, i N, in parallel with an equivalent resistance, R N. 11.3.3.2  Determination of Norton or Thévenin Equivalent Resistance The first step in computing a Thévenin or Norton equivalent circuit consists of finding the equivalent resistance presented by the circuit at its terminals. This is done by setting all sources in the circuit equal to zero and computing the effective resistance between terminals. The voltage and current sources present in the circuit are set to zero as follows: voltage sources are replaced by short circuits, current sources by open circuits

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  The Fundamentals of Electrical Engineering

  for Mechatronics

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

    (Publisher)

  Fig. 5.16 is Thévenin’s equivalent to the active two-terminal circuit on the left side. If we are just interested in the load circuit (here a single resistor) it can simplify the analysis if we use this theorem.

  Example: We want to determine the maximum power transfer to the load resistor. Using the original two-terminal circuit connected to the load resistor we have to calculate the power transfer for changing load resistor values. Depending on complexity of the two-terminal circuit this might be difficult. Using Thévenin equivalent immediately reveals the solution: maximum power is transferred to the load resistor if it is equal to the internal resistor of the Thévenin equivalent.

  How can we (easily) determine the two parameters (Uq and Ri ) of the Thévenin equivalent?

    Algorithm to determine the Thévenin equivalent

  1. The load network must not contain dependencies of the supply network
  2. Determine the open loop voltage at the terminals of the supply network (i.e. load resistor RL = ∞ Ω). This yields Uq = Urep = U
  3. Determine the inner resistance if the supply circuit Ri = Rrep . Two possibilities:
     • – First possibility:
       • Determine the short-circuit current Isc (i.e. RL = 0 Ω) with
     • – Second possibility:
       • Short-circuit of all ideal voltage sources
       • Remove all ideal current sources (open load)
       • Leave depending sources as is
       • Look from the outside into the modified arbitrary network and determine the resistance that you “see from the outside”
       • This resulting resistor will be the replacement resistor of the Thévenin equivalent

  An example for Thévenin’s theorem

  Fig. 5.17 : An example for a circuit of a supply network and a simple load network (top); supply network is to be replaced by the Thévenin equivalent (bottom).

  In this example (see Fig. 5.17 ) we are not interested in the internals of the supply network, but just in the behavior of the terminals 1 and 0 and we want to know the value of the load resistor for maximum power transfer from the source. Therefore we simplify the supply network by applying Thévenin’s theorem. First we regard the supply network without the load as depicted in Fig. 5.18

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  Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  With the current level of computational power available to us, we can solve the node or loop equations that define the network in a flash. With regard to the theorems, we have found that in some cases the theorems do not neces- sarily simplify the problem and a straightforward attack using node or loop analysis is as good an approach as any. This is a valid point, provided that we are simply looking for some particu- lar voltage or current. However, the real value of the theorems is the insight and understanding that they provide about the physical nature of the network. For example, superposition tells us what each source contributes to the quantity under investigation. However, a computer solu- tion of the node or loop equations does not tell us the effect of changing certain parameter values in the circuit. It does not help us understand the concept of loading a network or the ramifications of interconnecting networks or the idea of matching a network for maximum power transfer. The theorems help us to understand the effect of using a transducer at the input of an amplifier with a given input resistance. They help us explain the effect of a load, such as a speaker, at the output of an amplifier. We derive none of this information from a node or loop analysis. In fact, as a simple example, suppose that a network at a specific pair of terminals has a Thévenin equivalent circuit consisting of a voltage source in series with a 2-kΩ resistor. If we connect a 2-Ω resistor to the network at these terminals, the voltage across the 2-Ω resistor will be essentially nothing. This result is fairly obvious using the Thévenin theorem approach; however, a node or loop analysis gives us no clue as to why we have obtained this result. We have studied networks containing only dependent sources. This is a very important topic because all electronic devices, such as transistors, are modeled in this fashion. Motors in power systems are also modeled in this way.

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  Foundations of Electromagnetic Compatibility

  with Practical Applications

  • Bogdan Adamczyk(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  Note that where and V 1 can be obtained using the current divider and Ohm’s law, as follows I 1 can be obtained using current divider as and thus thus Therefore, the Thévenin voltage is Figure 9.26 Circuits for calculation of open‐circuit voltage. Thévenin resistance is the resistance between nodes A and B when the load is disconnected and the independent sources are deactivated. The resulting circuit is shown in Figure 9.27. Figure 9.27 Circuit for calculation of Thévenin Resistance. Thus the Thévenin resistance is The Thévenin equivalent circuit is shown in Figure 9.28. Figure 9.28 Thévenin equivalent for Example 9.3. When the circuit consists of independent and dependent sources and resistors, we use another approach to determine the Thévenin resistance. We deactivate the independent sources and drive the circuit with an external voltage or current source connected between nodes A and B. (We will use this approach later in this chapter when discussing the two‐port networks.) This approach is based on the following discussion. Consider a linear circuit with no independent sources (or the independent sources suppressed) and/or dependent sources, as shown in Figure 9.29 (a). Figure 9.29 Thévenin equivalent resistance. Figure 9.29 (b) shows its Thévenin equivalent resistance. The Thévenin resistance of this circuit can be obtained by applying an external voltage or current source as shown in Figures 9.30 (a) and (b). Figure 9.30 Calculation of Thévenin resistance. Now consider the circuits shown in Figure 9.31. Figure 9.31 Calculation of Thévenin resistance. Since the circuits to the left of nodes A and B are equivalent, it follows that in order to obtain the Thévenin resistance of an arbitrary linear circuit, we first deactivate the independent sources (if present)

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  Electronic and Electrical Engineering

  Principles and Practice

  • Lionel Warnes(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  So, thirdly, we combine them by the product-over-sum rule: And so the Thévenin equivalent of the network in Figure 1.23a is that of Figure 1.23c. It is important to realise that the circuits of Figures 1.23a and 1.23c are equivalent only for measurements of current and voltage made at terminals A and B , and nowhere else. Having found the Thévenin equivalent we can put it in the network’s place, connect any other network across AB, then go ahead and calculate all the currents and voltages in that network. However, we cannot calculate the power developed within a network by considering the power developed in its Thévenin equivalent (see Problem 1.11). Example 1.4 A second 4 resistance is connected across AB in Figure 1.23a. What is the current in it? We can calculate the current through it by using the Thévenin circuit of Figure 1.23c, as in Figure 1.24a. The 4 resistance is in series with the 1.333 Thévenin resistance, so the current through them is 2/(4 + 1.333) = 0.375 A, while the voltage across AB will be 4 × 0.375 = 1.5 V (by the voltage-divider rule it will be 2 × 4/(4 + 1.333) = 1.5 V too). Circuit analysis 20 10 E L Norton worked for Bell Telephone Laboratories in the USA. He published his theorem in 1926. Figure 1.24 (a) Using Thévenin’s equivalent circuit. The 4 resistance across AB is the same as (b) Connecting the 4 resistance across the original circuit Figure 1.25 Norton’s equivalent circuit. Any two-terminal network of sources and resistances may be replaced by this Looking at the original circuit in Figure 1.23a, we can connect the 4 resistance across AB (Figure 1.24b), and find the current from the source by combining the two parallel 4 resistances using the product-over-sum rule, making 2 (Figure 1.24c). This result can be added to the 2 resistance in series to give 4 , so the current from the 3V source is 3/4 = 0.75 A. This current passes through the 2 resistance and divides equally between the 4 resistances, 0.375 A each.

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  Circuits, Signals, and Speech and Image Processing

  • Richard C. Dorf(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Maximum power transfer theorem: In any electrical network that carries direct or alternating current, the maximum possible power transferred from one section to another occurs when the impedance of the section acting as the load is the complex conjugate of the impedance of the section that acts as the source. Here, both impedances are measured across the pair of terminals in which the power is transferred with the other part of the network disconnected.

  Norton theorem: The voltage across an element that is connected to two terminals of a linear, bilateral network is equal to the short-circuit current between these terminals in the absence of the element divided by the admittance of the network looking back from the terminals into the network with all generators replaced by their internal admittances.

  Principle of superposition: In a linear electrical network, the voltage or current in any element resulting from several sources acting together is the sum of the voltages or currents from each source acting alone.

  Reciprocity theorem: In a network consisting of linear, passive impedances, the ratio of the voltage introduced into any branch to the current in any other branch is equal in magnitude and phase to the ratio that results if the positions of the voltage and current are interchanged.

  Thévenin theorem: The current flowing in any impedance connected to two terminals of a linear, bilateral network containing generators is equal to the current flowing in the same impedance when it is connected to a voltage generator whose voltage is the voltage at the open-circuited terminals in question and whose series impedance is the impedance of the network looking back from the terminals into the network, with all generators replaced by their internal impedances.

  Reference

  J.D. Irwin, Basic Engineering Circuit Analysis, 7th ed., New York: Wiley, 2003.

  A.D. Kraus, Circuit Analysis, St. Paul: West Publishing, 1991.

  R.C. Dorf, Introduction to Electric Circuits, New York: Wiley, 2004.

  3.4    Power and Energy Norman Balabanian and Theodore A. Bickart

  The concept of the voltage, v, between two points was introduced in Section “Voltage and Current Laws” as the energy, w, expended per unit charge in moving the charge between the two points. Coupled with the definition of current, i, as the time rate of charge motion and that of power, p, as the time rate of change of energy, this leads to the following fundamental relationship between the power delivered to a two-terminal electrical component and the voltage and current of that component, with standard references (meaning that the voltage reference plus is at the tail of the current reference arrow) as shown in Figure 3.22

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