Norton Theorem

11 Key excerpts on "Norton Theorem"

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  eBook - ePub

  Foundations of Electromagnetic Compatibility

  with Practical Applications

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

    (Publisher)

  The Norton current, I N, is obtained by placing a short circuit across nodes A and B, as shown in Figure 9.36, and calculating the so‐called short‐circuit current flowing from A to B. Figure 9.36 Short‐circuit current. Note that the Norton equivalent can of course be obtained from the Thévenin equivalent by a source transformation, as shown in Figure 9.37. Figure 9.37 Thévenin and Norton Equivalence. Therefore, (9.24) or (9.25) It also follows that the Thévenin or Norton resistance can be obtained from (9.26) The following example illustrates the application of Norton’s theorem. Example 9.5 Norton equivalent circuit Determine the Norton equivalent with respect to nodes A and B, for the circuit shown in Figure 9.38. (This is the same circuit as we used for the Thévenin equivalent in Example 9.3.) Figure 9.38 Circuit for Example 9.4. Norton resistance is the same as the Thévenin resistance calculated in Example 9.3 : The Norton current is the current flowing through a short circuit from A to B when the load is disconnected (or not present), and is shown in Figure 9.39. Figure 9.39 Short‐circuit current. To calculate the short‐circuit current we could use any appropriate method of circuit analysis. For this particular circuit, mesh analysis would be well suited. Let’s assign mesh currents, as shown in Figure 9.39. Note that and the Norton current is Writing KVL around the first mesh results in Writing KVL around the second mesh results in This system of equations yields: and thus The Norton resistance can be now calculated from which, of course agrees with the result of Example 9.3. 9.5 Maximum Power Transfer 9.5.1 Maximum Power Transfer – Resistive Circuits When interfacing the driving circuitry to the load, it is important to consider the voltage, current, and power available at an interface between a fixed source and an adjustable load. For simplicity we will consider the case in which both the source and the load are linear resistive circuits

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  Essentials of Advanced Circuit Analysis

  A Systems Approach

  • Djafar K. Mynbaev(Author)
  • 2024(Publication Date)
  • Wiley

    (Publisher)

  Norton, 6 in developing his proposition in 1926, was inspired by Thevenin’s theorem. Reviewing the Thevenin model at the LHS of Figure 2.14b, we see that the voltage v, which this source supplies to a load, is given by v V v V iR Th R Th Th = − = − , (2.24a) as in (2.23). In addition, from a Norton model, we find i I i I v R N RN N N = − = − . (2.24b) Figure 2.14 Norton’s theorem: a) The concept; b) comparison of Thevenin’s and Norton’s theorems. 6 Edward L. Norton (1898–1983) worked for the famous Bell Laboratories most of his professional life. His areas of interest included acoustics and electrical engineering. Graduated from MIT with BS and Columbia University with MS, he was well known for applying theory masterly and his intuition to design a new electrical system for telephony and data transmission. In his work, he extensively used Thevenin’s theorem. The idea known today as Norton’s theorem was born when he had to design the current-driven circuit. His published legacy includes 19 patents, 3 papers, and numerous technical memorandums, one of which contains his theorem. 2 Methods of DC Circuit Analysis 116 These two equations describe the linkage among v V Th and and i I N and . To find the relationship between Thevenin’s and Norton’s parameters, we must bear in mind that these two models are equivalent because a load “sees” the same input regardless of which model—Thevenin’s or Norton’s—is used to replace an actual circuit. Refer to Figures 2.10a and 2.14a. Thus, at the open a b − terminals, Thevenin’s model has V V oc Th = , and Norton’s model shows V R I oc N N = , which gives V R I Th N N = . (2.25a) At the short-circuit terminals, the current for Thevenin’s model is I V R sc Th Th = , whereas Norton’s model has I I sc N = . Therefore, I V R N Th Th = . (2.25b) Equations 2.25a and 2.25b show the relationship between the parameters of Thevenin’s and Norton’s models.

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  Principles of Analog Electronics

  • Giovanni Saggio(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  In fact, suppose we have at our disposal a Norton’s equivalent cir-cuit of a certain network, as illustrated in Figure 6.19. To verify the equivalence, we need to connect a resistance R L to both circuits and calculate the current value i RL : i i R R R R R RL N N L N L L = + 1 ; i V R R i R R R RL TH N L N N N L = + = + both of which, obviously, are equal. A B Linear network (with sources and resistors) (a) R N i N A B (b) FIGURE 6.18 (a) Active linear network, (b) Norton’s equivalent circuit R N R L i N A B (a) A B + R N = R TH v TH = i N R N R L (b) FIGURE 6.19 Verification schemes of Thévenin–Norton equivalence. Principles of Analog Electronics 202 6.5 SUPERPOSITION THEOREM Within the context of linear circuit analysis, both for transient or steady-state conditions, the superposition theorem is of fundametal importance. It is really useful for calculating the voltages and currents in a circuit with two or more (volt-age and/or current) sources. It asserts that “if in a linear network more than one voltage and/or current source operates at the same time, the voltage between any two nodes in the net-work (or the current in any branch) is the algebraic sum of voltages (or currents) owing to each of the independent sources acting one at a time.” Observation To apply the superposition theorem to a network, it is fundamental to respect the linearity requirement. Take into consideration, for instance, that power in an resistor is a nonlinear function, varying either with the square of voltage across it or with the square of current flowing through it. So, the superposition theorem cannot be applied to determine the power associated with a resistor. Example Let us consider the following circuit for which V V 9 B 1 = ; V V 4.5 B 2 = ; = = = Ω R R R 3 1 2 3 We calculate the current that flows through R 3 .

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  Basic Engineering Circuit Analysis

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

    (Publisher)

  In addition, the independent sources do not have to assume their actual value or zero. However, it is mandatory that the sum of the different values chosen add to the total value of the source. Superposition is a fundamental property of linear equations and, therefore, can be applied to any effect that is linearly related to its cause. In this regard, it is important to point out that although superposition applies to the current and voltage in a linear circuit, it cannot be used to determine power because power is a nonlinear function. 5.3 Thévenin’s and Norton’s Theorems 153 5.3 Thévenin’s and Norton’s Theorems Thus far we have presented a number of techniques for circuit analysis. At this point we will add two theorems to our collection of tools that will prove to be extremely useful. The theo- rems are named after their authors, M. L. Thévenin, a French engineer, and E. L. Norton, a scientist formerly with Bell Telephone Laboratories. Suppose that we are given a circuit and that we wish to find the current, voltage, or power that is delivered to some resistor of the network, which we will call the load. Thévenin’s theorem tells us that we can replace the entire network, exclusive of the load, by an equivalent circuit that contains only an independent voltage source in series with a resistor in such a way that the current–voltage relationship at the load is unchanged. Norton’s theorem is identical to the preceding statement except that the equivalent circuit is an independent current source in parallel with a resistor. Note that this is a very important result. It tells us that if we examine any network from a pair of terminals, we know that with respect to those terminals, the entire network is equiva- lent to a simple circuit consisting of an independent voltage source in series with a resistor or an independent current source in parallel with a resistor.

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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)

  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 shown that the two forms are equivalent. An ideal constant-voltage generator is one with zero internal resistance so that it supplies the same voltage to all loads. An ideal constant-current generator is one with infinite internal resistance so that it supplies the same current to all loads. Norton's theorem states: 'The current that flows in any branch of a network is the same as that which would flow in the branch if it were connected across a source of electrical energy, the short-circuit current of which is equal to the current that would flow in a short-circuit across the branch, and the internal resistance of which is equal to the resistance which appears across the open-circuited branch terminals. ' The procedure adopted when using Norton's theorem is summarized below. To determine the current flowing in a resistance R of a branch AB of an active network: (i) short-circuit branch AB (ii) determine the short-circuit current I sc flowing in the branch (iii) remove all sources of emf and replace them by their internal resistance (or, if a current source exists, replace with an open-circuit), then determine the resistance r, 'looking-in' at a break made between A and B (iv) determine the current / flowing in resistance R from the Norton equivalent network shown in Fig 9, i.e. <7h> Fig 9 ÓB Note the symbol for an ideal current source (BS 3939, 1985) shown in Fig 9 (See Problems 9 to 13) 9 The Thévenin and Norton networks shown in Fig 10 are equivalent to each other. The resistance 'looking-in' at terminals AB is the same in each of the networks, i.e.

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  Circuits, Signals, and Systems for Bioengineers

  A MATLAB-Based Introduction

  • John Semmlow(Author)
  • 2005(Publication Date)
  • Academic Press

    (Publisher)

  7.3 THÉVENIN AND Norton TheoremS: NETWORK REDUCTION WITH SOURCES The Thévenin theorem states that any network of passive elements and sources can be reduced to a single voltage source and series impedance. Thus the reduced network would look like a Thévenin circuit such as that shown in Figure 7.12, except that the internal resistance, R T , would be replaced by a generalized imped-ance, Z – q . The Norton Theorem makes the same claim for Norton circuits, which is reasonable because Thévenin circuits can easily be converted to Norton circuits via Eq. 7.19 and Eq. 7.20. There are a few constraints on these theorems. The elements in the network being reduced must be linear, and if there are multiple sources in the network, they must be at the same frequency. As has been done in the past, the techniques for network reduction will be developed using phasor representation and hence limited to net-works with sinusoidal sources. However, the approach for generalizing this and other techniques involving phasor analysis to a wide range of signals is presented in the next chapter. There are two approaches to finding the Thévenin or Norton Equivalent of a circuit. One is based on the strategy used above to find the equivalence between Thévenin and Norton circuits: find the open-circuit voltage, v oc , and the short-circuit current, i sc . The other method evaluates only the open-circuit voltage, v oc , then deter-mines R T (or R N ) through network reduction. During network reduction, a source is replaced by its equivalent resistance, that is, open circuits ( R Æ • W ) substitute for voltage sources, and short circuits ( R = 0.0 W ) substitute for current sources. Both these approaches are straightforward to implement and are best shown though examples. Example 7.11: Find the Thévenin equivalent of the circuit below using both the v oc -i sc method and the v oc -network reduction technique.

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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)

  Z can be predicted:

  Figure 5.14 Open-circuit test.

  Figure 5.15 Short-circuit test.

  (5.30)

  It is important to note that the described experimental approach does not require any a priori knowledge concerning the active network. For this reason, the open-circuit and short-circuit tests are widely used for characterizations of many devices such as transformers, induction motors, and synchronous generators.

  5.5 Norton’s Theorem

  Norton’s theorem is very similar to Thevenin’s theorem. Basically it states that as far as the current through an impedance Z is concerned any active network can be replaced by a non-ideal current source. It is graphically represented in Figure 5.16 . This theorem is trivial to prove by using Thevenin’s theorem.

  Figure 5.16 Norton’s theorem.

  Proof: We know that according to Thevenin’s theorem the active network in the left circuit of Figure 5.16 can be equivalently replaced by a nonideal voltage source. Since any nonideal voltage source can be equivalently transformed to a nonideal current source (see Figure 5.17 ), Norton’s theorem is proven.

  Figure 5.17 “Proof” of Norton’s theorem.

  We know the equivalence between nonideal voltage and current sources means that

  (5.31)

  (5.32)

  Therefore, the current source in Norton’s theorem is equal to the short-circuit current, and the admittance is equal to the input admittance of the corresponding passive network:

  (5.33)

  (5.34)

  Norton’s theorem can be used in a manner similar to Thevenin’s theorem.

  EXAMPLE 5.3

  We want to find the Norton equivalent nonideal current source for the circuit shown in Figure 5.18 .

  Figure 5.18 Circuit for the first Norton example.

  First, we can find the equivalent impedance by setting the ideal sources to zero. For this example, that entails shorting the voltage source and opening the current source . The resulting passive circuit is given in Figure 5.19 . Because Z 1 is disconnected on the left side, it cannot have any current flow through it and thus has no effect on the equivalent impedance. Thus, the elements Z 2 and Z 3 are effectively in series and that combination is in parallel with Z 4

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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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  Dorf's Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Consider circuit A and determine its short-circuit current i sc at its terminals. Then the equivalent circuit of A is a current source i sc in parallel with a resistance R n , where R n is the resistance looking into circuit A with all its independent sources deactivated. We therefore have the Norton circuit for circuit A as shown in Figure 5.5-1. Finding the Th evenin equivalent circuit of the circuit in Figure 5.5-1 shows that R n R t and v oc R t i sc . The Norton equivalent is simply the source transformation of the Th evenin equivalent. a b R n R n i sc FIGURE 5.5-1 Norton equivalent circuit for a linear circuit A. E X A M P L E 5 . 5 - 1 Norton Equivalent Circuit Determine the Norton equivalent circuit for the circuit shown in Figure 5.5-2. 160 Ω 125 V 2 A a b 40 Ω + – FIGURE 5.5-2 The circuit considered in Example 5.5-1. Solution In Figure 5.5-3, source transformations and equivalent circuits are used to simplify the circuit in Figure 5.5-2. These simplifications continue until the simplified circuit in Figure 5.5-3d consists of a single current source in parallel with a single resistor. The circuit in Figure 5.5-3d is the Norton equivalent circuit of the circuit in Figure 5.5-2. Consequently i sc 1 125 A and R t R n 32 V 160 Ω 32 Ω 1.125 A a b a b (a ) (b ) 40 Ω 80 V (d ) 125 V 160 Ω a b – 40 Ω 45 V 160 Ω 1.125 A a b (c ) 40 Ω + – + – + FIGURE 5.5-3 Using source transformations and equivalent circuits to determine the Norton equivalent circuit of the circuit shown in Figure 5.5-2. Norton’ s Equivalent Circuit 187 E X A M P L E 5 . 5 - 2 Norton Equivalent Circuit of a Circuit Containing a Dependent Source Determine the Norton equivalent circuit for the circuit shown in Figure 5.5-4. 40 Ω 12 V a b i a + – 10 Ω 5 Ω 4.5 i a FIGURE 5.5-4 The circuit considered in Example 5.5-2. Solution We determined the Th evenin equivalent of the circuit shown in Figure 5.5-4 in Example 5.4-2.

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  N4 Communication Electronics

  • R Britz(Author)
  • 2013(Publication Date)
  • Future Managers

    (Publisher)

  N4 Communications Electronics | Hands-On! 21 I T(10) = 0,32 + 0,1428 = 0,46 A 6V 4V 2  10  3  I T T Norton’s Theorem This theorem is the dual of Thévenin's theorem. This theorem states that a linear circuit containing independent sources, dependent sources and passive elements can be replaced by a model containing a current source and an equivalent resistance. A two terminal network of generators and impedance may be replaced by a single constant current generator with an impedance across it, as in Figure 1.18. Figure 1.18 (a) (b) A B NETWORK OF GENERATORS A B I AB R AB The generator must produce a constant current equal to the current that flows through terminals AB when they are short circuited. The shunt impedance of the network “looking back” into terminals AB while the generators are replaced by their internal impedance is R AB . (Note that the internal resistance of a constant current generator is taken to be infinity while the internal resistance of a constant voltage generator is taken to be zero.) Module 1 • Network Theorems 22 Example 1 Obtain the Norton equivalent of the circuit in Figure 1.19(a). When a short circuit is placed across AB, (no current flows through the 200 ohm resistor as it is shorted) the short circuit current I AB is: I = V __ R = 100V ____ 100  = 1A Figure 1.19 (a) (b) A B IA 66,7  A B 200  100  100V The equivalent Norton resistance is simply the parallel connection of the 100 ohm and 200 ohm resistors. i.e. R AB = 100 × 200 _______ 300 = 66,7 ohms. Example 2 Obtain the Norton equivalent circuit for Figure 1.20 below. 15  5  Figure 1.20 R 1 R 2 V 2 = 20V V 1 = 10V A B i) Apply short circuit to AB. Current = V 1 __ R 1 + V 2 __ R 2 = 10 __ 5 + 20 __ 15 = 3,33 A N4 Communications Electronics | Hands-On! 23 ii) Impedance with e.m.f.’s set to zero. Z = R 1 × R 2 _____ R 1 + R 2 = 5 × 15 _____ 5 + 15 = 3,75 Ω iii) Equivalent circuit.

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  Engineering Circuit Analysis

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

    (Publisher)

  3.5 Thévenin’s and Norton’s Theorems 163 164 Chapter 3 Network Theorems 3.6 ❯ Maximum Power Transfer As we begin our discussion of maximum power transfer, it is instructive to examine what is known as a load line. This graphical technique is typically employed in (nonlinear) electronic circuits and used to determine the operating point of the network. Consider, for example, the network in Fig. 3.48a. The load line is used to represent the relationship between voltage and current in the linear portion of the circuit, represented in this case by the 12‑V source and 2‑kΩ resistor. This line is shown in Fig. 3.48b and defined by the two points along the axes. The load, which may be linear or nonlinear, has a characteristic curve that defines its voltage/current relationship. If the load is a 4‑kΩ resistor, as illustrated in Fig. 3.48c, its characteristic curve will appear as shown in Fig. 3.48b. The operating point is defined as the point at which the characteristic curve intersects the load line, because at this point both the voltage and current parameters for each circuit match. Thus, in this example, the operating point is at V o = 8 V and I o = 2 mA. In nonlinear circuits, such as when the load is a diode, the diode’s characteristic curve is not a straight line, and this tech‑ nique provides a useful mechanism for graphically determining the operating point of the circuit. There are situations in circuit design when we want to select a load so that the maximum power can be transferred to it. We can determine the maximum power that a circuit can supply and the manner in which to adjust the load to effect maximum power transfer by employing Thévenin’s theorem. In circuit analysis, we are sometimes interested in determining the maximum power that can be delivered to a load. By employing Thévenin’s theorem, we can determine the maximum power that a circuit can supply and the manner in which to adjust the load to effect maximum power transfer.

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