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Network theorems chapter 9

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This document contains chapters from the textbook "Introductory Circuit Analysis" by Boylestad on network theorems. It discusses the objectives of learning superposition theorem, Thevenin's theorem, and Norton's theorem. The chapter on superposition theorem explains that it allows analyzing circuits with multiple sources by considering each source independently and summing their effects. It provides examples of applying superposition theorem to determine currents and voltages in circuits. The chapter also discusses Thevenin's theorem, which allows reducing any two-terminal network to a single voltage source in series with a resistor.

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Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] Introductory Circuit Analysis, 12/e Boylestad Chapter 9 Network Theorems
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] OBJECTIVES • Become familiar with the superposition theorem and its unique ability to separate the impact of each source on the quantity of interest. • Be able to apply Thévenin’s theorem to reduce any two-terminal, series-parallel network with any number of sources to a single voltage source and series resistor. • Become familiar with Norton’s theorem and how it can be used to reduce any two-terminal, seriesparallel network with any number of sources to a single current source and a parallel resistor.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] OBJECTIVES • Understand how to apply the maximum power transfer theorem to determine the maximum power to a load and to choose a load that will receive maximum power. • Become aware of the reduction powers of Millman’s theorem and the powerful implications of the substitution and reciprocity theorems.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM • The superposition theorem is unquestionably one of the most powerful in this field. • It has such widespread application that people often apply it without recognizing that their maneuvers are valid only because of this theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM • In general, the theorem can be used to do the following: – Analyze networks such as introduced in the last chapter that have two or more sources that are not in series or parallel. – Reveal the effect of each source on a particular quantity of interest. – For sources of different types (such as dc and ac, which affect the parameters of the network in a different manner) and apply a separate analysis for each type, with the total result simply the algebraic sum of the results.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM • The superposition theorem states the following: – The current through, or voltage across, any element of a network is equal to the algebraic sum of the currents or voltages produced independently by each source.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.1 Removing a voltage source and a current source to permit the application of the superposition theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.2 Network to be analyzed in Example 9.1 using the superposition theorem. FIG. 9.3 Replacing the 9 A current source in Fig. 9.2 by an open circuit to determine the effect of the 36 V voltage source on current I2.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.4 Replacing the 36 V voltage source by a short-circuit equivalent to determine the effect of the 9 A current source on current I2. FIG. 9.5 Using the results of Figs. 9.3 and 9.4 to determine current I2 for the network in Fig. 9.2.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.6 Plotting power delivered to the 6Ω resistor versus current through the resistor.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.7 Plotting I versus V for the 6Ω resistor.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.8 Using the superposition theorem to determine the current through the 12Ω resistor (Example 9.2).
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.9 Using the superposition theorem to determine the effect of the 54 V voltage source on current I2 in Fig. 9.8.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.10 Using the superposition theorem to determine the effect of the 48 V voltage source on current I2 in Fig. 9.8.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.11 Using the results of Figs. 9.9 and 9.10 to determine current I2 for the network in Fig. 9.8.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.12 Two-source network to be analyzed using the superposition theorem in Example 9.3. FIG. 9.13 Determining the effect of the 30 V supply on the current I1 in Fig. 9.12.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.14 Determining the effect of the 3 A current source on the current I1 in Fig. 9.12. FIG. 9.15 Example 9.4.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.16 The effect of the current source I on the current I2.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.17 The effect of the voltage source E on the current I2.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.18 Example 9.5. FIG. 9.19 The effect of E1 on the current I.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.20 The effect of E2 on the current I1. FIG. 9.21 The effect of I on the current I1
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.22 The resultant current I1.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM • The next theorem to be introduced, Thévenin’s theorem, is probably one of the most interesting in that it permits the reduction of complex networks to a simpler form for analysis and design.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM • In general, the theorem can be used to do the following: – Analyze networks with sources that are not in series or parallel. – Reduce the number of components required to establish the same characteristics at the output terminals. – Investigate the effect of changing a particular component on the behavior of a network without having to analyze the entire network after each change.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM • Thévenin’s theorem states the following: – Any two-terminal dc network can be replaced by an equivalent circuit consisting solely of a voltage source and a series resistor as shown in Fig. 9.23. FIG. 9.23 Thévenin equivalent circuit.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM FIG. 9.25 Substituting the Thévenin equivalent circuit for a complex network.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • Preliminary: 1. Remove that portion of the network where the Thévenin equivalent circuit is found. In Fig. 9.25(a), this requires that the load resistor RL be temporarily removed from the network. 2. Mark the terminals of the remaining two- terminal network. (The importance of this step will become obvious as we progress through some complex networks.)
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • RTh: – 3. Calculate RTh by first setting all sources to zero (voltage sources are replaced by short circuits and current sources by open circuits) and then finding the resultant resistance between the two marked terminals. (If the internal resistance of the voltage and/or current sources is included in the original network, it must remain when the sources are set to zero.)
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • ETh: – 4. Calculate ETh by first returning all sources to their original position and finding the open-circuit voltage between the marked terminals. (This step is invariably the one that causes most confusion and errors. In all cases, keep in mind that it is the opencircuit potential between the two terminals marked in step 2.)
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • Conclusion: – 5. Draw the Thévenin equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit. This step is indicated by the placement of the resistor RL between the terminals of the Thévenin equivalent circuit as shown in Fig. 9.25(b).
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.26 Example 9.6. FIG. 9.27 Identifying the terminals of particular importance when applying Thévenin’s theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.28 Determining RTh for the network in Fig. 9.27.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.29 Determining ETh for the network in Fig. 9.27. FIG. 9.30 Measuring ETh for the network in Fig. 9.27.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.31 Substituting the Thévenin equivalent circuit for the network external to RL in Fig. 9.26. FIG. 9.32 Example 9.7.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.33 Establishing the terminals of particular interest for the network in Fig. 9.32. FIG. 9.34 Determining RTh for the network in Fig. 9.33.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.35 Determining ETh for the network in Fig. 9.33. FIG. 9.36 Substituting the Thévenin equivalent circuit in the network external to the resistor R3 in Fig. 9.32.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.37 Example 9.8.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.38 Identifying the terminals of particular interest for the network in Fig. 9.37.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.39 Determining RTh for the network in Fig. 9.38.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.40 Determining ETh for the network in Fig. 9.38.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.41 Network of Fig. 9.40 redrawn. FIG. 9.42 Substituting the Thévenin equivalent circuit for the network external to the resistor R4 in Fig. 9.37.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.43 Example 9.9.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.44 Identifying the terminals of particular interest for the network in Fig. 9.43.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.45 Solving for RTh for the network in Fig. 9.44.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.46 Determining ETh for the network in Fig. 9.44.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.47 Substituting the Thévenin equivalent circuit for the network external to the resistor RL in Fig. 9.43.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.48 Example 9.10.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.49 Identifying the terminals of particular interest for the network in Fig. 9.48. FIG. 9.50 Determining RTh for the network in Fig. 9.49.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.51 Determining the contribution to ETh from the source E1 for the network in Fig. 9.49. FIG. 9.52 Determining the contribution to ETh from the source E2 for the network in Fig. 9.49.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.53 Substituting the Thévenin equivalent circuit for the network external to the resistor RL in Fig. 9.48.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures • Measuring Eth • Measuring RTh
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.54 Measuring the Thévenin voltage with a voltmeter: (a) actual network; (b) Thévenin equivalent.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.55 Measuring RTh with an ohmmeter: (a) actual network; (b) Thévenin equivalent.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.56 Using a potentiometer to determine RTh: (a) actual network; (b) Thévenin equivalent; (c) measuring RTh.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.57 Determining RTh using the short-circuit current: (a) actual network; (b) Thévenin equivalent.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM • In Section 8.3, we learned that every voltage source with a series internal resistance has a current source equivalent. • The current source equivalent can be determined by Norton’s theorem. It can also be found through the conversions of Section 8.3. • The theorem states the following: – Any two-terminal linear bilateral dc network can be replaced by an equivalent circuit consisting of a current source and a parallel resistor, as shown in Fig. 9.59
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM FIG. 9.59 Norton equivalent circuit.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • Preliminary: – 1. Remove that portion of the network across which the Norton equivalent circuit is found. – 2. Mark the terminals of the remaining two-terminal network.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • RN: – 3. Calculate RN by first setting all sources to zero (voltage sources are replaced with short circuits and current sources with open circuits) and then finding the resultant resistance between the two marked terminals. (If the internal resistance of the voltage and/or current sources is included in the original network, it must remain when the sources are set to zero.) Since RN = RTh, the procedure and value obtained using the approach described for Thévenin’s theorem will determine the proper value of RN.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • IN: – 4. Calculate IN by first returning all sources to their original position and then finding the short-circuit current between the marked terminals. It is the same current that would be measured by an ammeter placed between the marked terminals.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • Conclusion: – 5. Draw the Norton equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.60 Converting between Thévenin and Norton equivalent circuits.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.61 Example 9.11. FIG. 9.62 Identifying the terminals of particular interest for the network in Fig. 9.61.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.63 Determining RN for the network in Fig. 9.62. FIG. 9.64 Determining IN for the network in Fig. 9.62.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.65 Substituting the Norton equivalent circuit for the network external to the resistor RL in Fig. 9.61.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.66 Converting the Norton equivalent circuit in Fig. 9.65 to a Thévenin equivalent circuit.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.67 Example 9.12. FIG. 9.68 Identifying the terminals of particular interest for the network in Fig. 9.67.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.69 Determining RN for the network in Fig. 9.68.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.70 Determining IN for the network in Fig. 9.68.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.71 Substituting the Norton equivalent circuit for the network external to the resistor RL in Fig. 9.67.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.72 Example 9.13.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.73 Identifying the terminals of particular interest for the network in Fig. 9.72. FIG. 9.74 Determining RN for the network in Fig. 9.73.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.75 Determining the contribution to IN from the voltage source E1. FIG. 9.76 Determining the contribution to IN from the current source I.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.77 Substituting the Norton equivalent circuit for the network to the left of terminals a-b in Fig. 9.72.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Experimental Procedure • The Norton current is measured in the same way as described for the short- circuit current (Isc) for the Thévenin network. • Since the Norton and Thévenin resistances are the same, the same procedures can be followed as described for the Thévenin network.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • When designing a circuit, it is often important to be able to answer one of the following questions: – What load should be applied to a system to ensure that the load is receiving maximum power from the system? • Conversely: – For a particular load, what conditions should be imposed on the source to ensure that it will deliver the maximum power available?
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.78 Defining the conditions for maximum power to a load using the Thévenin equivalent circuit. FIG. 9.79 Thévenin equivalent network to be used to validate the maximum power transfer theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM TABLE 9.1
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.80 PL versus RL for the network in Fig. 9.79.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • If the load applied is less than the Thévenin resistance, the power to the load will drop off rapidly as it gets smaller. However, if the applied load is greater than the Thévenin resistance, the power to the load will not drop off as rapidly as it increases.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.81 PL versus RL for the network in Fig. 9.79.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • The total power delivered by a supply such as ETh is absorbed by both the Thévenin equivalent resistance and the load resistance. Any power delivered by the source that does not get to the load is lost to the Thévenin resistance.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.82 Efficiency of operation versus increasing values of RL.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.83 Examining a circuit with high efficiency but a relatively low level of power to the load.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • If efficiency is the overriding factor, then the load should be much larger than the internal resistance of the supply. If maximum power transfer is desired and efficiency less of a concern, then the conditions dictated by the maximum power transfer theorem should be applied.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.84 Defining the conditions for maximum power to a load using the Norton equivalent circuit.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.85 Example 9.14.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.86 Example 9.15. FIG. 9.87 dc supply with a fixed 16Ω load (Example 9.16).
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.88 Example 9.17. FIG. 9.89 Determining RTh for the network external to resistor RL in Fig. 9.88.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.90 Determining ETh for the network external to resistor RL in Fig. 9.88.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM • Through the application of Millman’s theorem, any number of parallel voltage sources can be reduced to one. – In Fig. 9.91, for example, the three voltage sources can be reduced to one. • This permits finding the current through or voltage across RL without having to apply a method such as mesh analysis, nodal analysis, superposition, and so on.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.91 Demonstrating the effect of applying Millman’s theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.92 Converting all the sources in Fig. 9.91 to current sources.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.93 Reducing all the current sources in Fig. 9.92 to a single current source. FIG. 9.94 Converting the current source in Fig. 9.93 to a voltage source.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.95 Example 9.18.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.96 The result of applying Millman’s theorem to the network in Fig. 9.95. FIG. 9.97 Example 9.19.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.98 Converting the sources in Fig. 9.97 to current sources. FIG. 9.99 Reducing the current sources in Fig. 9.98 to a single source.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.100 Converting the current source in Fig. 9.99 to a voltage source.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.101 The dual effect of Millman’s theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM • The substitution theorem states the following: – If the voltage across and the current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.102 Demonstrating the effect of the substitution theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.103 Equivalent branches for the branch a-b in Fig. 9.102.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.104 Demonstrating the effect of knowing a voltage at some point in a complex network.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.105 Demonstrating the effect of knowing a current at some point in a complex network.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM • The reciprocity theorem is applicable only to single-source networks. • It is, therefore, not a theorem used in the analysis of multisource networks described thus far. The theorem states the following: – The current I in any branch of a network due to a single voltage source E anywhere else in the network will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM FIG. 9.106 Demonstrating the impact of the reciprocity theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM FIG. 9.107 Finding the current I due to a source E. FIG. 9.108 Interchanging the location of E and I of Fig. 9.107 to demonstrate the validity of the reciprocity theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM FIG. 9.109 Demonstrating the power and uniqueness of the reciprocity theorem.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Thévenin’s Theorem FIG. 9.110 Network to which PSpice is to be applied to determine ETh and RTh.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Thévenin’s Theorem FIG. 9.111 Using PSpice to determine the Thévenin resistance of a network through the application of a 1 A current source.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Thévenin’s Theorem FIG. 9.112 Using PSpice to determine the Thévenin voltage for a network using a very large resistance value to represent the open-circuit condition between the terminals of interest.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Maximum Power Transfer FIG. 9.113 Using PSpice to plot the power to RL for a range of values for RL.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Maximum Power Transfer FIG. 9.114 Plot resulting from the dc sweep of RL for the network in Fig. 9.113 before defining the parameters to be displayed.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Maximum Power Transfer FIG. 9.115 A plot of the power delivered to RL in Fig. 9.113 for a range of values for RL extending from 0Ω to 30Ω .
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim Superposition FIG. 9.116 Applying Multisim to determine the current I2 using superposition.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim Superposition FIG. 9.117 Using Multisim to determine the contribution of the 36 V voltage source to the current through R2.
Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim Superposition FIG. 9.118 Using Multisim to determine the contribution of the 9 A current source to the current through R2.

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Network theorems chapter 9

  • 1. Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] Introductory Circuit Analysis, 12/e Boylestad Chapter 9 Network Theorems
  • 2. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] OBJECTIVES • Become familiar with the superposition theorem and its unique ability to separate the impact of each source on the quantity of interest. • Be able to apply Thévenin’s theorem to reduce any two-terminal, series-parallel network with any number of sources to a single voltage source and series resistor. • Become familiar with Norton’s theorem and how it can be used to reduce any two-terminal, seriesparallel network with any number of sources to a single current source and a parallel resistor.
  • 3. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] OBJECTIVES • Understand how to apply the maximum power transfer theorem to determine the maximum power to a load and to choose a load that will receive maximum power. • Become aware of the reduction powers of Millman’s theorem and the powerful implications of the substitution and reciprocity theorems.
  • 4. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM • The superposition theorem is unquestionably one of the most powerful in this field. • It has such widespread application that people often apply it without recognizing that their maneuvers are valid only because of this theorem.
  • 5. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM • In general, the theorem can be used to do the following: – Analyze networks such as introduced in the last chapter that have two or more sources that are not in series or parallel. – Reveal the effect of each source on a particular quantity of interest. – For sources of different types (such as dc and ac, which affect the parameters of the network in a different manner) and apply a separate analysis for each type, with the total result simply the algebraic sum of the results.
  • 6. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM • The superposition theorem states the following: – The current through, or voltage across, any element of a network is equal to the algebraic sum of the currents or voltages produced independently by each source.
  • 7. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.1 Removing a voltage source and a current source to permit the application of the superposition theorem.
  • 8. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.2 Network to be analyzed in Example 9.1 using the superposition theorem. FIG. 9.3 Replacing the 9 A current source in Fig. 9.2 by an open circuit to determine the effect of the 36 V voltage source on current I2.
  • 9. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.4 Replacing the 36 V voltage source by a short-circuit equivalent to determine the effect of the 9 A current source on current I2. FIG. 9.5 Using the results of Figs. 9.3 and 9.4 to determine current I2 for the network in Fig. 9.2.
  • 10. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.6 Plotting power delivered to the 6Ω resistor versus current through the resistor.
  • 11. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.7 Plotting I versus V for the 6Ω resistor.
  • 12. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.8 Using the superposition theorem to determine the current through the 12Ω resistor (Example 9.2).
  • 13. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.9 Using the superposition theorem to determine the effect of the 54 V voltage source on current I2 in Fig. 9.8.
  • 14. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.10 Using the superposition theorem to determine the effect of the 48 V voltage source on current I2 in Fig. 9.8.
  • 15. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.11 Using the results of Figs. 9.9 and 9.10 to determine current I2 for the network in Fig. 9.8.
  • 16. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.12 Two-source network to be analyzed using the superposition theorem in Example 9.3. FIG. 9.13 Determining the effect of the 30 V supply on the current I1 in Fig. 9.12.
  • 17. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.14 Determining the effect of the 3 A current source on the current I1 in Fig. 9.12. FIG. 9.15 Example 9.4.
  • 18. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.16 The effect of the current source I on the current I2.
  • 19. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.17 The effect of the voltage source E on the current I2.
  • 20. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.18 Example 9.5. FIG. 9.19 The effect of E1 on the current I.
  • 21. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.20 The effect of E2 on the current I1. FIG. 9.21 The effect of I on the current I1
  • 22. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUPERPOSITION THEOREM FIG. 9.22 The resultant current I1.
  • 23. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM • The next theorem to be introduced, Thévenin’s theorem, is probably one of the most interesting in that it permits the reduction of complex networks to a simpler form for analysis and design.
  • 24. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM • In general, the theorem can be used to do the following: – Analyze networks with sources that are not in series or parallel. – Reduce the number of components required to establish the same characteristics at the output terminals. – Investigate the effect of changing a particular component on the behavior of a network without having to analyze the entire network after each change.
  • 25. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM • Thévenin’s theorem states the following: – Any two-terminal dc network can be replaced by an equivalent circuit consisting solely of a voltage source and a series resistor as shown in Fig. 9.23. FIG. 9.23 Thévenin equivalent circuit.
  • 26. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM FIG. 9.25 Substituting the Thévenin equivalent circuit for a complex network.
  • 27. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • Preliminary: 1. Remove that portion of the network where the Thévenin equivalent circuit is found. In Fig. 9.25(a), this requires that the load resistor RL be temporarily removed from the network. 2. Mark the terminals of the remaining two- terminal network. (The importance of this step will become obvious as we progress through some complex networks.)
  • 28. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • RTh: – 3. Calculate RTh by first setting all sources to zero (voltage sources are replaced by short circuits and current sources by open circuits) and then finding the resultant resistance between the two marked terminals. (If the internal resistance of the voltage and/or current sources is included in the original network, it must remain when the sources are set to zero.)
  • 29. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • ETh: – 4. Calculate ETh by first returning all sources to their original position and finding the open-circuit voltage between the marked terminals. (This step is invariably the one that causes most confusion and errors. In all cases, keep in mind that it is the opencircuit potential between the two terminals marked in step 2.)
  • 30. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure • Conclusion: – 5. Draw the Thévenin equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit. This step is indicated by the placement of the resistor RL between the terminals of the Thévenin equivalent circuit as shown in Fig. 9.25(b).
  • 31. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.26 Example 9.6. FIG. 9.27 Identifying the terminals of particular importance when applying Thévenin’s theorem.
  • 32. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.28 Determining RTh for the network in Fig. 9.27.
  • 33. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.29 Determining ETh for the network in Fig. 9.27. FIG. 9.30 Measuring ETh for the network in Fig. 9.27.
  • 34. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.31 Substituting the Thévenin equivalent circuit for the network external to RL in Fig. 9.26. FIG. 9.32 Example 9.7.
  • 35. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.33 Establishing the terminals of particular interest for the network in Fig. 9.32. FIG. 9.34 Determining RTh for the network in Fig. 9.33.
  • 36. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.35 Determining ETh for the network in Fig. 9.33. FIG. 9.36 Substituting the Thévenin equivalent circuit in the network external to the resistor R3 in Fig. 9.32.
  • 37. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.37 Example 9.8.
  • 38. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.38 Identifying the terminals of particular interest for the network in Fig. 9.37.
  • 39. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.39 Determining RTh for the network in Fig. 9.38.
  • 40. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.40 Determining ETh for the network in Fig. 9.38.
  • 41. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.41 Network of Fig. 9.40 redrawn. FIG. 9.42 Substituting the Thévenin equivalent circuit for the network external to the resistor R4 in Fig. 9.37.
  • 42. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.43 Example 9.9.
  • 43. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.44 Identifying the terminals of particular interest for the network in Fig. 9.43.
  • 44. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.45 Solving for RTh for the network in Fig. 9.44.
  • 45. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.46 Determining ETh for the network in Fig. 9.44.
  • 46. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.47 Substituting the Thévenin equivalent circuit for the network external to the resistor RL in Fig. 9.43.
  • 47. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.48 Example 9.10.
  • 48. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.49 Identifying the terminals of particular interest for the network in Fig. 9.48. FIG. 9.50 Determining RTh for the network in Fig. 9.49.
  • 49. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.51 Determining the contribution to ETh from the source E1 for the network in Fig. 9.49. FIG. 9.52 Determining the contribution to ETh from the source E2 for the network in Fig. 9.49.
  • 50. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Thévenin’s Theorem Procedure FIG. 9.53 Substituting the Thévenin equivalent circuit for the network external to the resistor RL in Fig. 9.48.
  • 51. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures • Measuring Eth • Measuring RTh
  • 52. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.54 Measuring the Thévenin voltage with a voltmeter: (a) actual network; (b) Thévenin equivalent.
  • 53. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.55 Measuring RTh with an ohmmeter: (a) actual network; (b) Thévenin equivalent.
  • 54. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.56 Using a potentiometer to determine RTh: (a) actual network; (b) Thévenin equivalent; (c) measuring RTh.
  • 55. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] THÉVENIN’S THEOREM Experimental Procedures FIG. 9.57 Determining RTh using the short-circuit current: (a) actual network; (b) Thévenin equivalent.
  • 56. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM • In Section 8.3, we learned that every voltage source with a series internal resistance has a current source equivalent. • The current source equivalent can be determined by Norton’s theorem. It can also be found through the conversions of Section 8.3. • The theorem states the following: – Any two-terminal linear bilateral dc network can be replaced by an equivalent circuit consisting of a current source and a parallel resistor, as shown in Fig. 9.59
  • 57. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM FIG. 9.59 Norton equivalent circuit.
  • 58. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • Preliminary: – 1. Remove that portion of the network across which the Norton equivalent circuit is found. – 2. Mark the terminals of the remaining two-terminal network.
  • 59. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • RN: – 3. Calculate RN by first setting all sources to zero (voltage sources are replaced with short circuits and current sources with open circuits) and then finding the resultant resistance between the two marked terminals. (If the internal resistance of the voltage and/or current sources is included in the original network, it must remain when the sources are set to zero.) Since RN = RTh, the procedure and value obtained using the approach described for Thévenin’s theorem will determine the proper value of RN.
  • 60. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • IN: – 4. Calculate IN by first returning all sources to their original position and then finding the short-circuit current between the marked terminals. It is the same current that would be measured by an ammeter placed between the marked terminals.
  • 61. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure • Conclusion: – 5. Draw the Norton equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit.
  • 62. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.60 Converting between Thévenin and Norton equivalent circuits.
  • 63. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.61 Example 9.11. FIG. 9.62 Identifying the terminals of particular interest for the network in Fig. 9.61.
  • 64. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.63 Determining RN for the network in Fig. 9.62. FIG. 9.64 Determining IN for the network in Fig. 9.62.
  • 65. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.65 Substituting the Norton equivalent circuit for the network external to the resistor RL in Fig. 9.61.
  • 66. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.66 Converting the Norton equivalent circuit in Fig. 9.65 to a Thévenin equivalent circuit.
  • 67. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.67 Example 9.12. FIG. 9.68 Identifying the terminals of particular interest for the network in Fig. 9.67.
  • 68. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.69 Determining RN for the network in Fig. 9.68.
  • 69. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.70 Determining IN for the network in Fig. 9.68.
  • 70. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.71 Substituting the Norton equivalent circuit for the network external to the resistor RL in Fig. 9.67.
  • 71. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.72 Example 9.13.
  • 72. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.73 Identifying the terminals of particular interest for the network in Fig. 9.72. FIG. 9.74 Determining RN for the network in Fig. 9.73.
  • 73. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.75 Determining the contribution to IN from the voltage source E1. FIG. 9.76 Determining the contribution to IN from the current source I.
  • 74. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Norton’s Theorem Procedure FIG. 9.77 Substituting the Norton equivalent circuit for the network to the left of terminals a-b in Fig. 9.72.
  • 75. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] NORTON’S THEOREM Experimental Procedure • The Norton current is measured in the same way as described for the short- circuit current (Isc) for the Thévenin network. • Since the Norton and Thévenin resistances are the same, the same procedures can be followed as described for the Thévenin network.
  • 76. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • When designing a circuit, it is often important to be able to answer one of the following questions: – What load should be applied to a system to ensure that the load is receiving maximum power from the system? • Conversely: – For a particular load, what conditions should be imposed on the source to ensure that it will deliver the maximum power available?
  • 77. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.78 Defining the conditions for maximum power to a load using the Thévenin equivalent circuit. FIG. 9.79 Thévenin equivalent network to be used to validate the maximum power transfer theorem.
  • 78. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM TABLE 9.1
  • 79. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.80 PL versus RL for the network in Fig. 9.79.
  • 80. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • If the load applied is less than the Thévenin resistance, the power to the load will drop off rapidly as it gets smaller. However, if the applied load is greater than the Thévenin resistance, the power to the load will not drop off as rapidly as it increases.
  • 81. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.81 PL versus RL for the network in Fig. 9.79.
  • 82. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • The total power delivered by a supply such as ETh is absorbed by both the Thévenin equivalent resistance and the load resistance. Any power delivered by the source that does not get to the load is lost to the Thévenin resistance.
  • 83. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.82 Efficiency of operation versus increasing values of RL.
  • 84. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.83 Examining a circuit with high efficiency but a relatively low level of power to the load.
  • 85. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM • If efficiency is the overriding factor, then the load should be much larger than the internal resistance of the supply. If maximum power transfer is desired and efficiency less of a concern, then the conditions dictated by the maximum power transfer theorem should be applied.
  • 86. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.84 Defining the conditions for maximum power to a load using the Norton equivalent circuit.
  • 87. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.85 Example 9.14.
  • 88. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.86 Example 9.15. FIG. 9.87 dc supply with a fixed 16Ω load (Example 9.16).
  • 89. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.88 Example 9.17. FIG. 9.89 Determining RTh for the network external to resistor RL in Fig. 9.88.
  • 90. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MAXIMUM POWER TRANSFER THEOREM FIG. 9.90 Determining ETh for the network external to resistor RL in Fig. 9.88.
  • 91. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM • Through the application of Millman’s theorem, any number of parallel voltage sources can be reduced to one. – In Fig. 9.91, for example, the three voltage sources can be reduced to one. • This permits finding the current through or voltage across RL without having to apply a method such as mesh analysis, nodal analysis, superposition, and so on.
  • 92. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.91 Demonstrating the effect of applying Millman’s theorem.
  • 93. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.92 Converting all the sources in Fig. 9.91 to current sources.
  • 94. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.93 Reducing all the current sources in Fig. 9.92 to a single current source. FIG. 9.94 Converting the current source in Fig. 9.93 to a voltage source.
  • 95. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.95 Example 9.18.
  • 96. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.96 The result of applying Millman’s theorem to the network in Fig. 9.95. FIG. 9.97 Example 9.19.
  • 97. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.98 Converting the sources in Fig. 9.97 to current sources. FIG. 9.99 Reducing the current sources in Fig. 9.98 to a single source.
  • 98. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.100 Converting the current source in Fig. 9.99 to a voltage source.
  • 99. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] MILLMAN’S THEOREM FIG. 9.101 The dual effect of Millman’s theorem.
  • 100. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM • The substitution theorem states the following: – If the voltage across and the current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.
  • 101. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.102 Demonstrating the effect of the substitution theorem.
  • 102. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.103 Equivalent branches for the branch a-b in Fig. 9.102.
  • 103. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.104 Demonstrating the effect of knowing a voltage at some point in a complex network.
  • 104. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] SUBSTITUTION THEOREM FIG. 9.105 Demonstrating the effect of knowing a current at some point in a complex network.
  • 105. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM • The reciprocity theorem is applicable only to single-source networks. • It is, therefore, not a theorem used in the analysis of multisource networks described thus far. The theorem states the following: – The current I in any branch of a network due to a single voltage source E anywhere else in the network will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured.
  • 106. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM FIG. 9.106 Demonstrating the impact of the reciprocity theorem.
  • 107. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM FIG. 9.107 Finding the current I due to a source E. FIG. 9.108 Interchanging the location of E and I of Fig. 9.107 to demonstrate the validity of the reciprocity theorem.
  • 108. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] RECIPROCITY THEOREM FIG. 9.109 Demonstrating the power and uniqueness of the reciprocity theorem.
  • 109. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Thévenin’s Theorem FIG. 9.110 Network to which PSpice is to be applied to determine ETh and RTh.
  • 110. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Thévenin’s Theorem FIG. 9.111 Using PSpice to determine the Thévenin resistance of a network through the application of a 1 A current source.
  • 111. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Thévenin’s Theorem FIG. 9.112 Using PSpice to determine the Thévenin voltage for a network using a very large resistance value to represent the open-circuit condition between the terminals of interest.
  • 112. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Maximum Power Transfer FIG. 9.113 Using PSpice to plot the power to RL for a range of values for RL.
  • 113. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Maximum Power Transfer FIG. 9.114 Plot resulting from the dc sweep of RL for the network in Fig. 9.113 before defining the parameters to be displayed.
  • 114. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS PSpice Maximum Power Transfer FIG. 9.115 A plot of the power delivered to RL in Fig. 9.113 for a range of values for RL extending from 0Ω to 30Ω .
  • 115. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim Superposition FIG. 9.116 Applying Multisim to determine the current I2 using superposition.
  • 116. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim Superposition FIG. 9.117 Using Multisim to determine the contribution of the 36 V voltage source to the current through R2.
  • 117. Introductory Circuit Analysis, 12/e Boylestad Copyright ©2011 by Pearson Education, Inc. publishing as Pearson [imprint] COMPUTER ANALYSIS Multisim Superposition FIG. 9.118 Using Multisim to determine the contribution of the 9 A current source to the current through R2.