Chapter 4. Techniques of Circuit Analysis

Chapter 4. Techniques of Circuit Analysis By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical Engineering, K.N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/electriccircuits1.htm Reference: ELECTRIC CIRCUITS, J.W. Nilsson, S.A. Riedel, 10 th edition, 2015.

Chapter Contents 4.1. Terminology 4.2. Introduction to the Node-Voltage Method 4.3. NVM and Dependent Sources 4.4. NVM: Some Special Cases 4.5. Introduction to the Mesh-Current Method 4.6. MCM and Dependent Sources 4.7. MCM: Some Special Cases 4.8. NVM Versus MCM 4.9. Source Transformations 4.10. Thévenin and Norton Equivalents 4.11. More on Deriving a Thévenin Equivalent 4.12. Maximum Power Transfer 4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 2

4.1. Terminology o To discuss the more involved methods of circuit analysis, we must define a few basic terms. o Planar circuits can be drawn on a plane with no crossing branches. o A circuit that is drawn with crossing branches still is considered planar if it can be redrawn with no crossover branches. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 3

4.1. Terminology o Nonplanar circuits cannot be redrawn in such a way that all the node connections are maintained and no branches overlap. o The node-voltage method is applicable to both planar and nonplanar circuits. o The mesh-current method is limited to planar circuits. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 4

4.1. Terminology Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 5

4.1. Terminology Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 6

4.1. Terminology Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 7

4.1. Terminology Simultaneous Equations How Many? o The number of unknown currents in a circuit equals the number of branches, b, where the current is not known. o We must have b independent equations to solve a circuit with b unknown currents. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 8

4.1. Terminology Simultaneous Equations How Many? o If n is the number of nodes in the circuit, we can derive n-1 independent equations by applying KCL to any set of n-1 nodes. o Application of KCL to the n th node does not generate an independent equation, because this equation can be derived from the previous n-1 equations. o Because we need b equations to describe a given circuit and we can obtain n-1 of these equations from KCL, we must apply KVL to loops or meshes to obtain the remaining b-(n-1) equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 9

4.1. Terminology Simultaneous Equations How Many? o Thus by counting nodes, meshes, and branches with unknown currents, we have established a systematic method for writing the necessary number of equations to solve a circuit. o We apply: KCL to n-1 nodes and KVL to b-(n-1) loops (or meshes). Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 10

4.1. Terminology Simultaneous Equations How Many? o These observations also are valid in terms of essential nodes and essential branches. o If n e is the number of essential nodes and b e is the number of essential branches with unknown currents, we can apply: KCL at n e -1 nodes and KVL around b e -(n e -1) loops or meshes. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 11

4.1. Terminology Simultaneous Equations How Many? o The number of essential nodes is less than or equal to the number of nodes. o The number of essential branches is less than or equal to the number of branches. o Thus it is often convenient to use essential nodes and essential branches, because they produce fewer independent equations to solve. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 12

4.1. Terminology The Systematic Approach o We write the equations on the basis of essential nodes and branches. o The circuit has 4 essential nodes and 6 essential branches, denoted i 1 -i 6, for which the current is unknown. o We derive 3 of 6 simultaneous equations by applying KCL to any 3 of 4 essential nodes. o Using nodes b, c, and e: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 13

4.1. Terminology The Systematic Approach o We derive the remaining 3 equations by applying KVL around 3 meshes. o Because the circuit has 4 meshes, we need to dismiss one mesh. o We choose R 7 I, because we don't know the voltage across I. o Using the other 3 meshes: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 14

4.1. Terminology The Systematic Approach o By introducing new variables, we can describe a circuit with just n-1 equations or b-(n-1) equations. o These new variables allow us to obtain a solution by manipulating fewer equations. o The new variables are known as node voltages and mesh currents. o Node-voltage method enables us to describe a circuit in terms of n e -1 equations. o Mesh-current method enables us to describe a circuit in terms of b e -(n e -1) equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 15

4.2. Introduction to Node-Voltage Method o We introduce node-voltage method by using essential nodes of circuit. o The first step is to make a neat layout of circuit so that no branches cross over and to mark clearly essential nodes on circuit diagram. o This circuit has n e =3essential nodes. o We need 2 or (n e -1) node-voltage equations to describe the circuit. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 17

4.2. Introduction to Node-Voltage Method o The next step is to select one of 3 essential nodes as a reference node. o Although theoretically the choice is arbitrary, practically the choice for the reference node often is obvious. o For example, the node with the most branches is usually a good choice. o We flag the chosen reference node with the symbol. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 18

4.2. Introduction to Node-Voltage Method o After selecting reference node, we define node voltages on circuit diagram, i.e., the voltage rise from the reference node to a non-reference node. o We are now ready to generate node-voltage equations. o We do so by first writing the current leaving each branch connected to a non-reference node as a function of the node voltages and then summing these currents to zero in accordance with KCL. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 19

4.2. Introduction to Node-Voltage Method o Node-voltage equation at node 1 is: o Node-voltage equation at node 2 is: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 20

4.2. Introduction to Node-Voltage Method o Once the node voltages are known, all branch currents can be calculated. o Once these are known, the branch voltages and powers can be calculated. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 21

4.2. Introduction to Node-Voltage Method Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 22

4.3. NVM and Dependent Sources o If the circuit contains dependent sources, the node-voltage equations must be supplemented with the constraint equations imposed by the presence of the dependent sources. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 24

4.3. NVM and Dependent Sources Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 25

4.4. NVM: Some Special Cases o When a voltage source is the only element between 2 essential nodes, the node-voltage method is simplified. o There are 3 essential nodes. o 2 simultaneous equations are needed. o A reference node has been chosen. o Two other nodes have been labeled. o The 100 V source constrains the voltage between node 1 and the reference node to 100 V. o There is only one unknown node voltage (v 2 ). Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 27

4.4. NVM: Some Special Cases o Knowing v 2, we can calculate the current in every branch. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 28

4.4. NVM: Some Special Cases o When you use NVM to solve circuits that have voltage sources connected directly between essential nodes, the number of unknown node voltages is reduced. Because the difference between the node voltages at these nodes equals the voltage of the source. o The circuit contains 4 essential nodes. o We anticipate writing 3 node-voltage equations. o 2 essential nodes are connected by an independent voltage source. o 2 other essential nodes are connected by a dependent voltage source. o There is only 1 unknown node voltage. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 29

4.4. NVM: Some Special Cases o We introduce current i because we cannot express it as a function of node voltages v 2 and v 3. o When a voltage source is between 2 essential nodes, we can combine those nodes to form a supernode. o KCL must hold for the supernode: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 30

4.4. NVM: Some Special Cases Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 31

4.4. NVM: Some Special Cases o When we used the branch-current method of analysis, we faced the task of writing and solving 6 simultaneous equations. o Nodal analysis can simplify our task. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 32

4.4. NVM: Some Special Cases o The circuit has 4 essential nodes. o Nodes a and d are connected by an independent voltage source as are nodes b and c. o The problem reduces to finding a single unknown node voltage. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 33

4.4. NVM: Some Special Cases Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 34

4.5. Introduction to Mesh-Current Method o Mesh-current method (MCM) describes a circuit in terms of b e -(n e -1) equations. o A mesh is a loop with no other loops inside it. o MCM is applicable only to planar circuits. o The circuit shown contains 7 essential branches with unknown currents and 4 essential nodes. o To solve it via the MCM, we must write 4 or 7-(4-1) mesh-current equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 36

4.5. Introduction to Mesh-Current Method o A mesh current is current that exists only in perimeter of a mesh. o On a circuit diagram, it appears as either a closed solid line or an almost-closed solid line that follows perimeter of appropriate mesh. o An arrowhead on solid line indicates reference direction for mesh current. o Mesh currents automatically satisfy KCL. At any node in circuit, a given mesh current both enters and leaves node. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 37

4.5. Introduction to Mesh-Current Method o Identifying a mesh current in terms of a branch current is not always possible. o i 2 is not equal to any branch current, whereas i 1, i 3, and i 4 can be identified with branch currents. o Measuring a mesh current is not always possible. o There is no place where an ammeter can be inserted to measure i 2. o The fact that a mesh current can be a fictitious quantity doesn't mean that it is a useless concept. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 38

4.5. Introduction to Mesh-Current Method Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 39

4.5. Introduction to Mesh-Current Method Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 40

4.5. Introduction to Mesh-Current Method o Above equations are identical in form, with mesh currents i a and i b replacing branch currents i 1 and i 2. o MCM of circuit analysis evolves quite naturally from branch-current equations. o MCM is equivalent to a systematic substitution of n e -1 current equations into b e -(n e -1) voltage equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 41

4.5. Introduction to Mesh-Current Method o The circuit has 7 branches (5 essential branches) where current is unknown and 5 nodes (3 essential nodes). o Thus, we need 3 or b-(n-1)=7-(5-1) or b e -(n e -1)=5-(3-1) mesh-current equations to describe circuit. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 42

4.5. Introduction to Mesh-Current Method Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 43

4.6. MCM and Dependent Sources o If the circuit contains dependent sources, mesh-current equations must be supplemented by appropriate constraint equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 45

4.6. MCM and Dependent Sources o Circuit has 6 branches with unknown currents and 4 nodes. o We need 3 mesh currents: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 46

4.6. MCM and Dependent Sources o What if you had not been told to use MCM? o Would you have chosen NVM? o It reduces problem to finding 1 unknownnodevoltage node because of the presence of 2 voltage sources between essential nodes. o We present more about making such choices later. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 47

4.7. MCM: Some Special Cases o When a branch includes a current source, MCM requires some additional manipulations. o We defined mesh currents i a, i b, and i c, and voltage across 5 A current source. o Circuit contains 5 essential branches with unknown currents and 4 essential nodes. o We need to write 2 or 5-(4-1) mesh-current equations to solve circuit. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 49

4.7. MCM: Some Special Cases o Presence of current source reduces 3 unknown mesh currents to 2 such currents. o It constrains difference between i a and i c to equal 5 A. o If we know i a, we know i c, and vice versa. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 50

4.7. MCM: Some Special Cases o For mesh a: o For mesh c: o Adding the above equations: o For mesh b: o For current source branch: o Solving equations: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 51

4.7. MCM: Some Special Cases o We can solve the problem without introducing unknown voltage v by using the concept of a supermesh. o To create a supermesh, we mentally remove current source from circuit by simply avoiding this branch when writing mesh-current equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 52

4.7. MCM: Some Special Cases o We express voltages around supermesh in terms of original mesh currents: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 53

4.7. MCM: Some Special Cases o Circuit has 4 essential nodes and 5 essential branches with unknown currents. o Circuit can be analyzed in terms of 5-(4-1) or 2 mesh-current equations. o Although we defined 3 mesh currents, dependent current source forces a constraint between mesh currents i a and i c. o We have only 2 unknown mesh currents. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 54

4.7. MCM: Some Special Cases Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 55

4.8. NVM Versus MCM o The greatest advantage of both NVM and MCM is that they reduce the number of simultaneous equations that must be solved. o They also require the analyst to be quite systematic in terms of organizing and writing these equations. o When is NVM preferred to MCM and vice versa? o There is no clear-cut answer. o Asking a number of questions, may help you identify the more efficient method. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 57

4.8. NVM Versus MCM o Does one of the methods result in fewer simultaneous equations to solve? o Does the circuit contain supernodes? If so, using NVM will permit you to reduce the number of equations to be solved. o Does the circuit contain supermeshes? If so, using MCM will permit you to reduce the number of equations to be solved. o Will solving some portion of the circuit give the requested solution? If so, which method is most efficient for solving just the pertinent portion of the circuit? o Some time spent thinking about the problem in relation to the various analytical approaches available is time well spent. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 58

4.8. NVM Versus MCM o Find the power dissipated in R = 300. o We need to find either current or voltage of resistor. o MCM yields current in the resistor. o MCM requires solving 5 simultaneous mesh equations. o In writing the 5 equations, we must include the constraint i = -i b. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 59

4.8. NVM Versus MCM o Find the power dissipated in R = 300. o The circuit has 4 essential nodes. o Only 3 NV equations are required. o Because of dependent voltage source between 2 essential nodes, we have to sum currents at only 2 nodes. o Problem is reduced to writing 2 NV equations and a constraint equation. o NVM requires only 3 simultaneous equations, it is the more attractive approach. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 60

4.8. NVM Versus MCM Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 61

4.8. NVM Versus MCM o Find the voltage v o in the circuit. o Circuit has 4 essential nodes and 2 voltage-controlled dependent sources. o NVM requires solving 3 NV equations and 2 constraint equations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 62

4.8. NVM Versus MCM o Find the voltage v o in the circuit. o Circuit contains 3 meshes. o We can use leftmost one to calculate v o. o If i a denotes leftmost mesh current, v o = 193-10i a. o Presence of 2 current sources reduces problem to solving a single supermesh equation and 2 constraint equations. o MCM is the more attractive technique here. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 63

4.8. NVM Versus MCM Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 64

4.8. NVM Versus MCM Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 65

4.9. Source Transformations o Even though NVM and MCM are powerful techniques, we are still interested in methods that can be used to simplify circuits. o Series-parallel reductions and -to-y transformations are already on our list of simplifying techniques. o We begin expanding this list with source transformations. o A source transformation, allows a voltage source in series with a resistor to be replaced by a current source in parallel with same resistor or vice versa. o A source transformation is bilateral. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 67

4.9. Source Transformations o We need to find relationship between v s and i s that guarantees 2 configurations are equivalent with respect to nodes a, b. o Equivalence is achieved if any R L experiences same current, and same voltage, if connected between nodes a, b: o If polarity of v s is reversed, orientation of i s must be reversed to maintain equivalence. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 68

4.9. Source Transformations o Circuit has 4 essential nodes and 6 essential branches with unknown currents. o We can find current in branch containing 6 V source by solving either 3 = [6-(4-1)] mesh-current equations, or 3 = [4-1] node-voltage equations. o We can also simplify circuit by using source transformations. o We must reduce circuit in a way that preserves identity of branch containing 6 V source. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 69

4.9. Source Transformations i s = (19.2-6)/16 = 0.825 A i s Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 70

4.9. Source Transformations o What happens if there is an R p in parallel with voltage source? o What happens if there is an R s in series with current source? o In both cases, resistance has no effect on equivalent circuit that predicts behavior with respect to terminals a, b. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 71

4.9. Source Transformations p 8A (developed) = -(-60)(8) = 480 W Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 72

4.9. Source Transformations o 125 and 10 resistors do not affect the value of v o but do affect the power calculations! Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 73

4.10. Thévenin and Norton Equivalents o Sometimes, we want to concentrate on what happens at a specific pair of terminals. o For example, when we plug a toaster into an outlet, we are interested primarily in voltage and current at terminals of toaster. o We have little or no interest in effect that connecting toaster has on voltages or currents elsewhere in circuit supplying outlet. o We then are interested in how voltage and current delivered at outlet change as we change appliances. o We want to focus on the behavior of circuit supplying outlet, but only at outlet terminals. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 75

4.10. Thévenin and Norton Equivalents o Thévenin and Norton equivalents are circuit simplification techniques that focus on terminal behavior. o They are extremely valuable aids in circuit analysis. o Here we discuss resistive circuits, but Thévenin and Norton equivalent circuits may be used to represent any circuit made up of linear elements. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 76

4.10. Thévenin and Norton Equivalents o Letters a and b denote pair of terminals of interest. o A Thévenin equivalent circuit is an independent voltage source V Th in series with a resistor R Th, which replaces an interconnection of sources and resistors. o If we connect same load across terminals a and b of each circuit, we get same voltage and current at terminals of the load. o This equivalence holds for all possible values of load resistance. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 77

4.10. Thévenin and Norton Equivalents o We must be able to determine V Th and R Th. o If load resistance is infinitely large, we have an open-circuit condition. The open-circuit voltage at the terminals a and b is V Th in circuit (b). It must be same as open-circuit voltage at terminals a and b in original circuit. o To calculate V Th, we simply calculate open-circuit voltage in original circuit. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 78

4.10. Thévenin and Norton Equivalents Reducing load resistance to zero gives us a short-circuit condition. o In circuit (b), short-circuit current directed from a to b is: o This short-circuit current must be identical to short-circuit current that exists in a short circuit placed across terminals a to b of original network. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 79

4.10. Thévenin and Norton Equivalents o Thévenin resistance is ratio of open-circuit voltage to short-circuit current: o If short-circuit current is directed from b to a is, a minus sign must be inserted in the equation. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 80

4.10. Thévenin and Norton Equivalents o When terminals a and b are open, there is no current in 4 resistor, and v ab is identical to v 1. o Thévenin voltage for circuit is 32 V. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 81

4.10. Thévenin and Norton Equivalents o The short-circuit current (i sc ) is found easily once v 2 is known. o Problem reduces to finding v 2 with the short-circuit in place: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 82

4.10. Thévenin and Norton Equivalents o If a 24 resistor is connected across terminals a and b in original circuit, voltage across resistor is 24 V and current in the resistor is 1 A, as would be the case with Thévenin circuit. o This same equivalence between circuits holds for any resistor value connected between nodes a and b. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 83

4.10. Thévenin and Norton Equivalents o A Norton equivalent circuit consists of: an independent current source in parallel with Norton equivalent resistance. o We can derive it from a Thévenin equivalent circuit simply by making a source transformation. o Norton current equals short-circuit current at terminals of interest. o Norton resistance is identical to Thévenin resistance. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 84

4.10. Thévenin and Norton Equivalents o Sometimes, we can make effective use of source transformations to derive a Thévenin or Norton equivalent circuit. o This technique is most useful when the network contains only independent sources. o The presence of dependent sources requires retaining identity of controlling voltages and/or currents. o This constraint usually prohibits continued reduction of circuit by source transformations. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 85

4.10. Thévenin and Norton Equivalents o Thévenin Equivalent: o Norton Equivalent: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 86

4.10. Thévenin and Norton Equivalents o Find the Thévenin equivalent for the circuit containing dependent sources. o Current i x must be 0. Note the absence of a return path for i x to enter left-hand portion of circuit. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 87

4.10. Thévenin and Norton Equivalents o Find the Thévenin equivalent for the circuit containing dependent sources. o With short circuit shunting 25, all current from dependent current source appears in short circuit: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 88

4.11. More on Deriving a Thévenin Equivalent o Technique for determining R Th we discussed is not always the easiest method available. o 2 other methods generally are simpler to use. o The first is useful if network contains only independent sources. o To calculate R Th for such a network, we: deactivate all independent sources, calculate resistance seen looking into network at designated terminal pair. o A voltage source is deactivated by replacing it with a short circuit. o A current source is deactivated by replacing it with an open circuit. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 90

4.11. More on Deriving a Thévenin Equivalent First alternative procedure to find R Th Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 91

4.11. More on Deriving a Thévenin Equivalent o If circuit or network contains dependent sources, an alternative procedure for finding R Th is as follows: We first deactivate all independent sources. We apply either a test voltage source or a test current source to Thévenin terminals a and b. Thévenin resistance equals ratio of voltage across test source to current delivered by test source. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 92

4.11. More on Deriving a Thévenin Equivalent Second alternative procedure to find R Th Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 93

4.11. More on Deriving a Thévenin Equivalent o We can use a Thévenin equivalent to reduce one portion of a circuit to greatly simplify analysis of larger network. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 94

4.11. More on Deriving a Thévenin Equivalent o This modification has no effect on branch currents i 1, i 2, i B, and i E. o We replace circuit made up of V CC, R 1, and R 2 with a Thévenin equivalent, with respect to terminals b and d. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 95

4.11. More on Deriving a Thévenin Equivalent Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 96

4.11. More on Deriving a Thévenin Equivalent Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 97

4.12. Maximum Power Transfer o We assume a resistive network containing independent and dependent sources and a designated pair of terminals a and b, to which a load R L, is to be connected. o Problem is to determine value of R L that permits maximum power delivery to R L. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 99

4.12. Maximum Power Transfer o Replacing original network by its Thévenin equivalent greatly simplifies task of finding R L. o Derivation of R L requires expressing power dissipated in R L as a function of 3 circuit parameters V Th, R Th, and R L : Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 100

4.12. Maximum Power Transfer Derivative is 0 and p is maximized when: Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 101

4.12. Maximum Power Transfer R L = 25 Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 102

4.12. Maximum Power Transfer Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 103

4.13. Superposition o A linear system obeys principle of superposition, i.e., whenever a linear system is excited, or driven, by more than one independent source of energy, total response is sum of individual responses. o An individual response is result of an independent source acting alone. o Because we are dealing with circuits made up of interconnected linear-circuit elements, we can apply superposition directly to analysis of such circuits. o At present, we restrict the discussion to simple resistive networks. o Principle is applicable to any linear system. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 105

4.13. Superposition o Superposition is applied in both analysis and design of circuits. o In analyzing a complex circuit with multiple independent voltage and current sources, there are often fewer, simpler equations to solve by applying superposition. o Applying superposition can simplify circuit analysis. o Sometimes, applying superposition actually complicates analysis, producing more equations to solve. o Superposition is required only if independent sources in a circuit are fundamentally different. o When all independent sources are dc sources, superposition is not required. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 106

4.13. Superposition o Superposition is applied in design to synthesize a desired circuit response that could not be achieved in a circuit with a single source. o If desired circuit response can be written as a sum of two or more terms, response can be realized by including one independent source for each term of response. o This approach to design of circuits with complex responses allows a designer to consider several simple designs instead of one complex design. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 107

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 108

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 109

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 110

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 111

4.13. Superposition o When applying superposition to linear circuits containing both independent and dependent sources, you must recognize that dependent sources are never deactivated. Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 112

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 113

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 114

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 115

4.13. Superposition Electric Circuits 1 Chapter 4. Techniques of Circuit Analysis 116