Ciletti, M.D., Irwin, J D, Kraus, A D, Balabanian, N, Bickart, T.A., Chan, S P, Nise NS"Linear Circuit Analysis The electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRc Press llc. 2000
Ciletti, M.D., Irwin, J.D., Kraus, A.D., Balabanian, N., Bickart, T.A., Chan, S.P., Nise, N.S. “Linear Circuit Analysis” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
3 Linear Circuit analysis 3. 1 Voltage and Current Laws Kirchhoff's Current Law. Kirchhoff's Current Law in the Complex Domain. Kirchhoffs Voltage Law. Kirchhoff's Voltage Law in the Complex Domain. Importance of KVL and KCL Michael D. Ciletti 3.2 Node and Mesh Analysis University of colorado Node analysis· Mesh Analysis· Summary J. David Irwin Linearity and Superposition. The Network Theorems of Thevenin and Norton· Tellegen's Theorem· Maximun Transfer·The Allan d. kraus ciprocity Theorem. The Substitution and Compensation Theorem Allan D. Kraus Associates 3.4 Power and Energy Tellegen's Theorem.AC Steady-State Power. Maximum Power Norman balabanian Transfer. Measuring AC Power and Energy 3.5 Three-Phase Circuits Theodore A. Bickart The k-Tree Approach. The Flowgraph Approach. The k-Tree Approach Versus the Flowgraph Approach. Some Topological Shu-Park Chan Applications in Network Analysis and Design 3.7 Two-Port Parameters and Transformations Modeling of Two-Port Networsk via z Parameters. Evaluating Two- orman Port Network Characteristics in Terms of z Parameters. An Example California State Polytechnic Finding z Parameters and Network Characteristics. Additional Two- Port parameters and Conversions Two Port Parameter Selection 3.1 Voltage and current laws Michael D. Ciletti ysis of linear circuits rests on two fundamental physical laws that describe how the voltages and currents in a circuit must behave. This behavior results from whatever voltage sources, current sources, and energy torage elements are connected to the circuit. A voltage source imposes a constraint on the evolution of the voltage between a pair of nodes; a current source imposes a constraint on the evolution of the current in a branch of the circuit. The energy storage elements(capacitors and inductors) impose initial conditions on currents and voltages in the circuit; they also establish a dynamic relationship between the voltage and the current at their terminals Regardless of how a linear circuit is stimulated, every node voltage and every branch current, at every instant of time, must be consistent with Kirchhoff's voltage and current laws. These two laws govern even the most complex linear circuits. (They also apply to a broad category of nonlinear circuits that are modeled by point models of voltage and current. A circuit can be considered to have a topological (or graph) view, consisting of a labeled set of nodes and a labeled set of edges. Each edge is associated with a pair of nodes. A node is drawn as a dot and represents a c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 3 Linear Circuit Analysis 3.1 Voltage and Current Laws Kirchhoff’s Current Law • Kirchhoff’s Current Law in the Complex Domain • Kirchhoff’s Voltage Law • Kirchhoff’s Voltage Law in the Complex Domain • Importance of KVL and KCL 3.2 Node and Mesh Analysis Node Analysis • Mesh Analysis • Summary 3.3 Network Theorems Linearity and Superposition • The Network Theorems of Thévenin and Norton • Tellegen’s Theorem • Maximum Power Transfer • The Reciprocity Theorem • The Substitution and Compensation Theorem 3.4 Power and Energy Tellegen’s Theorem • AC Steady-State Power • Maximum Power Transfer • Measuring AC Power and Energy 3.5 Three-Phase Circuits 3.6 Graph Theory The k-Tree Approach • The Flowgraph Approach • The k-Tree Approach Versus the Flowgraph Approach • Some Topological Applications in Network Analysis and Design 3.7 Two-Port Parameters and Transformations Introduction • Defining Two-Port Networks • Mathematical Modeling of Two-Port Networsk via z Parameters • Evaluating TwoPort Network Characteristics in Terms of z Parameters • An Example Finding z Parameters and Network Characteristics • Additional TwoPort Parameters and Conversions • Two Port Parameter Selection 3.1 Voltage and Current Laws Michael D. Ciletti Analysis of linear circuits rests on two fundamental physical laws that describe how the voltages and currents in a circuit must behave. This behavior results from whatever voltage sources, current sources, and energy storage elements are connected to the circuit. A voltage source imposes a constraint on the evolution of the voltage between a pair of nodes; a current source imposes a constraint on the evolution of the current in a branch of the circuit. The energy storage elements (capacitors and inductors) impose initial conditions on currents and voltages in the circuit; they also establish a dynamic relationship between the voltage and the current at their terminals. Regardless of how a linear circuit is stimulated, every node voltage and every branch current, at every instant of time, must be consistent with Kirchhoff’s voltage and current laws. These two laws govern even the most complex linear circuits. (They also apply to a broad category of nonlinear circuits that are modeled by point models of voltage and current.) A circuit can be considered to have a topological (or graph) view, consisting of a labeled set of nodes and a labeled set of edges. Each edge is associated with a pair of nodes. A node is drawn as a dot and represents a Michael D. Ciletti University of Colorado J. David Irwin Auburn University Allan D. Kraus Allan D. Kraus Associates Norman Balabanian University of Florida Theodore A. Bickart Michigan State University Shu-Park Chan International Technological University Norman S. Nise California State Polytechnic University
FIGURE 3.1 Graph representation of a linear circuit. connection between two or more physical components; an edge is drawn as a line and represents a path,or branch, for current flow through a component(see Fig 3.1) The edges, or branches, of the graph are assigned current labels, i,, iz,..., im. Each current has a designated direction, usually denoted by an arrow symbol. If the arrow is drawn toward a node, the associated current is said to be entering the node; if the arrow is drawn away from the node, the current is said to be leaving the node. The current i is entering node b in Fig 3. 1; the current is is leaving node e. Given a branch, the pair of nodes to which the branch is attached defines the convention for measuring voltages in the circuit. Given the ordered pair of nodes(a, b), a voltage measurement is formed as follows: where va and v, are the absolute electrical potentials(voltages)at the respective nodes, taken relative to some reference node. Typically, one node of the circuit is labeled as ground, or reference node; the remaining nodes re assigned voltage labels. The measured quantity, vab, is called the voltage drop from node a to node b. we is called the voltage rise from a to b. Each node voltage implicitly defines the voltage drop between the respective node and the ground node. The pair of nodes to which an edge is attached may be written as(a, b)or(b, a). Given an ordered pair of nodes(a, b), a path from a to b is a directed sequence of edges in which the first edge in the sequence contains node label a, the last edge in the sequence contains node label b, and the node indices of any two adjacent members of the sequence have at least one node label in common. In Fig. 3.1, the edge sequence (ej, e2, e is not a path, because e, and e do not share a common node label. The sequence le, e,) is a path from node a to node c a path is said to be closed if the first node index of its first edge is identical to the second node index of its last edge. The following edge sequence forms a closed path in the graph given in Fig. 3.1: leu, ex, e,, ea, e,, Note that the edge sequences fes) and fen, el are closed paths Kirchhoff's Current law Kirchhoff's current law(KCL) imposes constraints on the currents in the branches that are attached to each node of a circuit. In simplest terms, KCL states that the sum of the currents that are entering a given node c 2000 by CRC Press LLC
© 2000 by CRC Press LLC connection between two or more physical components; an edge is drawn as a line and represents a path, or branch, for current flow through a component (see Fig. 3.1). The edges, or branches, of the graph are assigned current labels, i1, i2, . . ., im. Each current has a designated direction, usually denoted by an arrow symbol. If the arrow is drawn toward a node, the associated current is said to be entering the node; if the arrow is drawn away from the node, the current is said to be leaving the node. The current i1 is entering node b in Fig. 3.1; the current i5 is leaving node e. Given a branch, the pair of nodes to which the branch is attached defines the convention for measuring voltages in the circuit. Given the ordered pair of nodes (a, b), a voltage measurement is formed as follows: vab = va – vb where va and vb are the absolute electrical potentials (voltages) at the respective nodes, taken relative to some reference node. Typically, one node of the circuit is labeled as ground, or reference node; the remaining nodes are assigned voltage labels. The measured quantity, vab, is called the voltage drop from node a to node b. We note that vab = –vba and that vba = vb – va is called the voltage rise from a to b. Each node voltage implicitly defines the voltage drop between the respective node and the ground node. The pair of nodes to which an edge is attached may be written as (a,b) or (b,a). Given an ordered pair of nodes (a, b), a path from a to b is a directed sequence of edges in which the first edge in the sequence contains node label a, the last edge in the sequence contains node label b, and the node indices of any two adjacent members of the sequence have at least one node label in common. In Fig. 3.1, the edge sequence {e1, e2, e4} is not a path, because e2 and e4 do not share a common node label. The sequence {e1, e2} is a path from node a to node c. A path is said to be closed if the first node index of its first edge is identical to the second node index of its last edge. The following edge sequence forms a closed path in the graph given in Fig. 3.1: {e1, e2, e3, e4, e6, e7}. Note that the edge sequences {e8} and {e1, e1} are closed paths. Kirchhoff’s Current Law Kirchhoff’s current law (KCL) imposes constraints on the currents in the branches that are attached to each node of a circuit. In simplest terms, KCL states that the sum of the currents that are entering a given node FIGURE 3.1 Graph representation of a linear circuit
must equal the sum of the currents that are leaving the node. Thus, the set of currents in branches attached to a given node can be partitioned into two groups whose orientation is away from(into) the node. The two groups must contain the same net current Applying KCL at node b in Fig 3. 1 gives i1(t)+i3()=2(t) A connection of water pipes that has no leaks is a physical analogy of this situation. The net rate at which water is flowing into a joint of two or more pipes must equal the net rate at which water is flowing away from the joint. The joint itself has the property that it only connects the pipes and thereby imposes a structure on ne flow of water, but it cannot store water. This is true regardless of when the flow is measured. Likewise, the nodes of a circuit are modeled as though they cannot store charge.( Physical circuits are sometimes modeled for the purpose of simulation as though they store charge, but these nodes implicitly have a capacitor that for storing the charge. Thus, KCL is ultimately KCL can be stated alternatively as: " the algebraic sum of the branch currents entering(or leaving) any node of a circuit at any instant of time must be zero. In this form, the label of any current whose orientation is away from the node is preceded by a minus sign. The currents entering node b in Fig 3. 1 must satisfy i1(t)-i2(t)+i3(t)=0 In general, the currents entering or leaving each node m of a circuit must satisfy ∑ 1km(D)=0 where ikm(t) is understood to be the current in branch k attached to node m The currents used in this expression are understood to be the currents that would be measured in the branches attached to the node, and thei values include a magnitude and an algebraic sign. If the measurement convention is oriented for the case where currents are entering the node, then the actual current in a branch has a positive or negative sign, depending on whether the current is truly flowing toward the node in question. Once KCL has been written for the nodes of a circuit, the equations can be rewritten by substituting into the equations the voltage-current relationships of the individual components. If a circuit is resistive, the resulting equations will be algebraic. If capacitors or inductors are included in the circuit, the substitution will produce a differential equation. For example, writing KCL at the node for v, in Fig 3. 2 prod +1-13 出+"- FIGURE 3.2 Example of a circuit containing energy storage c 2000 by CRC Press LLC
© 2000 by CRC Press LLC must equal the sum of the currents that are leaving the node. Thus, the set of currents in branches attached to a given node can be partitioned into two groups whose orientation is away from (into) the node. The two groups must contain the same net current. Applying KCL at node b in Fig. 3.1 gives i1(t) + i3 (t) = i2 (t) A connection of water pipes that has no leaks is a physical analogy of this situation. The net rate at which water is flowing into a joint of two or more pipes must equal the net rate at which water is flowing away from the joint. The joint itself has the property that it only connects the pipes and thereby imposes a structure on the flow of water, but it cannot store water. This is true regardless of when the flow is measured. Likewise, the nodes of a circuit are modeled as though they cannot store charge. (Physical circuits are sometimes modeled for the purpose of simulation as though they store charge, but these nodes implicitly have a capacitor that provides the physical mechanism for storing the charge. Thus, KCL is ultimately satisfied.) KCL can be stated alternatively as: “the algebraic sum of the branch currents entering (or leaving) any node of a circuit at any instant of time must be zero.” In this form, the label of any current whose orientation is away from the node is preceded by a minus sign. The currents entering node b in Fig. 3.1 must satisfy i1 (t) – i2 (t) + i3 (t) = 0 In general, the currents entering or leaving each node m of a circuit must satisfy where ikm(t) is understood to be the current in branch k attached to node m. The currents used in this expression are understood to be the currents that would be measured in the branches attached to the node, and their values include a magnitude and an algebraic sign. If the measurement convention is oriented for the case where currents are entering the node, then the actual current in a branch has a positive or negative sign, depending on whether the current is truly flowing toward the node in question. Once KCL has been written for the nodes of a circuit, the equations can be rewritten by substituting into the equations the voltage-current relationships of the individual components. If a circuit is resistive, the resulting equations will be algebraic. If capacitors or inductors are included in the circuit, the substitution will produce a differential equation. For example, writing KCL at the node for v3 in Fig. 3.2 produces i2 + i1 – i3 = 0 and FIGURE 3.2 Example of a circuit containing energy storage elements. i km  ( )t = 0 C dv dt v v R C dv dt 1 1 4 3 2 2 2 + 0 - - = R1 C2 R2 i 1 + – v + – v1 + 2 – i 2 C1 v3 vin v4 i 3
KCL for the node between C2 and RI can be written to eliminate variables and lead to a solution describing the capacitor voltages. The capacitor voltages, together with the applied voltage source, determine the remaining voltages and currents in the circuit Nodal analysis(see Section 3. 2)treats the systematic modeling and analysis of a circuit under the influence of its sources and energy storage elements Kirchhoff,s Current Law in the Complex Domain Kirchhoff's current law is ordinarily stated in terms of the real (time-domain) currents flowing in a circuit, because it actually describes physical quantities, at least in a macroscopic, statistical sense. It also applied, however, to a variety of purely mathematical models that are commonly used to analyze circuits in the so-called comp domain For example, if a linear circuit is in the sinusoidal steady state, all of the currents and voltages in the circuit are sinusoidal. Thus, each voltage has the form v(t)=Asin(ot+φ) and each current has the form i(t)=Bsin(ot+θ) where the positive coefficients A and B are called the magnitudes of the signals, and o and e are the phas angles of the signals. These mathematical models describe the physical behavior of electrical quantities, and instrumentation,such as an oscilloscope, can display the actual waveforms represented by the mathematical model. Although methods exist for manipulating the models of circuits to obtain the magnitude and phase coefficients that uniquely determine the waveform of each voltage and current, the manipulations are cumbe some and not easily extended to address other issues in circuit analysis Steinmetz [Smith and Dorf, 1992] found a way to exploit complex algebra to create an elegant framework for representing signals and analyzing circuits when they are in the steady state. In this approach, a model is developed in which each physical sign is replaced by a"complex" mathematical signal. This complex signal in polar, or exponential, form is represented as v(t)= Ae(jot +o) The algebra of complex exponential signals allows us to write this as v(t)=Aepejor and Euler's identity gives the equivalent rectangular form: v(t)=A[cos(ot +o)+j sin(ot +o) So we see that a physical signal is either the real(cosine)or the imaginary (sine) component of an abstrac complex mathematical signal. The additional mathematics required for treatment of complex numbers allows us to associate a phasor, or complex amplitude, with a sinusoidal signal. The time-invariant phasor associate V=A Notice that the phasor ve is an algebraic constant and that in incorporates the parameters A and o of the ding time-domain sinusoidal signal. Phasors can be thought of as being vectors in a two-dimensional plane. If the vector is allowed to rotate about the origin in the counterclockwise direction with frequency a, the projection of its tip onto the horizontal c 2000 by CRC Press LLC
© 2000 by CRC Press LLC KCL for the node between C2 and R1 can be written to eliminate variables and lead to a solution describing the capacitor voltages. The capacitor voltages, together with the applied voltage source, determine the remaining voltages and currents in the circuit. Nodal analysis (see Section 3.2) treats the systematic modeling and analysis of a circuit under the influence of its sources and energy storage elements. Kirchhoff’s Current Law in the Complex Domain Kirchhoff’s current law is ordinarily stated in terms of the real (time-domain) currents flowing in a circuit, because it actually describes physical quantities, at least in a macroscopic, statistical sense. It also applied, however, to a variety of purely mathematical models that are commonly used to analyze circuits in the so-called complex domain. For example, if a linear circuit is in the sinusoidal steady state, all of the currents and voltages in the circuit are sinusoidal. Thus, each voltage has the form v(t) = A sin(wt + f) and each current has the form i(t) = B sin(wt + q) where the positive coefficients A and B are called the magnitudes of the signals, and f and q are the phase angles of the signals. These mathematical models describe the physical behavior of electrical quantities, and instrumentation, such as an oscilloscope, can display the actual waveforms represented by the mathematical model. Although methods exist for manipulating the models of circuits to obtain the magnitude and phase coefficients that uniquely determine the waveform of each voltage and current, the manipulations are cumbersome and not easily extended to address other issues in circuit analysis. Steinmetz [Smith and Dorf, 1992] found a way to exploit complex algebra to create an elegant framework for representing signals and analyzing circuits when they are in the steady state. In this approach, a model is developed in which each physical sign is replaced by a “complex” mathematical signal. This complex signal in polar, or exponential, form is represented as vc(t) = Ae(jwt + f) The algebra of complex exponential signals allows us to write this as vc(t) = Aejfejwt and Euler’s identity gives the equivalent rectangular form: vc(t) = A[cos(wt + f) + j sin(wt + f)] So we see that a physical signal is either the real (cosine) or the imaginary (sine) component of an abstract, complex mathematical signal. The additional mathematics required for treatment of complex numbers allows us to associate a phasor, or complex amplitude, with a sinusoidal signal. The time-invariant phasor associated with v(t) is the quantity Vc = Ae jf Notice that the phasor vc is an algebraic constant and that in incorporates the parameters A and f of the corresponding time-domain sinusoidal signal. Phasors can be thought of as being vectors in a two-dimensional plane. If the vector is allowed to rotate about the origin in the counterclockwise direction with frequency w, the projection of its tip onto the horizontal
(real) axis defines the time-domain signal corresponding to the real part of v(n),i.e, A cos(ot projection onto the vertical (imaginary) axis defines the time-domain signal corresponding to the part of v(t), i.e., A sin(at +o]. The composite signal v(n) is a mathematical entity, it cannot be seen with an oscilloscope. Its value lies in the fact that when a circuit is in the steady state, its voltages and currents are uniquely determined by their corresponding phasors, and these in turn satisfy Kirchhoff's voltage and current laws! Thus, we are able to write ∑hm=0 where Ikm is the phasor of ikm(O), the sinusoidal current in branch k attached to node m. An equation of this form can be written at each node of the circuit. For example, at node b in Fig. 3.1 KCL would have the form Consequently, a set of linear, algebraic equations describe the phasors of the currents and In a circu in the sinusoidal steady state, i.e., the notion of time is suppressed(see Section 3. 2).The of the set of quations yields the phasor of each voltage and current in the circuit, from which the expressions can be extracted On It can also be shown that KCL can be extended to apply to the Fourier transforms and the Laplace transforms the currents in a circuit. Thus, a single relationship between the currents at the nodes of a circuit applies to all of the known mathematical representations of the currents [ Ciletti, 1988] Kirchhoff's Voltage Law Kirchhoff's voltage law(KVL) describes a relationship among the voltages measured across the branches in any closed, connected path in a circuit. Each branch in a circuit is connected to two nodes. For the purpose of applying KVL, a path has an orientation in the sense that in"walking"along the path one would enter one of the nodes and exit the other. This establishes a direction for determining the voltage across a branch in the path: the voltage is the difference between the potential of the node entered and the potential of the node at which the path exits. Alternatively, the voltage drop along a branch is the difference of the node voltage at the entered node and the node voltage at the exit node. For example, if a path includes a branch between node"a and node " b, the voltage drop measured along the path in the direction from node"a"to node"b"is denoted by vab and is given by vab=v-w Given vab, branch voltage along the path in the direction from node"b"to node "a"is vi=v Kirchhoff's voltage law, like Kirchhoff's current law, is true at any time. KVL can also be stated in terms of voltage rises instead of voltage drops. KVL can be expressed mathematically as"the algebraic sum of the voltages drops around any closed path of a circuit at any instant of time is zero. This statement can also be cast as an equation: ∑ vhm,()=0 where vm(t) is the instantaneous voltage drop measured across branch k of path m. By convention, the voltage drop is taken in the direction of the edge sequence that forms the path. The edge sequence le, e2, e,, e4, e, e,) forms a closed path in Fig 3. 1. The sum of the voltage drops taken around the path must satisfy KVL vab(t)+vc(t)+ va(t)+ vae(t)+ ve (t)+ ya(t)=0 Since va(n)=-(t), we can also write c 2000 by CRC Press LLC
© 2000 by CRC Press LLC (real) axis defines the time-domain signal corresponding to the real part of vc(t), i.e., A cos[wt + f], and its projection onto the vertical (imaginary) axis defines the time-domain signal corresponding to the imaginary part of vc(t), i.e., A sin[wt + f]. The composite signal vc(t) is a mathematical entity; it cannot be seen with an oscilloscope. Its value lies in the fact that when a circuit is in the steady state, its voltages and currents are uniquely determined by their corresponding phasors, and these in turn satisfy Kirchhoff’s voltage and current laws! Thus, we are able to write where Ikm is the phasor of ikm(t), the sinusoidal current in branch k attached to node m. An equation of this form can be written at each node of the circuit. For example, at node b in Fig. 3.1 KCL would have the form I 1 – I 2 + I 3 = 0 Consequently, a set of linear, algebraic equations describe the phasors of the currents and voltages in a circuit in the sinusoidal steady state, i.e., the notion of time is suppressed (see Section 3.2). The solution of the set of equations yields the phasor of each voltage and current in the circuit, from which the actual time-domain expressions can be extracted. It can also be shown that KCL can be extended to apply to the Fourier transforms and the Laplace transforms of the currents in a circuit. Thus, a single relationship between the currents at the nodes of a circuit applies to all of the known mathematical representations of the currents [Ciletti, 1988]. Kirchhoff’s Voltage Law Kirchhoff’s voltage law (KVL) describes a relationship among the voltages measured across the branches in any closed, connected path in a circuit. Each branch in a circuit is connected to two nodes. For the purpose of applying KVL, a path has an orientation in the sense that in “walking” along the path one would enter one of the nodes and exit the other. This establishes a direction for determining the voltage across a branch in the path: the voltage is the difference between the potential of the node entered and the potential of the node at which the path exits. Alternatively, the voltage drop along a branch is the difference of the node voltage at the entered node and the node voltage at the exit node. For example, if a path includes a branch between node “a” and node “b”, the voltage drop measured along the path in the direction from node “a” to node “b” is denoted by vab and is given by vab = va – vb. Given vab, branch voltage along the path in the direction from node “b” to node “a” is vba = vb – va = –vab . Kirchhoff’s voltage law, like Kirchhoff’s current law, is true at any time. KVL can also be stated in terms of voltage rises instead of voltage drops. KVL can be expressed mathematically as “the algebraic sum of the voltages drops around any closed path of a circuit at any instant of time is zero.” This statement can also be cast as an equation: where vkm(t) is the instantaneous voltage drop measured across branch k of path m. By convention, the voltage drop is taken in the direction of the edge sequence that forms the path. The edge sequence {e1, e2 , e3 , e4 , e6 , e7} forms a closed path in Fig. 3.1. The sum of the voltage drops taken around the path must satisfy KVL: vab (t) + vbc(t) + vcd (t) + vde(t) + vef(t) + vfa(t) = 0 Since vaf(t) = –vfa (t), we can also write I  km = 0 vkm  ( )t = 0
var(t)=vab(t)+ vbe(t)+ vea(t)+ vde(t)+ vef(t) Had we chosen the path corresponding to the edge sequence lep,e, e6, e) for the path, we would have obtained t)=vab(r)+ ve(t)+ ver(t) This demonstrates how KCL can be used to determine the voltage between a pair of nodes. It also reveals the fact that the voltage between a pair of nodes is independent of the path between the nodes on which the voltages are measured Kirchhoffs Voltage Law in the Complex domain Kirchhoffs voltage law also applies to the phasors of the voltages in a circuit in steady state and to the Fourier transforms and Laplace transforms of the voltages in a circuit. Importance of KVL and KCL Kirchhoff's current law is used extensively in nodal analysis because it is amenable to computer-based imple mentation and supports a systematic approach to circuit analysis. Nodal analysis leads to a set of algebraic equations in which the variables are the voltages at the nodes of the circuit. This formulation is popular in CAd programs because the variables correspond directly to physical quantities that can be measured easily Kirchhoff's voltage law can be used to completely analyze a circuit, but it is seldom used in large-scale circuit simulation programs. The basic reason is that the currents that correspond to a loop of a circuit do not necessarily correspond to the currents in the individual branches of the circuit. Nonetheless, KVL is frequently used to troubleshoot a circuit by measuring voltage drops across selected components. Defining Terms Branch: A symbol representing a path for current through a component in an electrical circuit. Branch current: The current in a branch of a circuit Branch voltage: The voltage across a branch of a circuit Independent source: A voltage(current) source whose voltage(current)does not depend on any other voltage or current in the circuit Node: A symbol representing a physical connection between two electrical components in a circuit. Node voltage: The voltage between a node and a reference node(usually ground Related Topic 3.6 Graph Theory eferences M.D. Ciletti, Introduction to Circuit Analysis and Design, New York: Holt, Rinehart and winston, 1988 R H. Smith and R.C. Dorf, Circuits, Devices and Systems, New York: Wiley, 1992 Further information Kirchhoff's laws form the foundation of modern computer software for analyzing electrical circuits. The interested reader might consider the use of determining the minimum number of algebraic equations that fully characterizes the circuit. It is determined by KCL, KVL, or some mixture of the two? c 2000 by CRC Press LLC
© 2000 by CRC Press LLC vaf(t) = vab (t) + vbc(t) + vcd (t) + vde(t) + vef(t) Had we chosen the path corresponding to the edge sequence {e1, e5 , e6 , e7} for the path, we would have obtained vaf(t) = vab (t) + vbe(t) + vef(t) This demonstrates how KCL can be used to determine the voltage between a pair of nodes. It also reveals the fact that the voltage between a pair of nodes is independent of the path between the nodes on which the voltages are measured. Kirchhoff’s Voltage Law in the Complex Domain Kirchhoff’s voltage law also applies to the phasors of the voltages in a circuit in steady state and to the Fourier transforms and Laplace transforms of the voltages in a circuit. Importance of KVL and KCL Kirchhoff’s current law is used extensively in nodal analysis because it is amenable to computer-based implementation and supports a systematic approach to circuit analysis. Nodal analysis leads to a set of algebraic equations in which the variables are the voltages at the nodes of the circuit. This formulation is popular in CAD programs because the variables correspond directly to physical quantities that can be measured easily. Kirchhoff’s voltage law can be used to completely analyze a circuit, but it is seldom used in large-scale circuit simulation programs. The basic reason is that the currents that correspond to a loop of a circuit do not necessarily correspond to the currents in the individual branches of the circuit. Nonetheless, KVL is frequently used to troubleshoot a circuit by measuring voltage drops across selected components. Defining Terms Branch: A symbol representing a path for current through a component in an electrical circuit. Branch current: The current in a branch of a circuit. Branch voltage: The voltage across a branch of a circuit. Independent source: A voltage (current) source whose voltage (current) does not depend on any other voltage or current in the circuit. Node: A symbol representing a physical connection between two electrical components in a circuit. Node voltage: The voltage between a node and a reference node (usually ground). Related Topic 3.6 Graph Theory References M.D. Ciletti, Introduction to Circuit Analysis and Design, New York: Holt, Rinehart and Winston, 1988. R.H. Smith and R.C. Dorf, Circuits, Devices and Systems, New York: Wiley, 1992. Further Information Kirchhoff’s laws form the foundation of modern computer software for analyzing electrical circuits. The interested reader might consider the use of determining the minimum number of algebraic equations that fully characterizes the circuit. It is determined by KCL, KVL, or some mixture of the two?
3.2 Node and mesh analysis . David irwin In this section Kirchhoffs current law(KCL) and Kirchhoff's voltage law(KVL) will be used to determine currents and voltages throughout a network. For simplicity, we will first illustrate the basic principles of both node analysis and mesh analysis using only dc circuits. Once the fundamental concepts have been explained and illustrated, we will demonstrate the generality of both analysis techniques through an ac circuit exampl Node analysis In a node analysis, the node voltages are the variables in a circuit, v1=4VV2=4v and KCl is the vehicle used to determine them. One node in the network is selected as a reference node and then all other node voltages are defined with respect to that particular node. This refer ence node is typically referred to as ground using the symbol (4),20 indicating that it is at ground-zero potential Consider the network shown in Fig. 3.3. The network has three nodes, and the nodes at the bottom of the circuit has been selected as the reference node. Therefore the two remaining nodes, labeled VI and V2, are measured with respect to this reference node. Suppose that the node voltages V, and V, have somehow been FIGURE 3.3 A three-node network. determined, i. e, V=4 V and v2=-4 V. Once these node voltages known, Ohms law can be used to find all branch currents. For example, 2A V-V2_4-(-4) 4A 3 4A 1 Note that KCL is satisfied at every node, i. e, l2+8+I3=0 +6-8-1=0 Therefore, as a general rule, if the node voltages are known, all branch currents in the network can be immediately determined. In order to determine the node voltages in a network, we apply KCL to every node in the network except the reference node. There- V, fore, given an N-node circuit, we employ N-1 linearly independent 20 simultaneous equations to determine the n-1 unknown node volt 19 ages. Graph theory, which is covered in Section 3.6, can be used to 1 prove that exactly N-1 linearly independent KCL equations are required to find the N-1 unknown node voltages in a network. Let us now demonstrate the use of KCL in determining the node FIGURE 3.4 A four-node network. voltages in a network. For the network shown in Fig 3.4, the bottom c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 3.2 Node and Mesh Analysis J. David Irwin In this section Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) will be used to determine currents and voltages throughout a network. For simplicity, we will first illustrate the basic principles of both node analysis and mesh analysis using only dc circuits. Once the fundamental concepts have been explained and illustrated, we will demonstrate the generality of both analysis techniques through an ac circuit example. Node Analysis In a node analysis, the node voltages are the variables in a circuit, and KCL is the vehicle used to determine them. One node in the network is selected as a reference node, and then all other node voltages are defined with respect to that particular node. This reference node is typically referred to as ground using the symbol ( ), indicating that it is at ground-zero potential. Consider the network shown in Fig. 3.3. The network has three nodes, and the nodes at the bottom of the circuit has been selected as the reference node. Therefore the two remaining nodes, labeled V1 and V2, are measured with respect to this reference node. Suppose that the node voltages V1 and V2 have somehow been determined, i.e., V1 = 4 V and v2 = –4 V. Once these node voltages are known, Ohm’s law can be used to find all branch currents. For example, Note that KCL is satisfied at every node, i.e., I1 – 6 + I2 = 0 –I2 + 8 + I3 = 0 –I1 + 6 – 8 – I3 = 0 Therefore, as a general rule, if the node voltages are known, all branch currents in the network can be immediately determined. In order to determine the node voltages in a network, we apply KCL to every node in the network except the reference node. Therefore, given an N-node circuit, we employ N – 1 linearly independent simultaneous equations to determine the N – 1 unknown node voltages. Graph theory, which is covered in Section 3.6, can be used to prove that exactly N – 1 linearly independent KCL equations are required to find the N – 1 unknown node voltages in a network. Let us now demonstrate the use of KCL in determining the node voltages in a network. For the network shown in Fig. 3.4, the bottom FIGURE 3.3 A three-node network. I V I V V I V 1 1 2 1 2 3 2 0 2 2 2 4 4 2 4 0 1 4 1 4 = - = = - = - - = = - = - = - A A A ( ) FIGURE 3.4 A four-node network
node is selected as the reference and the three remaining nodes, labeled Vi, v2 and V3, are measured with respect to that node. All unknown branch currents are also labeled. The KCl equations for the three nonref erence nodes are 1+4+12=0 4+l3+I4=0 -l1-4-2=0 Using Ohms law these equations can be expressed as 3+4+ +2 V-V3)(V2-V) Solving these equations, using any convenient method, yield V1=-8/3V,V2=10/3V, and V=8/3V. Applying Ohms law V a we find that the branch currents are I,=-16/6 A, I,=-8/6A ,=20/6A, and 14=4/6A. A quick check indicates that KCL is satisfied at every node. The circuits examined thus far have contained only current sources and resistors. In order to expand our capabilities, we next examine a circuit containing voltage sources. The circuit shown in Fig. 3.5 has three nonreference nodes labeled VI figure 3.5 A four-node network cont and V,. However, we do not have three unknown node volt- voltage soure ages. Since known voltage sources exist between the reference node and nodes Vi and V,, these two node voltages are known, i.e. Vi=12 V and V3=-4 V. Therefore,we have only one unknown node voltage, V2. The equations for this network are then V1=12 -l1+l2+1 The KCL equation for node V2 written using Ohms law is (12-V2),V2,V2-(-4) 0 2 Solving this equation yields V,=5V,I=7A,I=5/2A, and 1, =9/2A. Therefore, KCL is satisfied at every node. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC node is selected as the reference and the three remaining nodes, labeled V1, V2, and V3, are measured with respect to that node. All unknown branch currents are also labeled. The KCL equations for the three nonreference nodes are I1 + 4 + I2 = 0 – 4 + I3 + I4 = 0 –I1 – I4 – 2 = 0 Using Ohm’s law these equations can be expressed as Solving these equations, using any convenient method, yields V1 = –8/3 V, V2 = 10/3 V, and V3 = 8/3 V. Applying Ohm’s law we find that the branch currents are I1 = –16/6 A, I2 = –8/6 A, I3 = 20/6 A, and I4 = 4/6 A. A quick check indicates that KCL is satisfied at every node. The circuits examined thus far have contained only current sources and resistors. In order to expand our capabilities, we next examine a circuit containing voltage sources. The circuit shown in Fig. 3.5 has three nonreference nodes labeled V1, V2, and V3. However, we do not have three unknown node voltages. Since known voltage sources exist between the reference node and nodes V1 and V3, these two node voltages are known, i.e., V1 = 12 V and V3 = –4 V. Therefore, we have only one unknown node voltage, V2. The equations for this network are then V1 = 12 V3 = – 4 and –I1 + I2 + I3 = 0 The KCL equation for node V2 written using Ohm’s law is Solving this equation yields V2 = 5 V, I1 = 7 A, I2 = 5/2 A, and I3 = 9/2 A. Therefore, KCL is satisfied at every node. V V V V V V 1 3 1 2 2 3 2 4 2 0 4 1 1 0 - + + = - + + - = - - - - - = (V V ) (V V ) 1 3 2 3 2 1 2 0 FIGURE 3.5 A four-node network containing voltage sources. - - + + - - = ( ) 12 ( ) 1 2 4 2 0 V2 V2 V2
Thus, the presence of a voltage source in the network actually simplifies a node analysis. In an attempt to generalize this idea, consider the network in Fig. 3.6. Note that in this case V,= 12V and the difference between node voltages V, and V, is constrained to be V1 6 V. Hence, two of the three equations needed to solve for the node 12v0 1a b voltages e network are 29 V,=12 V3-V2=6 IGURE 3. 6 A four-node network used to a supernode. To obtain the third required equation, we form what is called a supernode, indicated by the dotted enclosure in the network. Just as KCl must be satisfied at any node in the network, it must be satisfied at the supernode as well. Therefore, summing all the currents leaving the sur 3×D =0 The three equations yield the node voltages V =12V, V2=5V, and V3=11 V, and therefore I =1 A,I=7 A,l3=5/2A,andl=11/2A Mesh analysis In a mesh analysis the mesh currents in the network are the variables and kvl is the mechanism used to determine them once all the mesh currents have been determined, Ohms law will yield the voltages anywhere in e circuit. If the network contains N independent meshes, 10 7a 2(2-②9 n graph theory can be used to prove that n independent linear simultaneous equations will be required to determine the N mesh currents The network shown in Fig. 3.7 has two independent meshes. They are labeled I and I,, as shown. If the mesh currents are known to be FIGuRE 3.7 A network containing two 1 =7 A and I2=5/2 A, then all voltages in the network can be independent meshes. calculated. For example, the voltage Vp, i.e., the voltage across the 1-12 resistor, is V,=-IR=-(7)(1)=-7 V Likewise V=(I-I)R=(7-5/2)(2)=9V. Furthermore, we can check our analysis by showing that KVl is satisfied around every mesh. Starting at the lower left-hand corner and pplying kVl to the left-hand mesh we obtain -(7)(1)+16-(7-5/2)(2)=0 where we have assumed that increases in energy level are positive and decreases in energy level are negative Consider now the network in Fig. 3.8. Once again, if we assume that an increase in energy level is positive and a decrease in energy level is 18 negative, the three KVl equations for the three meshes defined are 9和习 29 1(1)-6-(l1-12)(1)=0 +12-(l2-1)(1)-(l2-l3)(2)=0 l3-12)(2)+6-I3(2)=0 FiGURE 3. 8 A three- mesh network. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Thus, the presence of a voltage source in the network actually simplifies a node analysis. In an attempt to generalize this idea, consider the network in Fig. 3.6. Note that in this case V1 = 12 V and the difference between node voltages V3 and V2 is constrained to be 6 V. Hence, two of the three equations needed to solve for the node voltages in the network are V1 = 12 V3 – V2 = 6 To obtain the third required equation, we form what is called a supernode, indicated by the dotted enclosure in the network. Just as KCL must be satisfied at any node in the network, it must be satisfied at the supernode as well. Therefore, summing all the currents leaving the supernode yields the equation The three equations yield the node voltages V1 = 12 V, V2 = 5 V, and V3 = 11 V, and therefore I1 = 1 A, I2 = 7 A, I3 = 5/2 A, and I4 = 11/2 A. Mesh Analysis In a mesh analysis the mesh currents in the network are the variables and KVL is the mechanism used to determine them. Once all the mesh currents have been determined, Ohm’s law will yield the voltages anywhere in e circuit. If the network contains N independent meshes, then graph theory can be used to prove that N independent linear simultaneous equations will be required to determine the N mesh currents. The network shown in Fig. 3.7 has two independent meshes. They are labeled I1 and I2, as shown. If the mesh currents are known to be I1 = 7 A and I2 = 5/2 A, then all voltages in the network can be calculated. For example, the voltage V1, i.e., the voltage across the 1-W resistor, is V1 = –I1R = –(7)(1) = –7 V. Likewise V = (I1 – I2)R = (7 –5/2)(2) = 9 V. Furthermore, we can check our analysis by showing that KVL is satisfied around every mesh. Starting at the lower left-hand corner and applying KVL to the left-hand mesh we obtain –(7)(1) + 16 – (7 – 5/2)(2) = 0 where we have assumed that increases in energy level are positive and decreases in energy level are negative. Consider now the network in Fig. 3.8. Once again, if we assume that an increase in energy level is positive and a decrease in energy level is negative, the three KVL equations for the three meshes defined are –I1(1) – 6 – (I1 – I2)(1) = 0 +12 – (I2 – I1)(1) – (I2 – I3)(2) = 0 –(I3 – I2)(2) + 6 – I3(2) = 0 FIGURE 3.6 A four-node network used to illustrate a supernode. V2 V1 V2 V3 V1 V3 1 2 1 2 0 - + + - + = FIGURE 3.7 A network containing two independent meshes. FIGURE 3.8 A three-mesh network
