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