PART I ELECTRIC CIRCUIT CONCEPT AND ANALYSIS

Transcription

1 PRT I ELECTRIC CIRCUIT CONCEPT ND NLYSIS Circuits as modelling tools Techniques for solving DC circuits.c. Linear electric circuits Threephase circuits

2 uc( t)

3 Chapter 2 Circuits as modelling tools Table of Contents 2.1 Introduction Definitions The charge conservation and Kirchhoff s current law The charge conservation law Charge conservation and circuits The electric current Kirchoff s Current Law formulations The circuit potential and Kirchhoff s voltage law The electric field inside conductors Kirchhoff s Voltage Law formulations Solution of a circuit Determination of linearly independent Kirchhoff s equations (loop cuts method) Constitutive equations Number of variables and equations Example The substitution principle Kirchhoff s in comparison with electromagnetism laws Power in circuits Example Historical notes Kirchhoff s short biography Tellegen s short biography Reference list... 30

4 2.4 Chapter 2: Circuits as modelling tools For the teacher This chapter, in comparison with similar books approaches, contains the basic innovation of making a clear distinction between physical systems and models. Since no uniform terminology in the books exists, here the following was adopted: circuital systems was the name adopted for physical systems constituted by devices connected by wires, that are good candidates to be modeled using lumpedcomponent models circuits was adopted for are actual (lumpedcomponent) models. This way of thinking has several advantages: it helps the student, i.e. the future engineer, to get accustomed to the importance of the activity of modelling a physical system, and the corresponding need to evaluate the approximations done in the process. This activity will accompany their whole working life; it allows to clearly distinguish between the electromagnetism laws, i.e. Maxwell s equations written in more or less complicated forms, and Kirchhoff s laws, that are assumed to be valid, by definition, to all circuits. The authors took the decision to make this shift with respect to what most frequently found in books, after years of teaching and meeting everyday students needs and doubts. They hope it will be appreciated by the teachers. If so, it will be useful and appreciated by their students as well.

5 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering Introduction Electrical engineers spend a large part of their time in working with Electric Circuits, or simply circuits. The word circuits is used to indicate graphical/mathematical tools that are used in description of systems in which charges flow constantly (the socalled directcurrent circuits), or sinusoidally (the socalled alternatingcurrent circuits), or in which they vary in a general way. They are able to model adequately electronic boards, home and industrial electrical installations, the inner behaviour of electrical machines, and so on. The physical systems circuits are intended to model, whenever precision is necessary, will be called circuital systems. More in Depth It is very common in written text (textbooks, articles, brochures) not to evidence the difference between physical systems and their model. This is widely erroneous, because physical systems have a given behaviour that can be modelled at different levels of precision. For instance the wheel of a car can be just an ideal rigid rolling cylinder for some purposes, but if one wants to evaluate the forces generated by collisions, elasticity must be taken into account; to evaluate energy losses also rubber hysteresis must be considered, and if one wants to evaluate the response to lateral forces, further characteristics must be taken into account. Making a difference between physical systems and their models is very important because to draw a model from a system a certain set of assumptions is to be made, and the results of the model s analysis can be applied to the given system only if these assumptions are met to a sufficient degree of precision. That is why in this book a lot of care has been put in distinguishing between physical systems and their models, i.e. circuital systems and circuits. Consider for instance the simple system shown in fig G 1 2 Fig simple circuital system. It shows an electric sinusoidal generator feeding two lamps with the interposition of a couple of wires, that are represented thick, because in a physical system they have not only a length but also width and depth. Obviously, the analysis of this system would be greatly simplified if, instead of having to analyse simultaneously all points of space using the electromagnetics laws (the four laws quoted in ppendix, and that are often referred to as Maxwell s equations) it would be possible to write independent equations of the involved individual elements and linking them by some congruence additional equations. qualitative analysis of this figure shows that the generator connects with the lamps with long wires, while short connections are present at the two horizontal sides of the circuit. It is intuitively understood that, if the effects of space around the wires is not significant for what happens inside the system components, the system of fig. 2.1 can be studied as a system constituted by connection of its main elements, as indicated in fig The elements are the

6 2.6 Chapter 2: Circuits as modelling tools generator, the two lamps, and the two long wires, while the short wires used for connecting the lamps to the longer wires are neglected. ll elements have terminals, that are the points of them used to connect them to the other circuits elements. In the figure terminals are indicated as small white circles. Thin lines represent ideal wires, i.e. graphical symbols indicating that what happens in the circuit is exactly the same as if the components at the ends of them were directly connected to each other. Thin lines are therefore like ideal wires, that are able to transfer electric charge from point to another of space in an ideal way, i.e. in such a way that charges present at one end are immediately transferred to the other end. The passage from the system in figure 2.1 to that of figure in 2.2 is dramatically important: it implies that we have modelled a spatially distributed system (for which all the fields connected to electromagnetics phenomena, such as electric field E, magnetic field H, current density field J, et. have a value at each point of space) as a lumped elements system, whose behaviour is determined by the behaviour of the boxes connected by ideal wires, that will be expressed in terms of equations that will be called constitutive equations of the components (or elements), and by the effect of the connections made by the wires. More specifically, we have modelled the circuital system from which we started, that was a distributed parameter one, through a circuit, that is a lumped parameter model, and whose analysis is much simpler. upper conductor ideal wire G lamp 1 lamp 2 circuit element lower conductor node terminal Fig Circuital approximation of (=circuit modelling) the system of fig. 2.1 The ease of analysis of electromagnetical systems using circuits will become more and more clearer as far as the study of the book proceeds. But what kind of systems can be studied as circuits? In a very simplified way, it can be answered that systems that can be studied as circuits are those we called earlier as circuital systems: i.e. systems that are physically constituted by devices, (such as the generator, the lamps and the long wires of figure 2.1) that perform complex actions, others that could play a marginal role (such as the short wires), and some space around these elements, whose effects on the electrical phenomena inside them can be neglected. These qualitative considerations often are too simplified. It must be stressed that if a circuit does not model correctly a physical system, the results of its analysis (the calculated currents and voltages) will not be close to the actual values of the original circuit, and this could lead to big mistakes. Therefore in this chapter, along with the next two ones, discussion is reported on what the basic hypotheses allowing a circuital system to be modelled as a circuit are. The construction of a lumped parameter model of a physical system is not unique. For instance, if the current between upper and lower conductors of the system reported in fig. 2.1 cannot be neglected, the model reported in fig. 2.3 can be considered, in which the transmission line is a unique lumped component, that can take into account also phenomena occurring between its upper and lower conductors.

7 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.7 ideal wire G Transmission line lamp 1 lamp 2 circuit element terminal node Fig circuit modelling the system of fig. 2.1 different from that of figure Definitions To deal with circuits effectively, precise definitions of the terms used are needed. lthough there is some uniformity in textbooks about terms and definitions, some significant differences exist. Therefore it is useful to summarise here all the major definitions related to circuits used in this book. The authors have carefully consulted the International Electrotechnical Vocabulary (IEV) [4] that is the most authoritative source of terminology standardisation for all electrical engineers all over the world. Therefore the definitions reported here normally comply with the ones reported in the IEV. Only the wording reported here is original, since it has the purpose of best explaining the meanings given the book s scope and audience. Where significant differences exist from the IEV small noted are added explaining the reasons beyond the choice of partly deviating from standards. The following definitions are reported in alphabetical order for the reader s convenience. Branch branch is a circuit element having two terminals. Branchbased circuit branchbased circuit is an electric circuit in which all elements, except nodes, have two terminals The IEV does not give any definition for circuits having only two terminal components, except nodes. his has required a definition specific to this book, because of the importance of these circuit topology. Circuit (electric circuit) n electric circuit, or simply circuit, graphicalmathematical tool that constitutes a lumpedcomponent model of a circuital system, consisting of lumped components connected to each other. These components are the circuit elements. Since circuits are models of real systems, circuits of different levels of accuracy can be produced for the same physical system. NOTES: 1. See the note on the circuital systems definition. 2. The circuit behaviour is defined by the inner behaviour of circuit elements and by their interconnection: no influence is possible between what happens outside and what happens inside circuit wires and elements. 3 In this definition no constraints are reported on the inner structure or behaviour of the circuit elements. However ideal transformers, very special circuit elements that will be discussed in chapter 4, play a special role in circuits: they are elements of interconnection of circuits to form electric networks. Circuit element circuit element is a component of a circuit that is connected to other components by means of some connection points, called terminals.

8 2.8 Chapter 2: Circuits as modelling tools Circuital system circuital system is a physical system containing elements connected to each other through wires in such a way that one or more closed that loops are formed. circuital system is a spatiallydistributed (threedimension) system. IEV does not provide special terminology to distinguish between physical systems and mathematical models. This distinction is however very important in practice, and therefore the author felt obliged to introduce this new term to refer to the physical systems, while circuits (and networks) are used for lumpedcomponent mathematical models of circuital systems. Ideal wire n ideal wire is a branch, the terminals of which have the same potential value, i.e. such that the voltage across its terminals is zero. Node node is a point in a circuit in which three or more circuit elements are connected to each other. lthough this definition does not coincide with the one reported in the IEV, it is very commonly used in textbooks Network (electric network) n electric network, o simply network, is a set of circuits separated from each other by ideal transformers. Two circuits separated by ideal transformers are also normally called magnetically coupled circuits. NOTES 1. in IEV electric circuits and electric networks are considered to be equivalent. The availability of two names is exploited in this book to indicate two different concepts, thus easing the description of the circuit theory and analysis techniques 2. networks and ideal transformers will be introduced in this book in chapter The charge conservation and Kirchhoff s current law The charge conservation law The reader should be already acquainted with the idea of charge conservation. The charge conservation principle states that the charge in a given region of space remains constant over time. However, studies carried out mainly in the XIX century showed that the charge intended as sum of individual charged elements (electrons, ions) does not actually conserve, but can accumulate in elements of space. The issue was solved by Maxwell, who has defined a new form of charge, called displacement charge, that enables us to state the following: Law: total charge conservation The total charge (sum of conduction and displacement charge) in any given region of space remains perfectly constant over time Conduction charge is the charge that moves through conductor wires (electrons) or conductive solutions (ions), while the displacement current due to the variation of the displacement field over time. This is detailed in the following more in depth box, as well as in the next chapters.

9 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.9 More in Depth this block is intended to those that have some previous knowledge of the electromagnetic behaviour of capacitors. and some basic knowledge of the behaviour of a simple RC circuit in which a capacitor is charged Battery R C sw i L armature dielectric armature In the circuit shown aside, if the armatures of the capacitor C are initially without any charge, when switch sw is closed, some current flows in the circuit, and neat charge enters the closed curve L. Charge is thus accumulated inside the armature, and therefore the charge conservation principle does not apply to it. If however we hypothesise that a special current, called displacement current flows in the dielectric, i.e. the space between the armatures, and that this current is exactly equal to the current i flowing in the conductors outside the capacitor, then the total charge conservation applies, the total charge being due to both conduction and displacement currents. In this case the armature total charge will not change since the dielectric will absorb a displacement current that is perfectly equivalent to the conduction current entering the upper conducting wire Charge conservation and circuits In the qualitative analysis shown in section 2.1, it was said that circuital systems may be correctly modelled by means of circuits when the effects of space around the wires is not significant for what happens inside the system components. If the space around system components must not affect what occurs inside them, no charge must flow in that space. Consider again the system reported in fig. 2.1, reported also in fig G 1 2 Fig The system of fig. 2.1, with indication of some stray currents. The reader should already know that, although materials are commonly classified in conducting and insulating ones, a perfect conductive material does not exist 1, as well as a perfectly insulating one. Therefore same stray currents can flow also though the air surrounding the wires, e.g. flowing from the upper conductor towards the lower one (as indicated in fig. 3.4), and viceversa. If the system of fig. 2.1 is represented by the circuit of fig. 2.2, it should be possible to neglect these stray currents. Indeed this is very often acceptable although in some rare cases stray currents are considered and different models are used. There is more than just considering the air surrounding the conductors to be perfectly conductive in the passage from figure 2.1 to 2.2. It must be noted that the system of figure 2.1 is subject to variable quantities, since the generator is a component, that we have not discussed yet in depth, but that has to be imagined to be able to determine at its ends a variable voltage. therefore all the quantities of the system of fig. 2.1 (currents, voltages, fields at all the points of space) vary with time. When quantities vary with time displacement currents can flow between upper and lower wires even though the air could be considered to be perfectly insulating. Indeed 1 the case of superconductivity, that occurs at very low temperatures is very special, and is not considered here.

10 2.10 Chapter 2: Circuits as modelling tools upper and lower conductors can be considered as being as the two cylindrical armatures of a capacitor whose dielectric is the air between (fig. 2.5). G 1 2 upper armature lower armature view Fig Stray currents through perfectly insulating means can be due to capacitive effects, or displacement currents between upper and lower wires. dielectric gain, the lumpisation process that leads to circuits requires that not only conductive but also displacement currents must not flow through the air (or space) interspersed between the ideal wires of any circuit. Consider again one of the lumpedcomponent models proposed, for the system of fig. 2.1, i.e. the circuit reported in fig. 2.3 and in fig, 2.6 also. Here, for simplicity, the terminals are not evidenced anymore (the reader should keep in mind that they always exist). 3 1 G Transmission line lamp 1 lamp 2 2 Fig. 2.6: Possible closed curves around circuit parts. In this figure three different closed curves crossing the circuit are considered: a curve surrounding a circuit node (type 1), one surrounding a single lumped element (type 2), one surrounding a group of elements and wires (type 3). For them the charge conservation principle applies and therefore the global charge entering the curves, of either conductive or displacement types must be equal to zero. But it was also said that whenever a circuit is created, all displacement currents between wires must be neglected. That is equivalent to saying that displacement currents can occur only inside circuit elements. s a consequence of this it can be concluded that the total (conductive) charge entering through some of the wires that traverse any of the considered curves must be exactly equal to the sum of those that exit it. s a consequence of this it can be concluded that, for any of the considered curves, the total (conductive) charge entering through the wires that traverse it must be exactly equal to zero. This is equivalent to saying the sum of the charge entering from some of the wires is identically equivalent to the sum of the charge that exits from the others. This is the rationale behind so called Kirchhoff s law, that is the charge conservation law for circuits. In the next sections it will be expressed in a more formal way (and more useful for practical computations.)

11 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering The electric current From their basic electromagnetics knowledge the readers should be already aware that electric charges present inside conductor materials (namely the electrons) are not linked to atoms, but can freely move. If therefore a wire is made of conducting material, these charges can freely move from an end to another of it. lso in other conducting means, such as electrolytic solutions there are charge free to move; while in conductor media the charge carriers are just electrons, and are negatively charged, in an electrolytic solution the charges are carried by ions, that can be either positive or negative. Therefore, in general it may happen that in a conducting wire charge carriers move from left to right or right to left, and they can be positively or negatively charged. It is experimentally verified that a positive charge moving from left to right is totally equivalent to a charge having the same size but opposite sign moving form right to left (fig. 3.7a) 2, and that the combined effect of a positive charge Q and a negative charge Q is equivalent to a generic charge Q,, assumed to be positive, moving in the same direction of Q 3.7b). Q is equivalent to: Q a) is Q equivalent Q to: Q b) Q=Q Q Fig Charge flows equivalence: a): a positive Q from left is equivalent to the opposite charge Q from right b): a comprehensive charge Q=Q Q from left is equivalent to Q >0 from left and Q <0 from right. Therefore in the general case in which a conductor medium there are both positive charges and they move in either direction, the analysis can be performed computing just an equivalent positive charge defined as: QQ1 Q2 in which Q 1 is the algebraic sum of charges in the same direction as the direction assumed for Q Q 2 is the algebraic sum of charges moving in direction opposite to the direction assumed for Q Consider this equivalent charge Q (flowing in the time t through a cross section of a conductor (fig. 2.7b), the current I through flowing in the conductor is defined as: 2 s usual in textbooks, for simplicity it is often said also in this book that a charge moves instead of a charge carrier moves.

12 2.12 Chapter 2: Circuits as modelling tools Q I for continuous flow at constant rate t In case the flow of charge varies over time, an infinitesimal time interval dt can be considered in which the charge flowing through is infinitesimal as well and can be indicated as dq (remember, chapt.1, that by convention lowercase letters indicate quantities that vary with time). In this case, the current i(t), by definition is: d qt ( ) it () for any flow (constant or variable). dt So, the electric current is defined as the rate at which the charges flow. Since charges are measured in coulomb (symbol: C ) in the SI the electric current will therefore be measured in coulomb per second, Because of its importance, to the electric current has a unit of measure of its own, called ampere (symbol: ): one ampere is one coulomb per second. Definition: the ampere (unit of measure of current) The unit of measure of current is the ampere (symbol: ). One ampere is the a charge flow of one coulomb per second. 3 In formula: 1=(1C)/(1s) s a further clarification of the definition and its sign, it can be said that a current of 1 moving from left to right can be equivalently due to a one positive coulomb charge crossing a given surface in one second from left to right, or to one negative coulomb charge crossing a given surface in one second from right to left Kirchoff s Current Law formulations Consider a generic closed curve drawn in a circuit, in such a way that it does not cross any circuit lumped component, but only wires. general representation of this generic curve is reported in fig. 2.8 a, where just the curve and the traversing wires are reported (the parts of the considered circuit inside and outside the curve are omitted). Since we are considering a circuit, that was already defined as a lumped component representation of a physical system in which no charge can flow outside conductors and circuit lumped elements, only conductive charge can flow from the inner of the curve to the outer and viceversa, through wires. Needless to say, even though, for graphical ease, only five wires traverse the closed curve displayed in the figure, the reasoning we are going to carry is valid for any number of wires. i 5 (t) dq 5 dq 4 i 4 (t) dq 3 i 3 (t) i 5 (t) dq 5 dq 4 i 4 (t) dq 3 i 3 (t) i 1 (t) dq 1 dq 2 i 2 (t) i 1 (t) i 2 (t) a) dq 1 dq 2 b) i t) i ( t) i ( t) i ( t) i ( t) 0 i t) i ( t) i ( t) i ( t) i ( t) 0 1( ( Fig Some nterminal element charge conservation law (arrows may indicate actual or reference charge flow directions). 3 the student should know from the electromagnetics study that this is not the definition of ampere of the S.I., although, obviously non in contrast with it; here this is proposed because it is more effective for the student s understanding at this point of the book.

13 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.13 Imagine that charge (considered to be positive charge, as discussed earlier) flow from the outside of the loop toward the inside according to the arrows reported in figure near the currents. Let dq k, be the infinitesimal charge entering the curve in the infinitesimal time interval dt from wire k. The charge conservation for circuits (sect ) will therefore imply that: dq dq dq dq dq 0 (charge flows directed as per figure a) dq1dq2 dq3dq4 dq5 0 (charge flows directed as per figure b) and, dividing by the corresponding time interval dt: i t) i ( t) i ( t) i ( t) i ( t) 0 (currents directed as per figure a) 1( ( t) i2( t) i3( t) i4( t) i5( t) i 0 (currents directed as per figure b) Note that in some cases the dependence on time t was explicitly shown with the writing i(t), in other cases was implicit. In all cases, however, the equations indicating the charge conservation law, either expressed in terms or charges or currents it valid at any time. We want to express this concept in a general way, without having the need of knowing a priori the actual flow directions. To do that, first the following convention is set: Convention: Current sign. The number indicating a current i is a real number whose module refers to the absolute charge flow itself, and whose positive or negative sign indicates respectively that the flow direction is in agreement or not with an arrow reported near the current name i. The arrow is called reference direction of current i. Using this sign convention, instead of considering the actual charge flow directions, we can use the reference directions. Therefore the current equations of figs 2.8 are still valid considering the arrows not being the actual current directions, but just reference (or, by some authors, assumed) current directions. These equations correspond to other ones, related to differently reference direction, for example to i 1( t) i2( t) i3( t) i4( t) i5( t) 0 (flows using the flow variable sign convention and reference directions all entering the node) This is a very useful result: we can state the law describing the inability of the element to accumulate charge simply as follows: the sum of currents of all terminals of a circuit element, assumed entering, must always be identically zero. nd, equivalently, the sum of currents of all terminals of an element, assumed exiting, must always be identically zero. In the following, whenever this does not cause ambiguity, the word or assumed is omitted and therefore for a branch current indicated with a symbol the expression current assumed entering... and current assumed exiting will be substituted with current entering... and current exiting.... When analysing a circuit, it often occurs that current reference directions are not all entering or exiting a curve. So, alternative formulations must be issued to be with generic assumed directions. They can be as follows (subscript in indicating assumed sign entering the curve, out indicating exiting currents): n _ in n _ out 1) i k, in( t) ik, out( t) 0 k1 n _ in k1 n _ out 2) i k, in( t) ik, out( t) 0 k1 k1 the sum of all currents entering a closed curve in a circuit identically equals the sum of all currents exiting it. if all currents are considered to be entering, the algebraic sum of all currents entering a closed curve in a circuit identically equals zero. Note, that in formulation 2, we introduced the notion of algebraic sum and of current considered to be entering: it obviously means that when the assumed direction enters the closed

14 2.14 Chapter 2: Circuits as modelling tools curve, the corresponding current will be taken with a positive sign, otherwise will be preceded by a negative sign. When charge conservation introduced is applied to a curve of type 1, 2 and 3 in figure 2.6, specialised versions of it are immediately derived, stating that the sum of currents assumed to be entering any node, any circuit element, any subcircuit is always identically equal to zero. In the previous circuit examples, for instance those shown in figures 2.2 and 2.3 circuit elements had two or more terminals, by which to exchange charge with their exterior. It will be seen that, except for nodes, in the large majority of cases the considered circuit elements will have always two terminals. Elements with two terminals are therefore of the biggest importance, and will need a name of their own. They will be therefore called branches 4. When considering circuit branches, the charge conservation law is immediately taken into consideration from the very beginning stage of analysis, simply using a unique value for the currents at their two ends. Consider fig It is clear from it that the usage of a unique current flowing in the branch, e.g. the i, shown in part b) of the figure, is very convenient instead of using different names for the currents at the two sides of element, as indicated in part a), and complement this with the addition of equations stating their algebraic equality. i 1 i 2 i 1 =i 2 i i 1 i 2 i 1 =i 2 B a) b) Fig Charge conservation for branches: a) stated by equations; b) implicitly defined using a single branch current Now we can state Kirchhoff s current law in three forms that are used in practice in circuit analysis, all equivalent. Since in the large majority of cases in everyday work Kirchhoff s law is applied to nodes, it is expressed with reference to nodes, even though, as should be now clear, it is still valid substituting the word node with closed curve : 4 The reader is suggested to refer to sect. 2.2 for definitions of branches and branchbased circuits.

15 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.15 Law: Kirchhoff s Current Law (KCL) In any circuit it happens that form 1: the algebraic sum of all of the currents assumed entering a node in a circuit is identically null; form 2: the algebraic sum of all of the currents assumed exiting a node in a circuit is identically null; form 3: the sum of the currents assumed entering a node in a circuit identically equals the sum of the currents assumed exiting it. KCL applies to any circuit, that is a mathematicalgraphical model of a physical system. ny circuit models the behaviour of a physical system within some degree of approximation. 2.4 The circuit potential and Kirchhoff s voltage law The electric field inside conductors Consider the very simple circuital system reported in figure 2.10, containing just a battery, a conductor and an electric load in the form of a lamp. It is to be recalled that inside the conductor, as everywhere in space, proportionality exists between current density J and the total electrical field E t : Et J (E t =E b E c ) (2.1) where =1/ is the resistivity of the material existing where E t and J are evaluated. This should be already known to readers, and is however rapidly recalled in sect..7.7 of ppendix. The E b is due to the input of external power that the battery comes from chemical potential energy while electric field E c is due to charges distributed in the surfaces of conductor wires, and is therefore conservative and its work along the circuit is null. These surface charges might have the aspect of those shown in figure by plus () and minus () signs. I E c f b b E b E c battery c f a a d I E c Fig simplified version of the circuital system of figure 2.1: here the generator generates a constant voltage Integrating (2.1) around the loop constituting the circuit gives: lb lc ll Et dl J d l b c l I ( Rb Rc Rl) I (2.2) Sb Sc Sl In the previous equation the symbols R b, R c and R l indicated physical characteristics of the battery b, conductor c and of the lamp l, that will be further discussed in the next chapter and that are called resistances. Exploiting the conservativity of E c : (2.2) gives: I E c

16 2.16 Chapter 2: Circuits as modelling tools b b d ( Rb Rc Rl) I a E l The equation shows that the power generated by the battery is dissipated in the circuit elements (including battery inner resistance) by effect of their resistivity. The example given, however, has also the very important purpose of showing that in this circuit an electric field E c is generated all around the loop that is due to surface charges, and therefore has the nature of an electrostatic field, is conservative and therefore has a potential function V. Between any two points of the circuit a difference of potential or voltage U can be defined. For instance between a and b or c and d it is: U ba =V b V a U cd =V c V d voltage expressed as potential difference is often called voltage across terminals. For instance U ba is the voltage across terminals b and a. natural circuit representing the system of figure 2.10 is the one shown in figure b c b Battery a a upper conductor lower conductor Fig. 2.11: circuit representation of the system of figure The ideal wires are needed to permit the four basic elements to be parted from eachother, but obviously the potentials at the ends of ideal wires must be the same, e.g. V a =V a, etc. If the consecutive branch voltages are summed to each other it is: U ba U cb U dc U ad =(V b V a ) (V c V b ) (V d V c ) (V a V d )=0 (2.3) where the equality of zero is due to the fact that all the potential values appear twice, with opposite signs. Eq (2.3) says that the consecutive voltages across terminals (i.e. potential differences) in the loop constituted by the circuit has a sum that it is equal to zero. This is very similar to the well known law of physics stating that the work performed by a conservative field in any closed loop is zero: the considered sum of the voltages is indeed the work, per unit charge, performed by the electric field E c present in the conductor. Eq. (2.3) is interesting but its usefulness would be much greater if it can be expanded to larger families of circuital systems and the related circuits. question that arises immediately is: does this result depend on the fact that the electric source is a battery, i.e. a constant voltage, or is it valid also in case of timevarying sources, such as the sinusoidal voltage source considered in fig. 2.1? Indeed result (2.3) comes from the physical model of the system in figure 2.10 that has made us consider the presence of the electric field E c inside the conductor only created by the charges appearing on the surface of the circuit conductors. It was implicitly assumed that no other fields were induced by possible presence of other electromagnetic phenomena outside the system shown in figure 2.10, for instance time varying magnetic fields produced by other systems not shown, that, by Faraday s law would have induced additional contribution to the conductor inner electric field. d d c lamp

17 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.17 nalysis of this topic requires knowledge that will be developed in chapter 4. Here the result that will be obtained there is just anticipated. The result (2.3) will be valid also in cases in which the voltage source varies with time, given that the selfinduction phenomena have negligible effects, or: the consecutive voltages across terminals (i.e. potential differences) in a oneloop circuital system have a sum that it is equal to zero whenever induction and selfinduction phenomena in the loop have negligible effects. This is congruent with the initial rule that has been stated for creating circuits from circuital systems, i.e. that any action on the space around the wires on what happens in the wires themselves should be negligible: in this chase the induced electric field by the presence of timevarying magnetic field around the circuit, and in the space occupied by the circuit loop. second very important question that arises from the analysis made on the simple circuit reported in figure 2.10 is: Can the result (2.3) be extended also to more complex circuits, that contain more than a single loop, such as the system reported in figure 2.1? Consider again the system of figure 2.1, reported again in fig. 2.12, slightly modified. t a given instant the situation inside the conductors can be imagined as reported in the figure: the field inside the generator E G plays the same role of E b in the previous example, and at any given point of the conductors field E c is present. Note that the value of E c varies with the point of the conductor considered (it is an electric field having a value at any point of space) and both E G and E c vary with time. s in the previous example E c is due to the surface charges present in the conductors including the surfaces of the conductive parts f a and f b that interface G with the wires. E c E G b f b E c L 1 E c c L 2 e E c a f a d f L 3 Fig. 2.12: different view of the circuital system of figure 2.1. E c has therefore the nature of electrostatic field, even though it varies with time, and therefore will admit a potential v(p), function of the point p of the circuit considered, variable with time and therefore reported using a lowercase symbol). Since v the function of the point, the potential differences in any closed loop that might be considered sum up to zero in a way that is totally similar to the process that brought to eq. (2.3). For instance, for the loops L 1, L 2, L 3 that might be considered in the system of figure 2.12, it will be: L 1 : u ba u cb u dc u ad =(v b v a ) (v c v b ) (v d v c ) (v a v d )=0 L 2 : u cd u ec u fe u df =(v c v d ) (v e v c ) (v f v e ) (v d v f )=0 (2.4) L 3 : u ba u eb u fe u af =(v b v a ) (v e v b ) (v f v e ) (v a v f )=0 where the equality of zero is due to the fact that all the potential values appear twice, with opposite signs. The voltages (or potential differences) can be reported in the circuit corresponding to the circuital system, as shown for the generator, the lamps and the lower conductor in fig

18 2.18 Chapter 2: Circuits as modelling tools Whenever a voltage is reported in a circuit the reference polarities must be reported: a plus sign or a couple plusminus signs. They indicate the references towards which to measure the corresponding voltages: the number indicating a voltage is equal to the difference of potentials between the point of circuit marked and the one marked. For instance, in the figure it is u G (t)=v b (t)v a (t). Since the negative sign must always be at the opposite side of the positive one, it can be omitted, as done in the figure for the conductor voltages u uc and u lc. time varying voltage u(t) will typically assume, as time passes, positive and negative values; for what said, the part of circuit marked with will actually have a higher potential the one marked with when u(t) has a positive value. For its importance the voltage sign convention is reported in the following box: Convention: Voltage sign The number indicating a circuit voltage u is equal to the difference of potentials existing at the circuit wire carrying the mark and the one carrying the mark. u G upper conductor b u uc G u L1 =u L2 a u lc lower conductor lamp 1 lamp 2 L 3 Fig The circuit of figure 2.2, with indication of some voltages, along with the corresponding polarities Kirchhoff s Voltage Law formulations It should be now clear that the results obtained in analysing figures 2.10 and 2.12 can be extended to circuital systems and related circuits of any complexity and number of loops. These results are summarised by equations (2.3) and (2.4), telling that in all circuits a function V of terminals exists, called potential, such as the voltages across terminals are expressed as the difference of the corresponding potential values. The very existence of V implies that the sum of consecutive branch voltages around any loop is identically equal to zero. In this section these results are expressed in a more formal and practical way. Before doing this, the concepts of voltage rises and voltage drops must first be introduced. Consider again figure Before writing eqs (2.4) possible loops in the circuit were indicated, and reference directions in which to follow these loops were assumed (indicated by the arrows reported on the loopcharacterising lines). This reference direction is arbitrary. Then, the loop is fully followed, starting from a point of it and returning to the same point, moving according the reference direction assumed for that loop. When the loop is followed, branches are traversed. It may happen that the negatively marked terminal of a traversed branch is met before the positively marked one, or after. In the first case the branch voltage will be considered to be a voltage rise, in the second one a voltage drop. For instance, considering loop L 3, in 2.13, u G and u lc are a voltage rises, while u uc and u L2 are voltage drops. The third of eq. (2.4) could be written, using the symbols reported in fig. 2.13, as follows: L 3 : u G u uc u L2 u lc =0 or, equivalently:

19 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.19 L 3 : u G u lc =u uc u L2 Now we can express the rationale behind loop equations as those in eq. (2.4) in a general and formal way. This constitutes a fundamental law of the circuits, that is called Kirchhoff Voltage Law (KVL). Similarly to KCL, can be expressed in three possible forms, all equivalent: Law: Kirchhoff s Voltage Law (KCL). In any circuit, if any loop is traversed in an arbitrary direction (either clockwise or counterclockwise), it happens that form 1: the algebraic sum of voltages across terminals, all considered as voltage rises, is identically null; form 2: the algebraic sum of all voltages across terminals, all considered as voltage drops, is identically null; form 3: the sum of all voltage rises across terminals identically equals that of all voltage drops across terminals. KVL applies to any circuit, that is a mathematicalgraphical model of a physical system. ny circuit models the behaviour of a physical system within some degree of approximation. In chapter 4 the circuit concept will be expanded, and the possibility of having sectioned circuits, i.e. circuits containing more sections will be introduced. It will be seen there that KVL applies individually to circuit sections, but not to the whole circuit: i.e. each section will have a potential function of its own, and the difference between potential of terminals belonging to different sections is meaningless. 2.5 Solution of a circuit Solving a circuit normally means to find a value for all node potentials and wire currents. For branchbased circuits this means just to find the values of all branch voltages and currents. Some of these equations will be Kirchhoff s equations (KCL and KVL equations), others will give the description of the branches inner behaviour and will be referred to as constitutive equations. In this section information on how to write linearly independent Kirchhoff s equations and constitutive equations will be provided, as well some discussion on whether the equations written are in number sufficient to determine all currents and voltages Determination of linearly independent Kirchhoff s equations (loopcuts method) Let us consider a branchbased circuit having b branches and n nodes. to fix ideas, consider the circuit shown in fig with names for nodes and some possible loops. Note that the displayed loops are not all the possible ones for the circuit (for instance a loop traversing sequentially B, C, F, G, D could also be considered).

20 2.20 Chapter 2: Circuits as modelling tools u i L 1 N 2 C N 3 u B i B B N 1 L 4 i C i D L 2 u C D Fig circuit with nodes and loops evidenced. u D i E E N 4 u E L 3 F i FG u F G u G Consider the KCL for all the circuit nodes (written using form 1): N 1 : i ib id 0 N 2 : i ib ic 0 (2.5) N 3 : i D ie ifg 0 N 4 : i i i 0 D E FG It can immediately verified that these equations are not linearly independent, since it is: N 4 =N 1 N 2 N 3. It is also easy to verify that (using the linear algebra usual techniques to solve linear systems, e.g. computing the determinant of the system 2.5) that only three of the equations in (2.5) are linearly independent. The four KVL equations related to the loops shown in fig can also be written, using again form 1, as follows: L 1 : u ub 0 L 2 : u B uc u E ud 0 (2.6) L 3 : u E uf 0 L 4 : u u u u 0 C F D gain, it is easy to find that these equations are not linearly independent, since it is: L 4 =L 1 L 2 L 3 and it is easily verified that only four of the equations (2.6) are linearly independent. Let us now introduce a technique allowing to write only linearly independent Kirchhoff s equations. The simplest case of circuits without nodes has obviously no KCL equations and a single KVL equation. In the other cases the following simple procedure, that in this book will be called method of loopcuts, can be used: The KCL can be applied to any set of nodes amounting to the total number of nodes n diminished by one The KVL can be determined in the following recursive manner: a an equation is be written considering an arbitrary circuit loop; once it is written, a cross can be written (or imagined to be written) to a branch of that circuit b if more loops are present in the circuit, return to step a, but loops must not contain branches already crossed

21 M. Ceraolo D. Poli: Fundamentals of Electrical Engineering 2.21 c in case of branches in parallel, if a unique voltage name is considered across the branches, a cross must be put on all branches in parallel but one, before starting to write loop equations. It is apparent that the procedure proposed allows determination of linearly independent equations. If for instance the KVL is considered, crossing a branch ensures that the equation later chosen does not contain the voltage across the crossed branch, and therefore are linearly independent with the previous one. Since this reasoning can be repeated recursively, it can be concluded that the set obtained is constituted by linearly independent equations. With similar considerations it can also demonstrated that the number of equations that can this way be obtained is equal to bn1. Rule: Determination of independent Kirchhoff s equations (loopcuts method). 1) independent KCL equations can be obtained applying the law to all the circuit nodes except one (arbitrarily chosen) 2) independent KVL equations can be obtained by crossing, whenever a loop equation is considered, an arbitrary branch of the loop and choosing the next loop equation in such a way that no crossed branch is traversed. s an example, fig shows two possible ways to determine KVL sets, that are linearly independent. fter L 1 is determined, cross C 1 excludes branch from the subsequent loops, and after L 2 is determined crossing C 2 excludes branches B 1 C and D from the subsequent, final loop L 3, that excludes previously crossed or unconnected branches. u C 1 u B B C 2 u C C L 2 E u E L 1 L 3 F u F D u D u u C L 1 C N 3 C 2 u B B E C 1 D u D Fig Graphical procedure to determine linearlyindependent KVL equations. The corresponding sets of linearly independent equations are: L1 : u ub1 0 L1 : u u Set1: L2 : ub uc ue ud 0 Set2: L2 : u u L3 : ue u F 0 L3 : ue u The fact that the two sets define the same space of functions is confirmed by the fact that set 2 can be obtained from set 1 by substituting L 2 with L 1 L 2. L 2 u E L 3 F B C F u F 0 u 0 E u D 0

22 2.22 Chapter 2: Circuits as modelling tools Constitutive equations Clearly, Kirchhoff s equations themselves cannot be sufficient to determine the full behaviour of a circuit, i.e. all the currents and voltages, since information must be given also about how the circuit elements, (i.e., for branchbased circuits, the circuit branches) operate internally. It may be said that by Kirchhoff s laws only the topology of the circuit is defined; no information is given about the inner behaviour of the circuit branches. This behaviour is introduced by means of equations, called constitutive equations; they will be often referred to with their acronym. In the next two chapters constitutive equations for the most important components will be introduced and discussed. Here, it can just be said that a constitutive equation is a relation between the branch current of voltage 5. Examples of constitutive equations are therefore the ones reported in table 2.I. The reader might recognise the equations of typical circuit components they have already met in their previous studies; for constitutive equations in which a current and a voltage appear, for the coefficients to be positive (i.e. R, L, C) branch voltage and current must be associated according in such a way that the assumed direction of current enters the branch in the positively marked terminal. This combination of reference signs will be later referred to as the load sign convention: Convention: load and generator sign convention. References for current and voltage in a branch follow the load sign convention when the assumed current direction enters the branch from the positively marked terminal. If the current exits the branch from the positively marked terminal, the references follow the generator sign convention Table 2.I. The constitutive equation types used in this book (in equations containing both current and voltage, load sign convention is used). ELEMENT EQUTION DESCRIPTION resistive element u b =R b i b branch voltage and current proportional (the sign must be used if the current is assumed entering the branch from the positively marked terminal) voltage is equal to u voltage source u b =u s (subscript stands for source ) regardless of s any other circuit quantity. current is equal to i current source i b =i s (subscript stands for source ) regardless of s any other circuit quantity branch voltage proportional to time derivative of branch current self inductive element u b =L di b /dt (the sign must be used if the current is considered entering the branch from the positively marked terminal) capacitive element i b =C du b /dt branch current proportional to time derivative of branch voltage (the sign must be used if the current is assumed entering the branch from the positively marked terminal) Not all constitutive equations are possible in all positions in a circuit. It is unpractical to treat this issue in a general way. s the more significant examples, however, consider that: since two branches in series share the same current they cannot be characterised by current source constitutive equations since the two currents might differ even slightly and 5 Note that for any circuit it is assumed that the effects on the behaviour of any branch of terminal potentials at its terminals p 1 and p 2 are always and only through the potential difference u=v p1 v p2 ). This is coherent with the assumption, valid for any potential, that single potential values do not have a meaning of their own, only potential differences have measurable meaning.