* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter Three-phase circuits
Survey
Document related concepts
Resistive opto-isolator wikipedia , lookup
Flexible electronics wikipedia , lookup
Valve RF amplifier wikipedia , lookup
Integrated circuit wikipedia , lookup
Josephson voltage standard wikipedia , lookup
Electronic engineering wikipedia , lookup
Index of electronics articles wikipedia , lookup
Power MOSFET wikipedia , lookup
Standing wave ratio wikipedia , lookup
Radio transmitter design wikipedia , lookup
Surge protector wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Opto-isolator wikipedia , lookup
Transcript
Chapter 5 Three-phase circuits Table of Contents 5.1 Introduction .............................................................................................................................. 3 5.2 From single‐phase to three‐phase systems .............................................................................. 3 5.2.1 Modelling three‐phase lines when induced EMFs between wires cannot be neglected ................ 7 5.3 The single‐phase equivalent of the three‐phase circuit ......................................................... 10 5.4 Power in three‐phase systems ................................................................................................ 12 5.5 Historical notes ....................................................................................................................... 14 5.5.1 Tesla’s short biography ............................................................................................................... 14 5.6 References .............................................................................................................................. 14 5.7 Proposed exercises ................................................................................................................. 15 For the teacher In this chapter, coherently with the approach followed in chapter four, the three-phase system is introduced modeling the lines connecting generators and loads as purely resistive branches: i.e. neglecting cross conduction between wires (both conductive and displacement currents) and the effect of EMF induced in meshes containing the lines due to the alternating current circulating in the line conductors. This is due to the fact that in this book the tricky nature of R-L models of transmission lines is evidenced, and the effect of cross-conductor currents and induced EMFs are considered an “advanced” topic, that is dealt with in Appendix B. The teacher might have met books in which the conductor lines are substituted by an R-L couple, and may have experienced some discomfort due to the fact that in three-phase system, even more than in single phase ones, the meaning of those L’s is really questionable. For a practical point of view, limiting the analysis to resistive models of the lines has negligible effects, since in low-voltage lines, probably when only ones this book’s students will be required to analyse quantitatively, the line inductances are nearly all negligible or quasi negligible. M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.3 5.1 Introduction Every reader of this book has surely seen, in their lives several times overhead high-voltage transmission lines, such as the ones depicted in fig. 5.1. It can seen from the pictures that in these lines energy is transferred using three-wire lines: the left-hand pole carries three wires, the right hand one holds four wires, but the highest one has just the purpose of protecting the line from lightning and does not contribute to carrying electric energy. Fig. 5.1. Two three-phase overhead lines (left: medium voltage; right: high voltage). This is a very visible evidence of the fact that electric energy is used and transferred, when power overcome a few kW always in systems called three-phase systems, that are based on three wires for transferring electric power. This chapter shows the genesis of three-phase lines and systems and gives some information on how to make computations with these systems in the particularly important (and easy) cases of balanced three-phase systems. 5.2 From single-phase to three-phase systems Imagine we want to feed three load impedance Zu by means of a generator of sinusoidal EMF., situated relatively far from the load, at distance L (fig. 5.2 a). Its electrical behaviour, if induced EMF between wires can be neglected, can be analysed as represented in fig. 5.2 b). Rl + 3Iu + E Rl L a) Fig. 5.2. Single-phase transmission to three loads. b) Iu Iu Iu Zu Zu Zu 5.4 Chapter 5: Three-phase circuits Consider that the load requires a power S=P+jQ to be delivered at a voltage U (1). The current flowing though the line is I=S/U and the power lost through the line is Pl=2RlI2 Now it will be shown that there is a technique that allows to transfer the same power under the same voltage to the three loads while reducing the line losses, if the quantity of copper for the lines is the same, or reducing the quantity of involved copper at equal line losses. Consider the arrangement shown in fig. 5.3 The three electromotive forces have equal amplitudes, and are out of phase with each other by 2π/2 rad. Such set of voltages is called balanced three-phase set of voltages. It will be shown in the next chapter that these sets of voltages are very easily created in electrical machines. In fig. 5.3 b) a circuit modelling the system of figure 5.3a is reported, valid in case induced EMFs between wires can be neglected. Ea + + Eb + + Ec + + Rl Ia A IAC IAB Z ab Rl Rl Ec Ea Ib B IBC Z bc Ic C Z ca Eb Z ab=Z bc=Z ca L a) b) Fig. 5.3. Three-phase transmission to three loads. Using this arrangement, it can be easily demonstrated that the three loads are fed with the same active and reactive power of the arrangement of fig. 5.3 but with reduced losses in the lines. To do this, consider figure 5.4. The voltages between wires at the source terminals can be determined using the common rule for vector differences (figure 5.4 a); the voltages between the corresponding phases at the load terminals will be similar to them, and are reported with a common origin in fig. 5.4 b). Considering that by hypothesis the three load impedances are equal to each other, the currents absorbed by the three loads will have the same amplitudes and phase displacements with respect to the corresponding voltages. Therefore they will be of the type shown in fig. 5.4 b). Considering KCL at nodes A, B, C, the corresponding line currents are found by difference of the currents flowing in the load impedances, and will be of the type shown in fig. 5.4 b). The amplitudes of the line currents will therefore be equal to the load currents, multiplied by 3 , as shown by the construction reported in fig. 5.4c). U ca=E c-E a U CA Ia ICA U bc Ea Eb ICA IAB U ab U BC 3 / 2 I AB 60° IAB IBC a) Ia / 2 U AB Ec b) c) Fig. 5.4. Phasors relative to the three-phase transmission shown in fig. 5.3. 1 remember that when in alternating current a voltage value is quoted, in case of no other specification, it is always intended to refer to the r.m.s. value of the sinusoidal waveform. M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.5 The conclusion is that the total line losses will be P' jl 3Rl ( 3I u ) 2 9 Rl I u2 Compared with the solution of fig. 5.2: Pjl 2 Rl (3I u ) 2 2 9Rl I u2 The ratio of the two powers is therefore P' jl / Pjl 0.5 that demonstrates that the transmission of electric power shown in fig. 5.3, called three-phase transmission is more effective than the single-phase one (2). Naturally, in case one wants the power loss to be unchanged, he can reduce the total copper used for the three-phase transmission, In this case, instead of having a reduction in power loss one would have had reduction in line cost. Another very important advantage of three-phase systems consists in the fact that, differently from the single-phase transmission, the instantaneous power delivered by three-phase generators, and transferred to three-phase loads is constant. This will be demonstrated in the next section. The three load impedances discussed earlier, were connected between the line wires, and formed a triangle. This was done because equality of the voltage at the load terminals was searched in comparison with the single-phase transmission. Indeed also star connection is possible, and very frequently used. Consider the scheme reported in fig. 5.5. In this case the load is constituted by three starconnected impedances. As usual, the left part of the figure represents a physical system whose mathematical model is the circuit reported in the right part of the figure. Obviously, that circuit, although being evidently a three-phase circuit, can be studied using the usual rules for solving AC circuits. In particular the nodal analysis can be uses, assuming as reference node O. The following equation can be thus written: Ea VN Eb VN Ec VN 0 Rl Z a Rl Z a Rl Z a Or: Ea Eb Ec VN 3 (°) Rl Z a Rl Z a Ec Ea -E a Eb It must now be observed any balanced three-phase set of quantities (voltages, currents) has null sum, as can be immediately inferred from the scheme aside: evidently Ea Eb Ec Ea Eb Ec 0 As a consequence eq. (°) yields: 2 The analysis performed considered only the amount of copper, at equal line-to-line voltages. A more detailed analysis should have also considered the cost of insulating to each other three wires instead of two. However, it can be seen that even with this more detailed analysis the advantages of three-phase transmission remains sound. 5.6 Chapter 5: Three-phase circuits VN VN Rl Z a 0 03 This shows that in the circuit of fig. nodes O and N share the same potential. Points O and N are called star points or neutral points of the three-phase system. Ea + + Za Rl Ec + O Ea Eb + Rl Zb Ec + Rl Zc N Eb + Z a=Z b=Z c L a) b) Fig. 5.5. Three-phase transmission to a star-connected load. It is easy to convince ourselves that this is a rather general conclusion: Result: potential of neutral points in three-phase balanced systems in three-phase a system fed by a balanced set of voltages, if all the impedances present in the three phases are equal to each other, all the neutral points of the star-connected impedances and voltage sources share the same potential Therefore entire three-phase networks can be built, where several three-phase loads might be present, e.g. constituted by impedances star or delta connected. In case of balanced networks, in which the impedances present in the three phases are equal to each other and the sources are all constituted by balances three-phase sets, all the star centers share the same potential. In the example of figure 5.6, for instance, it is VN1=VN2=VN3=VO. When impedances are delta connected, they can be replaced by three, equivalent, star-connected equivalents. The star-delta conversion formulas are the same used in chapter 4 for generic AC circuits, that, in the case considered here, of impedances equal to each other become simply: Zstar=3*Zdelta. O Ea + Rl Eb + Rl Ec + Rl I2a + R2 Z4 R2 Z4 R2 U 2a Z1 Z1 Z1 N1 E Z1 Z2 Z2 Z4 R2 Z2 Z 3D a) Z 3D N2 Rl + Z4 Z 3D Z2 Z 3s N4 Z 3S Z 3D 3 Fig. 5.6. A three-phase system with several components and its unifilar representation. b) M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.7 It is apparent that when large balanced three-phase networks are to be represented it is superfluous to always replicate circuit parameters for the three phase. Therefore very often, in these cases, the so-called unifilar representation is reported, such as that shown in fig 5.6 b), where indication that a single line indicates indeed three elements is given by use of the three inclined small bars . Three-phase systems are characterised by two kinds of voltages: line-to-line and line-toneutral voltages. Line-to-lint voltages are the voltages that can be measured between two of the three wires, while line-to-neutral are the voltages between a common point, called neutral point. Both are expressed as r.m.s. values. There is always a very simple and important relation between line and phase voltages of a three-phase system: U line 3 U phase Ec Ea 60° Ea Eb U ab Eb 3 / 2 Ea This can be easily seen looking at the phasors of these voltages as in the drawing aside. For instance it is U ab Ea Eb ; U U 3 E 3 U line ab a phase The duality of line and phase voltages is so important that very commonly three-phase systems are named after two numbers, representing the two voltages. For instance in the European Union countries the voltage used for distributing energy at users needing slightly more power than homes is the following: 400/230 V The first number represents the line-to-line nominal voltage of this system, while the second one represents the phase voltage. Note that the ratio between the two is only approximately 3 : this was chosen to avoid decimal numbers in the representation of any of the two nominal values. Then a three-phase system is named using just a single value, it always refers to the r.m.s. value of the line-to-line voltages: for instances some standard voltage levels are 10 kV, 15 kV, 20 kV, 130 kV, 220 kV. All these values are line RMS voltages. 5.2.1 Modelling three-phase lines when induced EMFs between wires cannot be neglected Consider again the system reported in fig. 5.5a. Let now discuss what happens when the EMFs between wires are not neglected. To understand the situation a three-phase wire of the line must be considered such as that reported in figure 5.7a. There are fluxes that are linked with the three different meshes constituted by the different couples of conductors. 5.8 Chapter 5: Three-phase circuits ab ia a ib bc + eab ca eca b uba + ab + bc ca - eba + ic - c a) b) Fig. 5.7. A three-dimensional view of the conductors constituting a transmission line (left) and a cross-section representation to better show the mesh-coupling fluxes (right). The following equations can be written, in general for time-varying functions (left) and for phasors (right): dab eab uab Ra ia Rbib dt dbc ebc ubc Rbib Rcic dt dca eca uca Rcic Raia dt d ab Eab U ab Ra Ia Rb Ib dt d bc Ebc U bc Rb Ib Rc Ic dt d ca Eca U ca Rc Ic Ra Ia dt By detailed analysis of these equations it can be demonstrated that they can transformed in a more useful form, that, for the phasor formulation, is: d Ia Eab U ab Ra Ia Rb Ib Laeq dt d Ib Ebc U bc Rb Ib Rc Ic Lbeq dt d Ic Eca U ca Rc Ic Ra Ia Lceq dt The parameters Laeq, Lbeq, Lceq, can be shown to be an easy function of the line geometry, although details are not reported here. The system can therefore be studied by the metacircuit3 reported in fig. 5.8. O Ea + Rl Laeq Za Eb + Rl Lbeq Zb Ec + Rl Lceq N Zc Fig. 5.8. Metacircuit representing the system of fig. 5.5a. when the EMFs between wires are not neglected. 3 For the concept of metacircuit the reader is suggested to consult sect 4.3.2. M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.9 It has been seen in the previous section that three-phase systems are particularly effective when voltages and currents are constituted by balanced sets: this was shown to happen when all the supply voltages are constituted by a balanced set and all the impedances on different phases are equal to each other. It is natural to require, for perfect balancing of the system, that also transmission lines must have symmetry: all the conductors must have the same resistance and their spatial disposition is such that when the conductors are traversed by a balanced set of currents, ab, bc, ca constitute a balanced set of fluxes as well. This happens when the line has physical symmetry, i.e., considering the disposition reported in figure 5.7b, when the conductors are positioned at the vertexes of an equilateral triangle. When a transmission line, fed by a balanced set of voltages, feeds a balanced load (i.e. a load constituted by three equal impedances or a more complex balanced load), and the line is physically symmetrical, the three fluxes ab, bc, ca constitute a balanced set of fluxes. It must be added that in normal engineering practice it is possible to assume very often, with reasonable degree of approximation, that the three-phase system of currents flowing through a line is balanced even though, for practical reasons, the conductors are not disposed at the vertexes of an equilateral triangle. Under these conditions, the values of the inductances Laeq, Lbeq, Lceq are equal to each other and their value is: D Leq k ln (5.1) kr r where D is the distance between two conductors of the line, r is its radius, and k and kr are constant values. More details on the value of Leq are not supplied here, because they are well beyond the scope of this book, and because even electrical engineers tend to read them on manuals instead of computing them directly. Eq 5.1 is reported just to show that Leq increases when D increases, and decreases when r increases; the dependence is a lograritmic one. It the equality between all the quantities oeprating in the different conductors of the circuit is to be evidenced, the drawing can be transformed obtainig that reported in fig, 5.9. O Ea + Rl Leq Ia Z Eb + Rl Lbeq Ib Z Ec + Rl Lceq Ic Z Fig. 5.9. The metacircuit reported in fig. 5.8, when the system is fully balanced. As a final consideration of this section, it is worth while noticing that in low voltage lines (up to 1 kV), inductances Leq are very small and designers neclect them very often. In other terms, at the usual frequency, in low voltage lines it can often be assumed that Leq Rl . Only in case of very large LV cables, having cross section of copper larger than 95 square millimeters, consideration of Leq has some improtance. On the other hand, in medium voltage lines (say between 1 and 35 kV)4 this line inductance becomes important, and the corresponding reactance Xeq=Leq typically reaches the same order of magnitue of line resistances Rl. When much higher voltage lines are considered reactances Leq tend to become dominant with respect to line resistances. 4 The threshold of 35 kV has set here because it is considered in the International Standard IEC 60038. Individual countries might have slightly different thresholds: for instance in Italy this value is set to 30 kV. 5.10 Chapter 5: Three-phase circuits 5.3 The single-phase equivalent of the three-phase circuit For what seen in the previous chapter, in a balanced three-phase system, that occurs in phase a occurs in phase b also, only with a phase displacement of 120° and then occurs in phase c, with another phase displacement of 120 degrees. So it is sufficient, to understand the full behaviour of the three-phase system, to evaluate what happens on one of the three phase: all voltages and current phasors in the other phases will have the same amplitudes and will be just rotated by 120 or 240 degrees. By convention, the phase considered is the first of the three: phase a, if phases are named a, b, c, phase 1 if they are named 1, 2, 3, phase r if the naming convention considers as phase names r, s, t. The circuit constituted by only the first phase of a three-phase system is called the singlephase equivalent of the three-phase circuit. Just to make this approach more explicit, consider again the circuit shown in fig. 5.6. Its behaviour can be easily evaluated considering the single-phase equivalent shown in fig. 5.10. Rl R2 I2a + + Z1 Ea Z2 U 2a Z 3s O N1 N2 N4 Z4 O Fig. 5.10. Single-phase equivalent of the circuit of fig. 5.6. The process for determining the single-phase equivalent of a three-phase circuit should be now clear: since the potential of all neutral points are the same, they are connected to each other, creating the lower common node of the circuit. The upper nodes of the single equivalent represent the corresponding nodes in phase a of the three-phase circuit. To make things clearer, in both figures voltage U2a and current I2a are shown. The same process can be used when EMFs between wires of transmission lines be neglected. For instance, the metacircuit reported in figure 5.9 can be studied by its single-phase equivalent, reported in figure 5.11. The currents and voltages flowing in this single-phase equivalent are the ones flowing in conductor a of the actual three-phase system. Rl Leq I + + E Z U - Fig. 5.11. Single-phase equivalent odf the circuit of fig. 5.9. Example 1 Determine I1, I1, I2, I3 and UAB in the following circuit, where: e1(t)=2380sin(314t+62°) (balanced system of voltages) e2(t)=2216sin(314t+30°) (balanced system of voltages) Z1=1+2j M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.11 Z2=1.5+2.5j Z3=120+40j Z1 I2 A e1(t) + I1 e2(t) Z2 + I1 Z1 Z2 B + + + Z2 Z1 + I3 Z3 Z3 Z3 Let’s first determine the phasors of sources: E1=38062° E=21630° Now let’s transform all delta connections to equivalent star connections: I1 I2 A Z1 E2 Z2 + E1 + Z1 Z2 B + + Z2 Z1 + + I3 Z3 Z3 Z3 where: E1=E1/3e-j30°= 219.432° = 186.1+j116.7 Z3=Z3/3=40+j13.33 Since all the neutral points have the same potential (balanced generators and loads), the following single-phase equivalent circuit can be used: Z1 A I2 E2 Z2 + I1 E1 + I3 Z3 To solve this circuit, let us apply Thévenin’s theorem left side the red section: 5.12 Chapter 5: Three-phase circuits I2 Zth E2 Z2 + Eth + + U - where: E Z Eth= 1 3 =183.1+j105.4 Z1 Z 3 Zth=Z1||Z3=1.032+j1.890. Hence: E E th I2= 2 =0.824-j0.417 Z th Z 2 U=E2-I2Z2=184.8+j106.6=213.329.97° I3=U/Z3=4.957+j1.012 I1=(E-U)/Z1=4.133+j1.429=4.37319.08° To calculate the phase-to-phase voltage UAB and the phase current I1, U and I1 must be modified in amplitude and angle, according to what discussed in sect. 5.2. UAB=U3*ej30°=184.9+j319.9=369.559.97° I1=I1/3*ej30°=1.654+j1.908=2.52549.08° 5.4 Power in three-phase systems Consider the three-phase circuit reported in fig. 5.5 and fig. 5.12. How the three loads are generic linear boxes, and the electric quantities are indicated as time functions. Since the electromotive forces are sinusoidal also the currents are sinusoidal; because the three loads are equal (they share the same constitutive equation u=f(i)) and the three electromotive forces are constituted by a three-phase balanced system of voltages, also the currents constitute a threephase balanced system. ea + Rl ia + va O eb + Rl ib + vb ec + Rl ic + vc A + ua N B + ub - C + uc p(t) ea (t ) Eˆ sin t eb (t ) Eˆ sin t 2 / 3 ˆ ec (t ) E sin t 2 / 3 ia (t ) I sin t ia (t ) I sin t 2 / 3 ia (t ) I sin t 2 / 3 ua=f(ia) ub=f(ib) uc=f(ic) p(t)=vaia+vbib+vcic Fig. 5.12. Circuit showing computation of instantaneous three-phase power. In a way similar to what done in chapters 3 and 4, we can evaluate the power p(t) transferred to charge crossing the curved line of fig. 5.12. The charges flowing through the three phases are moved by the electric field present inside the conductor, whose potential values, written with reference to node O are v1(t), v2(t), v3(t). Therefore the power transferred to the load is the sum of the powers transferred though each wire and it is: p (t ) p a (t ) pb (t ) pc (t ) va ia vb ib vc ic u a ia ub ib u c ic (5.2) M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.13 the latter equality being justified by the fact that potentials of node N and O are the same. Consider the expression for the instantaneous power entering a single linear circuit branch: p UI cos (1 cos 2 t ) UI sin sin 2 t UI cos f (2 t ) where U and I RMS values of are branch voltage and current respectively and f ( x ) UI cos cos x UI sin sin x . f(x) has the following property: f ( x) f ( x 4 / 3 ) f ( x 4 / 3 ) UI cos cos x cos( x 4 / 3 ) cos( x 4 / 3 ) UI sin sin x sin( x 4 / 3 ) sin( x 4 / 3 ) UI cos cos x cos( x 2 / 3 ) cos( x 2 / 3 ) UI sin sin x sin( x 2 / 3 ) sin( x 2 / 3 ) 0 because the sum of three sinusoidal functions displaced from each other 2/3 is always zero. Therefore it is: p (t ) 3UI cos f (2 t ) f 2( t 2 / 3 ) f 2( t 2 / 3 ) 3UI cos f (2 t ) f 2 t 4 / 3 ) f 2 t 4 / 3 ) (5.3) 3UI cos Eq. (5.3) is a very important result: Result: Instantaneous power in three-phase balanced systems The instantaneous flowing in three-phase balances systems is constant This is different from what happens in single-phase circuits, and is therefore another big advantage of three-phase circuits over single-phase ones. It will seen in the next chapters that this implies very uniform transfer of power to mechanical shafts when three-phase machines are involved, since when the machine shafts rotate at constant speed, whey receive constant torque, therefore with no vibration induced. As far as active, reactive and apparent powers are concerned, their values can be obtained trivially considering that a three-phase phasor circuit is still an AC circuit: Ptrif 3UI cos Qtrif 3UI sin S trif U a I a* U c Ib* U c Ic* S trif 3UI (5.4) It is remembered again that in (5.4) U is the RMS value of the line-to-neutral voltages. The power transferred though a three-phase circuit are often expressed in terms of the line to line voltage Ul,. From the fundamental relation U l 3U immediately: Ptrif 3U l I cos Qtrif 3U l I sin Strif 3U l I Example 2 Consider the circuit of Example 1 and determine the active and reactive power: - delivered by the two three-phase generators - absorbed by the two lines - absorbed by the load Assuming the frequency to be 50 Hz, determine the capacity of the capacitors (before star connected and then delta connected) required to correct the load power factor to 1. Remembering that the three-phase powers can be obtained multiplying by 3 the powers calculated in the single-phase circuit, we obtain: 5.14 Chapter 5: Three-phase circuits Sgen1=3E1I1*=2805 W+j643.6 var Sgen2=3E2I2*=327.2 W+j501.3 var Sline1=3I12Z1=57.37 W+j114.7 var Sline2=3I22Z2=3.840 W+j6.400 var Sload=3I32Z3=3071 W+j1024 var Please note that the Boucherot’s theorem is respected: Sgen1+Sgen2=Sline1+Sline2+Sload The power factor correction can be carried out on the single-phase circuit, directly obtaining the capacity of each star-connected capacitor: L3 13.33 / (2 50) =23.88 F. 2 R X 3 402 13.332 Qsin gle phase Z3 1024 / 3 Also: C= =23.88 F 2 2 50 213.32 U The capacity of the three delta-connected capacitors can be easily calculated considering that in a balanced load: Z=3Zthus C=C/3=7.962 F. C= 2 3 5.5 Historical notes 5.5.1 Tesla’s short biography Nicolas Tesla (Croatia, 1865 - New York, USA 1943) is a scientist and engineer, that gave an outstanding contribution to the development and diffusion of electrification in the U.S. in the early years of the XX century. Born in Austria-Hungary, he moved in the United states in 1884. There, he participated to the so-called “war of the currents” between those, lead by Edison, that were promoting DC current for commercial electricity distribution and those promoting AC, lead by Westinghouse. One of the big advantages of AC in constituted by the existence of polyphase systems, especially three-phase systems, developed by Tesla, and whose patents were bought from Westinghouse. Tesla first worked under Edison then, because he felt himself underestimated, moved to Westinghouse’s, and then could deploy all his ideas and knowledge on AC and polyphase AC systems. Tesla is also credited to have discovered the rotating magnetic field, that will introduced and discussed in the next chapter, independently of Galileo Ferraris, that made the same discovery in the same period. The S.I. unit of magnetic flux density is the tesla, a name given in honour of him. 5.6 References M. Ceraolo - D. Poli: Fundamentals of Electrical Engineering 5.15 5.7 Proposed exercises 5.1. Determine Ia, Ib and Ic in the following circuit, where: Ea=10020° (balanced system of voltages) Zl=2+3j Z=30+10j Ea + Eb + Zl Ia Ib Zl Z=1+3j Zl=18+12j Z2=5+j + Zl Ic Z Eb Z Ec Z I2 A Z2 B + Z Z2 I1 Z Z1 5.2. Determine Ia, Ib, Ic and Iab in the following circuit, where: Eab=38030° (balanced system of voltages) Zl=5+10j Z=18+6j Zl + Eab Iab Ib Z Z Z1 5.5. Determine I1, I2, I3 and UAB in the following circuit, where: Ea=2200° (balanced system of voltages) Eb=22030° (balanced system of voltages) Zl=1+2j Z2=2+3j Z3=15+9j I1 E1 + Z1 Z1 I2 A + E2 Z2 + Eca Zl Ia Z2 Ia Z + Ec Ea + + Zl Ic Z Z1 Zl + Zl Eca Ib Iab + Z + Ebc Zl Ic I3 Z3 Z3 Z3 Ia Z Eab Z2 Z1 + + 5.3. Determine Ia, Ib, Ic and Iab in the following circuit, where: Eab=2200° (balanced system of voltages) Zl=2+4j Z=12+3j Z2 B + + Ebc Z 5.4. Determine Ia, I1, I2 and UAB in the following circuit, where: Ea=22015° (balanced system of voltages) 5.6. Using the method of nodal voltage, determine Ia, Ib and Ic in the following unbalanced system, where: Ea=2200°, Eb=210-120°, Ec=215115° Z=1+3j, Za=20, Zb=5+25j, Zc=30-8j Then calculate the current INO flowing in a resistor R=10 , positioned between O and N (use Thévenin’s theorem). 4.16 Chapter 4: Techniques for solving AC circuits Ea Z Eb Z Ia Za + O Ib Zb + Ec Z Ic N Zc + 5.7. Using the method of nodal voltage, determine Ia, Ib, Ic and I’ in the following unbalanced system, where: Ea=12730°, Eb=120-90°, Ec=130145° Z=1+j, Za=10+5j, Zb=8j, Zc=12j, Z’=15 Ea Z Ia Za + Eb Z Ec Z Ib Zb + Ic Zc + Z’ I’ Note: To solve the following problems, assume that the voltage phasors assigned in the previous exercises were expressed considering the voltage RMS values. 5.8. Consider exercise 5.1. Calculate the active and reactive power absorbed by the three-phase load and by the line; determine the active and reactive power delivered by the generating system and verify the Boucherot’s theorem. 5.9. Consider exercise 5.2. Calculate the active and reactive power absorbed by the three-phase load and by the line; determine the active and reactive power delivered by the generating system and verify the Boucherot’s theorem. 5.10. Consider exercise 5.3. Calculate the active and reactive power absorbed by the three-phase load and by the line; determine the active and reactive power delivered by the generating system and verify Boucherot’s theorem. 5.11. Calculate the active and reactive power delivered by the two three-phase generators of exercise 5.5 and the active power losses of the two lines. 5.12. Verify Boucherot’s theorem in exercise 5.7. 5.13. Calculate the power factor of the load of exercise 5.4 (Z1 and Z2). Determine the capacity of the three star-connected capacitors required to correct the power factor to 1. Repeat if the capacitors are delta connected.