Calculation of voltage and current in electric network (circuit)

Transcription

1 UNIVERSITY OF LJUBLJANA Calculation of voltage and current in electric network (circuit) Power distribution and Industrial Systems Alba Romero Montero 13/04/2018 Professor: Grega Bizjak

2 Content Background Electric field strength...3 Electric potential and potential difference...4 The electric current...5 Amperage....6 Electric network...9 Constituent elements...9 Passive components:...9 Active components:...15 Active, reactive and apparent power...16 METHODS...17 Ohm s Law...17 Kirchhoff laws...19 The first Kirchhoff Law:...20 The second Kirchhoff Law:...20 Application of Kirchhoff s laws:...21 CONCLUSION...27 QUESTIONS...28 HOMEWORK...29 References

3 Background The existing matter is constituted by groups of atoms. At the same time, atoms are constituted by elementary particles (that cannot be subdivided), such as electrons, neutrons and protons. Its distribution within the atom is detailed in Figure 1. Figure 1. Atom elements In the Figure 1 appears the electric charge. The electric charge is the property of a particle by which it may repel (or attract) other particles that have a charge of similar (or opposite) sign. By agreement, the sign of the charge is: Negative for electrons. Positive for protons. Neutrons do not have electric charge. Between the protons and electrons appear forces of attraction and repulsion, which are different from gravitational forces. They do not manifest themselves with neutrons, since they don t have electric charge. This force is known as electromagnetic force. The electromagnetic force is the force that arises between particles with electric charge; the second strongest of the four fundamental forces. The sense of the forces exerted between the electric charges is of attraction between the charges of opposite sign, and of repulsion between charges of the same sign, as shown in Figure 2: 2

4 Figure 2. Electric Forces in the atom. A well-known scientist (Charles Augustin de Coulomb) quantified electric attraction and repulsion quantitatively and deduced the law that governs (Coulomb law). As this law said: two point electric charges, q 1 and q 2, exert on each other a force that is proportional to the product of said charges, and inversely proportional to the square of the distance r that separates them. Considering the loads expressed in coulombs (C) and the distance in meters (m), the mathematical expression of this law will provide force in newton (N): The value of the constant K depends on the medium in which the charges are located. As Coulomb was concerned, the force exerted between two electric charges has a vectorial character, this means that it not only has a numerical value but that it is also characterized by its direction and sense (for example, when a magnet is placed in front of another, is quickly appreciated if they have the same polarity by the direction and sense in which they move). The region of the space where one or more electric charges are present is called electric field. Its points are subjected to a force of electrostatic origin caused by the presence of one or more electric charges. Each of these points are defined by: Field strength (vector magnitude). The potential (scalar magnitude). Electric field strength Definition: The electric field strength, at a given point, is the force that is exerted on a load of unit value. 3

5 Considering the force expressed in newton (N), and the charge in coulombs (C), the field strength will be expressed in newton/coulombs (N/C), or volts per meter (V/m). When several point charges are those that create a field on a point P, the total field strength on it, is the vector sum of all of them. This phenomenon is shown in the Figure 3. Figure 3. Electric field strength of several charges. Electric potential and potential difference Definition: The potential at a point (M) of an electric field is the potential energy that would have an electric charge of unit value at that point. In other words, the electric potential is the work that would have to be done, per unit of charge, to transfer an electric charge (q 2 ) from point M to the origin of the electric field (q 1 ), which is at a distance r from said point. Figure 4 shows how the electrical potential influences electrical charges. Figure 4. Electric potential. On the other hand, the potential difference between two points (M and N) of an electric field is the work that must be done, by electric charge unit, to transfer the charge (q 2 ) from one point to another. 4

6 Figure 5. Potential difference. Figure 5 shows graphically the potential difference between two points, does not depend on the distance between the two of them, but depends exclusively on the distances at which two points are located respect to the origin of the electric field. If the distance between point N and q 1 is the same as from L to q 1 (r N ), the potential difference of U MN and U ML is the same. Mathematically is represented by the following equation: The potential difference is known in the practice as electric tension or voltage (V). The electric current In the previous sections a series of concepts related to static electric charges has been represented. Now it will be proceed to observe the movement of these charges in a medium under the presence of an electric field. That movement constitutes the electric current. The electric current is the movement of electric charges through metallic conductors, semiconductors, electrolytes or gases and always under the action of an electric field. Figure 6. Potential sense of electric current. 5

7 Formerly it was believed that the electric current in the conductors constituted the movement of positive particles in the sense of the field (sense of decreasing potential). It is really the electrons (negative particles) that move in the opposite direction to the field (growing potential sense). However, the conventional sense of the current is that of positive particles, as explained in the Figure 6. Amperage. The intensity of current allows evaluating the amount of electric charge that circulates through a conductor to which a potential difference between its ends has been applied. In a more formal way: The electric current intensity is conscious between the load that a driver goes through and the time spent in it. Therefore, the mathematical expression of the current intensity results: Considering that the load is expressed in coulombs (C) and the time in seconds (s), the current intensity results in amperes (A). The name of the unit of intensity is due to the French physicist André Marie Ampère ( ) a mathematician and physicist specialized in the mathematical formulation of the electromagnetism, and father of the electrodynamics. When the behaviour of the load is linear with time, the intensity of current is constant for any time, the relationship between is shown in the Figure 8. Figure 8. The relationship between charge (q) and time (t). If the current intensity is not constant, its value will take different values for each time interval. The average value of an interval is: Whereas if what matters is to know the value at a certain moment of time: 6

8 The situation of a variable charge is shown in the Figure 9. Alba Romero Montero Figure 9. The variability of the charge versus time. Electricity is a type of energy transmitted by the movement of electrons through a conductive material. For example, metals are materials with high electrical conductance and allow the movement of electrons easily. Within the conductive material the electrons can move in one or two directions, depending on which two types of current can be distinguished, the direct and the alternating current. Direct current (DC): When the flow of electric current is given in only one direction. The intensity of the current can vary with time, but the general direction of movement stays the same at all times. In a DC circuit, electrons emerge from the negative, or minus, pole and move towards the positive, or plus, pole. Alternate Current (AC): Alternating current describes the flow of charge that changes direction periodically. As a result, the voltage level also reverses along with the current. The value of voltages and currents in the load varies with the pulsations as a function of time. Normally the wave acquires a sinusoidal shape, and a complete cycle takes a time of 2 radians. To measure the average and effective value of the characteristic curves of the intensity and voltage, the complete cycle of the wave must be taken. To calculate the working cycle time of a given average voltage value, we integrate the expression of the voltage with respect to time, For the intensity the equation has the same expression: The effective value of a voltage or current is also known as the mean quadratic value or rms (root mean square). The effective value of a periodic voltage wave is based on the average power delivered to a resistor. For a continuous voltage applied to a resistance, 7

9 For a periodic voltage applied to a resistor, the effective voltage is defined as a voltage that provides the same average power as the DC voltage. The effective tension can be calculated using the following equation: If we calculate the average power of a resistance from the expression of power: Or In the same way, the current develops from In the Figure 10 is shown the differences between DC and AC current. Figure 10. Current waves of DC and AC current. 8

10 Electric network An electrical circuit is a set of conductive elements connected in a way that they constitute a closed path through which an electric current circulates, or can circulate. Constituent elements Generator: Part of the circuit where electricity is produced, maintaining a voltage difference between its ends. Its symbol is shown in the Figure 10. Lead: Thread where the electrons driven by the generator circulate. Its symbol is shown in the Figure 10. Switch: Element that allows to open or close the passage of electric current. If the switch is open, the electrons do not circulate, and if it is closed, it allows its passage. Linear lumped elements: Elements that oppose the passage of electric current. They are resistors, capacitors, inductors or diodes. Its symbol is shown in the Figure 10. Figure 10. Electronic components symbols. Passive components: Passive elements are those components of the circuit, which dissipate or store electrical or magnetic energy and therefore constitute the receivers or loads of the circuit. They may have different properties, and depending on their placement in the circuit they will be associated in different ways. They are: Resistance: They dissipate electrical energy. Resistors can be connected in two different ways: o Association in series: The associated elements are placed one after the other. The electric current has only one way to go, having the same intensity in the whole circuit. However, the voltage supply is divided between the resistors. A simple example is the shown in the Figure 11: 9

11 Figure 11. Resistors connected in series. The equivalent resistance is the sum of the individual resistors in the electrical network. o Association in parallel: Derivations are created in the circuit. The electric current leaving the generator has different ways to go. However, the voltage applied to each branch is the same when having the input and output nodes in common with the voltage source. In the Figure 12, is shown a little example to find the equivalent resistance with this type of connection. Figure 12. Resistors connected in parallel. In the case of having four resistors in parallel, the equivalent resistance is calculated as: The unit of resistance in the International System is the ohm, which is represented by the Greek letter omega (Ω), in honour of the German physicist Georg Simon Ohm, who discovered the principle. Inductance: Inductance is typified by the behaviour of a coil of wire in resisting any change of electric current through the coil. Inductors can be connected in series and in parallel; these interconnections of inductors produce more complex networks whose overall inductance is a combination of the individual inductors. However, there are certain rules for connecting inductors in series or parallel and these are based on the fact that no mutual inductance or magnetic coupling exists between the individual inductors. 10

12 o Association in series: Inductors can be connected together in a series connection when they are daisy chained together sharing a common electrical current. Inductors in series are simply added together because the number of coil turns is effectively increased, with the total circuit inductance. Figure 13. Association in series of the inductance Seeing the example of the Figure 13 and applying the definition over the picture, is known that the total inductance, L T is: Association in parallel: Inductors are said to be connected together in Parallel when both of their terminals are respectively connected to each terminal of the other inductor or inductors. Here, like the calculations for parallel resistors, the reciprocal ( 1/Ln ) value of the individual inductances are all added together instead of the inductances themselves. But again as with series connected inductances, the above equation only holds true when there is NO mutual inductance or magnetic coupling between two or more of the inductors, (they are magnetically isolated from each other). Where there is coupling between coils, the total inductance is also affected by the amount of coupling. Seeing the example in the Figure 14, is easier to understand this association: Figure 14. Inductances associated in parallel. 11

13 The unit of magnetic inductance is the henry, named in honour of the 19th-century American physicist Joseph Henry, who first recognized the phenomenon of self-induction. o Difference between DC current and AC current. DC current: The behaviour of the inductance in DC networks is like a short-circuit. AC current: Resistors, capacitors, and inductors, called reactive components, oppose the current in AC circuits. The amount of opposition offered by an inductor to the alternating current is called the inductive reactance XL. The inductive reactance of a coil is not constant, but varies with the inductance L and the frequency f of the alternating current. XL can be calculated with the formula: Where f is the frequency of the network measured in Hz, L is the inductance of the component measured in H and XL is the inductive reactance measured in ohms (Ω). From the previous equation, some important facts are deduced: The first is that the inductive reactance varies directly with the frequency, that is, it is directly proportional to f. The second that can be deduced from the formula is that for direct current, that is when f = 0, XL = 0. This fact coincides with the definition of the inductance, since the characteristic of a coil is to oppose a current variation. Capacitance: property of an electric conductor, or set of conductors, that is measured by the amount of separated electric charge that can be stored on it per unit change in electrical potential. Capacitance also implies an associated storage of electrical energy. The capacitances when they are associated get the opposite behaviour than the resistors and the inductances. o Association in series: When capacitors are connected in series, the total capacitance is less than any one of the series capacitors individual capacitances. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacing of the individual capacitors. Following the Figure 15, the equivalent capacitance is obtained as: 12

14 Figure 15. Capacitances associated in series. o Association in parallel: When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors capacitances. If two or more capacitors are connected in parallel, the overall effect is that of a single equivalent capacitor having the sum total of the plate areas of the individual capacitors. Figure 16. Capacitances associated in parallel. In the Figure 16 is shown an example of capacitors connected in parallel. The equivalent capacitance is: The Faraday (F) is the SI derived unit of electrical capacitance, the ability of a body to store an electrical charge. It is named in honour of the English physicist Michael Faraday. o Difference between DC current and AC current. DC current: The behaviour of the capacitance in DC networks is like an open-circuit. AC current: The capacitor is opposed to alternating current. This opposition is not a resistance, since, in fact, the alternating current does not circulate through the capacitor. This reaction of the capacitor is called capacitive reactance or Xc, it is calculated by: 13

15 Where f is the frequency measured in Hz, C is the capacitance of the capacitor measured in F, and Xc is the capacitive reactance measured in ohms (Ω). The capacitor behaves in the following way: when a capacitor is charged, one of its armatures produces an excess of electrons. As equal charges repel, the more electrons there are in the armature, the more opposition there will be to the electrons that are arriving. This is not a resistance, but a reaction produced by the electrons that have already been accumulated in the armour. As the alternating current reverses the direction of its flow, the armour is periodically loaded and unloaded. Impedance: is the apparent resistance of a circuit equipped with capacity and selfinductance to the flow of an alternating electric current, equivalent to the effective resistance when the current is continuous. The expression is: Where, R is the resistance of the circuit, XL is the reactive inductance of the circuit and XC is the reactive capacitance of the circuit, all the measurements in ohms (Ω). The result of this operation is Z, the impedance measured in ohms (Ω). Unlike DC resistance, the impedance is represented by a complex number: o o R is the real part of the impedance and corresponds to the resistive value of the element. X is the imaginary part and corresponds to the total reactance, which is calculated as the difference of the inductive and capacitive reactance. Impedance triangle The impedance triangle of an element or a circuit is formed by representing the real part of the impedance (corresponding to the resistance) and the imaginary part (corresponding to the difference between the inductive and capacitive reactance) in the legs of a triangle. The hypotenuse is calculated in the same way as the module of a complex number, that is, by the Pythagorean Theorem. In circuits where the total reactance is negative, the triangle has the following form (Figure 17): 14

16 Figure 17. Impedance with negative reactance. In circuits where the total reactance is positive, the triangle is represented as follows (Figure 18): Figure 18. Impedance with positive reactance. The angle observed in the impedance triangle also corresponds to the phase shift angle between the voltage and current and can be calculated by trigonometry if the value of the impedance is known. This angle, also called power factor, is used to know the level of use of electrical energy in a power network. Active components: The main sources of an electric network are voltage and current sources. Voltage sources: An ideal voltage source is a two-terminal device that maintains a fixed voltage drop across its terminals. It is often used as a mathematical abstraction that simplifies the analysis of real electric circuits. It symbol is shown in the Figure 19. Figure 19. Voltage sources symbol. 15

17 In the Figure 19 are shown two types of symbols, the differences are explained in the questions. Current sources: A current source provides a constant current, as long as the load connected to the source terminals has sufficiently low impedance. An ideal current source would provide no energy to a short circuit and approach infinite energy and voltage as the load resistance approaches infinity (an open circuit). It symbol is shown in the Figure 20. Figure 20. Current Source. Active, reactive and apparent power In an AC circuit, what is distributed through the lines is the power, which is the product of the intensity by the voltage. But not all is exploited, so you can talk about: Active power: is used as useful power. It is also called average, real or true power and is due to resistive devices. Its unit of measurement in watt (W). Being : Reactive Power: it is the power that the coils and capacitors need to generate magnetic or electric fields, but that does not transform into effective work, but fluctuates through the network between the generator and the receivers. Its unit of measure is the reactive voltamperium (VAr). The reactive power is positive if the receiver is inductive and negative if the receiver is capacitive, coinciding in sign with the imaginary part of the impedance. Apparent Power: it is the total power consumed by the load and is the product of the effective values of voltage and intensity. It is obtained as the vector sum of the active and reactive powers and represents the total occupation of the installations due to the connection of the receiver. Its unit of measure is the voltamper (VA). Or 16

18 METHODS Alba Romero Montero Ohm s Law There are several laws that have studied electrical circuits. Among them stands out the year 1827 when, experimentally, Georg Simon Ohm found the relationship that could be expressed mathematically between the three most important magnitudes of an electrical circuit: difference of potential, current intensity and resistance. Ohm's law is the fundamental law of electric current that says: "In an electrical circuit, the intensity of the current that runs through it is directly proportional to the voltage applied and inversely proportional to the resistance it presents." It is expressed as follows: Analysing a resistive circuit in an example, this law is best explained. Example: A purely resistive circuit is powered by a direct current source of 12 V value. Knowing that its resistance is 5 ohm, what value will the current circulating in the circuit have? Solution: Figure 11. Resistive circuit. Applying the formula directly and clearing the unknown I, you get a value for the current of 2.4 A. Since power circuits are designed in alternating current, it is important to formulate the law of ohm and understand it for this case: The intensity of current flowing through a circuit of C. A. is directly proportional to the voltage V applied, and inversely proportional to the impedance Z. 17

19 The impedance Z is the difficulty that puts the circuit to the passage of alternating current due to passive elements such as: a resistance R, a coil L or a capacitor C. On the other hand, there are active elements that also oppose difficulty to the passage of current like: the motors, the transformers. Figure 12. Graphic of simple CA circuit. Taking the simple circuit of Figure 12 as a reference, is possible to find the relationship between the variables of the network. Being the absolute value of the impedance, applying the Pythagorean theorem to the triangle shown below in the Figure 13: Figure 13. Impedance triangle. Graphically, the relationship between intensity and voltage in an alternating circuit has a shape similar to the impedance triangle. The phase angle is the same and is corresponded to the power factor; in the Figure 14 the triangle is drawn: Figure 14. Voltage-Current relationship. 18

20 Kirchhoff laws The laws of Kirchhoff were formulated by Gustav Kirchhoff in 1845, while he was still a student. They are widely used in electrical engineering to obtain the values of current and potential at each point of an electrical circuit. They arise from the application of the law of conservation of energy. In the Law of Ohm the equivalent resistance is studied when two or more resistors are connected together in series, parallel or combinations of both, since these circuits obey Ohm's Law. However, sometimes in complex circuits such as bridge networks or T, they are not solved simply by using Ohm's Law to find the voltages or currents that circulate within the circuit. For this type of calculations, the laws of Kirchhoff are needed. When the circuits have a more complicated topology (they have more voltage or current sources distributed around the circuit), to analyse either DC circuits or AC circuits using Kichhoffs Circuit Laws, a number of definitions and terminologies are used to describe the parts of the circuit being analysed such as: node or junction, paths, branches, loops and meshes. In the Figure 15 are graphically explained: Figure 15. Definition and terminologies -Kirchhoff laws-. Circuit: a circuit is a closed loop conducting path in which an electrical current flows. Path: a single line of connecting elements or sources. Junction or node: a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. In the Figure are the points a, b, c, d, e and f. Branch: a branch is a single or group of components such as resistors or a source which are connected between two nodes. 19

21 Alba Romero Montero Loop: It is any closed path in the electric circuit in which no circuit element or node is encountered more than once. In the Figure 15, they are called Loop abefa and Loop ebcde. Mesh: a mesh is a single open loop that does not have a closed path. There are no components inside a mesh. Note: in some lectures is possible to find the word mesh used with the same meaning than loop. The first Kirchhoff Law: Kirchhoffs Current Law or KCL, states that the total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node. In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to zero, I(output) + I(input) = 0. This idea by Kirchhoff is commonly known as the Conservation of Charge. Figure 16. Kirchhoff Current Law. In the Figure 16, the three currents entering the node, I 1, I 2, I 3 are all positive in value and the two currents leaving the node, I 4 and I 5 are negative in value. Then, the equation results: The second Kirchhoff Law: Kirchhoff s Voltage Law or KVL, states that in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop which is also equal to zero. In other words, the algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchhoff is known as the Conservation of Energy. 20

22 Figure 17. Kirchhoff Voltage Law. In the Figure 17, the sum of all the voltage drops around the loop is equal to zero, it is written like: The equation should be written starting at any point in the loop continue in the same direction noting the direction of all voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anticlockwise or the final voltage sun will not be equal to zero. Application of Kirchhoff s laws: These two laws enable the Current and Voltages in a circuit to be found, and the basic procedure for using Kirchhoff s Circuit Laws is as follows: Assume all voltages and resistances are given. (If not label them,, ;,, etc.) Label each branch with a branch current. (,,, etc.) Find Kirchhoff s first law equations for each node. Find Kirchhoff s second law equations for each of the independent loops of the circuit. Use linear simultaneous equations as required to find the unknown currents. DC Applications Example 1: A typical DC circuit is going to be analysed. Note that: Components are said to be connected together in Series if the same current values flows through all the components. Components are said to be connected together in Parallel if they have the same voltage applied across them. The goal is to find the current flowing in the 40Ω Resistor, R 3. 21

23 Figure 17. Example 1 of Kirchhoff s Laws. The circuit has three branches, two nodes (A and B) and two independent loops. Using Kirchhoff Current Law (KCL),, the equations are given as: At node A: At node B: Using Kirchhoff Voltage Law (KVL), the equations are given as: Loop 1 is given as: 10= Loop 2 is given as: 20= Loop 3 is given as: 10-20= As is the sum of, the equations can be rewritten as: With these two equations the values of and can be obtained being: and As, the current flowing through in resistor is =0.286A. And the voltage across the resistor is given as: 0.286*40=11.44V The negative sign for means that the direction of current flow initially chosen was wrong, but nevertheless is still valid. In fact, the 20V battery is charging the 10V battery. Example 2: Now the task is to find the unknown currents and voltages in this circuit of the Figure

24 Figure 18. Example 2 of Kirchhoff s Laws. To start, it helps to give names to voltages, currents and nodes, and assign polarity to the voltage and current of each element. In the Figure 19, the elements are written. Figure 19. Example 2 elements. Figure 20. Example 2, simplified circuit. To emphasize there are only three nodes in this circuit, it is redrawn in the Figure 20 to highlight the junctions at nodes b and c. It is helpful to do a list of the circuit features and unknowns in order to choose an independent variable to solve the circuit: 5 elements. 3 nodes, labelled a, b and c. 3 meshes (inner loops). 1 source voltage, Vs, and 2 elements voltages, V1 and V2. 23

25 1 source current, Is, and 3 elements currents, I1, I2 and I3. Alba Romero Montero At this point, is necessary to choose the voltage v or the current I as independent variable. One good way to make this choice is to compare the number of unknown voltages to unknown currents. There are 2 unknown voltages, and 2 unknown currents. If the voltage is selected as the independent variable, there will be equations with 2 voltage terms as opposed to 3 current terms. 2 is simpler, so voltage will be the independent variable for this problem. Since there is two unknown voltages, is necessary two independent equations to solve for them. The left-most mesh is taken to apply the KVL around it: Figure 21. Example 2 KVL-. The left-most mesh is taken because it includes the all remaining circuit elements not fully controlled by node b, since this one is taken to do the KCL equation because has several connections, making it an interesting focal point of the circuit. KVL equation (the sum of the element voltages around the loop must add up to zero): The sign for v1 and v2 are because their + sign is encountered first during the clockwise tour around the loop, indicating we will see a voltage drop as we go through the component. KCL at node b (the current into a node must equal the currents flowing out of the node): Since it was decided to use v1 and v2 as the independent variables, Ohm s law is used to express the unknown currents in terms of voltage and resistance. Solving the equations, the unknown voltages v1 and v2, and unknown currents i1, i2 and i3 are found. 24

26 As well as using Kichhoffs Circuit Law to calculate the various voltages and currents circulating around a linear circuit, loop analysis can also be used to calculate the currents in each independent loop which helps to reduce the amount of mathematics required by using just Kirchhoff s laws. AC applications Circuits with sinusoidal excitation can be solved using complex impedances for the elements and complex peak or complex rms values for the currents and voltages. Using the complex values version of Kirchhoff's laws, nodal and mesh analysis techniques can be employed to solve AC circuits in a manner similar to DC circuits. Example 3: The goal is to find the amplitude and phase angle of the current i vs (t) if Figure 22. Example 3 AC Kirchhoff analysis-. To solve this problem, is good to start working with the complex impedances. For example, R, L and C 2 are connected in parallel, so the circuit can be simplified by computing their parallel equivalent. means the parallel equivalent of the impedances: In the Figure 23 the values numerically obtained are written: 25

27 Figure 23. Example 3 Impedance simplified-. Using the simplified circuit, the equations of KCL and KVL are ordered form: Using the Ohm Law: There are four unknowns variables: I, I z ; V C1 ; V z, and there 4 equations to solve the circuit. Express I after substituting the other unknowns from the equations: The final circuit is drawing numerically in the Figure 24: 26

28 Figure 24. Example 3 numerically solution-. CONCLUSION From the first laws that Coulomb raised about the electric charge until nowadays, electric power networks have been evolved so that the power lines that arrive to the houses every time have less losses. For this, it is necessary to calculate the voltage and current drops in the electrical networks in order to reduce them to the maximum. The most used theorems are the law of ohm and the laws of Kirchhoff. Once understood, any circuit that arises can be solved. 27

29 QUESTIONS a) What does it mean that the power factor has inductive character in the electrical network? It means that in the network there will be distortions produced by reactive power due to inductances. As a consequence, the drop in intensity and voltage will be negatively affected. The power factor can be compensated by adding capacitor banks in this case. b) Why is Alternating Current preferred than Direct Current? AC current is a specific type of electric current in which the direction of the current's flow is reversed, or alternated, on a regular basis. Direct current is no different electrically from alternating current except for the fact that it flows in the same direction at all times. Alternating current was chosen early in the 20th century as the North American standard because it presented fewer risks and promised higher reliability than competing DC systems of the day. c) When is recommended use DC current? In DC circuits, the electricity is always the same polarity, which means that in a two-wire circuit, one "wire", or side of the circuit, is always negative, and the negative side is always the one that sends the electricity. There is no hum because there is no cyclic change in current flow. DC current is more effective for long-distance, high-voltage transmission because it results in less energy lost in transmission, but the cost of converting DC current to AC is relatively high, so DC is typically cost-effective only for long-distance transmission. d) What is the main against having a low impedance in a circuit? Figure 25. Electric fault. As shown in the Figure 25, if a short-circuit occurs, the only limiting impedance is that of the power system. Since the power system impedance is designed to be as low as economical, the fault current levels can be very large. e) What is the difference between a voltage independent source and a voltage dependent one? Independent voltage sources are those that maintain the same voltage at their terminals, regardless of the amount of current flowing through it. And the voltage dependent sources 28

30 generate the voltage according to another variable that can be another voltage or current in some of the elements of the circuit. HOMEWORK In the circuit of Figure 26, the instantaneous values of the generators are known: The complex power supplied by the generator. is S=10+j40, find the instantaneous electromotive force of Figure 26. Homework. Nodes: Figure 27. Nodes names. Data in frequency domain: 29

31 Since the node 5 is taken as the ground: Alba Romero Montero -j= ) ) j= ) ) ( ) ( ) ( ) 10+j40=5( ) Being, References