Chapter 2 Alternating Current PDF

Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Chapter 2 ALTERNATING CURRENT (AC current) I. Introduction to AC Circuits When a loop of wire...

Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Chapter 2 ALTERNATING CURRENT (AC current) I. Introduction to AC Circuits When a loop of wire rotates inside a magnetic field, Faraday's law predicts that the changing magnetic flux induces an emf (electromotrice force) that oscillates sinusoidally in time with a frequency determined by the angular speed of the coil. This is a source of alternating current (AC). More precisely, the emf in the coil is a source of alternating voltage, which will create alternating current in whatever circuit is connected to the coil. The symbol used to represent an AC source is shown below. The AC voltage source is a sinusoidal function of time with a frequency (or pulsation: ω) and amplitude V0, determined by the characteristics of the electric generator (the rotating coil in the B-field). A plot of the voltage as a function of time, V(t), is shown in the figure below. The period and the amplitude V0 are shown. The AC source is connected to an electrical circuit where it can contain a resistor, a capacitor, an inductor, or a combination of these elements in series or in parallel. The current flows through the circuit. If the source is oscillating with an angular frequency ω , we expect the current to oscillate with the same frequency, therefore the current is also a sinusoidal function of time given by:. Academic year : 2024-2025 1  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity where I0 is the amplitude and ϕ the phase shift of the current with respect to the voltage. The choice of a minus sign will be explained below. The angular frequency is called the driving frequency, the voltage source is driving the circuit with frequency ω. Note:  Angular frequency (ω), also known as radial or circular frequency, measures angular displacement per unit time ( ). It’s units are therefore degrees (or radians) per second :.  Regular or linear frequency (f), sometimes also denoted by the Greek symbol "nu" (ν), counts the number of complete oscillations or rotations in a given period of time. Its units are therefore cycles per second (cps), also called hertz (Hz).  The current is not in phase with the voltage. Since the current peaks after the voltage, we say that it "lags" the voltage (the opposite shift is called "leading" the voltage).  The shift between the two curves is given by the phase shift ϕ. In the expression sin(ωt−ϕ), a positive value of ϕ will correspond to the peaks occurring at a larger value of t. The convention is that we think of a shift to later time as a "positive" shift that should correspond to a positive value of ϕ, and this is why the expression sin(ωt−ϕ) has a minus sign. In summary: In an AC circuit, the voltage oscillates with an angular frequency ω and if we use it as a reference signal we can describe it as V(t)=V0cos(ωt) or V(t)=V0sin(ωt) depending on our choice of zero for the time. We remark that the resulting current is not necessarily in phase with the driving voltage, therefore we will assume that is expressed as I(t)=I0cos(ωt−ϕ) or I(t)=I0sin(ωt−ϕ), with the choice of triangular function being the same for voltage and current. Example : Formally, a sinusoidal function of time is defined as one having the general form and is characterised by three parameters : amplitude A, angular frequency w and phase angle. The argument (angle) of the sine function viz., is measured in radians and increases at Academic year : 2024-2025 2  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity the rate of radians per second. Since a sine function repeats itself at intervals of radians of its angle. Waveform of a general sinusoidal voltage 𝒗 𝒕 𝑽,𝒎 𝒔𝒊𝒏 𝝎𝒕 𝜽)  Root Mean Square current (Effective current). The Root Mean Square (RMS) value of an alternating current (or alternating voltage) is defined as the square root of the average of the square of the intensity (or voltage) calculated over one period. It is written as : √ ∫ √ ∫ In the case of a sinusoidal alternating current, we obtain: √ √ and are known as the peak values of the alternating voltage and current respectively. The instantaneous value of such a current and voltage are then written : √ √ The mean value of a periodic signal of period is given by: ̅ ∫ ̅ ∫ II. Magnetic field and Electromagnetic induction Throughout this part of chapter, we'll be looking at the case of magnetic fields created by fixed wire-shaped circuits of simple geometry, through which permanent currents flow (electric charges are in motion, but the intensity of the electric current does not depend on time). Academic year : 2024-2025 3  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity The fields created are not time-dependent, and this area of physics is generally referred to as magnetostatics (by analogy with electrostatics, which is concerned with electric, or electrostatic, fields). However, to simplify the language, the term magnetostatic field is rarely used, in favor of magnetic field, which is also the case here. Question: The important question is: how do you generate a magnetic field from permanent currents? Experience 1: It all began with Oersted's experiment in 1820. He placed a conducting wire over a compass and passed a current through it. In the presence of a current, the compass needle was indeed deflected, unambiguously proving a link between electric current and magnetic field. He also observed: - If you reverse the direction of the current, the deflection changes direction. - The force deflecting the needle is non-radial Remember that Oersted in 1820, discovered that a steady current produce a steady magnetic field and that connected electricity with magnetism. A little time later, Faraday therefore suggest that maybe a steady magnetic field produce a steady current II.1. Magnetic field applications Electric and magnetic forces both act only on particles carrying charge. Moving electric charge create a magnetic field. A changing magnetic field creates a magnetic field, this elect is called magnetic induction. This links electricity and magnetism in a fundamental way. Magnetic induction is also the key to many practical applications  Induction charging :  Inductive recharging requires no physical connection between the charger and the device.  is based on the principle of electrical induction, whereby the circulation of an electric current in a copper coil (wire winding) creates a current in a nearby coil. When the device to be recharged is placed on the charger, their proximity is such that the magnetic field created induces an electric current in the “receiver” coil, powering its battery. Academic year : 2024-2025 4  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Electric motors : The electric motor consists of two main components:  The stator is a ferromagnetic frame containing electrical windings (coils). When an electric current passes through a coil, which becomes an electromagnet, it creates a magnetic field rotating inside the stator.  The rotor is the rotating element at the centre of the motor, and is subjected to the magnetic field created by the stator, transforming its power into mechanical power.  Magnetic levitation trains : A magnetic levitation train is a train that uses magnetic forces to levitate and move forward. II.2. ELECTROMAGNETIC INDUCTION The term electromagnetic induction refers to the production of currents, and therefore of emf, from magnetic fields; we speak of induced currents and induced emf. Electromagnetic induction is responsible for the operation of generators and transformers, and for the production of electromagnetic waves such as light and radio waves. So, the Electromagnetic induction is the phenomenon in which electric current induced in a conductor by varying magnetic field. 1- Lorentz force The total force, electrical and magnetic (known as electromagnetic) experienced by a particle of charge q and velocity measured in a Galilean frame of reference is: The expression of the Lorentz force can be regarded the definition of electric 𝐸 and magnetic 𝐵 fields. The magnetic field 𝐵 , unlike the electric field 𝐸 does not exert any force on a stationary charge. Academic year : 2024-2025 5  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity In the case where a charge at rest is surrounded by point charges also at rest, the charge q undergoes an electric force:. In the case where the various charges are in motion, we see the appearance of a force different from and which depends on the charge speed of the magnetic field created by all the moving charges other than. The expression of this force is : This force is called the magnetic force, or the magnetic part of the Lorentz force. The magnetic field is defined by its action on a charged particle, of electric charge , moving at speed in a reference frame. The magnitude is given by: Where is the electric charge (C), is the speed charge ( ), is the magnitude of magnetic field vector (Tesla: ) and formed by and. The SI unit of magnetic field is called the Tesla ( ): the Tesla equals a. A smaller unit, called the gauss (G) is sometimes used, where.  Direction of the Magnetic Field by the Right-Hand Rule (Fleming’s Right Hand Rule) The direction of magnetic Lorentz force is determined by the three-finger rule of the right hand (see figure below (a):  Thumb: direction of (= direction of , if ; = opposite direction to , if ),  Index finger: direction of  Middle finger: direction of (a) (b) So, the magnetic fields exert forces on moving charges. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B ( plane) and follows the right- Academic year : 2024-2025 6  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity hand rule (RHR-1) as shown (b). The magnitude of the force is proportional to q,v,B, and the sine of the angle between v and B. The representation of magnetic fields by magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. As shown in Figure below, each of these lines forms a closed loop, even if not shown by the constraints of the space available for the figure. The field lines emerge from the north pole (N), loop around to the south pole (S), and continue through the bar magnet back to the north pole. 2- Force on a current-carrying wire When a wire carrying an electric current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force). By combining the Lorentz force law above with the definition of electric current, the following equation results, in the case of a straight stationary wire in a homogeneous field: or The magnitude of the force on a wire carrying current I with length L in a magnetic field is given by the equation: When the magnetic force relationship is applied to a current-carrying wire, the right-hand rule is used to determine the direction of force on the wire. This force is perpendicular to the plane formed by the field and the element of current. Academic year : 2024-2025 7  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  2-1. Magnetic field created by a straight current  Definition of Magnetic field created by an electric current (H): BIOT AND SAVART'S LAW: An electric circuit (C), through which a current flows, causes, by “induction”, the appearance of an excitation magnetic field H at any point M in space such as M, located at a distance r from an element dl of the circuit, whose modulus is given by the BIOT and SAVART formula. r: distance between point M  and the portion dl In a magnetic medium subjected to magnetic excitation H, we can define a magnetic induction vector B (expressed in tesla, T). The magnetic excitation field H, created in the presence of a magnetic material, takes the magnetic medium into account. The two quantities are linked by the scalar relationship : 7 𝜇 4𝜋 : Magnetic permeability of vacuum. 𝜇𝑟 : Relative magnetic permeability of the material Academic year : 2024-2025 8  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity A rectilinear conductor with a current flowing through it creates a magnetic field. in space, such that : The field lines are circles centered on the conductor. The vector is perpendicular to the conductor. Its direction is given by the right-hand rule. where I: current intensity in A, d: distance between point M and the conducting wire, : Magnetic permeability of vacuum and B in tesla (T) and is measured with a teslameter.  2.2 Magnetic field created by a solenoid (Ampère Law) A solenoid is a long helical coil of wire through which a current is run in order to create a magnetic field. Inside a solenoid, the magnetic field is uniform. The current enclosed in the loop will be the number of turns N in the length L that go thru the loop multiplied by the current I in each coil. It given by: N Or B μ I L Academic year : 2024-2025 9  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Where : number of turns per meter N: total number of turns of the solenoid : length of solenoid. The absolute permeability for other materials can be expressed relative to the permeability of free space as: 𝜇⬚ 𝜇𝐫 𝜇𝟎 ; Where μr is the relative permeability which is a dimensionless quantity. Permeability and Relative Permeability of Materials 3- Induction Electromagnetic Up to now, we've been mainly interested in the creation of a magnetic field from a permanent current. This was motivated by Oersted's experiment. At the same time, the English physicist Faraday was preoccupied with the opposite question: since these two phenomena are linked, how can a current be produced from a magnetic field? He carried out a number of experiments, but failed because he was trying to produce a permanent current. In fact, he did notice some disturbing effects, but they were always transient. 3.1. Magnetic Flux The current induced by a variation in flux in a conductive loop generates a magnetic field whose effects are such as to oppose the movement that induced the current. Faraday's great insight lay in discovering a simple mathematical relationship to explain the series of experiments he carried out on electromagnetic induction. First, we need to explain the concept of magnetic flux,. Definition of magnetic Flux Magnets and electromagnets produce a magnetic field represented by imaginary lines known as magnetic field lines or magnetic lines of force. Magnetic flux is the number of lines passing Academic year : 2024-2025 10  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity through a given area, whether in air or vacuum or inside a magnetic material. The magnetic flux is analogous to the electric flux. The magnetic flux is a scalar quantity and can be quantified by establishing an imaginary surface represented by an area vector in the vicinity of a magnetic field. The magnetic flux φ is given by the dot product of the magnetic field and the area vector: ∫ ∫ Where the angle between the magnetic field vector and the area vector. If is uniform over the flat of area, then the magnetic flux is given by: The SI unit of magnetic flux is Weber (Wb), One weber is the amount of magnetic flux over an area of 1 meter held normal to a uniform magnetic field of one tesla. Thus: If the coil has N turns, total amount of magnetic flux linked with the coil is: 3.2. Induction laws (Faraday law and Lenz law) Academic year : 2024-2025 11  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity These are two laws of induction: (1) Faraday's law defining the induced emf, and (2) Lenz's law concerning the direction of the induced current. These two closely related laws are based on magnetic flux. A- FARADAY’S LAW OF INDUCTION From the experimental observations, Faraday arrived at a conclusion that an emf is induced in a coil when magnetic flux through the coil changes with time. The discovery and understanding of electromagnetic induction are based on a long series of experiments carried out by Faraday and Henry (see figure below). First experiences (a) : A coil C1 (or loop circuit C1) is connected to a galvanometer G.  When the North-pole of a bar magnet is pushed towards the coil, the pointer in the galvanometer deflects, indicating the presence of electric current in the coil. The deflection lasts as long as the bar magnet is in motion.  The galvanometer does not show any deflection when the magnet is held stationary.  When the magnet is pulled away from the coil, the galvanometer shows deflection in the opposite direction, which indicates reversal of the current’s direction.  Moreover, when the South-pole of the bar magnet is moved towards or away from the coil, the deflections in the galvanometer are opposite to that observed with the North-pole for similar movements. Further, the deflection (and hence current) is found to be larger when the magnet is pushed towards or pulled away from the coil faster. Second Experience (b): In this experiment, the bar magnet is replaced by a second coil C2 connected to a battery. The steady current in the coil C2 produces a steady magnetic field. As coil C2 is moved towards the coil C1, the galvanometer shows a deflection. This indicates that electric current is induced in coil C1. When C2 is moved away, the galvanometer shows a deflection again, but this time in the opposite direction. The deflection lasts as long as coil C2 is in motion. When Academic year : 2024-2025 12  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity the coil C2 is held fixed and C1 is moved, the same effects are observed. Again, it is the relative motion between the coils that induces the electric current. Interpretation of Faraday: 1- The change in magnetic flux induces emf in coil. It was this induced emf which caused electric current to flow in coil and through the galvanometer. 2- The common point in all these observations is that the time rate of change of magnetic flux through a circuit induces emf in it. Faraday stated experimental observations in the form of a law called Faraday’s law of electromagnetic induction. The law is stated below: The magnitude of the induced emf (Motional ElectroMotive Force) in a circuit is equal to the time rate of change of magnetic flux through the circuit. Mathematically, the induced emf is given by: Faraday’s Law indicates how to calculate the potential difference that produces the induced current. However, that direction is most easily determined The magnitude of the induced emf with a rule known as Lenz’s law, which we will discuss equals the rate of change of the magnetic flux shortly The induced electromotive force is expressed in volts and the magnetic flux in Weber. In the case of a closely wound coil of N turns, change of flux associated with each turn, is the same. Therefore, the expression for the total induced emf is given by : 𝜙𝑠𝑜𝑙𝑜𝑛𝑜𝑖𝑑 𝑁 𝜙𝑠𝑝𝑖𝑟𝑒 B- Lenz’s Law The current induced by a flux variation in a conductive loop generates a magnetic field whose effects are such as to oppose the movement that induced the current. The polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it. Academic year : 2024-2025 13  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Faraday's law can be used to determine the intensity of the induced emf and to deduce the intensity of the induced current using Ohm's law: where R is the resistance of | | the conductor loop Changing the current in the right-hand coil induces a current in the left-hand coil. The induced current does not depend on the size of the current in the right-hand coil. III. INDUCTANCE The inductance of an electrical circuit is a coefficient that reflects the fact that a current flowing through it creates a magnetic field across the section surrounded by the circuit. As a result, the magnetic field flows through the section bounded by the circuit. The inductance is equal to the quotient of the flux of this magnetic field and the intensity of the current flowing through the circuit or electric dipole. The unit of inductance is the Henry (H). These dipoles are generally coils, often called inductors or chokes. III.1. Self-inductance The emf is induced in a single isolated coil due to change of flux through the coil by means of varying the current through the same coil. This phenomenon is called self-induction. In this case, flux linkage through a coil of N turns is proportional to the current through the coil. The self- inductance L of the electric circuit is then defined as the ratio between the flux embraced by the circuit and the current and is expressed as: 𝑵𝝓𝑩 𝝓𝑩 So : 𝑳 𝒊𝒇 𝑵 𝟏 𝑳 𝑰 𝑰 When the current is varied, the flux linked with the coil also changes and an emf is induced in the coil. The induced emf is given by:  SELF INDUCTANCE OF A LONG SOLENOID Academic year : 2024-2025 14  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity N We have B μ I , The Total magnetic flux linked with the solenoid so lenght N2 the self-inductance of solenoid is given by: μ 𝐴. If core is of any other magnetic lenght N2 material, μ is replaced by μ, where μ μr lenght 𝐴 The symbol for an inductor : If the coil is wrapped around an iron core so as to enhance its magnetic effect, it is symbolised by putting two lines above it, as shown here III.2. Grouping of coils The grouping of inductances (coils) can be in series or in parallel (see following figure).  Coils in series: Coils equivalent to series coils are added together: in effect, the magnetic fields are added together (figure (a)).So the equivalence inductance is given by:. For N coils we obtain : ∑  Coils in parallel: The inverse of the equivalent inductance of parallel coils is the sum of the inverses of each of the inductances (figure (b)). So,. For N coils we obtain: ∑ IV. AC GENERATOR The phenomenon of electromagnetic induction has been technologically exploited in many ways. An exceptionally important application is the generation of alternating currents (ac). The Academic year : 2024-2025 15  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity main application of Faraday's law is undoubtedly the electric generator or Alternator (dynamo). transforms mechanical energy into electrical energy. The mechanical energy supplied to the generator rotates its axis⇒ turn drives a conductor (several, in fact) between the poles of a magnet. The result is a magnetic flux through the coil, and an e.m.f. and current are induced in the conductor. When the coil is rotated with a constant angular speed , the angle between the magnetic field vector B and the area vector A of the coil at any instant t is (assuming = 0° at t = 0). If the generator pivot is rotated with a constant angular velocity, ω, we have : The flux at any time t is given by : The emf induced in such a generator can be calculated using Faraday's law: Thus, the instantaneous value of the emf is: where = is the maximum value of the emf. The emf is an alternating emf that varies sinusoidally with time. The direction of the current changes periodically and therefore the current is called alternating current (ac). Since , So, we can be written as: where f is the frequency of revolution of the generator’s coil. The alternative voltage is Academic year : 2024-2025 16  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity V. AC circuits: phase shift, Fresnel representation, phasors and reactance. V.1. Complex Numbers in AC circuits The complex numbers may be used to analyze and compute currents and voltages in AC circuits. The resistance, the impedance of a capacitor and the impedance of an inductor are represented by complex numbers. It is also shown how the use of complex impedances allows the use of a law similar to Ohm's law in order to mathematically model AC circuits. Two main reasons that make the use of complex numbers suitable to model AC circuits, and many other sine wave phenomena in several branches of engineering, are: 1) The AC signals (and many other sine wave phenomena) are characterized by a magnitude and a phase that are, respectively, very similar to the modulus and argument of complex numbers. 2) The basic operations such as addition, subtraction, multiplication and division of complex numbers are easier to carry out and to program on a computer. Note: 1) Because the symbol is used for currents in AC circuits, here we use as the imaginary unit defined by or √. 2) The symbol Re represents the real part of a complex number and Im represents the imaginary part. A complex number in standard form may be written  In exponential form as follows with  In polar form as follows : where √ is the modulus of Z or Magnitude of Z and its argument. Take the real part, written as Re , of each side of a complex number in exponential form ( ) ( )  Phasor Diagramm : use complex numbers to represent the important information from the time functions (magnitude and phase angle) in vector form. Where: VRMS, IRMS = RMS magnitude of voltages and currents = phase shift in degrees for voltages and currents Academic year : 2024-2025 17  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Phasor Representation (Representation Fresnel) The advantage of the Fresnel representation is that it makes it easy to sum two sinusoidal quantities of the same pulsation. In electricity, this representation makes it easy to find:  a voltage by means of a loop law  or a current by means of a junction law. Example, consider two voltages √ and 4 √ whose waveforms are shown in Figure Two sinusoids with a phase difference of 60' (a) Waveforms (b) Phasors  Representation of a sinusoidal quantity Academic year : 2024-2025 18  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Definition of Phasor A phasor is a rotating line whose projection on a vertical axis can be used to represent sinusoidally varying quantities. To get at the idea, consider the line of length Vm shown in Figure (It is the phasor.). The vertical projection of this line (indicated in dotted line) is , We assume that the phasor rotates at angular velocity of rad/s in the counterclockwise direction. The geometric relationship between various forms of the sine and cosine functions can be derived from Figure below.     The phase shift between two waveforms indicates which one leads or lags, and by how many degrees or radians. Phase difference refers to the angular displacement between different waveforms of the same frequency. If the angular displacement is 0°, the waveforms are said to be in phase, otherwise, they are out of phase. i(t) leads v(t) by 40° v(t) and i(t) in phase V.2. VOLTAGE APPLIED TO A RESISTOR Academic year : 2024-2025 19  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity The Figure shows a resistor connected to a source ε of ac voltage. The symbol for an ac source in a circuit diagram is. We consider a source which produces sinusoidally varying potential difference across its terminals. Let this potential difference, also called ac voltage, be given by: where is the amplitude of the oscillating potential difference and ω is its angular frequency. To find the value of current through the resistor, we apply Kirchhoff’s loop rule 𝜀 𝑡. So 𝑣𝑚 𝜔𝑡 𝑅 𝑖 𝑡 or 𝑣𝑚 𝑣 𝑖 𝑡 𝜔𝑡 𝑖𝑚 𝜔𝑡. Since R is a constant, 𝑖𝑚 = 𝑅𝑚 𝑅 the peak amplitude of current. This equation is Ohm’s law, which for resistors, works equally well for both AC and DC voltages. The voltage across a pure resistor and the current through it, are plotted as a function of time as below: We note, that both (t) and reach zero, minimum and maximum values at the same time. So, the voltage and current are in phase with each other. There is Joule heating and dissipation of electrical energy when an AC current passes through a resistor. The instantaneous power dissipated in the resistor is: The average value of p over a cycle is: ̅ ∫ where the bar over a letter (here, p) denotes its average value and denotes taking average of the quantity inside the bracket. Since, are constants, so: ̅ ∫ ∫ ∫ ( ) Academic year : 2024-2025 20  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity We have < cos2t > = 0, finally we obtain: 𝐼𝑚𝑎𝑥 With : 𝐼𝑟𝑚𝑠 √ ̅  REPRESENTATION OF AC CURRENT AND VOLTAGE BY ROTATING VECTORS — PHASORS In order to show phase relationship between voltage and current in an AC circuit, we use the notion of phasors. The analysis of an AC circuit is facilitated by the use of a phasor diagram. A phasor is a vector which rotates about the origin with angular speed , as shown in figure. In the phasor diagram, peak values of alternating current and alternating e.m.f. are represented by arrows called phasors. They are inclined to horizontal axis at angle and rotate in the anticlockwise direction.(a): phasor diagram (Fresnel representation) and waveform of (t) and (t). V.3. VOLTAGE APPLIED TO AN INDUCTOR The figure below shows an AC source connected to an inductor. Usually, inductors have appreciable resistance in their windings, but we shall assume that this inductor has negligible resistance. Thus, the circuit is a purely inductive ac circuit. Let the AC source voltage, be given by:. Using the Kirchhoff’s loop rule, (t ) = 0 , and since there is no resistor in the circuit. where the second term is the self-induced Faraday emf in the inductor; and L is the self- inductance of the inductor. The negative sign follows from Lenz’s law. So we obtain : To obtain the current, we integrate with respect to time: ∫ ∫ ⇒ Academic year : 2024-2025 21  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Using ( ), we obtain finaly ( ) ( ) where is the amplitude of the current (Peak value of amplitude). The quantity is analogous to the resistance and is called inductive reactance, denoted by The unit of inductive reactance is the same as that of resistance and its SI unit is ohm ( ). The inductive reactance limits the current in a purely inductive circuit. The inductive reactance is directly proportional to the inductance and to the frequency of the current. The source voltage and the current in an inductor show that the current lags the voltage by /2 or one-quarter (1/4) cycle. The figure shows the voltage and the current phasors in the present case at instant t1. The current phasor is /2 behind the voltage phasor. When rotated with frequency counterclockwise. We see that the current reaches its maximum value later than the voltage by one-fourth of periode * +. The instantaneous power supplied to the inductor is ( ) So: Since the average of 𝒔𝒊𝒏 𝟐𝝎𝒕 over a complete cycle is zero. The average value of power over one cycle is given by : ̅ Thus, the average power supplied to an inductor over one complete cycle is zero. Academic year : 2024-2025 22  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity V.4. AC VOLTAGE APPLIED TO A CAPACITOR The figure below shows an AC source e generating AC voltage 𝑣 𝑡 𝑣𝑀 𝑠𝑖𝑛𝜔 𝑡 connected to a capacitor only, a purely capacitive ac circuit. Using the Kirchhoff’s loop rule, 𝜀 (t ) = 0 When the capacitor is connected to an ac source, it limits or regulates the current, but does not completely prevent the flow of charge. The capacitor is alternately charged and discharged as the current reverses each half cycle. Let be the charge on the capacitor at any time t. The instantaneous voltage v across the capacitor is: From the Kirchhoff’s loop rule, the voltage across the source and the capacitor are equal: To find the current, we use the relation Using relation, , we have : ( ) Where the amplitude of oscillating current is. So, we can rewrite it as:. The quantity is analogous to the resistance and is called capacitive reactance, denoted by The unit of capacitive reactance is the same as that of resistance and its SI unit is ohm ( ).The capacitive reactance limits the amplitude of the current in a purely capacitive circuit in the same way as the resistance limits the current in a purely resistive circuit. But it is inversely proportional to the frequency and the capacitance. The Figure shows the phasor diagram at an instant. Here the current phasor is π/2 ahead of the voltage phasor as they rotate counterclockwise. We see that the current reaches its maximum value earlier than the voltage by one-fourth (1/4) of a period. Academic year : 2024-2025 23  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Thus, in the case of a capacitor, the current leads the voltage by. The instantaneous power supplied to the capacitor is: So, as in the case of an inductor, the average power: Since the average of 𝒔𝒊𝒏 𝟐𝝎𝒕 over a complete cycle is zero. Unlike a resistor, a capacitor does not dissipate energy over a half-integer number of periods. This is due to the phase shift between current and voltage. During a quarter cycle, the capacitor stores energy by accumulating charges on its armatures, the product is positive, energy which it returns entirely to the source during the next quarter of the cycle, the product product is negative VI. IMPEDANCE CONCEPT In practice, we represent circuit elements by their impedance, and determine magnitude and phasor relationships in one step. Before we do it, however, we need to learn how to represent circuit elements as impedance. The relation between current flow in a circuit element and the voltage across the element can be expressed as a relation between the complex numbers that represent the voltage and the current. Certain circuit elements oppose current (coils) or voltage (capacitors) fluctuations without consuming energy as a simple resistor would. These components have a reactance that opposes current fluctuations. In particular, this reactance is responsible for the phase shift between current and voltage. As with resistors, impedance is expressed in Ohm. It is represented by the letter. For example, Ohm's law can be written as But don't confuse impedance Z with resistance R:  Resistance R does not depend on the nature of the current (AC or DC).  Impedance Z is only to be considered for alternating current, and is frequency dependent. Academic year : 2024-2025 24  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Phasors : Complex impedances Impedance is the modulus of the complex impedance. The angle of the complex impedance is the phase shift of current 𝒊 with respect to voltage 𝑣. Academic year : 2024-2025 25  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Electrical impedance, or simply impedance, describes a measure of opposition to alternating current (AC). Electrical impedance extends the concept of resistance to AC circuits, describing not only the relative amplitudes of the voltage and current, but also the relative phases. Academic year : 2024-2025 26  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Equivalent complex impedances  Impedances connected in series  Impedances connected in parallel. VI. Analysis of a series RLC circuit Figure shows a series RLC circuit connected to an ac source. As usual, we take the voltage of the source to be. If q is the charge on the capacitor and the current, at time t, we have, from Kirchhoff’s loop rule: Academic year : 2024-2025 27  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity The phasor relation whose vertical component gives the above equation is : Academic year : 2024-2025 28  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity VII. Resonance An interesting characteristic of the series RLC circuit is the phenomenon of resonance. The phenomenon of resonance is common among systems that have a tendency to oscillate at a particular frequency. This frequency is called the system’s natural frequency. At impedance Z is minimum and equals R; current is maximum VIII. POWER IN AC CIRCUIT: THE POWER FACTOR Electrical power is the quantity of energy supplied or received during one second. There are three types of AC power:  Active power;  Reactive power;  Apparent power  Active power Active power is referred to as (P). Its unit is W (Watt). The expression of active power (P) depends on the rms voltage ( ), the rms electric current ( ) and the phase shift ( ). The symbol (φ) is the phase shift of the current with respect to the voltage. Its expression is : Academic year : 2024-2025 29  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity { La puissance active (P) est mesurée à l’aide d’un wattmètre. The average power delivered by the source to the circuit is given by: The is called power factor and is given by:  Apparent power Apparent power is denoted by (S). Its unit is VA (VoltAmpere). It depends on the rms voltage (Vrms) and the rms current (Irms). Its expression is:  Reactive power Reactive power is designated (Q). Its unit is (VAR: stands for Voltampere Reactive). It is absorbed by inductive receivers such as motors, electric welders, etc., or supplied by capacitive receivers such as batteries; electric welders, etc., or supplied by capacitive receivers such as capacitor banks; capacitor banks. Its expression is: The power factor ( ) is also defined as the ratio of active power to apparent power. Its expression is : Academic year : 2024-2025 30  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Summary  Case of an RLC series circuit : We can associate i(t) and v(t) with their complex notations, giving : We find the impedance of the series RLC dipole : The modulus and phase of are : Academic year : 2024-2025 31  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Parallel RLC circuit : We can associate i(t) and v(t) with their complex notations, giving : Academic year : 2024-2025 32  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Active Power  Reactive Power Academic year : 2024-2025 33  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Apparent Power The power consumption of each basic element is shown below Academic year : 2024-2025 34