4m electromagnetic induction__alternating_currents

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4m electromagnetic induction__alternating_currents

  1. 1. ELECTROMAGNETIC INDUCTION1. Magnetic Flux2. Faraday’s Experiments3. Faraday’s Laws of Electromagnetic Induction4. Lenz’s Law and Law of Conservation of Energy5. Expression for Induced emf based on both laws6. Methods of producing induced emf a) By changing Magnetic Field b) By changing the Area of the Coil (Motional emf) c) By changing the Relative Orientation of the coil with the Magnetic Field7. Eddy Currents8. Self Induction and Self Inductance9. Mutual Induction and Mutual Inductance10. Additional Information
  2. 2. Magnetic Flux (Φ):Magnetic Flux through any surface is the number of magnetic lines of forcepassing normally through that surface.It can also be defined as the product of the area of the surface and thecomponent of the magnetic field normal to that surface. dΦ = B . ds = B.ds. n Direction of ds is along the normal to B cos θ n dΦ = B . ds cos θ the surface and n ds θ is unit normal B vector. Φ = B . A = B.A.n Φ = B . A cos θPositive Flux:Magnetic Flux is positive for 0° ≤ θ < 90° & 270°< θ ≤ 360°Zero Flux:Magnetic Flux is zero for θ = 90° & θ = 270° Flux is maximum when θ = 0° and isNegative Flux: Φ=B.AMagnetic Flux is negative for 90°< θ < 270°
  3. 3. Φ = B . A cos θMagnetic Flux across a coil can be changed by changing :1) the strength of the magnetic field B2) the area of cross section of the coil A3) the orientation of the coil with magnetic field θ or4) any of the combination of the above* Magnetic flux is a scalar quantity.* SI unit of magnetic flux is weber or tesla-metre2 or ( wb or Tm2).* cgs unit of magnetic flux is maxwell.* 1 maxwell = 10-8 weber* Magnetic flux (associated normally) per unit area is called Magnetic Flux Density or Strength of Magnetic Field or Magnetic Induction (B).
  4. 4. Faraday’s Experiment - 1: S N N S N S G G N S S N S N G G
  5. 5. S N N S G Magnetic flux linked with the coil changes relative to the positions of the coil and the magnet due to the magnetic lines of force cutting at different angles at the same cross sectional area of the coil.
  6. 6. Observe: i) the relative motion between the coil and the magnet ii) the induced polarities of magnetism in the coil iii) the direction of current through the galvanometer and hence the deflection in the galvanometer iv) that the induced current (e.m.f) is available only as long as there is relative motion between the coil and the magnetNote: i) coil can be moved by fixing the magnet ii) both the coil and magnet can be moved ( towards each other or away from each other) i.e. there must be a relative velocity between them iii) magnetic flux linked with the coil changes relative to the positions of the coil and the magnet iv) current and hence the deflection is large if the relative velocity between the coil and the magnet and hence the rate of change of flux across the coil is more
  7. 7. Faraday’s Experiment - 2: When the primary circuit is closed N S S N current grows from zero to maximum value. P S During this period changing, current induces changing magnetic flux across the primary coil. E K G This changing magnetic flux is linked across the secondary coil N S N S and induces e.m.f (current) in the secondary coil. P S Induced e.m.f (current) and hence deflection in galvanometer lasts only as long as the current in the primary coil and hence the E K G magnetic flux in the secondary coil change.
  8. 8. When the primary circuit is open current decreases from maximum value tozero.During this period changing current induces changing magnetic flux across theprimary coil.This changing magnetic flux is linked across the secondary coil and inducescurrent (e.m.f) in the secondary coil.However, note that the direction of current in the secondary coil is reversedand hence the deflection in the galvanometer is opposite to the previous case.Faraday’s Laws of Electromagnetic Induction: I Law: Whenever there is a change in the magnetic flux linked with a circuit, an emf and hence a current is induced in the circuit. However, it lasts only so long as the magnetic flux is changing. II Law: The magnitude of the induced emf is directly proportional to the rate of change of magnetic flux linked with a circuit. E α dΦ / dt E = k dΦ / dt E = dΦ / dt E = (Φ2 – Φ1) / t (where k is a constant and units are chosen such that k = 1)
  9. 9. Lenz’s Law:The direction of the induced emf or induced current is such that it opposesthe change that is producing it.i.e. If the current is induced due to motion of the magnet, then the inducedcurrent in the coil sets itself to stop the motion of the magnet. If the current is induced due to change in current in the primary coil, theninduced current is such that it tends to stop the change.Lenz’s Law and Law of Conservation of Energy:According to Lenz’s law, the induced emf opposes the change that producesit. It is this opposition against which we perform mechanical work in causingthe change in magnetic flux. Therefore, mechanical energy is converted intoelectrical energy. Thus, Lenz’s law is in accordance with the law ofconservation of energy.If, however, the reverse would happen (i.e. the induced emf does not opposeor aids the change), then a little change in magnetic flux would produce aninduced current which would help the change of flux further therebyproducing more current. The increased emf would then cause further changeof flux and it would further increase the current and so on. This would createenergy out of nothing which would violate the law of conservation of energy.
  10. 10. Expression for Induced emf based on both the laws: E = - dΦ / dt E = - (Φ2 – Φ1) / t And for ‘N’ no. of turns of the coil, E = - N dΦ / dt E = - N (Φ2 – Φ1) / tExpression for Induced current: Note: I = - dΦ / (R dt) Induced emf does not depend onExpression for Charge: resistance of the circuit where as the induced current and dq / dt = - dΦ / (R dt) induced charge depend on dq = - dΦ / R resistance.Methods of producing Induced emf:1. By changing Magnetic Field B: Magnetic flux Φ can be changed by changing the magnetic field B and hence emf can be induced in the circuit (as done in Faraday’s Experiments).
  11. 11. 2. By changing the area of the coil A available in Magnetic Field: Magnetic flux Φ can be changed by changing the area of the loop A which is acted upon by the magnetic field B and hence emf can be induced in the circuit. B P’ P Q’ Q dA v l I S’ v.dt S R’ R dΦ = B.dAThe loop PQRS is slided into uniform and perpendicular = B.l.v.dtmagnetic field. The change (increase) in area of the coilunder the influence of the field is dA in time dt. This E = - dΦ / dtcauses an increase in magnetic flux dΦ. E = - BlvThe induced emf is due to motion of the loop and so it is called ‘motional emf’.If the loop is pulled out of the magnetic field, then E = BlvThe direction of induced current is anticlockwise in the loop. i.e. P’S’R’Q’P’ byFleming’s Right Hand Rule or Lenz’s Rule.
  12. 12. According Lenz’s Rule, the direction of induced current is such that itopposes the cause of changing magnetic flux.Here, the cause of changing magnetic flux is due to motion of the loop andincrease in area of the coil in the uniform magnetic field.Therefore, this motion of the loop is to be opposed. So, the current is settingitself such that by Fleming’s Left Hand Rule, the conductor arm PSexperiences force to the right whereas the loop is trying to move to the left.Against this force, mechanical work is done which is converted into electricalenergy (induced current).NOTE: If the loop is completely inside the boundary of magnetic field, thenthere will not be any change in magnetic flux and so there will not be inducedcurrent in the loop. Magnetic ForceFleming’s Right Hand Rule: Field (F)If the central finger, fore finger and thumb (B)of right hand are stretched mutuallyperpendicular to each other and the forefinger points to magnetic field, thumb Electricpoints in the direction of motion (force), Currentthen central finger points to the direction of (I)induced current in the conductor.
  13. 13. 3. By changing the orientation of the coil (θ) in Magnetic Field: Magnetic flux Φ can be changed by changing the relative orientation of the loop (θ) with the magnetic field B and hence emf can be induced in the circuit. Φ = N B A cos θ At time t, with angular velocity ω, ω θ = ωt (at t = 0, loop is assumed to S be perpendicular to the magnetic field and θ = 0° ) Φ = N B A cos ωt P θ B Differentiating w.r.t. t, n dΦ / dt = - NBAω sin ωt E = - dΦ / dt E = NBAω sin ωt R E = E0 sin ωt (where E0 = NBAω is the maximum emf) Q
  14. 14. The emf changes continuously in Emagnitude and periodically indirection w.r.t. time giving rise to E0alternating emf. 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T tIf initial position of the coil is takenas 0° i.e. normal to the coil is at 90° ,with the magnetic field, thenθ becomes θ + π/2 or ωt + π/2 E E = E0 cos ωt E0So, alternating emf and 0consequently alternating current π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωtcan be expressed in sin or cos T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2Tfunction. tThis method of inducing emf is the basic principle of generators.
  15. 15. Eddy Currents or Foucault Currents:The induced circulating (looping) currents produced in a solid metal due tochange in magnetic field (magnetic flux) in the metal are called eddy currents. Applications of Eddy Currents: B 1. In induction furnace eddy currents are used for melting iron ore, etc. 2. In speedometer eddy currents are used to measure the instantaneous speed of the vehicle. 3. In dead beat galvanometer eddy currents Metallic Block Eddy Currents are used to stop the damping of the coil in a shorter interval. 4. In electric brakes of the train eddy currents are produced to stop the rotation of the axle of the wheel. 5. In energy meters (watt – meter) eddy currents are used to measure the consumption of electric energy. 6. In diathermy eddy currents are used for localised heating of tissues in human bodies.
  16. 16. Self Induction:Self Induction is the phenomenon of inducing emf in the self coil due tochange in current and hence the change in magnetic flux in the coil.The induced emf opposes the growth or decay of current in the coil andhence delays the current to acquire the maximum value.Self induction is also called inertia of electricity as it opposes the growth ordecay of current.Self Inductance:ΦαI or Φ = LI (where L is the constant of proportionality and is known as Self Inductance or co-efficient of self induction)If I = 1, then L= ΦThus, self inductance is defined as the magnetic flux linked with a coilwhen unit current flows through it.Also, E = - dΦ / dt or E = - L (dI / dt)If dI / dt = 1, then L=EThus, self inductance is defined as the induced emf set up in the coilthrough which the rate of change of current is unity.
  17. 17. SI unit of self inductance is henry (H).Self inductance is said to be 1 henry when 1 A current in a coil linksmagnetic flux of 1 weber.orSelf inductance is said to be 1 henry when unit rate of change of current(1 A / s) induces emf of 1 volt in the coil.Self inductance of a solenoid: AMagnetic Field due to the solenoid isB = µ0nI I lMagnetic Flux linked across one turn of thecoil isΦ per turn = B A = µ0nIA = µ0NIA / l Energy in Inductor:Magnetic Flux linked across N turns of the Small work done dW incoil is establishing a current I in the coil in time dt is dW = - EI dt Φ = µ0N2IA / l dW = LI dI (since E = -L(dI / dt) I0But, Φ = LI W = ∫ L I dI = ½ LI 02 So, L = µ0N2A / l = µ0n2Al 0
  18. 18. Mutual Induction:Mutual Induction is the phenomenon of inducing emf in the secondary coildue to change in current in the primary coil and hence the change inmagnetic flux in the secondary coil.Mutual Inductance: (where M is the constant of proportionality and isΦ21 α I1 or Φ21 = MI1 known as Mutual Inductance or co-efficient of mutual induction)If I1 = 1, then M= ΦThus, mutual inductance is defined as the magnetic flux linked with thesecondary coil when unit current flows through the primary coil.Also, E2 = - dΦ21 / dt or E 2= - M (dI1 / dt)If dI1 / dt = 1, then M=EThus, mututal inductance is defined as the induced emf set up in thesecondary coil when the rate of change of current in primary coil is unity.SI unit of mututal inductance is henry (H).Mutual inductance is said to be 1 henry when 1 A current in the primary coillinks magnetic flux of 1 weber across the secondary coil. orMutual inductance is said to be 1 henry when unit rate of change of current(1 A / s) in primary coil induces emf of 1 volt in the secondary coil.
  19. 19. Mutual inductance of two long co-axial solenoids:Magnetic Field due to primary solenoid isB1 = µ0n1I1Magnetic Flux linked across one turn of the Gsecondary solenoid is SΦ21 per turn = B1 A = µ0n1I1A = µ0N1I1A / l AMagnetic Flux linked across N turns of the secondary Psolenoid is I1 l Φ21 = µ0N1N2I1A / lBut, Φ21 = M21I1 M21 = µ0N1N2A / l = µ0n1n2Al lllly M12 = µ0N1N2A / l = µ0n1n2Al For two long co-axial solenoids of same length and cross-sectional area, the mutual inductance is same and leads to principle of reciprocity. M = M12 = M21
  20. 20. Additional Information:1) If the two solenoids are wound on a magnetic core of relative permeability µr, then M = µ0 µr N1N2A / l2) If the solenoids S1 and S2 have no. of turns N1 and N2 of different radii r1 and r2 (r1 < r2), then M = µ0 µr N1N2 (πr12)/ l3) Mutual inductance depends also on the relative placement of the solenoids.4) Co-efficient of Coupling (K) between two coils having self-inductance L1 and L2 and mutual inductance M is K = M / (√L1L2) Generally, K < 15) If L1 and L2 are in series, then L = L1 + L26) If L1 and L2 are in parallel, then (1/L) = (1/L1) + (1/L2)
  21. 21. ALTERNATING CURRENTS1. Alternating EMF and Current2. Average or Mean Value of Alternating EMF and Current3. Root Mean Square Value of Alternating EMF and Current4. A C Circuit with Resistor5. A C Circuit with Inductor6. A C Circuit with Capacitor7. A C Circuit with Series LCR – Resonance and Q-Factor8. Graphical Relation between Frequency vs XL, XC9. Power in LCR A C Circuit10. Watt-less Current11. L C Oscillations12. Transformer13. A.C. Generator
  22. 22. Alternating emf:Alternating emf is that emf which continuously changes in magnitude andperiodically reverses its direction.Alternating Current:Alternating current is that current which continuously changes in magnitudeand periodically reverses its direction. E = E0 sin ωt E = E0 cos ωt E ,I I = I0 sin ωt E ,I I = I0 cos ωt E0 E0 I0 I0 0 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T t tE, I – Instantaneous value of emf and current Symbol ofE0, I0 – Peak or maximum value or amplitude of emf and current AC Source ω – Angular frequency t – Instantaneous time ωt – Phase
  23. 23. Average or Mean Value of Alternating Current:Average or Mean value of alternating current over half cycle is that steadycurrent which will send the same amount of charge in a circuit in the time ofhalf cycle as is sent by the given alternating current in the same circuit inthe same time. dq = I dt = I0 sin ωt dt T/2 q = ∫ I0 sin ωt dt 0 q = 2 I0 / ω = 2 I0 T / 2π = I0 T / π Mean Value of AC, Im = Iav = q / (T/2) Im = Iav = 2 I0 / π = 0.637 I0 = 63.7 % I0Average or Mean Value of Alternating emf: Em = Eav = 2 E0 / π = 0.637 E0 = 63.7 % E0 Note: Average or Mean value of alternating current or emf is zero over a cycle as the + ve and – ve values get cancelled.
  24. 24. Root Mean Square or Virtual or Effective Value ofAlternating Current:Root Mean Square (rms) value of alternating current is that steady currentwhich would produce the same heat in a given resistance in a given time asis produced by the given alternating current in the same resistance in thesame time. dH = I2R dt = I02 R sin2 ωt dt T H = ∫ I02 R sin2 ωt dt 0 H = I02 RT / 2 (After integration, ω is replaced with 2 π / T) If Iv be the virtual value of AC, then H = Iv 2 RT Iv = Irms = Ieff = I0 / √2 = 0.707 I0 = 70.7 % I0Root Mean Square or Virtual or Effective Value ofAlternating emf: E =E = E = E / √2 = 0.707 E = 70.7 % E v rms eff 0 0 0Note:1. Root Mean Square value of alternating current or emf can be calculated over anyperiod of the cycle since it is based on the heat energy produced.2. Do not use the above formulae if the time interval under the consideration is less thanone period.
  25. 25. Relative Values Peak,Virtual and Mean Values ofAlternating emf: E0 Ev Em Em = Eav = 0.637 E0 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T t Ev = Erms = Eeff = 0.707 E0 Tips: 1. The given values of alternating emf and current are virtual values unless otherwise specified. i.e. 230 V AC means Ev = Erms = Eeff = 230 V 2. AC Ammeter and AC Voltmeter read the rms values of alternating current and voltage respectively. They are called as ‘hot wire meters’. 3. The scale of DC meters is linearly graduated where as the scale of AC meters is not evenly graduated because H α I2
  26. 26. AC Circuit with a Pure Resistor: RE = E0 sin ωt I=E/R = (E0 / R) sin ωt E = E0 sin ωt I = I0 sin ωt (where I0 = E0 / R and R = E0 / I0) Emf and current are in same phase. E = E0 sin ωt yE ,I I = I0 sin ωt E0 E0 I0 I0 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T t 0 ωt x
  27. 27. AC Circuit with a Pure Inductor: E = E0 sin ωt Induced emf in the inductor is - L (dI / dt) L In order to maintain the flow of current, the applied emf must be equal and opposite to E = E0 sin ωt the induced emf. I = ∫ (E0 / L) sin ωt dt E = L (dI / dt) E0 sin ωt = L (dI / dt) I = (E0 / ωL) ( - cos ωt ) dI = (E0 / L) sin ωt dt I = I0 sin (ωt - π / 2)(where I0 = E0 / ωL and XL = ωL = E0 / I0) Current lags behind emf by π/2 rad.XL is Inductive Reactance. Its SI unit is ohm. y E 0 E = E0 sin ωt E ,I I = I0 sin (ωt - π / 2) E0 I0 ωt 0 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt π/2 x T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T t I0
  28. 28. AC Circuit with a Capacitor:E = E0 sin ωt Cq = CE = CE0 sin ωt E = E0 sin ωtI = dq / dt = (d / dt) [CE0 sin ωt] (where I0 = E0 / (1 / ωC) andI = [E0 / (1 / ωC)] ( cos ωt ) XC = 1 / ωC = E0 / I0) XC is Capacitive Reactance.I = I0 sin (ωt + π / 2) Its SI unit is ohm.Current leads the emf by π/2 radians. y E0 E = E0 sin ωtE ,I I = I0 sin (ωt + π / 2) I0 E0 I0 π/2 0 ωt π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt 0 x T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T t
  29. 29. Variation of XL with Frequency: XLI0 = E0 / ωL and XL = ωLXL is Inductive Reactance and ω = 2π fXL = 2π f L i.e. XL α f 0 fVariation of XC with Frequency: XCI0 = E0 / (1/ωC) and XC = 1 / ωCXC is Inductive Reactance and ω = 2π fXC = 1 / 2π f C i.e. XC α 1 / f 0 f TIPS: 1) Inductance (L) can not decrease Direct Current. It can only decrease Alternating Current. 2) Capacitance (C) allows AC to flow through it but blocks DC.
  30. 30. L RAC Circuit with L, C, R in Series CCombination: VR VLThe applied emf appears as VCVoltage drops VR, VL and VCacross R, L and C respectively.1) In R, current and voltage are in E = E0 sin ωt phase. VL VL2) In L, current lags behind voltage by - VC π/2 π/2 π/23) In C, current leads the voltage by 0 π/2 π/2 I VR I VRE = √ [VR2 + (VL – VC)2] VC VC EI= E √ [R2 + (XL – XC)2] VL - VC ΦZ = √ [R2 + (XL – XC)2] I VR E = √ [VR2 + (VL –Z=√ [R2 + (ω L – 1/ωC)2] VC)2] XL – XC ω L – 1/ωC tan Φ = or tan Φ = R R
  31. 31. XL – XC ω L – 1/ωC tan Φ = or tan Φ = R RSpecial Cases:Case I: When XL > XC i.e. ω L > 1/ωC, tan Φ = +ve or Φ is +ve The current lags behind the emf by phase angle Φ and the LCR circuit is inductance - dominated circuit.Case II: When XL < XC i.e. ω L < 1/ωC, tan Φ = -ve or Φ is -ve The current leads the emf by phase angle Φ and the LCR circuit is capacitance - dominated circuit.Case III: When XL = XC i.e. ω L = 1/ωC, tan Φ = 0 or Φ is 0° The current and the emf are in same phase. The impedance does not depend on the frequency of the applied emf. LCR circuit behaves like a purely resistive circuit.
  32. 32. Resonance in AC Circuit with L, C, R:When XL = XC i.e. ω L = 1/ωC, tan Φ = 0 or Φ is 0° and Z = √ [R2 + (ω L – 1/ωC)2] becomes Zmin = R and I0max = E / R i.e. The impedance offered by the circuit is minimum and the current is maximum. This condition is called resonant condition of LCR circuit and the frequency is called resonant frequency.At resonant angular frequency ωr, R1 < R2 < R3ωr L = 1/ωrC or ωr = 1 / √LC or fr = 1 / (2π √LC) I0Resonant Curve & Q - Factor: I0max Band width = 2 ∆ ω Quality factor (Q – factor) is defined as the R1 ratio of resonant frequency to band width. I0max / √2 Q = ωr / 2 ∆ ω It can also be defined as the ratio of potential R2 drop across either the inductance or the R3 capacitance to the potential drop across the resistance. 0 ωr ω Q = VL / VR or Q = VC / VR ωr - ∆ ω ωr + ∆ ω or Q = ωr L / R or Q = 1 / ωrCR
  33. 33. Power in AC Circuit with L, C, R:E = E0 sin ωtI = I0 sin (ωt + Φ) (where Φ is the phase angle between emf and current)Instantaneous Power = E I = E0 I0 sin ωt sin (ωt + Φ) = E0 I0 [sin2 ωt cosΦ + sin ωt cosωt cosΦ] If the instantaneous power is assumed to be constant for an infinitesimally small time dt, then the work done is dW = E0 I0 [sin2 ωt cosΦ + sin ωt cosωt cosΦ] Work done over a complete cycle is T W = ∫ E0 I0 [sin2 ωt cosΦ + sin ωt cosωt cosΦ] dt 0 W = E0I0 cos Φ x T / 2 Average Power over a cycle is Pav = W / T Pav = (E0I0/ 2) cos Φ (where cos Φ = R / Z Pav = (E0/√2) (I0/ √2) cos Φ = R /√ [R2 + (ω L – 1/ωC)2] is called Power Factor) Pav = Ev Iv cos Φ
  34. 34. Pav = Ev Iv cos Φ Wattless Current or Idle Current:Power in AC Circuit with R: EvIn R, current and emf are in phase. Iv cos Φ IvΦ = 0° ΦPav = Ev Iv cos Φ = Ev Iv cos 0° = E v Iv 90°Power in AC Circuit with L: Iv sin ΦIn L, current lags behind emf by π/2. The component Iv cos ΦΦ = - π/2 generates power with Ev.Pav = Ev Iv cos (-π/2) = Ev Iv (0) = 0 However, the component Iv sin Φ does notPower in AC Circuit with C: contribute to power along Ev and hence powerIn C, current leads emf by π/2. generated is zero. ThisΦ = + π/2 component of current is called wattless or idlePav = Ev Iv cos (π/2) = Ev Iv (0) = 0 current.Note: P = Ev Iv sin Φ cos 90°= 0Power (Energy) is not dissipated in Inductor and Capacitor and hence theyfind a lot of practical applications and in devices using alternating current.
  35. 35. L C Oscillations: L L +++++ + + + L C C C - - - - - - - -At t = 0, UE=Max. & UB=0 At t = T/8, UE = UB At t = 2T/8, UE=0 & UB=Max. L L L - - - - - - - - - - - C C C + + + +++++ + + + At t =3T/8, UE = UB At t = 4T/8, UE=Max. & UB=0 At t =5T/8, UE = UB L L + + + +++++ L C C - - - - - - - -CAt t = 6T/8, UE=0 & UB=Max. At t =7T/8, UE = UB At t =T, UE=Max. & UB=0
  36. 36. q0 q0 qq0 0 t t Undamped Oscillations Damped Oscillations If q be the charge on the capacitor at any time t and dI / dt the rate of change of current, then L dI / dt + q / C = 0 The final equation represents Simple or L (d2q / dt2) + q / C = 0 Harmonic Electrical Oscillation with ω as angular frequency. or d2q / dt2 + q / (LC) = 0 So, ω = 1 / √LC Putting 1 / LC = ω2 1 or f= d2q / dt2 + ω2 q =0 2π √LC
  37. 37. Transformer:Transformer is a device which converts lower alternating voltage at highercurrent into higher alternating voltage at lower current. Principle:Transformer is based onMutual Induction.It is the phenomenon ofinducing emf in the P S Loadsecondary coil due tochange in current in theprimary coil and hence thechange in magnetic flux inthe secondary coil.Theory:EP = - NP dΦ / dt For an ideal transformer, Efficiency (η):ES = - NS dΦ / dt Output Power = Input Power η = ESIS / EPIPES / EP = NS / NP = K ESIS = EPIP For an ideal transformer η(where K is called ES / EP = IP / IS is 100%Transformation Ratio ES / EP = IP / IS = NS / NPor Turns Ratio)
  38. 38. Step - up Transformer: Step - down Transformer: P S Load P S Load NS > NP i.e. K > 1 NS < NP i.e. K < 1 ES > EP & IS < IP ES < EP & IS > IPEnergy Losses in a Transformer:1. Copper Loss: Heat is produced due to the resistance of the copper windings of Primary and Secondary coils when current flows through them. This can be avoided by using thick wires for winding.2. Flux Loss: In actual transformer coupling between Primary and Secondary coil is not perfect. So, a certain amount of magnetic flux is wasted. Linking can be maximised by winding the coils over one another.
  39. 39. 3. Iron Losses: a) Eddy Currents Losses: When a changing magnetic flux is linked with the iron core, eddy currents are set up which in turn produce heat and energy is wasted. Eddy currents are reduced by using laminated core instead of a solid iron block because in laminated core the eddy currents are confined with in the lamination and they do not get added up to produce larger current. In other words their paths are broken instead of continuous ones. b) Hysteresis Loss: When alternating current is passed, the iron core is magnetised and demagnetised repeatedly over the cycles and some energy is being lost in the process. Solid Core Laminated Core This can be minimised by using suitable material with thin hysteresis loop.4. Losses due to vibration of core: Some electrical energy is lost in the form of mechanical energy due to vibration of the core and humming noise due to magnetostriction effect.
  40. 40. A.C. Generator: Q R S S R Q N N P S R1 S P R1 B1 B1 R2 R2 B2 B2 Load Load A.C. Generator or A.C. Dynamo or Alternator is a device which converts mechanical energy into alternating current (electrical energy).
  41. 41. Principle:A.C. Generator is based on the principle of Electromagnetic Induction.Construction:(i) Field Magnet with poles N and S(ii) Armature (Coil) PQRS(iii) Slip Rings (R1 and R2)(iv) Brushes (B1 and B2)(v) LoadWorking:Let the armature be rotated in such a way that the arm PQ goes down andRS comes up from the plane of the diagram. Induced emf and hencecurrent is set up in the coil. By Fleming’s Right Hand Rule, the directionof the current is PQRSR2B2B1R1P.After half the rotation of the coil, the arm PQ comes up and RS goes downinto the plane of the diagram. By Fleming’s Right Hand Rule, the directionof the current is PR1B1B2R2SRQP.If one way of current is taken +ve, then the reverse current is taken –ve.Therefore the current is said to be alternating and the corresponding waveis sinusoidal.
  42. 42. Theory: Φ = N B A cos θ ω At time t, with angular velocity ω, R θ = ωt (at t = 0, loop is assumed to be perpendicular to the magnetic field and θ = 0° ) Q θ B Φ = N B A cos ωt nDifferentiating w.r.t. t,dΦ / dt = - NBAω sin ωt SE = - dΦ / dtE = NBAω sin ωtE = E0 sin ωt (where E0 = NBAω) PE0 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ = ωt T/4 T/2 3T/4 T 5T/4 3T/2 7T/4 2T t

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