NEET Physics Notes - Chapter 24 - Alternating Current PDF

60 1350 Alternating Current Chapter E3 24 Alternating Current (1) An alternating quantity (current i or voltage V) is one whose magnitude changes continuously with time between zero and a maximum value and whose direction reverses periodically. i or V + + t D YG i or V t – – Sinusoidal Triangular i...

60 1350 Alternating Current Chapter E3 24 Alternating Current (1) An alternating quantity (current i or voltage V) is one whose magnitude changes continuously with time between zero and a maximum value and whose direction reverses periodically. i or V + + t D YG i or V t – – Sinusoidal Triangular i or V + + t – t Rectangular ac super imposed on dc U Fig. 24.1 current or voltage varying as (3) Equation for i and V : Alternating sine function can be written as 2 i = i sint = i sin 2 t = i sin t T 0 and V  V0 sint  V0 sin 2t  V0 sin where i and V are Instantaneous values of current and voltage, i and V are peak 0 values of voltage 0 current and  = Angular frequency in rad/ sec,  = Frequency in Hz and T = time period 2 t T V0 or i0 i or V T Fig. 24.2 (1) Peak value (i or V ) : The maximum value of alternating quantity (i or V) is defined as peak value or amplitude. 0 0 (2) Mean square value ( V 2 or i 2 ) : The average of square of instantaneous values in one cycle is called mean square value. It is always positive for one complete cycle. e.g. V 2  1 T  T V 2 dt  0 V02 i2 or i2  0 2 2 rms i12  i22 ...  i  2 n 2 – T/2 Important Values of Alternating Quantities irms   i dt   dt 0 T 2 i0 = 0.707 i = 70.7% i 0 2 0 t or  T/4 value. T  0 T from the beginning, i or V reaches to their maximum 4 rms Positive half cycle + (v) At t  (3) Root mean square (r.m.s.) value : Root of mean of square of voltage or current in an ac circuit for one complete cycle is called r.m.s. value. It is denoted by V or i 0 ST 0 (iii) The value of alternating quantity is zero or maximum 2  times every second. The direction also changes 2 times every second. (iv) Generally sinusoidal waveform is used as alternating current/voltage. U (2) Some graphical representation for alternating quantities i or V (i) The time taken to complete one cycle of variations is called the periodic time or time period. (ii) Alternating quantity is positive for half the cycle and negative for the rest half. Hence average value of alternating quantity (i or V) over a complete cycle is zero. ID Alternating Quantities (i or V) Negative half cycle Similarly Vrms  V0 2  0.707 V0  70.7 % of V 1  2 2  sin ( t)  cos ( t)  2    0 0 Alternating Current (i) The r.m.s. value of alternating current is also called virtual value or effective value. Similarly Vav  (ii) In general when values of voltage or current for alternating circuits are given, these are r.m.s. value. Physical quantity which represents both the instantaneous value and direction of alternating quantity at any instant is called it's phase. It's a dimensionless quantity and it's unit is radian. av E3 If an alternating quantity is expressed as X  X 0 sin( t   0 ) then the argument of sin( t   ) is called it's phase. Where  t = instantaneous phase (changes with time) and w.r.t. time) T /2  T /2 dt  0.637i0  63.7 % of i ,  0 0 ID 0 60 Phase The average value of ac over half cycle (t = 0 to T/2) iav 0 (6) Form factor and peak factor : The ratio of r.m.s. value of ac to it's average during half cycle is defined as form factor. The ratio of peak value and r.m.s. value is called peak factor 2  200  311V. 2i0 0  2 2 Vrms  2.828 Vrms (4) Mean or Average value (i or V ) : The average value of alternating quantity for one complete cycle is zero. i dt 0 0 (v) r.m.s. value of ac is equal to that value of dc, which when passed through a resistance for a given time will produce the same amount of heat as produced by the alternating current when passed through the same resistance for same time.     0.637 V0  63.7% of V.  Peak to peak value = V + V = 2V (iv) In our houses ac is supplied at 220 V, which is the r.m.s. value of av  (5) Peak to peak value : It is equal to the sum of the magnitudes of positive and negative peak values (iii) ac ammeter and voltmeter are always measure r.m.s. value. Values printed on ac circuits are r.m.s. values. voltage. It's peak value is 2 V0 1351 0 = initial phase (constant Table 24.1 : Some important values Nature of wave form value i or V  i0 2 D YG + average value 0 Form factor U Sinusoidal r.m.s. Wave form 2 Rf  2 i0  r.m.s. value Average value   1.11 2 2 Peak factor Rp  Peak value r.m.s. value 2  1.41 – Half wave rectified + + ST  Square or Rectangular i0  i0 2i0  2  1.57 2 i or V U Full wave rectified i0 2 2  2  2 2 i0 i0 1 2 i or V + – (1) Phase difference (Phase constant) : The difference between the phases of currents and voltage is called phase difference. If alternating voltage and current are given by V  V0 sin( t  1 ) and i  i0 sin( t   2 ) then phase difference  =  –  (relative to current) 1 or    2  1 (relative to voltage) 1 2 (2) Time difference : If phase difference between alternating current and voltage is  then time difference between them is given as T.D.  T  2 1352 Alternating Current (3) Phasor diagram : A diagram representing alternating current and alternating voltage (of same frequency) as vectors (phasors) with the phase angle between them is called a phasor diagram. While drawing phasor diagram for a pure element (e.g. R, L or C) either of the current or voltage can be plotted along X-axis. But when phasor diagram for a combination of elements is drawn then quantity which remains constant for the combination must be plotted along X-axis so we observe that In dc circuits power is given by P = Vi. But in ac circuits, since there is some phase angle between voltage and current, therefore power is defined as the product of voltage and that component of the current which is in phase with the voltage. Thus P  V i cos  ; where V and i are r.m.s. value of voltage and current. (1) Instantaneous power : Suppose in a circuit V  V0 sin t and (ii) In parallel circuits voltage has to be plotted along X-axis. (2) Average power (True power) : The average of instantaneous power in an ac circuit over a full cycle is called average power. It's unit is watt i.e. Measurement of Alternating Quantities Alternating current shows heating effect only, hence meters used for measuring ac are based on heating effect and are called hot wire meters (Hot wire ammeter and hot wire voltmeter) ac measurement dc measurement (1) All dc meters read average value (2) All ac meters are based on heating effect of current. (2) All dc meters are based on magnetic effect of current (3) Deflection in hot wire meters (3) Deflection in dc meters  i 2   irms. i0 cos   V2 R 1 2 V0 i0 cos   irms R  rms2 2 Z 2 2 (3) Apparent or virtual power : The product of apparent voltage and apparent current in an electric circuit is called apparent power. This is Vi always positive Papp  Vrms irms  0 0 2 Power Factor (1) It may be defined as cosine of the angle of lag or lead ( i.e. cos ) (2) It is also defined as the ratio of resistance and impedance ( i.e. R ) Z ID (1) All ac meters read r.m.s. value. V0 E3 Table 24.2 : Measurement of ac and dc Pav  Vrms irms cos   60 (i) In series circuits current has to be plotted along X-axis. i  i0 sin( t   ) then Pinstantane ous  Vi  V0 i0 sin t sin( t   ) True power W kW    cos  Apparent power VA kVA U (3) The ratio Resistive Circuit (R-Circuit) (Linear scale) (non-linear scale) R (1) Current : i  i0 sin t D YG Impedance, Reactance, Admittance and Susceptance (1) Impedance (Z) : The opposition offered by ac circuits to the flow of ac through it is defined it's impedance. It’s unit is ohm(). (2) Reactance (X) : The opposition offered by inductor or capacitor or both to the flow of ac through it is defined as reactance. It is of following two type (i) Inductive reactance (X ) : Offered by inductive circuit X L  L  2L  dc  0 so for dc, X = 0. (2) Peak current : i0  V0 R i (3) Phase difference between voltage and current :  = 0 V  V0 sin t o (4) Power factor : cos   1 Fig. 24.3 L L Capacitive reactance (X ) : 1 1 for dc X = .   C 2C XC Offered U C by capacitive circuit 1  known as admittance  Y  . It’s unit is mho Z  (4) Susceptance (S) : the reciprocal of reactance is defined as 1  susceptance  S  . It is of two type X   (i) inductive susceptance S L  1 1  and X L 2 L (ii) Capacitive susceptance, S C  Power in ac Circuits (6) Time difference : T.D. = 0 V Reciprocal of impedance is ST V0 Vrms  i0 irms V0 i0 2 (7) Phasor diagram : Both are in same phase C (3) Admittance (Y) : Z  (5) Power : P  Vrms irms  1   C  2 C. XC i Fig. 24.4 Inductive Circuit (L-Circuit) L   (1) Current : i  i0 sin   t   2  (2) Peak current : i0  V0 V V0  0  XL L 2L (3) Phase difference between i V  V0 sin t Fig. 24.5 Alternating Current  voltage and current   90 o (or  2 (4) Peak current i0  ) (4) Power factor : cos   0 V0  Z V0 R  2  2 (7) Leading quantity : Voltage Resistive, Capacitive Circuit (RC-Circuit) V V 90o or 90o i R VR   (1) Current : i  i0 sin  t   2  C ID U (4) Peak current : i0  ) D YG T 4 i 90o 90o V0 R  2 ST i R XC 1  tan 1 R CR (6) Power factor : cos   R L C VL VC i R  X C2 VL V = iX , V = iX L L C C (1) Applied voltage : V  (1) Applied voltage : V  VL  VC VR  VL2 (2) Impedance : Z  R 2  X L2  R 2   2 L2  R 2  4 2 2 L2 (3) Current : i  i0 sin  t    90o Fig. 24.11 L Fig. 24.9 VR2 V= (VL – VC) V  V0 sin  t V VL  V  V0 sin  t 1 4  C 2 2 2 VC V = iR , V = iX L R2  (5) Phase difference :   tan 1 L VL V0  X C2 V Resistive, Inductive Circuit (RL-Circuit) VR V0  Z Inductive, Capacitive Circuit (LC-Circuit) Fig. 24.8 R 2 (7) Leading quantity : Current or U V Fig. 24.10 2 (7) Phasor diagram : Current leads the voltage by /2 i V (3) Current : i  i0 sin  t    Fig. 24.7 (4) Power factor : cos   0 (6) Time difference : TD  VC  1  (2) Impedance : Z  R 2  X C2  R 2     C  V  V0 sin t (3) Phase difference between 2 i C (1) Applied voltage : V  VR2  VC2 V0  V0 C  V0 (2 C) XC  C V  V0 sin  t i (2) Peak current : VC  E3 i voltage and current :   90 o (or  VR V = iR, V = iX R Capacitive Circuit (C-Circuit) (5) Power : P = 0 C i Fig. 24.6 i0  R 2  X L2 60 (7) Phasor diagram : Voltage leads the current by R  4 2 2 L2 R (6) Power factor : cos   T 4 V0 2 L XL  tan 1 R R (5) Phase difference :   tan 1 (5) Power : P = 0 (6) Time difference : T.D.   X L2 1353 i (2) Impedance : Z  X L  X C  X   (3) Current : i  i0 sin  t   2  (4) Peak current : i0  V0 V0 V0   1 Z X L  XC  L C (5) Phase difference :  = 90 o i 1354 Alternating Current (6) Power factor : cos   0 (10) Half power frequencies and band width : The frequencies at which the power in the circuit is half of the maximum power (The power at resonance), are called half power frequencies. (7) Leading quantity : Either voltage or current Series RLC-Circuit L R i VL VC Pmax i VR V = V0 sint i (1) Equation of current : i  i0 sin( t   ) ; where i0  1 E3 (b)  2 called upper half power frequency. It is greater than  0. At this frequency the circuit is inductive. 2 (iii) Band width () : The difference of half power frequencies  1 and  2 is called band width () and   2  1. For series resonant 2 R circuit it can be proved     L (4) Phase difference : From phasor diagram (11) Quality factor (Q-factor) of series resonant circuit 1 1  L 2 L  C 2 C   R R (i) The characteristic of a series resonant circuit is determined by the quality factor (Q - factor) of the circuit. U VL  VC X  XC  L VR R ID 1   Z  R 2  ( X L  X C )2  R 2   L   C   (5) If net reactance is inductive : Circuit behaves as LR circuit D YG (6) If net reactance is capacitive : Circuit behave as CR circuit (ii) It defines sharpness of i -  curve at resonance when Q - factor is large, the sharpness of resonance curve is more and vice-versa. (iii) Q - factor also defined as follows (7) If net reactance is zero : Means X  X L  X C  0 Q - factor  2   X = X. This is the condition of resonance (8) At resonance (series resonant circuit) (i) X = X  Z = R i.e. circuit behaves as resistive circuit L C min (ii) V = V  V = V i.e. whole applied voltage appeared across the resistance L C R (iii) Phase difference :  = 0  p.f. = cos  = 1 U o (iv) Power consumption P = V i  rms rms  Max. energy stored Energy dissipation 2 Max. energy stored Resonant frequency  0    T Mean power dissipated Band width  (iv) Q - factor   Q - factor  1 V0 i0 2 VL V  L 1 or C  0 or VR VR R 0 CR 1 R L C R=0 Q - factor = Infinity i (v) Current in the circuit is maximum and it is i0  ST  (a)  1 called lower half power frequency. At this frequency the circuit is capacitive. V0 Z (3) Impedance of the circuit : C 0 2 Fig. 24.13 (2) Equation of voltage : From phasor diagram L 2 (ii) There are two half power frequencies Fig. 24.12 tan   V0 R R = Very low Q- factor = Large R = low Q- factor = Normal R = High Q- factor = Low  (vi) These circuit are used for voltage amplification and as selector circuits in wireless telegraphy. (9) Resonant frequency (Natural frequency) 1 2 LC 1 0 C  0  0 1 rad LC s e c Hz (or cps) (Resonant frequency doesn't depend upon the resistance of the circuit) Resonance curve Parallel RLC Circuits iR  V0  V0 G R V = V0 sint At resonance X L  X C   0 L   0  Pmax 60 Phasor diagram  (VL  VC ) P P VC VR = iR, VL = iXL, VC = iXC V or 0.707 or 70.7% of maximum current (current at resonance). V (VL – VC)  VR2 1 2 C VL VR (i) The current in the circuit at half power frequencies (HPF) is Fig. 24.14 i iR R iL L Fig. 24.15 iC C Alternating Current V0  V0 S L XL (e) Quality factor of the circuit  (1) Current and phase difference : From phasor diagram current i  iR2  (iC  iL )2   tan 1 and phase difference iC (iC  iL ) (S  S L )  tan 1 C iR G (ii) If inductance has no resistance : If R = 0 then circuit becomes parallel LC circuit as shown i L  C  V iL (2) Admittance (Y) of the circuit : From equation of current iR Fig. 24.18 2 Condition of resonance : iC  iL  2    G 2  (S L  S C )2   ID av or (i) The average of wattless component over one cycle is zero (ii) Amplitude of wattless current = i sin 0 and r.m.s. value of wattless current = irms sin   C i0 sin. 2 V i cos L R Hz U D YG 1 2 LC (4) Parallel LC circuits : If inductor has resistance (R) and it is connected in parallel with capacitor as shown (i) At resonance 2 LC The component of current which does not contribute to the average power dissipation is called wattless current (iv)   0  p.f. = cos = 1 = maximum (v) Resonant frequency    1 In an ac circuit R = 0  cos = 0 so P = 0 i.e. in resistance less circuit the power consumed is zero. Such a circuit is called the wattless circuit and the current flowing is called the wattless current.  S C  S L  S  0 V R iR impedance is infinite. Resonant frequency :  0  Wattless Current (3) Resonance : At resonance (i) iC  iL  imin  iR (iii) Z max  V V  XC XL  X C  X L. At resonance current i in the circuit is zero and  1 1 1 1  Y        Z R  XL XC 2 V V  (ii) XC X L V = V0 sint Fig. 24.16 V iL E3 2 V V      0  0  X X  C   L i iC iR V0 V   0 Z  R 1. In the state of R2 1  LC L2 resonance the quality factor of the circuit is equivalent to the current amplification of the circuit. V0  V0 S C XC iC  1. CR 60 iL  1355 i U i 1 Y min  L CR It is quadrature (90 ) with voltage. o (b) Current through the circuit is minimum and imin (c) S L  S C  (d) 0  1 2 Resonant Fig. 24.19 Fig. 24.17 ST (a) Z max  Choke Coil V CR  0 L Choke coil (or ballast) is a device having high inductance and negligible resistance. It is used to control current in ac circuits and is used in fluorescent tubes. The power loss in a circuit containing choke coil is least. 1 1   X  XL XC frequency i sin  V = V0 sint 0  1 R 2 rad  2 LC L sec R2 1  2 Hz (Condition for parallel resonance is R  LC L or Iron core L ) C Starter Coil of Cu wire Application of choke coil Choke coil Fig. 24.20 Choke coil L, R 1356 Alternating Current (1) It consist of a Cu coil wound over a soft iron laminated core. (2) Thick Cu wire is used to reduce the resistance (R) of the circuit. (3) Soft iron is used to improve inductance (L) of the circuit. (4) The inductive reactance or effective opposition of the choke coil is given by X =  L = 2 L L (5) For an ideal choke coil r = 0, no electric energy is wasted i.e. average power P = 0. 60 (6) In actual practice choke coil is equivalent to a R – L circuit. (7) Choke coil for different frequencies are made by using different substances in their core.  ID E3 For low frequency L should be large thus iron core choke coil is used. For high frequency ac circuit, L should be small, so air cored choke coil is used. If ac is produced by a generator having a large number of poles then it's frequency  Number of poles  rotation per second P n  2 2 U Where P is the number of poles; n is the rotational frequency of the coil.  Alternating current in electric wires, bulbs etc. flows 50 times in D YG one direction and 50 times in the opposite direction in 1 second. Since in one cycle the current becomes zero twice, hence a bulb lights up 100 times and is off 100 times in one second (50 cycles) but due to persistence of vision, it appears lighted continuously.  ac is more dangerous than dc.  The rate of change of ac is minimum at that instant when they are near their peak values.  ac equipments such as electric motors, are more durable and convenient compared to dc equipments. U  Skin Effect A direct current flows uniformly throughout the cross-section of the conductor. An alternating current, on the other hand, flows ST mainly along the surface of the conductor. This effect is known as skin effect. the reason is that when alternating current flows through a conductor, the flux changes in the inner part of the conductor are higher. Therefore the inductance of the inner part is higher than that of the outer part. Higher the frequency of alternating current, more is the skin effect.  I =0 ac The depth upto which ac current flows through a wire is called skin depth ().