Unit 4bb kho tài liệu bách khoa

VERY SHORT AND SHORT-ANSWERS QUESTIONS 46 Draw the graphs for instantaneous voltage and current against time for RC series circuit 47 What is the impedence of a coil having resistance of Ω and reactance of Ω ? 48 Show graphically the variation of inductive and capacitive reactances with the frequency of applied voltage in a.c circuits (AISSCE 1994) 49 A bulb and a capacitor are connected in series to a source of alternating current What will happen on increasing the frequency of the source ? 50 A choke coil and a bulb are connected in series to an a.c source If an iron core is inserted in the choke coil, is there any change in the brightness of the bulb ? (AISSCE 1995) 51 In the above problem if the a.c source is replaced by a d.c source, than what is the change in brightness of the source ? 52 (a) How does the impedance of an L-R circuit change with the increase in the frequency of the ac source ? (b) What is the effect of increase in the source frequency on the impedence of a C-R circuit 53 A coil of area A and number of turns N rotates with a constant angular velocity ω, with its axis perpendicular to a magnetic field B Write an expression for the instantaneous e.m.f induced in the coil 54 What is the working principle of an a.c generator ? 55 What is the source of electrical energy obtained from a dynamo ? 56 For which position of a coil, rotating in a uniform magnetic field, is the induced e.m.f maximum ? 57 What is the use of a starter with a motor ? 58 What is the efficiency of a d.c motor ? S Chand & Company Limited 59 What is the principle of induction coil ? 60 An ordinary moving coil ammeter used for d.c can not be used to measure a.c even if its frequency is low Explain why ? (AISSCE 1994) 61 What is the condition of resonance in an LCR series circuit ? What is the impedance of the circuit at resonance ? 62 What is a choke ? 63 The domestic electric supply is 220 V-50 Hz What are the rms and the peak values of the voltage ? 64 Name the effect on which a.c meters are based ? 65 The instantaneous value of alternating current in a circuit is I = 14.14 sin 2πt What is the rms value of the current ? 66 A pure inductance is connected to a 220 V, 50 Hz source What is the phase difference between the current and the e.m.f in the circuit ? 67 What is the minimum value of power factor ? When does it occur ? 68 An ideal inductor, when connected to an a.c source, does not produce any heating effect Yet it reduces the current in the circuit Explain 69 A capacitor is used in the primary circuit of an induction coil Why ? 70 In an inductor, does the current rise to a steady value at a constant rate 71 When the current through an inductor is switched off, will the induced current be in the direction of the main current or opposite to it ? 72 Can we use a capacitor of suitable capacitance instead of a choke coil in an a c circuit ? 73 What is the effect on the current if the frequency of the a.c source is increased in the following circuits ? S Chand & Company Limited 74 75 76 77 78 79 80 81 (a) An a.c circuit containing a pure inductor (b) An a.c circuit containing a pure capacitor Why is a choke preferred to a rheostat in controlling the current in an a.c circuit ? How will the inductive reactance change on doubling the frequency of a.c ? Give the phase difference between the applied a.c voltage and the current in an LCR circuit at resonance The impedence of a coil in an a.c circuit is 141.4 Ω and its resistance is 100 Ω What is its reactance ? (AISSCE 1997) What is the flux linked with a coil rotating in a magnetic field when the current induced in the coil is maximum What energy transformation occurs in a d.c motor? What is the function of a starter in a motor ? (a) Show that the average value of a.c over one complete cycle is zero (b) Find the average value of a.c for the positive half cycle 82 Show that for an alternating current I rms = I0 83 An a.c voltage E = E0 sin ωt is applied across an inductance L Obtain an expression for the current I (AISSCE 1990) 84 An a.c voltage E = E0 sin ωt is applied across a capacitor of capacitance C Obtain an expression for the current 85 Derive an expression for the impedance of an a.c circuit with a capacitor and a resistor in series (AISSCE 93) S Chand & Company Limited 86 An inductor L and a resistor R are connected in series in an a.c circuit Derive an expression for the impedance Z of circuit Also derive an expression for the phase angle 87 In an a.c circuit, an inductance L, a capacitance C and a resistance R are connected in series Derive an expression for the impedance and the phase angle 88 Show that average power transferred to an a.c circuit is in general given by P = Erms Irms cos φ where the symbols have their usual meanings 89 What is a choke ? Explain its action in a.c circuits (AISSCE 1992 C) 90 What you understand by wattless current ? Show that the current flowing in an ideal choke coil is wattless ? ANSWERS 46 V = V0 sin ωt I = I0 sin (ωt + φ) V I V and I t S Chand & Company Limited 47 Z= R +X 2 L For R = Ω and XL = Ω, Z = +4 =5Ω 2 48 We know that XL = ωL = 2π ν L ⇒ XL ∝ ν [Fig B.5] XC = ωC ⇒ XC ∝ = 2π ν C [Fig B.6] ν 49 The impedance of the circuit is Z = R2 + (ω C ) With the increase in the frequency of the source, Z will decrease, which will result in the increase of current So the bulb will glow more brightly 50 When an iron core is inserted in the choke coil, its inductance L increases, so the current in the circuit decreases Therefore the brightness of the bulb decreases 51 No change Choke coil offers reactance to a.c only S Chand & Company Limited 52 (a) Impedance of L-R circuit is given by Z = 2 R +ω L With increase in frequency, ω increases and hence the impedance will increase (b) Impedance of C-R circuit is ( 2 Z = R +1/ ω C ) With increase in frequency of a.c impedance will decrease 53 Instantaneous induced e.m.f is given by E = E0 sin ωt, 54 55 56 57 where E0 = NBA ω Working principle of a.c generator When a coil is rotated in a magnetic field about an axis perpendicular to the field, the flux linked with the coil changes Due to this change in the magnetic flux, an e.m.f is induced and current flows in the coil The mechanical energy expanded in rotating the coil is obtained in the form of electrical energy The induced e.m.f in a rotating a coil is maximum when the plane of the coil is parallel to the magnetic field When a motor is switched on, the back e.m.f is negligible initially and so a large current would flow This would burn the windings To control this current, the starter, which is a variable resistance, is put in series with the motor 58 Efficiency of d.c motor, η = Back e.m.f Applied e.m.f 59 Induction coil is based on the principle of mutual induction 60 Due to inertia the needle will not follow the variation of alternating current even if the frequency S Chand & Company Limited is low 61 Resonance in LCR circuit takes place when the inductive reactance becomes equal to the capacitive reactance, i.e., XL = XC or ωL = 1/ωC At resonance, the impedance is purely resistive, i.e., Z = R 62 A choke is an inductor coil having large value of self inductance 63 RMS value of voltage = 220 V; Peak value of voltage = 200 V 64 Heating effect of current is used in a.c meters because it does not depend on the direction of current 65 I = 14.14 sin 2π t Peak value of current, I0 = 14.14 A RMS value of current, I rms = I0 = 14.14 14.14 = 10 A 66 In an a.c circuit containing a pure inductor, the current lags behind the voltage in phase by π/2 67 The minimum value of power factor is zero It occurs in a pure inductive or pure capacitive circuit 68 Heat is produced due to resistance An inductor offers nonresistive opposition to the flow of current, called reactance Therefore no heat is produced 69 The high induced voltage when the circuit is broken charges the capacitor This avoids sparking in the circuit 70 No, in an inductor the current rises as I = (E/R) (1– e –Rt/L) S Chand & Company Limited 71 The induced current will flow in the direction of the main current 72 Yes, like a choke coil, a pure capacitor consumes no power Therefore, it can be used to control the current without any appreciable power loss 73 (a) The current will decrease on increasing the frequency of a.c (b) The current will increase on increasing the frequency of a.c 74 A choke is preferred over a rheostat to control the current in an a.c circuit because it reduces the current without consuming any power 75 Inductive reactance XL = ωL = 2π ν L, i.e., XL ∝ ν Thus, on doubling the frequency, the reactance will be doubled 76 The phase difference is zero 77 Impedence Z = R + X L ∴ Z = 141.4 Ω, R = 100 Ω XL = ZL – R2 = (141.4)2 – 1002 = ( × 100) − 100 78 79 80 81 = 1002 or XL = 100 Ω The flux is zero In a d.c motor, electrical energy is converted into mechanical energy When the motor is switched on, the starter, which is a veriable resistance, does not allow the current to increase beyond safe limit Thus it prevents the burning of the armature winding of a motor (a) Average value of a.c for one complete cycle : The instantaneous value of a.c is given by S Chand & Company Limited I = I0 sin ωt I= Average value = T T ∫ I dt T ∫ I sin ωt dt T T I  cos ωt  I = −  = [cos 0° − cos ωT ] ω 0 ωT T  I0  π   = − cos  T    T  ωT  I = [1 − 1] ωT = Zero Thus, the average value of a.c over one complete cycle is zero (b) Average value of a.c over half cycle : = (T / 2) T /2 ∫ I sin ωt dt      cos ωt T / −2 T = I − I cos  2π T  − cos 0° T  T 2π  ω 0  T 2  2I = π = 82 Irms = I S Chand & Company Limited Now I = = T T ∫ I dt T 2 ∫ T (1 − cos ωt ) dt T {∫ 2T I0 ∫ I sin ωt dt T I = = T T dt − ∫ cos 2ωt dt } I  T sin ωt T  = (t ) −    2T   2ω 0  I0 = ⇒ 2T I rms {(T − 0) − 0} = I0 2 I = = 0.707 I 83 E = E0 sin ωt If I is the current in the circuit at some instant then the induced e.m.f is – L S Chand & Company Limited dI dt E−L dI or =0 dt E E dI = dt = sin ωt dt L L E0 I= ∫ sin ωt dt L E = − cos ωt ωL E = sin ( ωt − π / 2) ωL or I = I0 sin (ωt – π / 2), We have or where I0 = E0 / ωL This is the required expression 84 E = E0 sin ωt Let I be the current in the circuit and q be the charge on the capacitor at some instant At that instant q the potential difference across the capacitor is Thus, C E= q C or q = CE = CE0 sin ωt Differentiating both sides w.r.t time S Chand & Company Limited dq dt = d dt (CE0 sin ωt ) = CE0 ω cos ωt or I = CE0 ω sin (ωt + π / 2) or I = I0 sin (ωt + π / 2) where I0 = ωC E0 85 The fig B.7 shows a source of alternating e.m.f connected to a series combination of a capacitor C and a resistor R Let Irms be root mean square value of current in the circuit, VR be the rms voltage across the resistance which is in phase with the current, and VC be the rms value of voltage across the capacitor which lags behind the current in phase by π/2 In the given phase diagram (Fig B.8), OA and OB represent VR and VC respectively, and OC gives the resultant of VR and VC, i.e., Vrms Thus, 2 Vrms = VR + VC 2 2 or I rms = R + XC where Xc = V VR O Vrm s VC B ωC A φ = I rms R + I rms X C = I rms R + X C Vrms R C S Chand & Company Limited C Vrms / Irms is the impedance Z of the circuit Thus Z = R + 86 Fig 4B.9 shows a source of alternating e.m.f connected to a series combination of an inductor L and a resistor R Let Irms be the rms value of the current in the circuit, VR and VL be the rms values of voltages across R and L respectively In the phase diagram (Fig B.10) VR = Irms R is the voltage across the resistor It is in phase with the current and is represented by OA VL = Irms XL is the voltage across the inductor It leads the current in phase by π and hence it is represents by OB The resultant of OA and OB is represented by OC Thus OC represents Vrms So, 2 2 2 ω C L R Fig B.9 C B s VL V rm φ O Vrms = VR + V = I rms R + I rms X L , L where XL = ωL Thus, S Chand & Company Limited VR Fig B.10 A Vrms = I rms R + ω L or 2 Vrms / I rms = R + ω L 2 Vrms / Irms is the impedance of circuit Z.So, 2 Z = R +ω L Also, from the phase diagram, tan φ = ⇒ tan φ = AC VL = OA VR ωL R 87 Fig B.11 shows a source of alternating e.m.f connected to a series combination of an inductor L, a capacitor C and a resistor R Let Irms be the root mean square value of the current in the circuit and VL , VC and VR be the rms values of voltages across L, C and R, respectively L C VR and Irms are in same phase, so VR is represented by OA VL leads the current by π /2, so it is represented by OB S Chand & Company Limited Fig B.11 R VC lags behind the current by π /2, so it is represented by OC B VL F D (VL – VC) Let VL > VC Then OD represents (VL – VC) OF is the resultant of OD and OA From the figure we have φ O Vrms = VR + (VL − VC ) where VC C VR = Irms R, Fig B.12 VL = Irms XL, VC = Irms XC, XL = ωL and XC = Vrms So, ωC 2 = I rms R + ( I rms X L − I rms X C ) = I rms VR A R 2 + ( X L − X C )2 Vrms / I rms = R + (ωL − / ωC ) Vrms / Irms is called the impedance of the circuit, Z Thus, S Chand & Company Limited Z = R2 + (ωL − 1/ ωC)2 If φ is the phase angle, then tan φ = AF OA = VL − VC VR  ωL − 1/ ωC     R tan φ =  or 88 The instantaneous e.m.f in an a.c circuit is given by E = E0 sin ωt Instantaneous current I = I0 sin (ωt – φ) Instantaneous Power P = EI Average power P = = = = T T ∫0 E0 I T E0 I T T T ∫0 EI dt E sin ωt I sin (ωt − φ) dt 0 T ∫ sin ωt sin (ωt − φ) dt T ∫ sin ωt (sin ωt cos φ − cos ωt sin φ) dt .(1) S Chand & Company Limited T Now ∫ sin ωt (sin ωt cos φ − cos ωt sin φ) dt T T = ∫ sin ωt cos φ dt − ∫ sin ωt cos ωt sin φ dt 0 T = cos φ ∫ sin ωt dt − = cos φ ∫ T sin φ T ∫ sin ωt cos ωt dt (1 − cos ωt ) sin φ dt − 2 T ∫ sin 2ωt dt T cos φ  T sin φ T dt − ∫ cos 2ωt dt  − ∫ ∫ sin ωt dt  0  T T cos φ  T  = dt since ∫ cos ωt dt = ∫ sin ωt dt = ∫   0 cos φ T cos φ (t ) = = T 2 = Eq (1) becomes E I cos φ T T E I = 0 cos φ = E I cos φ rms rms 2 P= 89 A coil having a high value of inductance and low resistance is called a choke It is made of a large number of turns of thick insultated coper wire S Chand & Company Limited It is used in a.c circuits to reduce the value of current because it consumes very little power We know that the power consumed by a pure inductor is zero If instead of a choke coil we use a resistor to reduce the current, then there will be substantial power loss due to joule heating 90 Wattless current - See Q 41 Average power consumed in an ideal choke coil Instantaneous Power P = VI In a pure inductor the current lage behind the voltage by π/2 Therefore, P = V0 I0 sin ωt sin (ωt – π/2) = – V0 I0 sin ωt cos ωt =– V0 I sin 2ωt Average power p= =− T T ∫0 P dt V0 I T T ∫0 sin ω tdt = This shows that the current in an ideal choke is wattless S Chand & Company Limited