UNESCO-NIGERIA UNESCO TECHNICAL & VOCATIONAL EDUCATION REVITALISATION PROJECT-PHASE PROJECT II NATIONAL DIPLOMA IN ELECTRICAL ENGINEERI ENGINEERING NG TECHNOLOGY ELECTRICAL CIRCUITS (II II) COURSE CODE: CODE EEC 249 YEAR III SEMESTER IV THEORY Version 1: December 2008 TABLE OF CONTENTS Department Electrical Engineering Technology Subject Electric Circuit Theory(I I) Year 2 Semester 4 Course Code EEC 249 Credit Hours 2 Theoretical 1 Practical 2 CHAPTER • 1 : Power in A.C. Circuit Assessment CHAPTER 2 Assessment • Laboratory • Mid term Test1 3 Assessment • Laboratory 2 4 2 Weeks 4 – 7 : Time Domain Analysis • CHAPTER Week 1-3 : Three Phase Systems • CHAPTER 1 3 Weeks 8 - 11 : Magnetic Coupling • Assessment • Laboratory 3 4 Weeks 12 –15 2 This Page is Intentionally Left Blank 3 Week 1 1.Power in A.C Circuit At the end of this week, the students are expected to: ♦ Calculate power in a.c circuits containing: ♦ Resistance ♦ Inductance ♦ Capacitance ♦ Combination of the resistance, inductance and capacitance 1.1 POWER IN A.C CIRCUIT 1.1.1 Power in A.C circuit containing Resistance only + V mI m i P(t) Average power V mI m 2 + e R _ t 0 i(t) _ v(t) Fig 1.1: (a) pure resistive circuit, (b) power wave form. (b) Inspection of the power waveform of fig 1.1 shows that its average value lies half way between zero and its peak value of VmIm. That is P = VmIm/2 Since V (the magnitude of the r.m.s value of voltage) is Vm/√2 and I (the magnitude of the r.m.s value of current) is Im/√2, this can be written as P = VI. Thus, power (average power) to a purely resistive load is P = VI (watts). (1.1) Example 1.1: Calculate the power dissipated by the circuit of fig 1.2 + 100V I R = 25Ω _ Fig 1.2 4 Solution I = 100V/25Ω = 4A ∴ P = VI = 100 x 4 = 400W 1.1.2 Power in A.C Circuit Containing Inductance only For a purely inductive load as in fig 1.3(a), current lags voltage by 900. A sketch of P versus time (obtained by multiplying V times i) then looks as shown in fig 1.3(b). + i PL(t) VI i(t) v(t) + PL V _ 0 t (a) Energy stored _ Energy released Energy stored Energy released Fig 1.3: (a) pure inductive circuit (b) power waveform(b) for a purely inductive circuit Consider fig 1.3. Energy stored during each quarter-cycle is returned during the next quarter cycle. Thus, the average power is zero. Consequently, the only power flowing in the circuit is reactive power. This is given by QL = VI (VAr) (1.2) Example 1.2: For the circuit of fig 1.4, determine the reactive power + 100V I XL = 20Ω _ Solution I = 100V/20Ω = 5A Fig 1.4 ∴ QL = VI = 100 x 5 = 500VAr 5 1.1.3 Power in A.C circuit containing capacitance only For a purely capacitive load current leads voltage by 900. Multiplications of V times i yield the power curve of figure 1.5 + PC(t) i v(t) VI i(t) + PC _ 0 V (a) -VI t (b) Energy released Energy released Energy stored Energy stored (b) Fig 1.5: (a) pure capacitive circuit (b) power waveform for a pure capacitive circuit. Consider fig 1.5. Energy stored during each quarter-cycle is returned during the next quarter cycle. Thus, the average power is zero. Consequently, the only power flowing in the circuit is reactive power. This is given by QC = I2XC (1.3) Example 1.3: With regard to fig 1.6, determine average and reactive power. + 100V I XC = 40Ω _ Fig 1.6 Solution I = 100V/40Ω = 2.5A. QC = VI = 100 x 2.5 = 250VAr. P = 0W. 6 1.1.4 Calculations of power in A.C circuits containing R, L and C Example 1.4: For the RL circuit of figure 1.7, I = 5A. Find the power and reactive power. XL = 4Ω R = 3Ω P 5A Q Fig 1.7 Solution P = I2R = (5)2 (3) = 75W QL = I2XL = (5)2 (4) = 100VAr Example 1.5: For the RC circuit of figure 1.8, determine the power consumed and the reactive power. + 40V _ 20Ω R 80Ω XC Fig 1.8 Solution P = V2/R = (40)2/20 = 80W QL = V2/XC = (40)2/80 = 20VAr Example 1.6: A 10Ω resistor, a 100µF capacitor and an inductor of 0.15H are connected in series to a supply at 230V, 50Hz. Calculate the power consumed by the circuit. Solution XL = 2πf L = 2π x 50 x 0.15 = 47.1Ω XC = 1 = 1 = 31.83Ω 2πfC 2π x 50 x 100 x 10-6 7 Z = √{R2 + (XL – XC)2} = √{102 + (47.1 – 31.83)2} = 18.28Ω I = VS = 230 = 12.58A Z 18.28 Power consumed, P = I2R = (12.58)2 x 10 = 1582W 8 Power in A.C Circuit Week 2 At the end of this week, the students is expected to: ♦ Explain power factor and factors affecting its value ♦ Explain methods of power factor correction 1.2 POWER FACTOR AND FACTORS AFFECTING ITS VALUE 1.2.1 Power Factor Power factor is the ratio of the real power dissipated in the load to the apparent power of the load. Thus, P.f = cosθ = P/S (1.4) Angle θ is the angle between voltage and current. For a pure resistance, therefore, θ = 00. For a pure inductance, θ = 900; for a pure capacitance, θ = -900. For a circuit containing resistance and inductance, θ will be somewhere between 00 and 900; for a circuit containing resistance and capacitance, θ will be somewhere between 00 and 900. 1.2.2 Factors Affecting the Value of Power Factor 1. Increase in supply voltage Due to increase in supply voltage, which usually occurs during low loads period, the magnetizing current of inductive reactance increase and power factor of the plant as a whole comes down. 2. Improper Maintenance and Repairs The power factor at which motors operates falls due to improper maintenance and repairs of motors. 3. Industrial Heating of Furnance Induction furnances operates at a very low lagging power factor due to heating of the furcance. 1.3 METHODS OF POWER FACTOR CORRECTION 1. Synchronous Motors These machines draw leading KVAr when they are over excited and, especially, when they are running idle. They are employed for correcting of the power factor in bulk and have the special advantage that the amount of correction can be varied by changing their excitation. 9 2. Static Capacitors They are installed to improve the power factor of a group of a.c motors and are practically loss-free. Since their capacitances are not variable, they tend to overcompensate on light loads, unless arrangements for automatic switching off the capacitor bank are made. 3. Phase Advancers The power factor of induction motors is being improved by equipping the machines with a phase advancer, which supply exciting current to the motor circuit. With the arrangement, the phase angle between the supply voltage and current would be decrease, thereby increasing the power factor. 10 Week 3 Power Calculation in A.C Circuit At the end of this week, the students are expected to: ♦ Explain apparent power, reactive power and active power ♦ Solve problems on power factor, active power apparent power reactive power and power factor 1.4 APPARENT POWER, REACTIVE POWER AND ACTIVE POWER 1.4.1 Apparent power I + S load V _ Fig 1.9: Apparent power S = VI When a load has voltage V across it and current I through it as in figure 1.9, the power that appears to flow to it is VI. However, if the load contains both resistance and reactance, this product represents neither active power nor reactive power. Since VI appears to represent power, it is called apparent power. Apparent power is given the symbol S and units of volt-amperes (VA). Thus S = VI (VA) (1.5) where V and I are the magnitude of the r.m.s voltage and current respectively. 1.4.2 Reactive Power Consider figure 1.3. During the intervals that instantaneous power PL(t) is negative, power is being returned from the load. (This can only happen if the load contains reactive elements: L or C.) The portion of power that flows into the load then back out is called reactive power. Reactive power is given by the symbol Q and units of voltampere reactive (VAr). Thus Q = VIsinθ (1.6) 1.4.3 Active power Consider again figure 1.3. Since PL represents the power flowing to the load; its average will be the average power to the load. Denote this average by the letter P. If P has a positive value, it represents the power that is really dissipated by the load. For 11 this reason, P is called Real power. In modern terminology, real power is also called active power. The unit of active power is in watt (W). Thus P = VIcosθ (W) 1.5 (1.7) SOLVED PROBLEMS ON POWER FACTOR, ACTIVE POWER, APPARENT POWER, REACTIVE POWER AND POWER FACTOR CORRECTIONS Example 1.7: (a) In figure 1.10, find the following: The active power (b) the apparent power (c) the reactive power and (d) the power factor. R = 8Ω XL = 7Ω XC = 15Ω V = 100 Fig 1.10 Solution Z = √{R2 + (XC – XL)2} = √{82 + (15 – 7)2} = 11.314Ω I = E = 100 = 8.84A Z 11.314 From the phasor diagram of RLC series circuit shown in fig 1.11 VL V V L - VC φ VR I Fig 1.11 VC φ = cos-1 (VR/V), where VR = IR = 8.84 x 8 = 70.72V 12 ∴ φ = cos-1 (70.72/100) = 450 (a) P = VIcosφ = 100 x 8.84 x cos450 = 625W (b) S = VI = 100 x 8.84 = 884VA (c) Q = VIsinφ = 100 x 8.84 x sin450 = 625VAr (d) P.f = cosφ = cos450 = 0.707 Example 1.8: A resistor of 40Ω is connected in parallel with a 67.6mH inductor, the combination being supplied by a 120V, 50Hz supply. Calculate (a) the power factor (b) the active power (c) the apparent power (d) the reactive volt-amperes consumed Solution IR = VS/R = 120/40 = A 120 = 5.65A IL = VS = -3 2πfL 2π x 50 x 67.6 x 10 I = √(IR2 + IL2) = √(32 + 5.652) = 6.397A From the phasor diagram of RL parallel circuit (fig 1.12) IR VS φ IL I Fig 1.12 (a) P.f = cosφ = IR/I = 3/6.397 = 0.4690 (b) P = VIcosφ = 120 x 6.397 x 0.4690 = 360W (c) S = VI = 120 x 6.397 = 767.64VA (d) Q = VIsinφ, φ = cos-1(0.469) = 620 = 120 x 6.397 x sin620 = 677.8VAr Example 1.9: A load of P = 1000KW with p.f = 0.5lagging is fed by a 5KV, 50Hz source. A capacitor is added in parallel such that the power factor is improved to 0.8. Find the value of the shunt capacitance needed to improve the power factor 13 Solution The value of the shunt capacitance is given by C = P(tanφ1 - tanφ2) 2πfV2 Before improvement. P = 1000KW, cosφ1 = 0.5, ⇒ φ1 = cos-1(0.5) = 600 ∴ Ptanφ1 = 1000K x tan600 = 1732.05KVAr ∴ C = 1732.05 – 750.82 ≅ 125µF 2π x 50 x (5000)2 14 Week 4 Three Phase Systems At the end of this week, the students are expected to: ♦ Define polyphase system ♦ Explain the basic differences between single phase and three phase systems ♦ Explain phase sequence of a three phase system ♦ State the advantages of three phase circuits. 2.1 POLYPHASE SYSTEM Circuits or system in which the ac sources operate at the same frequency but different phases are known as polyphase systems. 2.2 BASIC DIFFERENCES BETWEEN SINGLE PHASE AND THREE PHASE SYSTEMS 2.2.1 Single-Phase Systems A single phase a.c power system consists of a generator connected through a pair of wires (a transmission line) to a load. Figure 2.1 depicts a single phase two wire system, where VP is the magnitude of the source voltage and φ is the phase. The power in a single phase system is pulsating (not constant). For large motors, pulsating power supply causes excessive vibration. Besides, it has neither delta nor star type of connection. Vp ∠Φ +_ ZL Fig 2.1: single-phase system two-wire type. 2.2.2 Three Phase Systems Figure 2.2 shows a three phase four wire systems. As distinct from a single phase system, a three phase system is produced by a generator consisting of three sources having the same amplitude and frequency but out of phase with each other by 1200. When one phase or two phase inputs are required, they are taken from the three phase system rather than generated independently. The instantaneous power in a three phase system can be constant. This result in uniform power transmission and less vibration of three phase machines. Generally, three-phase systems have two type of connection, which are star and delta connections 15 Vp 00 _ + Vp -12 00 _ + a A ZL1 b B ZL2 c C ZL3 Vp +12 00 _ + n N Fig 2.2: Three phase four wire-systems 2.3 PHASE SEQUENCE OF A THREE PHASE SYSTEM Consider the waveform of three phase e.m.f shown in fig 2.3. From the waveform it is seen that the e.m.f of phase A lead that of B by 1200, and in similar way, the e.m.f of phase B lead that of C by 1200. Hence, the order in which the e.m.fs of phases A, B and C attain their maximum values is ABC. It is called phase sequence ABC. Therefore, Phase sequence is the order in which the three phases (A,B and C) reach their maximum value. e A B C ωt 120 0 2400 Fig 2.3: three phase waveforms 16 2.4 ADVANTAGES OF 3-PHASE CIRCUITS 1. The instantaneous power in a three phase system can be held constant. This results in uniform transmission. 2. Three phase systems are more economical than single phase systems. 3. Three phase systems are more efficient 17 Week 5 Three Phase Systems At the end of this week, the students are expected to: ♦ Explain how 3-phase e.m.fs are produced ♦ Distinguish between star and delta three-phase system ♦ Derive the relationship between line and phase values of voltages and currents in a star and delta connected windings 2.5 GENERATION OF THREE PHASE E.M.F’s B1 A C1 N S C B A1 Fig 2.4: Generation of 3-φ e.m.f’s In figure 2.4, three similar coils (A,B, and C) are displaced from one another by 120 electrical degrees. If the coils are rotated within the magnetic field, emf would be induced or generated in the three coils. It is evident that counterclockwise rotation results in coil sides A, B, and C in the order A-B-C. The result for the three coils is as shown in fig 2.3. Voltage B is 120 electrical degrees later than A, and C is 2400 later. Changing the direction of rotation would result in A-C-B, which is called the ACB phase sequence. 2.6 DIFFERENCE BETWEEN STAR AND DELTA 3–PHASE SYSTEM 2.6.1 Star-Connected 3-Phase System 1. Neutral wire is available 2. Phase current = line current 3. Phase voltage = line voltage √3 4. It can handle both lighting and power loads 18 2.6.2 Delta-Connected 3-Phase system 2.7 1. Neutral wire is not present 2. Phase current = line current √3 3. Phase voltage = line voltage 4. It can handle power loads only DERIVATION OF THE RELATIONSHIP BETWEEN LINE AND PHASE VALUES OF VOLTAGES AND CURRENT IN A STAR AND DELTA CONNECTED WINDINGS 2.7.1 Line voltages and phase voltages in a star connected windings Line 1 VR VRY VB VBR Vy Line2 VYB Line3 (a) Fig 2.5: (a) Star connection of a 3-Φ circuit VRY VR -VY -VB 300 VYB VB VY -VR VBR 19 Fig 2.6: phasor diagram of a star connected load Consider the star connection of a three phase circuit shown in fig 2.5. It phasor diagram is as shown in fig 2.6. To obtain the line voltages we proceed as under: Let VRY = line voltage between red phase and yellow phase VYB = line voltage between yellow phase and blue phase VBR = line voltage between blue phase and red phase VR = voltage across the red phase VY = voltage across the yellow phase VB = voltage across the blue phase Thus, VYB, VBR and VYB are called line voltages, while VB, VR and VY are called phase voltages. The p.d between line 1 and 2 in (fig 2.5) is VRY. Hence, VRY is found by compounding VR and VY reversed and its value is given by the diagonal of the parallelogram of fig 2.6. Obviously, the angle between VR and VY reversed is 600. The parallelogram is shown in fig 2.7 below. VY VRY x o 300 VR Fig 2.7: parallelogram of fig 2.6 From fig 2.7, ox = ½ VRY (2.1) Also, ox = VR cos 300 (2.2) Equating (2.2) to equation (2.1) gives VRY = 2 (VRcos300) = √3 VR = √3 Vph Considering that the system is balanced, ∴ VYB = √3 Vph also, VBR = √3 Vph Now VRY = VYB = VBR = line voltage, say VL. Hence, in star connection VL = √3Vph (2.3) 20 2.7.2 Line currents and phase currents in a star connected windings R IR N IB Iy Y B Fig 2.8: star connection of three-phase circuit Consider fig 2.8. Let IR = current flowing through the red phase IY = current flowing through the yellow phase and IB = current flowing through the blue phase IR, IY and IB are called phase current (Iph). It is seen that each of the phase currents is equal to the current flowing through the respective lines. Thus, the current flowing through the respective lines is known as the line current (IL) Hence, in star connection, line current = phase current. ∴ IL = Iph (2.4) 2.7.3 Line currents and phase currents in a delta connected windings 1 (IR – IB) R IB IR VRY VBR 2 IY (IY – IR) Y VYB (IB – IY) 3 B Fig 2.9 Delta connection of a three phase circuit 21 VRY (IR – IB) IR -IY Φ -IB (IB – Iy) Φ Φ IY IB VYB -IR VBR (IY – IR) Fig 2.10 Phasor diagram of a delta connection of a three phase circuit Consider the delta connection of a three phase circuit shown in fig 2.9. It phasor diagram is as shown in fig 2.10. To obtain the line currents we proceed as under: Let I1 = IR – IB I2 = IY - IR I3 = IB - IY I1, I2 and I3 are called line currents Line current I1 is also found by compounding IR and IB reversed as shown in fig 2.10. Its value is given by diagonal of fig 2.10. The parallelogram of fig 2.10 is as shown in fig 2.11. IR I1 x o 300 IB Fig 2.11: parallelogram of fig 2.10 From fig 2.11, Ox = ½ I1 Also, ox = IB cos30 (2.5) 0 (2.6) 22 From equation (2.6) and (2.6), ½ I1 = √3/2 IB If IR = IY = IB = phase current (Iph), then Current in line no.1 is I1 = √3Iph Current in line no.2 is I2 = √3Iph and Current in line no.3 is I3 = √3 Iph Since all the line currents are equal in magnitude, i.e I1 = I2 = I3 = IL ∴ IL = √3Iph (2.7) 2.7.4 Line voltages and phase voltages in a delta connected windings It is seen from fig 2.9 that there is only one phase winding completely included between any pair of terminals. Hence, in delta connection, the voltage between any pair of lines is equal to the phase voltage of the phase winding connected between the two lines considered. Hence, for a balanced system, VRY = VYB = VBR = line voltage VL. Then, it is seen that VL = Vph (2.8) 23 Week 6 Three Phase Systems At the end of this week, the students are expected to: ♦ Derive an expression for power in a three-phase circuit (balanced and unbalanced) ♦ Explain the two wattmeter and single wattmeter methods of measuring three-phase power 2.8 DERIVATION FOR POWER IN A 3-PHASE CIRCUIT (BALANCED AND UNBALANCED) 2.8.1 Power in a 3-Phase circuit (balanced-delta connection) Consider the delta connection of a three phase circuit (fig 2.9). The total active power in a delta connected circuit is the sum of the three phase powers. Hence, Power per phase = VphIphcosφ Total power = 3 x VphIph cos φ However, Vph = VL and Iph = IL/√3. In terms of line values, the above expression for power becomes P = 3 x VL x IL x cosφ = √3 VLILcosφ √3 (2.9) Similarly, total reactive power is given by Q = √3 VLILsinφ (2.10) and total apparent power of the three phase is S = √3 VLIL (2.11) 2.8.2 Power in a three phase circuit (unbalanced star connection) IR Vph ZR In IY Vph ZY N ZB Vph IB Fig 2.11: unbalanced three phase star connected load Consider the unbalanced three phase star connected load, where it is assumed that the phase voltages (Vph) are equal. The line currents are obtained from ohms law. 24 Power in phase R, PR = phase voltage x Phase current x cosφR, where φR is the angle between phase voltage and phase current. ∴PR = Vph IRcosφR (2.12) Similarly, power in phase Y is obtained as PY = Vph IY cosφY, (2.13) Power in phase B is PB = Vph IB cosφB (2.14) Total power in the three phase is PT = PR + PY + PB = Vph IRcosφR + Vph IYcosφY + Vph IBcosφB 2.9 (2.15) 2-WATTMETER AND SINGLE-WATTMETER METHODS OF MEASURING 3-PHASE POWER 2.9.1 2-Wattmeter Method In this method, two wattmeters are connected in two phases and their potential coils are connected to the remaining phase. As shown in fig 2.12, wattmeter W1 is inserted in phase a and wattmeter W2 in phase c. Their potential coils are connected across a and b, and c and b respectively. The current coil (c.c) of each wattmeter measures the line current, while the respective potential coil (p.c) measures the line voltage. The algebraic sum of the two wattmeter readings equals the total power absorbed by the loads. Thus W1 + W2 = Total three phase power (√3 VLILcosφ) (2.16) The following are important with regard to two wattmeter method. 1. For a lagging Power factor, tanφ = √3 W2 – W1 W2 + W1 2. (2.17) For a leading power factor tanφ = _ √3 (W1 – W2) (2.18) W1 + W2 25 a c.c W1 ia p.c ZA b ZB ZC p.c ic c c.c Fig 2.12: schematic diagram of a two wattmeter method 2.9.2 Single- Wattmeter Method In this method, one wattmeter is used to get two readings which are obtained by two wattmeter in the two wattmeter method. The total power is 3 times the reading of that one wattmeter. This method is used only when the load is balanced. Figure 2.13 shows the schematic diagram of a single wattmeter method. a W1 c.c Z p.c Z Z b c Fig_ 2.13: schematic diagram of a single-wattmeter method 26 Week 7 Three Phase Systems At the end of this week, the students are expected to: ♦ Solve problems on Line and phase voltages and current in a star and delta connected windings Power in a three phase circuit 2-wattmeter and single wattmeter methods of measuring 3-phase circuits 2.10 SOLVED PROBLEMS ON LINE AND PHASE VOLTAGES AND CURRENT IN A STAR AND DELTA CONNECTED WINDINGS, POWER IN THREE PHASE CIRCUIT AND 2-WATTMETER AND SINGLE WATTMETER METHOD OF MEASURING 3-PHASE POWER Example 2.1The impedance per phase of a delta connected load is 10Ω per phase, the line voltage being 230V. Determine (a) the phase current (b) the line current (c) the apparent power of the load. Solution (a) Iph = Vph/Zph = 230/10 = 23A (b) IL = √3 Iph = √3 x 23 = 39.8A (c) S = √3 VL IL = √3 x 230 x 39.8 = 15.9KVA The circuit arrangement is shown in fig 2.14. Iph = 39.8A VL = 230V Zph = 10Ω Iph = 23A Zph = 10Ω Zph = 10Ω Fig 2.14 for example 2.1 27 Example 2.2 An unbalanced star-connected load has balanced phase voltage of 120V and the a.b.c sequence. The line currents are given as Ia = 10∠00A, Ib = 20∠-400A and Ic = 30∠1550A. Find the power consumed by the circuit. Solution Ia = 10∠50A, Vpa = 120∠00V Ib = 20∠400A, Vpb = 120∠-1200V Ic = 30∠1550A, VPc = 120∠-2400V Pa = VPIacosφa = 120 x 10cos(0 – 0) = 1200W Pb = VpIbcosφb = 120 x 20cos{-120 –(-40)} = 416.76W Pc = VPIccosφc = 120 x 30cos{-240 – (-155)} = 313.76W P = Pa + Pb + Pc = 1200 + 416.76 + 313.76 ≅ 1.93KW Example 2.3: Line voltage and current of a delta connected inductive load is 160V and 40A respectively. Power factor of load is 0.8lag. Assuming that power is being measured using two wattmeters, find the readings of the wattmeters. Solution VPh = VL = 160V, IL = 40A Total power, P = √3 VLILcosφ = √3 x 160 x 40 x 0.8 = 8868W ∴ W1 + W2 = 8868 (2.4.1) cosφ = 0.8, ⇒ φ = cos-1(0.8) = 36.90 tanφ = 0.75 For a lagging P.f, tanφ = √3 (W1 - W2) W1 + W2 ⇒ 0.75 = √3 (W1 – W2), ⇒ W1 – W2 = 8868 x 0.75 = 3840 8868 √3 ∴ W1 – W2 = 3840 (2.4.2) Adding eqtn. (2.4.1) and (2.4.2) together gives 2W1 = 12708 ⇒ W1 = 12708/2 = 6354W From eqtn. (2.4.1), W2 = 8868 – 6354 = 2514W 28 Example 2.4: The current coil of a wattmeter is connected to the red phase of a star connected load, while the potential coil is connected between the line and the neutral. If the phase voltage and current are respectively 20∠800V and 30∠500A, find the total power consumed by the load. Solution The circuit diagram is shown in fig 2.15 W1 c.c IR R Vph Z p.c Z Z Y B Fig 2.15: schematic diagram of a single-wattmeter method P = √3 VLILcosφ where IR = IL and VL = √3 VPh = √3 x 20∠800 = 34.64∠800V P = √3 x 34.64 x 30 cos(800 – 500) = 1558.8W 29 Week 8 Time Domain Analysis At the end of this week, the students are expected to: ♦ Explain the meaning of transient ♦ Derive formulae for current and voltage growth and decay in RC circuits 3.1 TRANSIENT when a circuit possesses energy storing elements such as inductance and capacitance, the energy state of the circuit can be disturbed by changing the position of the switch (fig 3.1) connecting the elements to the source. There is a transitional period during which the branch current and element voltages change from their former values to new ones. This period is called the transient. 3.2 DERIVATIONS OF FORMULAE FOR CURRENT AND VOLTAGE GROWTH AND DECAY IN RC CIRCUIT 3.2.1 Current and Voltage Growth in RC Circuit Let us consider the circuit shown in figure 3.1 R C S V Fig 3.1: RC series circuit Let the current in the circuit at any instant t be i, and capacitor voltage be v. When switch S is closed and Kirchhoff’s law is being applied, this yield Ri + v = V (3.1) Since i = C(dv/dt), equation (3.1) becomes V = RC (dv/dt) + v ∴ V – v = RC (dv/dt) so that dt/RC = dv/(V – v) Integrating both sides, we have t = -ln (V – v) + A RC where A is the constant of integration when t = 0, v = 0 30 ∴ A = ln V V . So that t = ln RC V–v ∴ and V = e t/RC V–v v = V (1 – e -t/RC ) volts (3.2) Also i = C dv = CV . d (1 – e-t/RC) dt dt ∴ i = V/R . e -t/RC (3.3) 3.2.2 Current and Voltage Decay in RC Circuit Let the capacitor C in fig 3.1 be charged initially to V volts and then be discharged through a resistor R as shown in figure 3.2 C R Fig 3.2: Decay of current and voltage in RC circuit By Kirchhoff’s law iR + v = 0 or -v = C dv R dt { i = C dv/dt} So that dt/RC = -dv/v Integrating both sides, we have t = - ln v + A RC When t = 0, v = V, so that A = ln V. Hence t = ln V/v RC So that V/v = et/RC and v = Ve -t/RC (3.4) Also i = -v/R The negative sign indicates that the direction of the discharge current is the reverse of that of the charging current ∴ i = -V/R e-t/RC (3.5) 31 Week 9 Time Domain Analysis At the end of this week, the students are expected to: ♦ Sketch the growth and decay curves in RC circuits ♦ Define time constant in RC circuit ♦ Solve problems involving transient in RC circuit 3.3 SKETCH FOR GROWTH AND DECAY CURVES OF CURRENT AND VOLTAGE IN RC CIRCUIT The sketch for growth of current and voltage in RC circuit is shown in figure 3.3 while the sketch for decay of current and voltage in RC circuit is as shown in fig 3.4 i,v V Charging current Voltage across the capacitor Voltage across the capacitor T t Discharging current t i Fig 3.3 3.4 Fig 3.4 TIME CONSTANT IN RC CIRCUIT Consider fig 3.3 above. T is known as the time constant in RC circuit. It is given by the expression T = RC, (seconds) (3.6) Time constant of an RC circuit is the time required for the voltage across capacitor to increase from zero to its final value if it continued increasing at its initial rate. 32 3.5 SOLVED PROBLEMS INVOLVING TRANSIENT IN RC CIRCUIT Example 3.1: A 20µF capacitor is charged to a p.d of 400V and then discharged through a 100,000Ω resistor. Determine: (a) the time constant (b) the initial value of discharged current Solution (a) T = RC = 100 000 x 20 = 2s 100 000 (b) Initial value of discharge current is V/R = 400/100 000 = 4mA Example 3.2: An 8µF capacitor is connected in series with a 0.5MΩ resistor across a 200V d.c supply. Calculate: (a) the time taken for the P.d across the capacitor to grow to 160V: (b) the current and the P.d across the capacitor 4s after it is connected to the supply Solution (a) -6 T = RC = 0.5 x 106 x 8 x 10 = 4s From v = V(1 – e-t/RC) -t/4 ⇒e = 0.2 From mathematical table t/4 = 1.61 ∴ t = 6.44s (b) At t = 4s -4/4 v = 200(1 – e and i = v/R e ) = 200( 1 – 0.68) = 126.4V -t/RC = 400 e-1 = 147µA Example 3.3: A 15µF capacitor in series with a 10KΩ resistor is connected across a 300V d.c supply. The fully charged capacitor is disconnected from the supply and is discharged by connecting a 11200Ω resistor across it terminals. Determine (i) the initial value of the charging current, and (ii) the initial value of the discharge current. 33 Solution (i) Initial value of charging current, I0 = V/R = 300 = 30mA 10 x 103 (ii) The initial value of discharging current is given by i = V/R, where R = 11200Ω ∴ i = 300 = 26.7mA 11200 34 Week 10 Time Domain Analysis At the end of this week, the students are expected to: ♦ Derive expression for the growth and decay of voltage and current in RL circuit ♦ Sketch curves for growth and decay of current and voltage in RL circuit ♦ Define time constant in RL circuit 3.6 DERIVATIONS OF FORMULAE FOR CURRENT AND VOLTAGE GROWTH ANF DECAY IN RL CIRCUIT 3.6.1 Current and Voltage Growth in RL Circuit Consider the circuit shown in figure 3.5 and suppose i amperes to be the current t seconds after the switch is closed, and di amperes to be the increase of current in dt seconds. Then rate of change of current is di/dt ampere per second and induced e.m.f is L di/dt volts i Switch L V R Fig 3.5: RL series circuit Applying KVL yield V = Ri + L di/dt (3.7) so that V – Ri = L di/dt and V/R – i = L/R . di/dt But V/R = final value of current = I ∴ R dt = di . L I–i Integrating both sides, we have Rt/L = -ln (I – i) + A Where A is the constant of integration 35 When t = 0, i = 0, so that, A = ln I. ∴ Rt/L = -ln(I – i) + ln I = ln I . I–i Hence I – i = e I -Rt/L ∴ i = I(1 – e -Rt/L ) (3.8) The voltage across the inductance, VL = L di/dt = L d/dt [ I (1 – e VL = IR e -Rt/L )] -R/L . t (3.9) 3.6.2 Current and Voltage decay in RL circuit Consider the circuit shown in figure 3.6. When the switch is moved from position X to Y, the battery voltage is short circuited. Hence, since battery is cut off from the circuit, applying Kirchoff’s law yield. R X L Switch Y V Fig 3.6 0 = Ri + L di/dt ∴ Ri = -L di/dt The negative sign indicate a decaying current, hence (R/L) dt = -1/t dt Integrating both sides, we have (R/L) t = -ln i + A where A is the constant of integration At the instant of closing switch, t = 0 and i = I, so that 0 = -ln I + A ∴ (R/L) t = ln I – ln i Hence I/i = e-Rt/L and i = I e -Rt/L (3.10) Also, VL = L di/dt 36 = L [d/dt {Ie -Rt/L }] VL = -IR e-Rt/L 3.7 (3.11) SKETCH FOR GROWTH AND DECAY CURVES OF CURRENT AND VOLTAGE IN RL CIRCUIT The sketch for growth of current and voltage in RL circuit is shown in figure 3.7 while figure 3.8 depicts the sketch for decay of current and voltage in RL circuit i,vL i Voltage across the inductance Decaying current Growing current in an inductive circuit t Voltage across the inductor t VL Fig 3.7 3.8 Fig 3.8 TIME CONSTANT IN RL CIRCUIT The time constant of an RL circuit is define as the time during which current would have reached its maximum value had it maintained its initial rate of rise. The expression for the time constant in an RL circuit is given by T = L/R (3.12) 37 Week 11 Time Domain Analysis At the end of this week, the students are expected to: ♦ Explain the need for connecting a resistor in parallel with an inductor ♦ Solve problems involving transient in RL circiuits 3.9 ESSENCE OF CONNECTING A RESISTOR IN PARALLEL WITH AN INDUCTOR A S R L B E RD Fig 3.9: Decay of current in an RL series circuit Let us consider the circuit shown in figure 3.9. A discharge resistor RD is connected in parallel with the inductive circuit. When the switch S is connected to A, the current becomes E/R. When the switch is moved to position B, the circuit is disconnected from the supply. When the position of the switch is change from A to B, the circuit in the inductive circuit is switched off and a voltage is induced in it. Arcing is produced at the time of charging the position of the switch and there is a possibility of damaging the insulation because the induced voltage may be many times greater than the applied voltage. To avoid this arcing, a discharge resistor RD is connected as shown in figure 3.9 3.10 SOLVED PROBLEMS INVOLVING TRANSIENT IN RL CIRCUIT Example 3.4: A coil having a resistance of 4Ω and a constant inductance of 2H is switched across a 20V d.c supply. Calculate: (a) the time constant (b) the final value of the current Solution (a) T = L/R = 2/4 = 0.5s 38 (b) I = V/R = 20/5 = 5A Example 3.5 For the network shown in fig 3.10: (a) Determine the mathematical expression for the variation of the current in the inductor following the closure of the switch at t =0 on to position 1 (b) The switch is closed on to position 2 when t = 100ms, determine the new expression for the inductor current 1 2 s R1 = 10Ω R = 15Ω 10V L = 0.1H Fig 3.10 Solution (a) For the switch in position 1, the time constant is T1 = L/R1 = 0.1/10 = 10ms ∴i1 = I(1 – e-t/T) = 10/10 (1 – e-t/10x10 -t/10x10 -3 = (1 – e (b) -3 ) ) amperes For the switch in position 2, the time constant is T2 = L = 0.1 = 4ms R1 + R 10 + 15 2 t/4x10-3 ∴i2 = Ie-t/T = 10/10 (e=e Example 3.6: -t/4x10 ) -3 amperes A coil having 10Ω resistance and 14H inductance is connected across a d.c voltage of 140V. After the current has reached its final value, how much time it would take for the current to reach a value of 8A after switching off the supply. 39 Solution i = Ie -t/T where i = 8A, and T = L/R = 14/10 = 1.4s and I = V/R = 140/10 = 14A ∴ 8 = 14 e -t/1.4 ⇒ e-t/1.4 = 8/14 ∴ t = 0.783s 40 Week 12 Magnetic Coupling At the end of this week, the students are expected to: ♦ Describe magnetic coupling ♦ Define mutual inductance ♦ Derive an expression for mutual inductance between two coils 4.1 MAGNETIC COUPLING R1 i1 R2 I2 φ12 φ2 φ1 Coil 1 Fig 4.1: Coil 2 Magnetic Coupling Figure 4.1 shows magnetic coupling between two coils. In this case, a portion of the magnetic flux established by one coil interlinks the other. In this case, energy is transferred from one circuit to the other through the medium of magnetic flux that is common to both. From fig 4.1, φ1 is the total flux produced by i1, and φ2 is the total flux produced by i2, and φ12 is the magnetic flux that linked both coils 4.2 MUTUAL INDUCTANCE The mutual inductance between two coils is defined as the Weber-turns in one coil due to current through the other coil 41 4.3 DERIVATION FOR MUTUAL INDUCTANCE BETWEEN TWO COILS Consider fig 4.1. The induced voltage in coil 2 due to a change in flux can be written as (by faraday’s law) e2 = N2dφ12 dt (4.1) where N2 is the number of turns in coil 2 Also, e2 is proportional to the rate of change of i1, that is e2 α di1 or e2 = M di dt dt (4.2) In equation (4.2), M is the constant of proportionality known as mutual inductance between two coils. It is measured in Henry (H) Equating equation (4.1) to (4.2) we have N2 dφ12 = M di1 dt dt ⇒ M = N2 dφ12 di1 (4.3) 42 Week 13 Magnetic Coupling At the end of this week, the students are expected to: ♦ Describe the polarity of coupled coils ♦ Define coefficient of coupling ♦ Derive an expression for the coupling coefficient 4.4 POLARITY OF COUPLED COILS Polarity of coupled coils could be determined by dot convention. Dot convention is very useful to determine the nature of mutually induced e.m.f instead of showing the actual mode of the winding. If the dotted terminals of the winding correspond to each other, the mutually induced e.m.f is positive otherwise it is negative. Figure 4.2 clearly explains the sign or polarity of mutually induced e.m.f. M positive M negative M negative M negative Fig 4.2: Dot Convention 43 4.5 COEFFICIENT OF COUPLING Coefficient of coupling is defined as the ratio of mutual inductance M to the square root of the product of inductances of coil 1 and coil 2. 4.6 DERIVATION FOR COEFFICIENT OF COUPLING In figure 4.1, let the two inductively coupled coils 1 and 2 have the number of turns N1 and N2 respectively. Their individual coefficients of self induction are, L2 = N2 and L2 = N2 . L/µ0µrA L/µ0µrA The flux φ1 produced in coil 1 due to a current I1 ampere is φ1 = N1 I1 . L/µ0µrA Suppose a fraction K1 of this flux i.e K1φ1 is linked with coil 2. Then M = K1 φ1 x N2 I1 where K ≤ 1. Substituting the value of φ1, we have, M = K1 x N1 N2 L/µ0µrA (4.4) Similarly, the flux φ2 produced in coil 2 due to I2 ampere is φ2 = N2 I2 . L/µ0µrA Suppose a fraction K2 of this flux i.e K2φ2 is linked with coil 2 then M = K2 φ2 x N1 I1 = K2 x N1 N2 L/µ0µrA (4.5) Multiplying eqtn. (4.4) and (4.5), we get M2 = K1K2 N12 x L/µ0µrA N22 = K1K2L1L2 L/µ0µrA Putting √(K1K2) = K, we have M = K√(L1L2) ∴K= M √(L1L2) where K is the coefficient of coupling. (4.6) 44 Week 14 Magnetic Coupling At the end of this week, the students are expected to: ♦ Define an ideal transformer ♦ Draw the equivalent circuit of an ideal transformer ♦ Explain with the aid of a diagram an equivalent circuit of a practical transformer 4.7 IDEAL TRANSFORMER An ideal transformer is one with perfect coupling (K = 1). 4.8 EQUIVALENT CIRCUIT OF AN IDEAL TRANSFORMER I1 P I2 S V1 V2 load Fig 4.5 : Equivalent circuit of an ideal transformer 4.9 EQUIVALENT CIRCUIT OF A PRACTICAL TRANSFORMER Z1 I1 Z2 X1 R1 I11 I0 I1 Z1 R V1 P E1 X2 S E2 R2 I2 I2 Z2 X To load IC V2 Img Ideal transformer Fig 4.6: Equivalent circuit of a practical transformer Fig 4.6 is an equivalent circuit of a practical transformer. P and S represent the primary and secondary windings of the ideal transformer, R1 and R2 are resistances equal to the resistances of the primary and secondary windings of the practical 45 transformer. Similarly, inductive reactances X1 and X2 represent the reactance of the windings due to leakage flux in the practical transformer. The inductive reactor X is such that it takes a reactive current equal to the magnetizing current Img of the practical transformer. The core losses due to hysteresis and eddy currents are allowed for, by a resistor R of such value that it takes a current Ic equal to the core loss of the practical transformer. The resultant of Img and Ic is Io, which is the current the transformer take on no load. 46 Week 15 Magnetic Coupling At the end of this week, the students are expected to: ♦ Solve problems involving magnetic coupling s 4.10 SOLVED PROBLEMS INVOLVING MAGNETIC COUPLING Example 4.1: If a coil of 150turns is linked with a flux of 0.01wb when carrying current of 10A, calculate the inductance of the coil. Solution L = Nφ = 150 x 0.01 = 0.15H i 10 Example 4.2: Two coils having 30 and 600turns respectively are wound side by side on a closed iron circuit. If the mutual inductance between the coils is 0.226H, and the change of current in the first coil is 20A in 0.02s, find the e.m.f induced in the second coil Solution e2 = M di/dt = 0.226 x 20/0.02 = 226V Example 4.3: The coefficient of coupling between two coils is 0.75. There are 250turns in coil 1. The total flux linking coil 1 is 0.4mWb, when the current in this coil is 3A. When i1 is changed from 3A to zero linearly in milliseconds, the voltage induced in coil 2 is 70V. Calculate: (a) the self inductance of coil 1 (b) the mutual inductance between the two coils. Solution (a) L1 = N1φ1 = 250 x 0.4 x 10-3 = 3.4 x 10-3H i1 (b) 3 e2 = M di, ∴ M = e2 dt = 70 x 3x 10-3 = 70 x 10-3H dt di 3 Example 4.4: Two coils X and Y have self inductance of 12H and 20H respectively, the mutual inductance between the coils being 8.5H. Find the coupling coefficient between the two coils Solution K= M = 8.5 = 0.55 √(LXLY ) √(2 x 20) 47 Example 4.5: The current in the primary winding of a pair of mutually coupled coil is 5A, and the primary flux of 6mWb links with the secondary winding. If the secondary winding has 950turns and the primary winding current is reduced to zero in 4ms, determine: (a) the average value of the e.m.f induced in the secondary winding (b) the mutual inductance between the two coils Solution (a) e2 = N2 dφ2 = 950 x 6 x 10-3 = 1425V dt 4 x 10-3 (b) M = e2 dt, whee dI1 = 5 – 0 = 5A dI1 ∴ M = 1425 x 4 x 10-3 = 1.14H 5 48