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ELECTRICAL TECHNOLOGY EET 103/4 Define and explain sine wave, frequency, amplitude, phase angle, complex number Define, analyze and calculate impedance, inductance, phase shifting Explain and calculate active power, reactive power, power factor Define, explain, and analyze Ohm’s law, KCL, KVL, Source Transformation, Thevenin theorem. 1 BASIC ELEMENTS AND PHASORS (CHAPTER 14) 2 14.2 The Derivative To understand the response of the basic R, L, and C elements to a sinusoidal signal, you need to examine the concept of the derivative. The derivative dx/dt is defined as the rate of change of x with respect to time. If x fails to change at a particular instant, dx = 0, and the derivative is zero. For the sinusoidal waveform, dx/dt is zero only at the positive and negative peaks (wt = p/2 and 3p/2) since x fails to change at these instants of time. 3 14.2 The Derivative The derivative dx/dt is actually the slope of the graph at any instant of time. The greatest change in x will occur at the instants wt = 0, p, and 2p. For various values of wt between these maxima and minima, the derivative will exist and will have values from the minimum to the maximum inclusive. The derivative of a sine wave is a cosine wave; it has the same period and frequency as the original sinusoidal waveform. 4 14.2 The Derivative 5 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Resistor For power-line frequencies and frequencies up to a few hundred kilohertz, resistance is, for all practical purposes, unaffected by the frequency of the applied sinusoidal voltage or current. For a purely resistive element, the voltage across and the current through the element are in phase, with their peak values related by Ohm’s law. 6 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current 7 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Inductor The magnitude of the voltage across the element is determined by the opposition of the element to the flow of charge, or current i. The inductive voltage, therefore, is directly related to the frequency (or, more specifically, the angular velocity of the sinusoidal ac current through the coil) and the inductance of the coil. For an inductor, vL leads iL by 90°, or iL lags vL by 90°. 8 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current iL I m sin wt diL d vL L L I m sin wt wLI m cos wt dt dt Or; vL wLI m sin wt 90 Vm sin wt 90 Where; Vm wLI m Hence; Vm wL X L Inductive reactance Im 9 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Inductive reactance is the opposition to the flow of current, which results in the continual interchange of energy between the source and the magnetic field of the inductor. 10 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Capacitor Since capacitance is a measure of the rate at which a capacitor will store charge on its plates, for a particular change in voltage across the capacitor, the greater the value of capacitance, the greater will be the resulting capacitive current. 11 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current The fundamental equation relating the voltage across a capacitor to the current of a capacitor [i = C(dv/dt)] indicates that for a particular capacitance, the greater the rate of change of voltage across the capacitor, the greater the capacitive current. 12 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current vC Vm sin wt dvC d iC C C Vm sin wt wCI m cos wt dt dt Or; iC wCI m sin wt 90 I m sin wt 90 Where; I m wCVm Hence; Vm 1 X C Capative reactance I m wC 13 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current 14 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Capacitive reactance is the opposition to the flow of charge, which results in the continual interchange of energy between the source and the electric field of the capacitor. If the source current leads the applied voltage, the network is predominantly capacitive, and if the applied voltage leads the source current, it is predominantly inductive. 15 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) The voltage across a 10- resistor is given by the expression; v 100 sin 377t Find the expression for the current i through the resistor and sketch the curves for v and i. 16 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) – solution v Vm sin wt 100 sin 377t Vm 100 V and w 377 rad/s 2pf Vm 100 Im 10 A R 10 Hence; im I m sin wt 10 sin 377t A 17 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) – solution (cont’d) The waveform; 18 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(a) – solution (cont’d) The alternative waveform; v, i 100 V v In phase 10 A 0 i 8.35 t (ms) 16.7 19 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) The voltage across a 10- resistor is given by the expression; v 25 sin 377t 60 Find the expression for the current i through the resistor and sketch the curves for v and i. 20 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) – solution v Vm sin wt 25 sin 377t 60 Vm 25 V; w 377 rad/s 2pf and 60 Vm 25 Im 2.5 A R 10 Hence; im I m sin wt 2.5 sin 377t 60 A 21 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) – solution (cont’d) The waveform; 22 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.1(b) – solution (cont’d) The alternative waveform; v, i v 25 V i In phase 8.33 12.5 2.5 A -4.17 2.78 0 4.17 16.67 t (ms) 23 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) The current a 0.1-H coil is given by the expression; i 10 sin 377t Find the expression for the voltage v across the coil and sketch the curves for v and i. 24 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution The voltage across a coil leads the current throuh it by 90. Hence, if the current is; iL I m sin wt the voltage will be; vL Vm sin wt 90 where; Vm I m X L and; X L wL 2pfL 25 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution (cont’d) From the expression; i 10 sin 377t we obtain; and; Also; I m 10 A; w 377 rad/s; 377 f 60 Hz; 2p 1 1 T 16.67 ms; f 60 X L wL 377 0.1 37.7 26 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution (cont’d) Vm I m X L 10 37.7 377 V Hence the expression for the voltage is; v Vm sin wt 90 377 sin 377t 90 27 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(a) – solution (cont’d) The waveforms; 16.67 t (ms) -4.17 4.17 8.33 12.5 28 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) The current a 0.1-H coil is given by the expression; i 7 sin 377t 70 Find the expression for the voltage v across the coil and sketch the curves for v and i. 29 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) – solution The voltage across a coil leads the current throuh it by 90. Hence, if the current is; the voltage will be; iL I m sin wt vL Vm sin wt 90 where; Vm I m X L and; X L wL 2pfL 30 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) – solution (cont’d) From the expression; i 7 sin 377t 70 we obtain; I m 7 A; w 377 rad/s; 1 1 T 16.67 ms; f 60 Also; 377 f 60 Hz; 2p and; 70 rad/s; X L wL 377 0.1 37.7 31 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) – solution (cont’d) Vm I m X L 7 37.7 263.9 V Hence the expression for the voltage is; v Vm sin wt 90 263.9 sin 377t 70 90 263.9 sin 377t 20 V 32 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.3(b) – solution (cont’d) The waveforms; 16.67 4.17 0.93 8.33 12.5 t (ms) 3.24 v leads i by 90 4.17 ms 33 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 The voltage across a 1-F capacitor is given by the expression; v 30 sin 400t V Find the expression for the current i through the capacitor and sketch the curves for v and i. 34 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution The current through a capacitor leads the voltage across it by 90. Hence, if the voltage is; the current will be; where; Vm Im XC vC Vm sin wt iC I m sin wt 90 and; 1 1 XC wC 2pfC 35 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution (cont’d) From the expression; v 30 sin 400t we obtain; Vm 30 V; w 400 rad/s; and; Also; 400 f 63.7 Hz; 2p 1 1 T 15.7 ms; f 63.7 1 1 XC 2500 6 wC 400 110 36 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution (cont’d) Vm 30 Im 12 mA X C 2500 Hence the expression for the current is; i I m sin wt 90 12 sin 400t 90 mA 37 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.5 – solution (cont’d) The waveforms; t (ms) 7.85 3.93 3.93 11.78 15.7 i leads v by 3.93 ms 90 38 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 The current through a 100-F capacitor is given by the expression; i 40 sin 500t 60 V Find the expression for the voltage v across the capacitor. 39 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 – solution The voltage across a capacitor lags the current through it by 90. Hence, if the current is; the voltage will be; iC I m sin wt vC Vm sin wt 90 where; Vm I m X C and; 1 1 XC wC 2pfC 40 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 – solution (cont’d) From the expression; i 40 sin 500t 60 we obtain; I m 40 A; w 500 rad/s; 1 1 T 12.57 ms; f 79.6 Also; 500 f 79.6 Hz; 2p and; 60 1 1 XC 20 6 wC 500 100 10 41 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.6 – solution (cont’d) Vm I m X C 40 20 800 V Hence the expression for the current is; v Vm sin wt 90 800 sin 500t 60 90 V 800 sin 500t 30 V 42 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(a) Determine the type of element in the box (C, L or R) and calculate its value if; v 100 sin wt 40 V and i 20 sin wt 40 A 43 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(a) – solution v 100 sin wt 40 V and i 20 sin wt 40 A The voltage and current are in phase – the element is a resistor (R). Vm 100 R 5 Im 20 44 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(b) Determine the type of element in the box (C, L or R) and calculate its value if; v 1000 sin 377t 10 V and i 5 sin 377t 80 A 45 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(b) – solution v 1000 sin 377t 10 V and i 5 sin 377t 80 A The voltage leads the current by 90 – the element is an inductor (L). Vm 1000 XL 200 Im 5 XL 200 L 0.53 H w 377 46 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(c) Determine the type of element in the box (C, L or R) and calculate its value if; v 500 sin 157t 30 V and i 1sin 157t 120 A 47 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(c) – solution v 500 sin 157t 30 V and i 1sin 157t 120 A The voltage lags the current by 90 – the element is a capacitor (C). Vm 500 XC 500 Im 1 1 1 C 12.74 F wX C 157 500 48 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(d) Determine the type of element in the box (C, L or R) and calculate its value if; v 50 cos wt 20 V and i 5 sin wt 110 A 49 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(d) – solution v 50 cos wt 20 V 50 sin wt 20 90 V 50 sin wt 110 V i 5 sin wt 110 A The voltage and current are in phase – the element is a resistor (R). 50 14.3 Response of Basic R, L and C Elements to a Sinusoidal Voltage or Current Example 14.7(d) – solution (cont’d) Vm 50 R 10 Im 5 51 14.4 Frequency Response of Basic Elements Resistance In the real world, each resistive element has stray capacitance levels and lead inductance that are sensitive to the applied frequency. Usually the capacitive and inductive levels involved are so small that their real effect is not noticed until the frequency is in the megahertz range. Frequency does have impact on the resistance of an element, but for our purpose the resistance level of a resistor is independent of frequency (below 15 MHz). 52 14.4 Frequency Response of Basic Elements As the applied frequency increases, the resistance of a resistor remains constant 53 14.4 Frequency Response of Basic Elements As the applied frequency increases, the reactance of an inductor increases linearly. X L wL 2pfL X L kf k 2pL 54 14.4 Frequency Response of Basic Elements As the applied frequency increases, the reactance of a capacitor decreases nonlinearly. 1 1 XC wC 2pfC k XC f 1 k 2pC 55 14.5 Average Power and Power Factor For any load in a sinusoidal ac network, the voltage across the load and the current through the load will vary in a sinusoidal nature. The average power (real power) is the power delivered to and dissipated by the load. It corresponds to the power calculations performed for dc networks. The angle (v - i) is the phase angle between v and i. The magnitude of average power delivered is independent of whether v leads i or i leads v, since cos(a) = cos a. 56 14.5 Average Power and Power Factor Power versus time for a purely resistive load. 57 14.5 Average Power and Power Factor For a purely resistive load; Vm I m Pav 2 2Vrms 2 I rms 2 Pav Vrms I rms 2 Pav I rms R 2 Vrms Pav R 58 14.5 Average Power and Power Factor Instantaneous power; p vi Vm sin wt v I m sin wt i Vm I m sin wt v sin wt i i = I m sin(w t + i ) v = V m sin(w t + v ) cos v i cos2wt v i sin wt v sin wt i 2 59 14.5 Average Power and Power Factor Hence; Vm I m cos v i cos2wt v i p 2 Vm I m Vm I m cos v i cos2wt v i 2 2 Fixed value Time-varying 60 14.5 Average Power and Power Factor 61 14.5 Average Power and Power Factor The average value of the term; Vm I m cos2wt v i 2 is zero Therefore the average power delivered to the load is contributed by the first term only i.e.; Pav Vm I m cos v i 2 v i is the angle difference between v and i 62 14.5 Average Power and Power Factor Defining v i , the average power delivered to the load becomes; Vm I m Pav cos 2 watt, W Pav Vrms I rms cos watt, W or; 63 14.5 Average Power and Power Factor For purely resistive load 0 and cos 1 Pav Vrms I rms cos Vrms I rms cos is known as the power factor (Fp) For purely inductive or purely capacitive load 90 and cos 0 Pav Vrms I rms cos 0 64 14.5 Average Power and Power Factor Power factor can be leading or lagging A power factor is said to be leading if the current leads the corresponding voltage When the current lags the voltage, the corresponding power factor is known as a lagging power factor In terms of the average power and terminal voltage and current; P Fp cos Vrms I rms 65 14.5 Average Power and Power Factor Example 4.10 Find the average power dissipated in a network whose input current and voltage are; i 5 sin wt 40 A v 10 sin wt 40 V 66 14.5 Average Power and Power Factor Example 4.10 – solution Since the current and voltage are in phase; 0 and cos 1 Vm I m 10 5 P 25 W 2 2 Or; Vm 10 R 2 Im 5 and I rms Im 5 2 2 67 14.5 Average Power and Power Factor Example 4.10 – solution (cont’d) 2 PI Or; Vrms 2 rms 5 R 2 25 W 2 Vm 10 2 2 2 V 10 1 P 25 W R 2 2 2 rms 68 14.5 Average Power and Power Factor Example 4.11(a) Determine the average power delivered to a network whose input voltage and current are; v 100 sin wt 40 V i 20 sin wt 70 A 69 14.5 Average Power and Power Factor Example 4.11(a) – solution Given; v 100 sin wt 40 V i 20 sin wt 70 A Hence; 70 40 30 cos cos 30 0.866 Vm I m 100 20 P cos 0.866 866 W 2 2 70 14.5 Average Power and Power Factor Example 4.11(b) Determine the average power delivered to a network whose input voltage and current are; v 150 sin wt 70 V i 3 sin wt 50 A 71 14.5 Average Power and Power Factor Example 4.11(b) – solution Given; v 150 sin wt 70 V i 3 sin wt 50 A Hence; 70 50 20 cos cos 20 0.9397 Vm I m 150 3 P cos 0.9397 211.4 W 2 2 72 14.5 Average Power and Power Factor Example 4.12(a) Determine the power factor and indicate whether it is leading or lagging 73 14.5 Average Power and Power Factor Example 4.12(a) – solution i 2 sin wt 40 A v 50 sin wt 20 V Therefore; 40 20 60 Fp cos 60 0.5 (i leads v) (leading) 74 14.5 Average Power and Power Factor Example 4.12(b) Determine the power factor and indicate whether it is leading or lagging 75 14.5 Average Power and Power Factor Example 4.12(b) – solution i 5 sin wt 30 A v 120 sin wt 80 V Therefore; 30 80 50 Fp cos 50 0.64 (i lags v) (lagging) 76 14.5 Average Power and Power Factor Example 4.12(c) Determine the power factor and indicate whether it is leading or lagging 77 14.5 Average Power and Power Factor Example 4.12(c) – solution P 100 W Veff I eff 20 5 100 W P 100 Fp 1 Veff I eff 100 (neither leading nor lagging) 78