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164 GEOMETRY AND TRIGONOMETRY Figure 15.35 Figure 15.36
Now try the following exercise. Exercise 72 Further problems on harmonic synthesis with complex waveforms 1. A complex current waveform i comprises a fundamental current of 50A rms and frequency 100 Hz, together with a 24% third harmonic, both being in phase with each other at zero time. (a) Write down an expression to represent current i. (b) Sketch the complex waveform of current using harmonic synthesis over one cycle of the fundamental. [ ] (a) i = (70.71 sin 628.3t + 16.97 sin 1885t) A 2. A complex voltage waveform v is comprised of a 212.1V rms fundamental voltage at a frequency of 50 Hz, a 30% second harmonic component lagging by π/2 rad, and a 10% fourth harmonic component leading by π/3 rad. (a) Write down an expression to represent voltage v. (b) Sketch the complex voltage waveform using harmonic synthesis over one cycle of the fundamental waveform. [ ] (a) v = 300 sin 314.2t + 90 sin (628.3t − π/2) + 30sin(1256.6t + π/3) V 3. A voltage waveform is represented by: v = 20 + 50 sin ωt + 20sin(2ωt − π/2) volts. TRIGONOMETRIC WAVEFORMS 165 Draw the complex waveform over one cycle of the fundamental by using harmonic synthesis. 4. Write down an expression representing a current i having a fundamental component of amplitude 16A and frequency 1 kHz, together with its third and fifth harmonics being respectively one-fifth and one-tenth the amplitude of the fundamental, all components being in phase at zero time. Sketch the complex current waveform for one cycle of the fundamental using harmonic synthesis. [ ] i = 16 sin 2π10 3 t + 3.2 sin 6π10 3 t + 1.6 sin π10 4 t A 5. A voltage waveform is described by ( v = 200 sin 377t + 80 sin 1131t + π ) 4 ( + 20 sin 1885t − π ) volts 3 Determine (a) the fundamental and harmonic frequencies of the waveform (b) the percentage third harmonic and (c) the percentage fifth harmonic. Sketch the voltage waveform using harmonic synthesis over one cycle of the fundamental. ⎡ ⎤ (a) 60 Hz, 180 Hz, 300 Hz ⎣ (b) 40% ⎦ (c)10% B
Page 1 and 2: Geometry and trigonometry 12 Introd
Page 3 and 4: INTRODUCTION TO TRIGONOMETRY 117 Fi
Page 5 and 6: INTRODUCTION TO TRIGONOMETRY 119 5.
Page 7 and 8: Hence 129.9 + x = 75 tan 20 ◦ = 7
Page 9 and 10: ( ) 1 (c) cot −1 2.1273 = tan −
Page 11 and 12: INTRODUCTION TO TRIGONOMETRY 125 or
Page 13 and 14: INTRODUCTION TO TRIGONOMETRY 127 Ap
Page 15 and 16: INTRODUCTION TO TRIGONOMETRY 129 th
Page 17 and 18: from which, AO sin 50◦ sin B = =
Page 19 and 20: Geometry and trigonometry 13 Cartes
Page 21 and 22: 4. (−5.4, 3.7) 5. (−7, −3) 6.
Page 23 and 24: Geometry and trigonometry 14 The ci
Page 25 and 26: (a) Since π rad = 180 ◦ then 1 r
Page 27 and 28: For example, Fig. 14.8 shows a circ
Page 29 and 30: The unit of angular velocity is rad
Page 31 and 32: THE CIRCLE AND ITS PROPERTIES 145 P
Page 33 and 34: 60° 12 cm 12 cm h ASSIGNMENT 4 147
Page 35 and 36: 180° X Figure 15.2 90° Y Quadrant
Page 37 and 38: θ = tan −1 1.7629 = 60 ◦ 26
Page 39 and 40: TRIGONOMETRIC WAVEFORMS 153 B Figur
Page 41 and 42: where α is a phase displacement co
Page 43 and 44: TRIGONOMETRIC WAVEFORMS 157 Figure
Page 45 and 46: (c) When t = 10 ms ( then v = 340 s
Page 47 and 48: TRIGONOMETRIC WAVEFORMS 161 B Figur
Page 49: TRIGONOMETRIC WAVEFORMS 163 B Figur
Page 53 and 54: tan x + sec x LHS = ( sec x 1 + tan
Page 55 and 56: TRIGONOMETRIC IDENTITIES AND EQUATI
Page 57 and 58: 3. 2 cosec 2 t − 5 cosec t = [ 12
Page 59 and 60: Geometry and trigonometry 17 The re
Page 61 and 62: THE RELATIONSHIP BETWEEN TRIGONOMET
Page 63 and 64: Hence = ( tan x + π ) ( tan x −
Page 65 and 66: COMPOUND ANGLES 179 B Figure 18.3 H
Page 67 and 68: COMPOUND ANGLES 181 ◦ ′ ◦ ′
Page 69 and 70: COMPOUND ANGLES 183 Now try the fol
Page 71 and 72: From equation (7), cos 6x + cos 2x
Page 73 and 74: COMPOUND ANGLES 187 p i v p v + i B
Page 75 and 76: Geometry and Trigonometry Assignmen

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