HNC/D Engineering Science

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Introduction to Alternating Current and Voltage

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HNC/D Engineering Science Complex Waveforms HNC/D Engineering Science

Radian 360° = 2π 180° = π 90° = π/2 45° = π/4

Example

Harmonics Three things can specify a sine wave Sine waves – pure Amplitude Frequency Phase Sine waves – pure Distortion For most instances we can assume that sine waves are pure, but in some cases distortion is present and needs to be considered. Distortion, due to harmonic waves occur in circuits containing devices such as rectifiers, thermionic valves, discharge lamps, and anything with a magnetic core. We image a wave form as being pure, however all sine waves have harmonics, which causes some distortion, heavy distortion, due to harmonics can give a complex wave.

Harmonic If 50 Hz is the fundamental frequency Then each integer multiple will be an harmonic 50 Hz = Fundamental 50 Hz x 2 = 100 Hz = Second Harmonic 50 Hz x 3 = 150 Hz = Third Harmonic Etc…

Complex Wave A waveform which is not sinusoidal is termed a complex waveform These comprise of the fundamental, plus a number of harmonics, each of which will have a specific amplitude and phase

Formula’s V1 = A sin ωt (Fundamental Harmonic) A = maximum voltage value ω = angular frequency (2πf) t = time V1 = value of waveform at time t and A maximum valve

Formula’s V2 = A sin 2ωt – (second harmonic) V3 = A sin 3ωt – (third harmonic) However if the maximum voltage changes with each harmonic V1 = A1 sin ωt V2 = A2 sin 2ωt V3 = A3 sin 3ωt

Formula’s The other element we have to take into consideration is when the second and third harmonic is out of phase. (Does not start at t = 0) V1 = A1 sin (ωt + φ1) V2 = A2 sin (2ωt+ φ2) V3 = A3 sin (3ωt+ φ3)

Formula’s These formulas help describe the complex waveform. As Jean Baptiste Fourier in 1822 proposed that any periodic waveform can be made up of a combination of sinusoidal waveforms. v = A1 sin (ωt + φ1) + A2 sin (2ωt+ φ2) + A3 sin (3ωt+ φ3) + ………

Complex Wave (Calculation) m/s 1 50 hz =6*SIN(100*PI()*B4) =6*SIN(100*PI()*C4) 100 hz =3*SIN(200*PI()*B4) =3*SIN(200*PI()*C4) 150 hz leading π 45 =1.5*SIN(300*PI()*(B4+0.0025)) =1.5*SIN(300*PI()*(C4+0.0025)) complex Wave =SUM(B6:B8) =SUM(C6:C8) 150 hz =1.5*SIN(300*PI()*B4) =1.5*SIN(300*PI()*C4)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)