Complex Waveforms HNCD Engineering Science Radian o o

Complex Waveforms HNC/D Engineering Science

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

Example

Harmonics o Three things can specify a sine wave n n n o o Amplitude Frequency Phase Sine waves – pure Distortion

Harmonic o o o 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 o o 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 o V 1 = A sin ωt (Fundamental Harmonic) o A = maximum voltage value ω = angular frequency (2πf) t = time V 1 = value of waveform at time t and A maximum valve o o o

Formula’s o o o V 2 = A sin 2ωt – (second harmonic) V 3 = A sin 3ωt – (third harmonic) However if the maximum voltage changes with each harmonic V 1 = A 1 sin ωt V 2 = A 2 sin 2ωt V 3 = A 3 sin 3ωt

Formula’s o o o 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) V 1 = A 1 sin (ωt + φ1) V 2 = A 2 sin (2ωt+ φ2) V 3 = A 3 sin (3ωt+ φ3)

Formula’s o o o 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 = A 1 sin (ωt + φ1) + A 2 sin (2ωt+ φ2) + A 3 sin (3ωt+ φ3) + ………

Complex Wave (Calculation) m/s 0 1 50 hz =6*SIN(100*PI()*B 4) =6*SIN(100*PI()*C 4) 100 hz =3*SIN(200*PI()*B 4) =3*SIN(200*PI()*C 4) =1. 5*SIN(300*PI()*(B 4+0. 0025)) =1. 5*SIN(300*PI()*(C 4+0. 002 5)) =SUM(B 6: B 8) =SUM(C 6: C 8) =1. 5*SIN(300*PI()*B 4) =1. 5*SIN(300*PI()*C 4) 150 hz leading π 45 complex Wave 150 hz

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)

Complex Wave (Calculation)