1. HNC/D Engineering Science
Complex Waveforms
HNC/D Engineering Science

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

3. Example

4. 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.

5. 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…

6. 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

7. 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

8. 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

9. 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)

10. 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) + ………

11. 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()*(B ))
=1.5*SIN(300*PI()*(C ))
complex Wave
=SUM(B6:B8)
=SUM(C6:C8)
150 hz
=1.5*SIN(300*PI()*B4)
=1.5*SIN(300*PI()*C4)

12. Complex Wave (Calculation)

13. Complex Wave (Calculation)

14. Complex Wave (Calculation)

15. Complex Wave (Calculation)

16. Complex Wave (Calculation)

17. Complex Wave (Calculation)