Understanding Complex Waveforms in Electrical Systems

The General Equation of a Complex Waveform

As a form of alternating energy often found in electrical industry, A.C supply is always considered to be sinusoidal. nonetheless, several AC waveforms are not sinusoidal. For instance, sawtooth provide waveforms for ramps, and multivibrators can generate rectangular waveforms. hence, a complex wave   It may be demonstrated that this type of waveform consists of a succession of sinusoidal waves with various periodic associated times. A f(t) function is considered to be regular if f(t+T), f(t) is regarded to be the interval between two successive repeats for every value of t, and it's termed function time f (t) (Capozzoli et al., 2018).  

The figure below depicts a typical complex waveform with period T seconds as well as frequency f.

A complicated wave like this can be translated into several sinusoidal waveforms, with a variable wavelengths, amplitude and phase. Originally, the main sine wave component had a frequency f equivalent to the complex wave frequency and is termed the fundamental frequency. The remaining sine wave components are considered as harmonics, and their frequencies are integrals of frequency f. The proceeding harmonic is therefore 2f frequency, while the third harmonic 3f frequency etc. Consequently, if the fundamental frequency of a complex wave is 50 Hz, the proceeding harmonic frequency will be 150 Hz, etc

The General Equation of a Complex Waveform.

A complex wave form is denoted by the below equation;

v = V_1m sin(wt + y₁) + V_2m sin(2wt + y₂)+….…+ V_nm sin(nwt + y_n)volts                  

where;   V_1m sin(wt + y₁) is  the fundamental component

 V_1m is the maximum or peak value,

frequency, f = w/2p 

y₁ is the phase angle with respect to time, t = 0.

How complex waveforms are produced from combining two or more sinusoidal waveforms.

A number of sine wave frequencies called "Harmonics" , and are built  by "attaching" complex wave forms. Harmonic is the widespread term used by different frequency waveforms to express the deformation of a sinusoidal shape. In other terms, "harmonics" can be said to be multiples of basic frequency, hence stated as shown, i.e 2f, 3f (Kriek et al., 2017)....

In addition to the number and amplitude of the existing harmonic frequencies, the overall complex wave form also rely on the phase connection between the fundamental frequencies as well as the specific harmonic frequencies. We can observe that a complex wave consists of a basic wave shape plus harmonics with its own maximum value and phase angle. For instance, if the basic frequency is indicated as; E = Vmax(25-0)

For the wave form given by:   V=10 sin⁡〖(100πt)+V1 sin⁡(100πt+π/2) 〗

Determine either by plotting a graph and adding ordinates or by calculation

Solution

Considering the general equation;

1. The amplitude of both wave forms

Amplitude of wave 1 = 10

Amplitude of wave 2 = 48

1. The frequency of both voltages

Frequency of wave 1 = reciprocal of  2

                          = 75.4

 Frequency of wave 2= 75.4

1. The phase angle of both voltages

Phase angle in wave 1 = 0

Phase angle in wave 2 =

              = 90°

1. The resultant and its phase angle.

 Will apply the parallelogram law

Hence;

V1 =10 sin 48

       = -7.68

   V2= 48 sin (48+π/2)

        = 48 sin (138)

         = -10.94

Vt = 13.37 V

Angle = 60°

1. Plot both waves and V (complex waveform)
   
   

Reference

Capozzoli, A., Curcio, C. and Liseno, A., 2018, November. Complex Waveforms Generation by SVO. In 2018 AMTA Proceedings (pp. 1-4). IEEE.

Kriek, N., Groeneweg, J.G., Stronks, D.L., De Ridder, D. and Huygen, F.J.P.M., 2017. Preferred frequencies and waveforms for spinal cord stimulation in patients with complex regional pain syndrome: a multicentre, doubleâ€Âblind, randomized and placeboâ€Âcontrolled crossover trial. European Journal of Pain, 21(3), pp.507-519.