Sinusoidal Waveform Phasor Method

Single-Phase System Alternating signal is a signal that varies with respect to time Alternating signal can be categories into ac voltage and ac current This voltage and current have positive and negative value There are many waveform in alternating circuit such as (i) Sinus waveform (ii) cosines waveform (iii) square waveform (iv) triangular waveform

Sinus and cosines waveform is an important because it is a basic waveform for electric supply in transmission system. Example of waveform is shown in figure below:

There are four type of alternating waveform: Symmetry Non Symmetry Periodic Non Periodic Example of signal is shown below: Symmetry and periodic

Sinus and cosines waveform Voltage and current value is represent by vertical axis and time represent by horizontal axis. In the first half, current or voltage will increase into maximum positive value and come back to zero. Then in second half, current or voltage will increase into negative maximum voltage and come back to zero. One complete waveform is called one cycle.

There are several specification in sinusoidal waveform: 1. Period 2. Frequency 3. instantaneous value 4. peak value 5. peak to peak value 6. average value 7. effective value

Period Period is defines as the amount of time is take to go through one cycle It symbol as T for time Period for sinusoidal waveform is equal for each cycle.

Frequency Frequency is defines as number of cycles in one seconds. It symbol as f and the units is Hertz ( Hz) It can derives as Signal with lower frequency Signal with higher frequency

Instantaneous value Instantaneous value is magnitude value of waveform at one specific time. Symbol for Instantaneous value of voltage is v(t) and current is i(t). Example of Instantaneous value for voltage is shown:

Peak Value Peak value is a maximum value from reference axis into maximum value of waveform. For one complete cycle, there are two peak value that is positive peak value and negative peak value. It symbol is

Peak to peak value Peak to peak value is a maximum amplitude value of waveform that been calculate from negative peak value into positive peak value It symbol is

Average value Average value is average value for all instantaneous value in half or one complete waveform cycle It can be calculate in two ways: 1. Calculate the area under the graph: Average value = area under the function in a period period 2. Use integral method

Effective value The most common method of specifying the amount of sine wave of voltage or current by relating it into dc voltage and current that will produce the same heat effect. It is called root means square value, rms The formula of effective value for sine wave waveform is where Im & Vm are peak values

A more general expression for the sinusoid (as shown in the figure): v(t) = Vm sin (wt + q) where q is the phase angle

Phase angle Phase angle is a shifted angle waveform from the reference origin Phase angle is been represent by symbol θ or Φ Units is degree ° or radian Two waveform is called in phase if its have a same phase degree or different phase is zero Two waveform is called out of phase if its have a different phase or different phase is not zero

- sin ωt = sin (ωt ± 180o) - cos ωt = cos (ωt ± 180o) A sinusoid can be expressed in either sine or cosine form. When comparing two sinusoids, it is expedient to express both as either sine or cosine with positive amplitudes. We can transform a sinusoid from sine to cosine form or vice versa using this relationship: - sin ωt = sin (ωt ± 180o) - cos ωt = cos (ωt ± 180o) cos ωt = sin (ωt + 90o) sin ωt = cos (ωt - 90o)

The sine and cosine may be useful in manipulation of sinusoidal functions:

Sinusoids are easily expressed in terms of phasors Phasor Diagram Sinusoids are easily expressed in terms of phasors Phasor is a complex number that represent magnitude and angle for a sine wave Phasor diagram is a vector line that represent magnitude and phase angle of a sine wave The magnitude of the phasor is equal to rms value

Phasor Diagram For example, if given a sine wave waveform It can be represent by a phasor diagram

Phasor Diagram From the phase diagram above, it can be conclude that: I leading V for θ° degree or V lagging I for θ° degree V leading V1 for Φ° degree or V1 lagging V for Φ° degree I leading V1 for (Φ° + θ° ) degree or V1 lagging I for (Φ° + θ° ) degree

Phasor Diagram Adding phasors is equivalent to adding the corresponding time function for each phasor One way is to dissolve the phasor to complex numbers and then adding then up according to the real & imaginary values Each phasor can be represented by a complex number.  Break each phasor into real and imaginary parts. V1 = V1cos(f1) + jV1sin(f1) V2 = V2cos(f2) + jV2sin(f2)

Phasor Diagram So, the sum of the two phasors can be computed by adding the real and the imaginary parts separately, giving: = V1cos(f1) + V2 cos(f2) + j[V1sin(f1)+ V2 sin(f2)] Then, we can note the real part and the imaginary part are the real and imaginary parts of the sum. = Vsum,real + jVsum,imaginary And, the phasor for the sum voltage can also be represented with a magnitude-angle representation. = Vsumcos(fsum) + jVsumsin(fsum)

Phasor Diagram whenever you deal with complex numbers or variables.  If we have two phasors that we are adding, we visualize the situation as shown below. Vsum = V1 + V2

Phasor Diagram We can add the two phasors any way possible.  That includes doing it graphically by hand, breaking the phasors into components and summing the real and imaginary components - as we did above - or any other way you can imagine to sum two vector-like quantities.

Phasor Diagram It is now clear that all the laws & rules applicable to DC can be generalised and used in AC circuits analysis. However, due to the alternating property of current & voltage, analysing it requires phasor method. In other words, apart of amplitude, we do need to consider the frequency of voltage or current.

Phasor Diagram To achieve this, using phasor and adopting complex number in calculation we can analyse the steady state AC circuits. Thus with these methods of analysis, the power supplied to the load in the circuit can be obtained.