Phasors

A Phasor is a complex number that represents the amplitude and phase of a sinusoid.

In our advanced mathematics we know that a complex number z can be written in rectangular form as z = x + iy where i = √-1 the imaginary number. For the whole discussion we will let i = j for our convention for the next topics. The real part  is x and y is Imaginary.

z = r ∠ θ = r(cos θ + j sin θ )   is the polar form.

z = x+jy                                    is rectangular form.

 To convert rectangular to polar form we use: r = √(x² + y²) and θ = tan^-1(y/x).

To convert polar to rectangular form we use:  x = r cos θ and y = r sin  θ.

Phase difference of voltage and current

Before we start, let’s consider first what is the meaning of LEADING and LAGGING in engineering world. They will say current is leading or current is lagging. What do you mean by that? The arrow in red represents the red phase of the sinusoidal of either current or voltage and the blue arrow represents the blue phase of the sinusoidal of either current or voltage of the same frequency.

We represent our phasors are stationary but in reality this phasors are rotating.

Now, the red phasor is leading the blue phasor and the blue phasor is lagging the red phasor at some angle. In other words;

Red leads blue by θ.

Blue lags red by θ.

There are some cases that blue leads red. It is by 360-θ degrees. To understand see figure below.

IF the given is, v(t)= 12 cos (50t + 10º)

We have to consider this sinusoidal voltage v(t) = Vm sin ωt were:

Vm = the amplitude of the sinusoid

 ω = the angular frequency in radians per second.

ωt = the argument of the sinusoidal

Example:

        

   10∠-30 + (3-j4)       /       (2+j4)(3-j5)'

                     

Solution:

            In the numerator our first term is in the form of polar form. In adding and subtracting complex numbers, it is better to perform it in rectangular form. So, we have to transform the polar form to rectangular form.

So,

               10 (cos (-30) + jsin (-30)) = 8.66-j5

  numerator;                        (8.66-j5) + (3-j4), (add like term)

                              Yields,

                                        

                                         11.6-j9  ---->  14.73∠-37.66 

              

             In the denominator the two terms are expressed in rectangular form but in multiplying and dividing complex number, it is better to perform the operation using the polar form. So, we have to transform rectangular form to polar form.

             (3-j5)' has prime, it could be Asterisk or Bar. This means that we have to express this into its conjugate.  (3-j5)' ----->  (3+j5).

So,

          (2+j4)(3+j5) =   26.08∠122.47

our new equation is :  

                        (14.73∠-37.66)  /     26.08∠122.47

Yields,

                        (0.56∠-160.13)

            

               I've learned that in analyzing AC circuit, complex number can be use and satisfies the conditions every circuit that we used to solve. The voltage and current impedance can be expressed into complex number which can be in rectangular form, polar form and exponential form.