Electronic – Phasor Representation of Sinusoidal Forcing Functions

acpassive-networksphasorvoltage

Lets take a phasor Vm<\$\theta\$ which is equivalent to Vmcos(\$\omega\$t + \$\theta\$)

Let us Assume that a forcing function on a network is Vmcos(\$\omega\$t + \$\theta\$) +j Vmsin(\$\omega\$t + \$\theta\$)

This can be written as Vm ej(\$\omega\$t + \$\theta\$) by Euler's relation

simplifying this gives Vme j\$\theta\$ .e jwt

which further can be written as V e j \$\omega\$ t where V is a defined by Vmej\$\theta\$

My question is : are both V and Vm<\$\theta\$ Equivalent?

My book implies that they are equal.

Correct Me

2nd Question How are Phasors Complex numbers?

Best Answer

They are equivalent in that they represent the same sinewave with amplitude \$V_m\$, phase \$\theta\$ and frequency \$\omega\$ (implicit at this level). Mind that, although they are often represented by exponentials, sinusoidal signals are purely real. Therefore you always have to take the real part of the exponential form, and that's how you obtain the cosine.

The phasor helps you in expressing the phase as an exponential, therefore making simpler to operate with different signals and reactive components.

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