Solving Question on Phasors: Transform into Exponential & Back to Sinusoidal

[Teczero]

Hi everyone!

I understand the general concept but I came across a function such as:

E(t)= 0.1 sin[10(pi)x]cos[6(pi)10^9t - Bz]

I'm supposed to combine them using phasors and I'm really confused how to

a) Transform it into the exponential function
b) Transform it back in the sinusoidal function

Thank you for the help =)

 

[Mr.Green]

First let us review how to covert a quantity written in phasor notation to time domain. As soon as you learn that it will be very simple to reverse the operation.

Assume that we have the phasor:

E(x, y, z) = A(x, y, z) exp^jф

Where:
-A is any real function of x, y, and z.
- ф is the phase reference.

The time domain of this phasor is e(x, y, z, t) and is given by:

e(x, y, z, t) = Re [ E(x, y, z) * exp^jωt] = Re [A(x, y, z) exp^j(ωt+ф)]
So,
e( x, y, z) = A(x, y, z) cos(ωt + ф)

Now to convert a time domain quantity to a phasor just remove the term [cos(ωt + ф)] and multiply A(x, y, z) by exp^jф