I'm doing an assignment on circuit analysis with phasors and it's brought up a point of confusion for me on how Phasors convert to rectangular form.
My textbook defines phasors as $$v(t) = V_M\text{cos}(\omega t + \phi) = \text{Re}[V_Me^{j(\omega t + \phi)} ]$$
and says that they can be written in phasor notation as $$ V_M\angle \phi $$
That makes plenty of sense to me. However, in my homework, when I have to add phasors and convert them to rectangular, the solution has them being represented as
$$ V_M\angle \phi = V_M\text{cos}(\phi) + jV_M\text{sin}(\phi) $$
This difference seems contradictory to me. The book seems to imply that phasors are only expressing the real component of the function, but when it comes to problems, phasors are now both the real and imaginary components? I know I have some conceptual misunderstanding somewhere, but I just can't identify it.
Best Answer
A phasor is a vector representation of a sinusoidal voltage (or current) frozen in time. Let's say the time domain equation is V(t) = A * cos(omega * t + phi). Please note that even though the wave is a sinusoid, there is magnitude (A) and an angle involved. For example, when you freeze the signal at time t = 0, then the sinusoidal voltage actually has an angle associated with it. In our case, that angle is phi (since t = 0). The magnitude is A.
Since the phasor has a magnitude and an angle or direction, it is actually a vector. When we convert it from polar coordinates to rectangular coordinates, we let one of the axes be the imaginary axis. That is how imaginary numbers come into the picture. When we freeze a rotating vector in time, the imaginary part provides information about the angle.
The instantaneous voltage is the real part of the vector. That is just how polar to rectangular coordinate conversions work.
You could, if you want, say that a sinusoidal voltage is really a snapshot of a mysterious rotating object that we can't see. It traces out a sine wave onto the real plane that we are capable of observing, but somewhere we can't see, there is another sine wave being traced out on an imaginary plane we cannot see. The imaginary sinusoid is out of phase by 90 degrees to the real one.
Later you may get into the concepts of in-phase and quadrature mixing of signals. Hilbert transforms and the Analytic Signal. It is all very interesting. My career didn't end up exposing me to that stuff in a deep way but it is abstract and hard to understand unless you spend some time struggling with it.