In a sinusoid, what does the real and imaginary parts refer to?

Answer 1

The real and imaginary parts are not really mathematical “abstractions” of the magnitude and phase of the sinusoid because they describe the sinusoid at the same bottom-level details, as do the magnitude and the phase. It is analogous to temperature in degree F is not an abstraction of temperature in degree C.

The real part is also not the magnitude of the voltage, or whatever physical measure, the sinusoid represents. And the imaginary part is not the phase.

The real part is the cosine of the phasor of the voltage. The imaginary part is the sine of the phasor. The phasor is the vector in a polar coordinate having its length equal the magnitude and its angle equal the phase of the sinusoid.

For example, consider a sinusoid A * cos(a * t + b), where A is the magnitude, a the angular frequency (the count of degrees, or radians, the sinusoid rotates per unit time), t the time elapsed, and b the phase. When written in complex number, r +jx, the magnitude A = (r^2 + x^2)^0.5, and the angle (at + b) (modulo 360 deg) = arctan(x/r).

The phasor seems more intuitively (once you are familiar with it) written as the exponential: A * e ^ j(at +b). In this notation, we can read the magnitude and the phase angle directly, and can use Euler’s formula, e^j(a) = cos(a) + j * sin (a), to go back to the complex number.

Answer 2

The real part is the actual voltage as measured with an oscilloscope or voltmeter. The imaginary part is a result of the phase angle. For example if the real part has gone to zero, but the imaginary part is +10 Then you have a 10 volt peak sine wave, but you are currently at the zero crossing and will read a zero voltage.

Answer 3

Complex arithmetic is used in science and engineering for calculations involving things which rotate. Trigonometry could also be used for these same calculations, but it's much more cumbersome. In such a calculation, the real and imaginary parts are mathematical abstractions created for the purposes of the calculation. Taken together, they abstractly represent the position of the rotating part. Taken separately, they are meaningless.