The simple complex algebra that I do on my I and Q values is easier than using trigonometry on the sine and cosine waveforms, which is also valid mathematics.
What you see on an oscilloscope is a function of time.
Often, as in frequency response or impedance calculations, what you want is a function of frequency, where complex algebra is useful.
Yes, tricky - "valid mathematics" I think being the sticking point (or question) for me.
Time domain, everything is real-valued, and therefore "real" in my mind.
Frequency domain is an abstraction which is much further removed from the direct analogue "this point 'has' this quantity of voltage right now" (itself a short abstraction away from potential difference of an electric field). Decomposing an arbitrary signal into frequency components is a transform which has no physical significance whatsoever (in the sense that inventing a fairy tale to describe some physical phenomenon is no more valid than some other description which works - a point made by penfold a while back). I like to keep this fact (it's a fairy tale) in mind where possible. Do I believe it? Yes - it takes one set of real quantities and converts it into another, isomorphically. The fairy tale gains physical meaning when we lose the arbitrariness of the signal and begin dealing with sinusoids - RF, sound, bandlimiting, synchronous demodulation... I am happy to think in terms of reactance and the give and take of energy - not a complex number in sight. But I still like to check myself when talking about "frequencies" so as not to get too carried away by the fairy tale.
Complex numbers are where my belief in a fairy tale ends. Not because they don't work, but because the mathematical validity seems to be based on a leap of faith - a circular definition (no pun intended). Not entirely, but not 100% convincing. The fact that most 'proof' seems to consist of wildly gesticulating at my paragraph above saying "but but phasors" suggests that the proponents of the fairy tale have become so under its spell that they have lost the ability to reason.
The time domain is more appropriate for some measurements, and the frequency domain is more appropriate for others.
Some clarification about I and Q measurements on my two-phase lock-in amplifier and (now more common) vector network analyzers:
The two values I and Q are functions of frequency, and the front-panel outputs from the lock-in amplifier are not suitable for seeing on an oscilloscope.
They are bipolar (positive or negative) non-sinusoidal voltages.
They represent the
amplitude of the in-phase and quadrature components that result from synchronous demodulation with respect to the reference input, one in-phase and the other in-phase-quadrature.
On the lock-in amplifier, the averaging time to extract the amplitudes is switchable, and provides noise filtering at the expense of slowing the response to changes in the signal.
(The low-pass filters are mandatory for synchronous demodulators, to extract the DC value for magnitude and reject the second harmonic of the reference frequency and other spurious outputs.)
Thus, they are suitable for measuring behavior in the frequency domain, so long as that behavior is not varying too quickly.
If you apply slightly different frequencies to the input and the reference, where the frequency difference is smaller than the low-pass filter cut-off, you will see the signal "rotate" between the I and Q outputs at the difference frequency.
Of course, in a free country, you are not required to use mathematics with which you are uncomfortable, but I assure you that when dealing with the results from these analyzers, complex algebra is logically consistent, gives physically correct results, and is very convenient.
What else is required to justify the use of a given mathematical method on a physical problem?