AC circuits - Complex vs Ordinary notation

[Nikitin]

When doing calculations on AC circuits, what is in your opinion the best notation? Here they mostly use complex notation. For example: V(t) = V_maxe^iωt. In US textbooks (at the non-EE level), they only use ordinary notation. Example: V(t) = V_maxcos(ωt).

I have a feeling that complex notation is faster, but also less intuitive. So which should I learn properly (and no, I don't want to be an expert in both)? The quicker complex notation, or the much more intuitively pleasing ordinary notation?

BTW: I don't study electrical engineering, but engineering physics.

 

[vanhees71]

The exponential form is far more convenient and becomes intuitive with its use. This becomes clear particularly when you want to study arbitrary signals, not only pure harmonic ones. Then the exponential Fourier integral or series is also far more convenient than the trigonometric ones.

 

[AlephZero]

So which should I learn properly (and no, I don't want to be an expert in both)?

Both, if you want to be an engineer as well as a physicist.

And you need to be clear about the difference between "phase and quadrature" components of a signal described by $P \cos \omega t + Q \sin \omega t$, and $A e^{i\omega t}$ where $A$ is complex. (If you haven't come across this little elephant trap, let $A = P + iQ$, multiply out the real part of $A e^{i\omega t}$, and note the minus sign!)

The exponential notation is very nice mathematically, but you can't measure the imaginary part of a signal in the time domain!

 

[Nikitin]

Does the inaginary part represent some kind of useful information?

 

[anorlunda]

Consider a circuit with only a source and a single capacitor (or inductor) the current will be 90 degrees out of phase with the voltage. Now compute the instantaneous power P(t)=V(t)*I(t). You should see that P(t) is positive for half of a 360 degree cycle and negative for the other half. The integrated value of P (or summed value if you take multiple samples) over a whole cycle is zero.

So what's happening? We have voltage and we have current flow, but the net power over any integral number of whole cycles is zero. That is what we call imaginary power. Ignore the philosophical implications of the word imaginary.

So, when V and I are in-phase with each other, the power is all real. When they are 90 degrees out of phase, it is all imaginary. For all other cases of phase shift, the complex notation with real and imaginary components describes exactly how things behave.

The poorly understood part of this (because it is seldom stated) is that this nomenclature applies only to purely sinusoidal wave forms and only when the analysis considers only whole cycles, and only after an AC steady state is reached after a number of whole cycles. In reality, AC analysis is nothing more than an enormously convenient approximation.

In power systems, those three conditions are approximately met almost all the time, so complex arithmetic makes complicated problems simpler.

 

[BruceW]

The exponential notation is very nice mathematically, but you can't measure the imaginary part of a signal in the time domain!

Does the inaginary part represent some kind of useful information?

The whole idea of the use of complex numbers (as far as I know) is that we originally have a real differential equation. And if we want to, we can solve it using just real numbers. But it is much easier to use complex numbers, since they have both a 'direction' and magnitude. So we are abstracting our differential equation to the complex numbers, so that we can solve it, and relate that result to the real-numbers case.

To do this, we just say that the component of the complex number in one particular direction is the result for the actual voltage (or current, or whatever). And just by convention, people often choose the real number line as this 'particular direction'. So if we have V(t) the complex voltage at some time t. Then RE(V(t)) is our solution for the actual voltage at that time.