Beer Analogy for Electric Reactive Power Not Adequate?

This comparison has been popularized on social media and is in such a way that it has gained the status of a didactic example for explaining the theory of power systems. Was there any truth in this comparison? This is what we will see here in this text. We will try to write in an impartial way, seeking rationality.

What is certain is that, in general, the term power is born in the classical mechanics of physics. Power will be useful to the extent that its application will perform mechanical work, or heat. In electrical systems, useful power will be that which is converted into useful mechanical power.

Through the Principle of Conservation of Energy, it is possible to obtain the power formula in terms of instantaneous electrical system parameters, that is,

where e(t) corresponds to the electric voltage or potential and i(t) corresponds to the electric current intensity in the circuit.

These quantities, in power systems, come from systems or machines that, for the most part, are of the rotating type, and, with this, their values ​​are mathematically modeled as trigonometric functions, as follows:

Or, in compact notation, using Euler’s formula,

More precisely, machine technology provides the quality that energy can be generated or transported by a set of conductors, thus defining the polyphase system. Then the electrical voltages, perhaps, start to be called phases and they will be electrically distanced by a phase shift. So we have, for electric 360º the phase shift, in a three-phase system, will be:

or,

The lags are inherent to electrical systems, even in direct current systems, since most of these voltages come from alternating current systems. This account above demonstrates a specific condition called the symmetric condition. Thus, for three voltages, we will have a symmetrical three-phase system, or symmetrical three-phase system, described by the expressions:

Still in connection with these three expressions, the following aspect is established: that rotating electrical systems, insofar as they work with the frequency ω, theoretically a constant (in conditions of stability or steady state), are then well represented by these exponential factors present in them, which are independent of frequency, called phasors. That is:

where

And everything from then on is described by a phasor algebra whose domain is within the set of Complex Numbers. So, there are voltage, current, impedance and power phasors.

Specifically to the aspect of the power phasor, this, if written in the conventional form of complex numbers, then becomes:

which, if applied to the complex analytical plane, or the Argand-Gauss plane, suggests the figure of a triangle, which is now called a power triangle, where the module S (phasor) will be given by the Pythagorean theorem formula , that is, correspond to the hypotenuse, P corresponds to the base leg (X axis), and Q corresponds to the quadrature leg (Y axis). Like this,

S, so it’s a geometric mean between P and Q.

It should also be noted, and by theorem of Euclidean geometry, that

Another important aspect is that, once the complex product between the voltage and the electric current has been carried out, we will have the power, that is,

If we do,

then we will have,

and, already in complex number notation, we have

which would be an expression analogous to the expression of the triangle of powers. So we can make a geometric representation out of it.

A suitable example more adhering to the power triangle would be like a car traveling on a perfectly horizontal track so all machine availability will be used as P.

However, if the same car has to travel a road with an irregular profile, then part of its machinery will have to be directed towards the moments of movement in quadrature.

So the vertical movements will be “transactional” with a power balance, Q, always zero. Quadrature motion tends to delay the motion of the automobile.

Comparing all this to the main point of the “beer” argument, we see that that analogy does not contain a “universal” explanation, as it has been put on social media.

Even so, the “beer glass” model has a didactic application, also as an analogy and in knowledge of electricity. See that the foam that forms when filling the glass resembles the skin effect that occurs in electrical conductors when they are traversed by an electric current, and it is something inherent like beer foam. Inside the beer in the glass, the foam density tends to zero, it goes all the way to the top.