RMS current, RMS value: introduction to RMS value, general and specific formula

Introduction to the rms value of a current and a voltage. Calculations of average power, active power, reactive power, apparent power and complex power, including the power factor or COS(φ) in single-phase, are ALL based on RMS values.
Note: the famous RMS value of a signal is not its mean value, otherwise the 2 words ‘RMS value’ would be meaningless. In addition, an average value cannot be an integral function below the root of 2, there's no such thing. See the video entitled {rms voltage, ‘EX9 - Video 1/2’ } for more details.

The instantaneous power formula is by definition: P(t) = V(t).i(t) = V(t)²/R = R.i²(t) .
For a sinusoidal signal, Pca(t) = Vca(t).Ica(t) = R.Ica²(t).
If we take a resistor R supplied by a constant DC voltage source Vcc(t), Ohm's law leads to Vcc(t) = R.Icc(t) and therefore P(t) = R.Icc²(t) .

Since the source Vcc(t) and R are constant in time and do not depend on time t, which implies that Icc(t) is also constant, let's call Icc(t) = Icc = constant. Hence P(t) = R.Icc² .

For t varying [ from 0 to infinity ] we obviously have Icc² = Icc² because Icc is independent of the variable t . So for t = 2.Pi [second], for example, we always have Icc² = Icc² ,
let's call Icc²(2.Pi) = Icc².
Hence P(t=2.Pi) = R.Icc²(2.Pi).
Knowing that the mean value of the power of a sinusoidal periodic signal, Pcamoy, is defined over a period of T = 2.Pi [second] via an integral :
Pcamoy per period whose duration is time T is: (1/T). Integral {Pca(t).dt} where T = 2.Pi [second].
In other words Pcamoy/T = (1/T). Integral {Pca(t).dt} .
In concrete terms, this means that during a period where t varies from t = 0 to t = 2.Pi we have : Pcamoy(t=2.Pi) = (1/T). Integral {Pca(t).dt} .

we now try to equalise the two expressions which are :
P(t=2.Pi) = R.Icc²(2.Pi) and Pcamoy(t=2.Pi) = (1/T). Integral {Pca(t).dt} .

But before we equalise them, we notice that Icc(2.Pi) is constant and we don't know what to call it, so let's call it ‘Ieff’ without trying to understand what the suffix eff means, i.e. Ieff² = Icc²(2.Pi) ,
so P(t=2.Pi) = R.Ieff² and we deduce Pca(t), i.e. :
R.Ieff² = (1/T). Integral {Pca(t).dt} .
By replacing Pca(t) by its expression Pca(t) = R.Ica²(t) we obtain :
R.Ieff² = (1/T). Integral {R.Ica²(t).dt}.
this leads to: Ieff = Square root of [ (1/T). { integral from 0 to T of Ica²(t).dt} ]. {A}

The expression {A} is a general formula for calculating the value of Ieff. Ieff does not represent an average value at all.

Let's now try to work out what Ieff means. What does the suffix eff mean?

Now if instead of a sinusoidal periodic signal Ica(t), we consider a constant signal Icc and replace it in the expression {A} and after integration we obtain Ieff = Icc .
the mean value of a constant, such as Icc, is the constant itself.
Ieff therefore represents, in this particular case, the mean value of the constant current Icc.

The first conclusion to be drawn is that the rms value of a signal only represents the mean value of the signal if the amplitude of the signal remains constant, which is the case for a direct current or a direct voltage.
Otherwise, for a sinusoidal signal of the form Ica(t) = cos(W.t) or Ica(t) = sin(W.t), the value Ieff resulting from the expression {A} cannot be the mean value of cos(W.t) or sin(W.t) since it is zero over one period. Supporting evidence: when we replace Ica(t) by cos(W.t) , or Ica(t) by sin(W.t) , in expression {A} we end up with Ieff = Imax/(Root of 2) , Ieff is therefore not the mean value of cos(W.t) or sin(W.t) which must be zero. {B}

The second conclusion is that an RMS value of a signal is only equal to the maximum value (maximum amplitude) of the signal divided by the root of two if the signal is in the form of a SINGLE sinusoid, otherwise the formula {B} will no longer be valid and we will have to use the formula {A} to determine the RMS value, this is the general formula.
If, for example, i(t) = cos(W.t + 30°) + sin(2W.t) , Ieff will not be equal to Imax/(Root of 2)
or if V(t) = 5.t - 2 , Veff will not be equal to Vmax/(Root of 2).

Note: When a signal does not represent/reflect a SINGLE sinusoid, it can be broken down into several terms in sinusoidal form called harmonics (Fourier series development), and in this case the general formula {A} is used to calculate its RMS value. Harmonic currents and voltages will be dealt with later in the analysis of filter circuits.