Electronic – Calculate the active power

circuit analysispower

schematic and input waveform

I'm asked to determine the active power (P). (I don't know if that is actually called active power in English?)

The formula of P is = U * I * sin(alpha)

How can we determine the alpha in this case, so that we can calculate the P?

Best Answer

Well, we have the following circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

The input voltage is given in the following diagram:

enter image description here

Assuming an ideal diode, the negative parts are cut of for the voltage across the resistor. That is shown in the following diagram:

enter image description here

Now, we know that the power in a resistor is given by:

$$\text{P}_\text{R}\left(t\right)=\frac{\text{V}_\text{R}^2\left(t\right)}{\text{R}}\tag1$$

It is not hard to show that \$\text{V}_\text{R}\left(t\right)\$, is given by:

$$\text{V}_\text{R}\left(t\right)=\frac{\text{V}}{\text{T}}\sum_{\text{n}\ge0}\theta\left(t-\text{Tn}\right)\mathcal{I}_\text{n}\left(t,\text{T}\right)\tag2$$

Where:

$$\mathcal{I}_\text{n}\left(t,\text{T}\right)=\left(\text{T}\left(1+2\text{n}\right)-2t\right)\left(\theta\left(t-\text{T}\left(\frac{1}{2}+\text{n}\right)\right)+\theta\left(t-\text{T}\left(1+\text{n}\right)\right)\right)\tag3$$

Now, we can look at the average power and the RMS power using the following two formulas:

  • Average power: $$\overline{\text{P}}_{\text{R}}=\frac{1}{\text{T}}\int_0^\text{T}\text{P}_\text{R}\left(t\right)\space\text{d}t\tag4$$
  • RMS-power: $$\text{P}_{\text{R}|\text{RMS}}=\sqrt{\frac{1}{\text{T}}\int_0^\text{T}\text{P}_\text{R}^2\left(t\right)\space\text{d}t}\tag5$$

I used Mathematica to find them:

Average power:

In[1]:=Integrate[(1/
    T)*(((Sum[(V/
           T)*((HeavisideTheta[
             t - T*n])*(((-2 t + T + 
               2 n T) (-HeavisideTheta[t - T/2 - n T] + 
               HeavisideTheta[t - (1 + n) T]))/1)), {n, 0, 
         Infinity}])^2)/R), {t, 0, T}, Assumptions -> T > 0]

Out[1]=V^2/(6 R)

RMS-power:

In[2]:=FullSimplify[
 Sqrt[(1/T)*
   Integrate[((((Sum[(V/
               T)*((HeavisideTheta[
                 t - T*n])*(((-2 t + T + 
                   2 n T) (-HeavisideTheta[t - T/2 - n T] + 
                   HeavisideTheta[t - (1 + n) T]))/1)), {n, 0, 
             Infinity}])^2)/R))^2, {t, 0, T}, Assumptions -> T > 0]], 
 Assumptions -> V > 0 && R > 0]

Out[2]=V^2/(Sqrt[10] R)
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