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There are two ideal voltage sources \$E_1= \angle 0°\$ and \$E_2=\angle 30°\$, \$E_1\$ and \$E_2\$ are not in phase, and there is an impedance \$z=5+0\mathrm{j~Ω}\$. The schematic is as below:

enter image description here

  • \$S_1=EI^*=268+1000\mathrm{j ~VA}\$, so the real power is \$\mathrm{Re}(S_1)=268\mathrm{~W}\$, and reactive power is \$\mathrm{Im}(S_1)=1000\mathrm{~VAr}\$.

  • \$S_2=EI^*=-268+1000\mathrm{j~VA}\$
    The real power is \$\mathrm{Re}(S_2)=-268\mathrm{~W}\$, and reactive power is \$Im(S_2)=1000\mathrm{~VAr}\$.

  • \$W=I^2R=536\mathrm{~W}\$.

We can say:

  1. Both \$E_1\$ and \$E_2\$ generate the real power \$268\mathrm{~W}\$ for each.
    For this answer, I want to ask why we can say both of them generate the real power, instead of saying \$E_1\$ generates the real power for 268 W, and \$E_2\$ uses this real power for 268 W? Because, $$S_1=EI^*=268+1000\mathrm{j~VA}$$ and $$S_2=EI^*=-268+1000\mathrm{j~VA}$$

2.\$E_1\$ provides \$1000\mathrm{~VAr}\$, and \$E_2\$ uses the \$1000\mathrm{VAr}\$
Same question: Why can we say \$E_1\$ provides \$1000\mathrm{~VAr}\$, and \$E_2\$ uses the \$1000\mathrm{~VAr}\$ instead of both of them generating the reactive power for \$1000\mathrm{~VAr}\$? Because, $$ S_1=EI^*=268+1000\mathrm{j~VA} $$ and, $$ S_2=EI^*=-268+1000\mathrm{j~VA}$$.

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    \$\begingroup\$ We can't say anything until you have shown a circuit of how things are connected and explain how the reactive VA is generated. \$\endgroup\$
    – Andy aka
    Commented Mar 21, 2023 at 13:43
  • \$\begingroup\$ @Andyaka I have edited the question,and the calculation of polar form is not my problem,my problem is how to define which machine or source use the power from others ,and which supply the power to others if the real /reactive power is negative or positive \$\endgroup\$ Commented Mar 21, 2023 at 23:41
  • \$\begingroup\$ I didn't say anything about the form in which you presented complex entities. I have no problem with the representation of complex numbers. \$\endgroup\$
    – Andy aka
    Commented Mar 22, 2023 at 0:12

2 Answers 2

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Please edit and present the original question, or I cannot understand English.

But there is just chance that, \$E_1\$ and \$E_1\$ are not in phase

On seeing what I see here I guess,

$${S_1}={1035.289332e^{j\cdot{74.997283}} (Approx.)}$$

$${S_2}={1035.289332e^{j\cdot{107.002716}} (Approx.)}$$

Total \$W\$ is just, $${\overrightarrow{S_1}}+{\overrightarrow{S_2}}=538$$

You can imagine the vectors and also imagine the answer. I will try to edit and upload image later.

TIP: If you wanted to any kind of mathematics in Electrical Engineering, just go polar form (or what I say as euler form.) Makes calculation either manipulatively simpler and normal addition as vector.

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  • \$\begingroup\$ The calculation of polar form is not my problem,my problem is how to define which machine or source use the power from others ,and which supply the power to others if the real /reactive power is negative or positive \$\endgroup\$ Commented Mar 21, 2023 at 23:41
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I am pretty sure made a mistake on your conventions somewhere. Most evident by the lack of polarity labels on your sources in your schematic.

You need to assign polarities to your AC source terminals in your schematic and then your equation must be consistent with that to have meaning. Just like passive sign convention in mesh or nodal analysis with DC. Otherwise you can't tell the difference between 0 and 180 degrees between AC sources.

For example, if your two sources had a phase of any of the four possible combination of 0 deg and 180 deg, you wouldn't know if they are constructively or destructively interferring (reinforcing or opposing). Without that, all the signs and polarities in your equations are meaningless.

Trying to solve with undefined voltage and current directions in the schematic leads to inconsistent sign usage and nonsensical results. Your circuit is symmetrical so signs that make no sense, but magnitudes that do, are exactly what you would expect if you did this.

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