I calculated the question and I did not get the same answer as in the book. Could someone please explain why my method is not correct?
Question:
My solution:
Book solution:
The problem is from the book "PPI FE Electrical and Computer Review Manual" by Michael R. Lindeburg PE.
Could some one please explain why my method is not correct?
The mistakes I see are that you have calculated total apparent power and not individual reactive powers. You have not recognized that the reactive power of a leading power factor load cancels with (subtracts from) the reactive power of a lagging power factor load. In other words, adding the apparent powers gets you nowhere close to solving the problem.
When you divided actual power (12 kW) with power factor (0.7) you actually arrived at apparent power (volts x amps): -
Image from here.
Power Factor is \$\cos(\theta)\$ and this equals real divided by apparent and not real divided by reactive power. So, you need to calculate the phase angle using \$\arccos(PF)\$ then use \$\tan\$ to calculate the ratio of reactive to real power.
Then, remember that it is the difference between the leading and lagging reactive powers that is important.
@andyaka or I don't always get a notification when you leave a comment.
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You made a big conceptual error in adding the two apparent powers together as scalars rather than as complex quantities. If you calculate your S values as complex values, then you can add them together to get the total S. The total reactive power is the imaginary part of the total S. The key is to think of S as a complex quantity with real and imaginary components. It is similar to a vector with vertical and horizontal components, for example. Just like you can't add the magnitude of two vectors together to get the magnitude of their sum, you can't just add apparent power magnitudes together.
$$S_i = P_i + j Q_i$$ $$ S_T = \sum{S_i} = \sum{P_i} + j \sum{Q_i} = P_T + j Q_T$$ $$ \left|S_T\right| = \sqrt{P_T^2 + Q_T^2} \ne \sum{\left|S_i\right|}$$
The only time that you can add apparent power magnitudes together is when all the apparent power values have the same power factor. In that situation, it's like adding vector magnitudes when their directions are all aligned.
