# Power Factor Correction Example Problem



## cdcengineer

Anyone want to take a crack at this example?

A 5kW, 0.8 PF lagging induction motor is connected in parallel with a 2 kVA synchronous motor which operates with a leading PF. Find the PF of the synchronous motor if the overall PF is 0.9 leading.

FYI - the solution is 10.9 degrees 0.982 PF leading

I have no idea how this was done. I tried with impedance and power triangles. I'm thinking that they're might be a trig identity which I'm not thinking of. Suggestions??


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## tonyb

cdcengineer said:


> Anyone want to take a crack at this example?
> A 5kW, 0.8 PF lagging induction motor is connected in parallel with a 2 kVA synchronous motor which operates with a leading PF. Find the PF of the synchronous motor if the overall PF is 0.9 leading.
> 
> FYI - the solution is 10.9 degrees 0.982 PF leading
> 
> I have no idea how this was done. I tried with impedance and power triangles. I'm thinking that they're might be a trig identity which I'm not thinking of. Suggestions??



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get S,Q from one that you know pf.

solve Cos(Atan((3750-Q)/(5000+P)))=0.9

plug that into either P or Q into

P^2 + Q^2 = 2000^2

P/2000=pf

get Q

then since synchronus is leading pf =&gt; is negative Q, it will subtract from the induction motor's +Q and then the P's for both add.

back substitute to check

above trig relationship.

if i typed something wrong, sorry; this got me P=1964 and Q=378


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## DK PE

Would you please recheck the problem statement. You have 3750 VARs to correct for the induction motor and you want to end up with a leading overall PF by adding only a 2kVA synchronous motor. Something seems wrong.


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## tonyb

yeah, that wouldn't work unless the combined leading was actually combined lagging.

Good catch!!!


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## DK PE

This took at bit of thinking but here is my opinion.... The question should be stated such that the aggregate load is 0.9 PF LAGGING. So then you draw one power triangle S= P+jQ of 6.25 = 5kw + 3.75kVAR for induction motor. Then add to that synch motor of 2KVA at unknown but leading angle so just draw vector at say approx -20 - 30 degrees at length 2kVA. You know aggregate PF of load is 0.9 so so draw that from origin at angle of + 26 degrees. The answer is the intersection. It is a triangle formed by the 6.25KVA at 36.5 degrees from induction motor combined with 2kVA at slightly unknown leading angle and line of unknown length at +26 degrees. I solved using law of sines since you know the length of two sides and one angle (angle you know is 11 = 37 - 26 degrees). Once you get that angle you can get rest of triangle and by extending a couple lines arrive at 11 degrees for solution. Let me know if you're still stuck and I'll see if I can post a diagram.


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## cdcengineer

Got it thanks to all.



DK PE said:


> This took at bit of thinking but here is my opinion.... The question should be stated such that the aggregate load is 0.9 PF LAGGING. So then you draw one power triangle S= P+jQ of 6.25 = 5kw + 3.75kVAR for induction motor. Then add to that synch motor of 2KVA at unknown but leading angle so just draw vector at say approx -20 - 30 degrees at length 2kVA. You know aggregate PF of load is 0.9 so so draw that from origin at angle of + 26 degrees. The answer is the intersection. It is a triangle formed by the 6.25KVA at 36.5 degrees from induction motor combined with 2kVA at slightly unknown leading angle and line of unknown length at +26 degrees. I solved using law of sines since you know the length of two sides and one angle (angle you know is 11 = 37 - 26 degrees). Once you get that angle you can get rest of triangle and by extending a couple lines arrive at 11 degrees for solution. Let me know if you're still stuck and I'll see if I can post a diagram.


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## tonyb

could I have a look at your diagram


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## tonyb

tonyb said:


> could I have a look at your diagram



This sketch is pretty weak but I think I understand how to look at this type of problem a little better. Thank You!


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## DK PE

Here is a diagram to help out. At this point you may be frustrated with "you've got to be kidding me, the obscure law of sines, should I bring my HS algebra and trig book"! If this isn't clear ask as this is a fairly important topic.

Remember the problems are multiple choice (I could give you an "unlike the old days but it would sound like I walked 10 miles to school, uphill both ways...) and it may be easier to set up the diagram as I've shown and plug in a couple of the potential solutions to see what angle of 2kVA leading synch motor produced the desired total power factor. Just my $0.02 worth but being able to draw a diagram something like I've shown or whatever really makes sense to you is valuable.


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## DK PE

I thought I had attached diagram... another attempt


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