# Power factor angle (leading)

#### mofet

Joined Oct 5, 2010
6
Hi

I was reading Guru's book on Electric machinery (page 38, 2nd edition) and I stuck here in this:
"The sign of the power factor angle teta can be easily determined from
the type of load. The power factor angle teta must be negative for an inductive load (R + jX) and positive for a capacitive load (R - jX)."

Well, Sadiku's book (fundamentals of electric circuits) tells the opposite...

So when the current leads the voltage then we have θI>θV, which of course implies that the argument of the cosine function, θV−θI, is negative.... (leading pf is seen for capacitive loads...) So the power factor angle for capacitive load should be NEGATIVE and not positive as Guru tells...

I wonder why Guru wrote that power factor angle is positive for capacitive load... well I'm seeing this image...
http://oee.nrcan.gc.ca/industrial/equipment/vfd-ref/images/figure-23.jpg (as Sadiku shows in his book) and it is clear that phi (the angle of power factor) is positive for inductive loads , and negative for capacitive loads!
I got confused.. hope you can help to solve this apparent dilemma..

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#### t_n_k

Joined Mar 6, 2009
5,455
For an inductive load Z=R+jX supplied by a source V with reference phase angle, the load current I is given by

$$I_{load}=\frac{V\angle0^o}{R+jX}$$

$$I_{load}=\frac{V\angle0^o}{Z\angle \theta^o}$$

Where

$$Z=\sqrt{(R^2+X^2)}$$

and

$$\theta=arctan(\frac{X}{R})$$

Hence

$$I_{load}=\frac{V}{Z}\angle -\theta^o$$

So the power factor is cos(-θ) and the power factor angle is negative - which agrees with Guru's statement.

Remember that the cosine is positive in the 4th quadrant so cos(-θ)=cos(θ)

One normally denotes the power factor as leading or lagging - rather than positive or negative.

#### mofet

Joined Oct 5, 2010
6
Thanks.
t_n_k, I must say that the angle of -theta in I(=current) cannot be the power factor angle, why? Because we are using theta V - theta Z... and not theta V - theta I... am I wrong?

I really got confused with the power triangle that shows a power factor angle positive for inductive loads and positive for capacitive loads (seems contradictory with what you told which I agree!).. as the forementioned link shows.. How can we conciliate this dilemma!?

(Note: I used to think that the terms "leading" and "lagging" pertain to the relationship that the current has with respect to the voltage... in the graph I presented... they seem not use this relationship as PF of capacitive loads is said to be lagging?! Maybe they use the opposite relationship but there is no statement that points it out.).

This graph - http://www.nepsi.com/images/CDIA150.gif - shows the one that Sadiku uses...

Thanks for your input, t_n_k.

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#### t_n_k

Joined Mar 6, 2009
5,455
I believe there is no dilemma to be resolved. I think you may have misinterpreted what the various authors are specifically saying - which may simply be related to the terminology used by them. The issue for you seems to center around the load impedance triangle angle and the angular displacement between the load voltage and current.

I doubt this is really a particularly critical issue for you (given your good understanding of the relevant circuit theory) as long as you are confident in correctly resolving the circuit conditions for any given problem. Trust to your own ability if you are consistently getting the right answers.

true. Thanks once again.