Correct power factor formua - Electrical Engineering Stack Exchange most recent 30 from electronics.stackexchange.com 2019-09-21T09:52:44Z https://electronics.stackexchange.com/feeds/question/220082 https://creativecommons.org/licenses/by-sa/4.0/rdf https://electronics.stackexchange.com/q/220082 1 Correct power factor formua delmadord https://electronics.stackexchange.com/users/24435 2016-02-29T19:11:38Z 2016-03-01T13:49:43Z <p>My instructor asked me why have I written the power factor formula as</p> <blockquote> <p>If \$\varphi\$ is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, \$cos\,\varphi\$: $$|P| = |S|\,\cdot\,cos\,\varphi$$</p> </blockquote> <p>with his reasoning that \$P\$ must always be positive with the exception when the resistance is negative, so I should not have the absolute value there.</p> <p><del>Now I cannot remember the reason why, but I am pretty sure I have found it somewhere, but since I did a poor job with references, now I cannot find the origin of this formula.</del> </p> <p>I have found it on <a href="https://en.wikipedia.org/wiki/Power_factor#Definition_and_calculation" rel="nofollow">wikipedia</a>.</p> <p>Is this formula wrong? If yes, how can it be written correctly in this form. If no, can you forward me to some more reading on this? Many thanks.</p> https://electronics.stackexchange.com/questions/220082/-/220108#220108 2 Answer by Roger C. for Correct power factor formua Roger C. https://electronics.stackexchange.com/users/59075 2016-02-29T21:47:49Z 2016-02-29T21:47:49Z <p>I would say that the most general formula is:</p> <p>$$P=Re\{\underline{S}\}=Re\{\underline{V}·\underline{I}^{\star}\}=S\cos{\theta}$$</p> <p>where I'm using underline to identify phasors. </p> <p>S is the module of \$\underline{S}\$ and it is always positive (it is simply the voltage amplitude times the current amplitude). </p> <p>\$\theta\$ is the angle between voltage and current. If this angle is bigger than 90º it means that the load is in reality providing active power to the source and not the other way around. </p> <p>Therefore P can be positive or negative.</p> https://electronics.stackexchange.com/questions/220082/-/220111#220111 1 Answer by Adam Haun for Correct power factor formua Adam Haun https://electronics.stackexchange.com/users/56411 2016-02-29T22:04:00Z 2016-03-01T13:49:43Z <p>The real power (<strong>P</strong>) should not be an absolute value. Let's look at the math. Power factor is defined as the ratio of real power (<strong>P</strong>) to apparent power (<strong>S</strong>). But <strong>S</strong> is a complex number, so a simple ratio is meaningless. We need to use the magnitude of <strong>S</strong>:</p> <p>$$pf = \frac {P} {|S|}$$</p> <p>If we only have <strong>S</strong>, we can use it to get a formula for <strong>P</strong>. <strong>P</strong> is the <em>real part</em> of <strong>S</strong>:</p> <p>$$P = Re\{S\} = |S| \cos \varphi$$</p> <p>where \$\varphi\$ is the <em>argument</em> of <strong>S</strong> -- the angle between the complex vector <strong>S</strong> and the positive real axis. If <strong>S</strong> has a negative real component, then <strong>P</strong> should be negative. And indeed, \$\cos \varphi\$ is negative when \$-90^\circ &lt; \varphi &lt; 90^\circ\$.</p> <p>This is a bit confusing because power triangles are almost always drawn with positive real power, which makes it look like the angle is between <strong>P</strong> and <strong>S</strong>. But it's not -- the angle is a property of the complex number <strong>S</strong> alone.</p> <p><a href="https://i.stack.imgur.com/cZWpl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cZWpl.jpg" alt="Power triangle from Wikipedia"></a></p> <p>So a negative power factor tells you that the load is supplying power to the generator. If you don't care about that, you can take the absolute value of the cosine to keep the power factor positive:</p> <p>$$|pf| = \frac {|P|} {|S|} = |\cos \varphi|$$</p> <p><strong>UPDATE:</strong> The Wikipedia article is incorrect. It contradicts itself:</p> <p><a href="https://i.stack.imgur.com/hSq4V.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hSq4V.png" alt="Screenshot of a small part of the Wikipedia article on power factor"></a></p> <p>The formula contains the absolute value of <strong>P</strong>, but then immediately says the power factor can be negative.</p>