# Reactive power equation

#### Moxica23

Joined Jun 23, 2014
12
Hi,

Here it says that the equation for Reactive power is

$$Q=\frac{E^2}{X}$$​

but in many other places I found that the correct equation is:

$$Q=\frac{E^2}{X}*sin phi$$​

where "phi" is the phase delay between current and voltage.

So, which is the correct version of that equation ?

Regards

#### crutschow

Joined Mar 14, 2008
27,755
The first equation is correct if X is just a reactive impedance.
The second equation is correct for X being any impedance (often indicated by the complex number Z).

#### Moxica23

Joined Jun 23, 2014
12
So the first equation describe the situation when in that circuit we have only reactance (capacitive or/and inductive) and no resistance ?

And the second equation describe the situation when in our circuit we have both reactance and inductance ?

If so, why the first equation is included in the calculation methodology for power factor corection for the case when in our circuit we have an electric motor ? An electric motor have both rectance and resistance. They shouldn't use the second equation ?

#### crutschow

Joined Mar 14, 2008
27,755
If the measured voltage is directly across the reactance and not the complete circuit than the first equation works.

#### Moxica23

Joined Jun 23, 2014
12
Yes, I agree. But on this page they speak about a circuit in which an electric motor is the only load. Considering that an electric motor has also some electric resistance, I ask myself why they used the equation for the case when the load is only a reactance.

#### MrAl

Joined Jun 17, 2014
8,479
Hi,

Could that be because they just want to calculate the reactive power, not the total power?

#### t_n_k

Joined Mar 6, 2009
5,455
The OP's link to the AAC page reveals this is in relation to power factor correction (pfc).

The object is to achieve a load side unity power factor. The value of Q as shown in that case is the leading compensation VARS required from a pfc capacitor placed in parallel with a lagging power factor load connected to a supply E. Since a pure capacitance exhibits a 90° phase shift in its current and applied terminal voltage, the sin(phi) value is unity.

I'm also querying the supposedly "correct" equation

$$Q=\frac{E^2}{X}sin(\phi)$$

I would rather have

$$Q=\frac{E^2}{|Z|}sin(\phi)$$

where E is the voltage applied to a general impedance Z = R±jX and

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

$$\phi=\mp \arctan(\frac{X}{R})$$

Last edited:
• Moxica23