# Why isn't Reactive Power defined as Q = S - P ?

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1. Jan 20, 2016

### vintageplayer

The instantaneous power in an AC circuit is given by: p = i(t)⋅v(t) = VrmsIrmscosφ + VrmsIrmscos(2ωt -φ).

The average power P = VrmsIrmscosφ is often a useful quantity to know. For example, it can tell me the work being done by a motor.

The apparent power S = VrmsIrms is also a useful quantity to know. For example, it can tell me the maximum work the motor could potentially do (when the current and voltage are in phase φ=0).

However... the reactive power Q = VrmsIrmssinφ = sqrt(S2 - P2) doesn't really tell me anything useful. All I know is that it is some unintuitive measure of the difference between the apparent and average power. Why isn't reactive power simply defined as Q = S - P instead?

2. Jan 20, 2016

### Hesch

Because P and Q have perpendicular directions. Thus Pythagoras must be used.

3. Jan 20, 2016

### tech99

The definition you give is fine; the reactive power is the energy stored in the system.

4. Jan 20, 2016

### vintageplayer

This is circular reasoning. We defined it to be that way!

5. Jan 20, 2016

### Hesch

Q = S - P →
S = P + Q , which doesn't make sense.

A phase shifted current, I / φ, can be represented by the complex value , I / φ = I*cos φ + j*I*sin φ.

Thus S = V * I = V * ( I*cos φ + j*I*sin φ ) = P + jQ ,
j = 1 / 90°

So the only "definition" comes from the complex representation of the phase shifted current, I / φ.

6. Jan 20, 2016

### jim hardy

When you study generators
peaks of reactive current and 'real' current occur with rotor directly or perpendicular to stator winding
so it was kinda natural for the old timers' thought processes to go with that complex representaion
figuring it out the first time from physical hands-on is different from trying to back into it from algebra.

7. Jan 21, 2016

### vintageplayer

S = VrmsIrms
P = VrmsIrmscosφ
Q = VrmsIrms - VrmsIrmscosφ

Then S = P + Q would make sense. This definition for reactive power is more intuitive because you can easily tell how much power in Watts is being stored, and how much power you are using for useful work. For example, 100VAr would mean you are missing out on 100W of extra useful power.

You're right that it comes out naturally with: VI* = S = P + jQ as long as you define reactive power to be Q = VrmsIrmssinφ. The single complex number contains a lot of information: P = Re{VI*}, Q = Im{VI*}, S = mag{VI*}, φ = arg{VI*} which is probably why Q was defined this way. On the other hand, you're now stuck with an ugly definition for reactive power. For example, 100VAr does not tell you how much power is being stored in Watts.

Do you have an example (or link to one) of this?

8. Jan 21, 2016

### Hesch

How would you calculate:

100 [W] + 100 [VAr] = 200 [unit ?] ?

9. Jan 21, 2016

### jim hardy

10. Jan 21, 2016

### cnh1995

Cosider a box floating in space in the absence of gravity.

If you give some velocity 'v' to the block at an angle Φ, it will move horizontally with velocity vcosΦ and vertically with velocity vsinΦ. If you know the vertical velocity vsinΦ and horizontal velocity vcosΦ and you are asked to find the total velocity v, would you say it is vsinΦ+vcosΦ? No, because velocity is a vector, you MUST consider it's direction and therefore, the angle it makes with the reference. Hence, you'll have to use Pythagoras theorem here for vector addition. Similarly, voltage and current are PHASORS. Hence, you must consider phase angles as Hesch has demonstrated in #2 and #5. VrmsIrmscosΦ is the power dissipated in the circuit, hence, VrmsIrmssinΦ turns out to be the reactive power in the circuit. Apparent power is the phasor addition of the powers and not simple arithmatic addition.
Also, you can verify this by drawing some waveforms. Take a simple RL ac circuit with some convenient component values and plot the waveforms of active power(power dissipated in R) and reactive power(power absorbed by L)separately for a complete cycle and then calculate their corrosponding average values.

Last edited: Jan 21, 2016
11. Jan 22, 2016

### Sergey_KGB

Once, I supervized installation of meteting equipment on Power Substation 220KV in Bashkortostan (Russia). Our device showed that nearby plant consumes only 0.5MW of active power but generates 10MVAR of reactive power (due to heating inductors that were switched off just before the measurement).
As for the question then let's look at the speed in horizontal and vertical directions. No one would mechanically sum them. Formula
V=SQRT(Vh*Vh+Vv*Vv)
is almost obvious.
The situation with active and reactive powers is essentially the same. They are (in specific meaning) orthogonal to each other.

12. Jan 22, 2016

### tech99

But is it not true that power is a scalar quantity and does not therefore have a phase angle, as do voltage and current?

13. Jan 22, 2016

### Sergey_KGB

If {Vn} and {In} are series of samples (voltage and current) then
P - active power is a sclar multiplication P=ΣVn*In
and thus is a scalar value while voltage and current are vectors. Though RMS values of both voltage and current are scalar values (length of respective vectors).

14. Jan 22, 2016

### sophiecentaur

Yes. Power must be a scalar quantity because Energy is scalar and power is energy per unit time. Imo, "reactive Power" is really an oxymoron which has been introduced into the vernacular for convenience. In a purely reactive component, energy is stored and not dissipated so the 'lost power' must be there because of resistive components in the supply impedance. If people stick with volts and current for their calculations then they can't go wrong and will get the right answers (arithmetic permitting).

15. Jan 22, 2016

### tech99

Agree, so it seems incorrect to show vector diagrams of P and Q.

16. Jan 22, 2016

### sophiecentaur

I'm not a Power Engineer so I would have to be more careful with my 'shorthand', which is what that diagram is. I would do any calculations the 'formal' way and stand a better chance of getting the right answer. A PQ vector diagram has got to be wrong but it 'may' be forgiveable if the people who use it are aware that they are being sloppy.

17. Jan 22, 2016

### jim hardy

Phasors aren't vectors i've been told by folks who are better at maths than i..

I'd have to go back to Steinmetz - i think it was he who came up with idea of complex numbers to represent AC current....

18. Jan 24, 2016

### vintageplayer

Assume a voltage Vcos(wt) across an impedance of R + jX = Z∠φ

The total instantaneous power is:
p(t) = v(t).i(t) = Vcos(wt).Icos(wt - φ) = VI.cos(wt).[cos(wt)cosφ + sin(wt)sinφ] = (VI/2)cosφ[1 + cos(2wt)] + (VI/2).sinφ.sin(2wt)

The expression above is the total power consumed, and it does not tell you the power consumed by the resistor, or the power consumed by the reactance. The power consumed by the resistor is actually NOT (VI/2)cosφ[1 + cos(2wt)], nor is the power consumed by the reactance (VI/2).sinφ.sin(2wt).

To find the power consumed by the individual resistor and reactance, you have to find the voltage across these individually (e.g. by voltage divider).
vR(t) = (VR/Z).cos(wt - φ) = Vcosφcos(wt - φ)
vX(t) = (VX/Z).cos(wt + 90 - φ) = Vsinφcos(wt + 90 - φ)

Hence:
pR = vR(t).i(t) = Vcosφcos(wt - φ).Icos(wt - φ) = (VI/2).cosφ.[1 + cos(2wt - 2φ)]
pX = vX(t).i(t) = Vsinφcos(wt + 90 - φ).Icos(wt - φ) = (VI/2).sinφ.sin(2φ - 2wt)

Note that adding these two terms together gives the total instantaneous power above (after some further trig identities).

The average power consumed is P = VrmsIrmscosφ.
The peak power consumed by the reactance is Q = VrmsIrmssinφ.

Although both of these quantities are scalars and mean different things (P is the average power, and Q is the peak power consumed by the reactance), you can artificially construct a vector of magnitude VrmsIrmsand angle φ. The x-component of this vector will be the average power P, and the y-component will be the reactive power Q.

I could also define another scalar quantity, Q = S - P, and call it the "reactive power", but it would not be the peak power consumed by the reactance. If you want the "reactive power" to be the peak power consumed by the reactance, then it must be equal to VrmsIrmssinφ.

19. Jan 24, 2016

### sophiecentaur

If we are trying to get to the bottom of something as fundamental as this, we could at least get the basics right. How can any power be "consumed" by a reactance? The Current and Volts are in quadrature. The reason that Power Factor is relevant is that the I and V are greater when there are reactive elements in the circuit. The excess current is passing through the Resisitive components of the supply circuit, which 'consume' extra Power. Also, the extra Volts may be relevant.
There is more power consumed when the PF is not unity but there is no way it can be dissipated in reactances. Reactances can only store energy and return it (twice) per cycle. The 'Peak' power involved, (rate that energy is being stored, instantaneously) can be given a value but it is not very meaningful

20. Jan 24, 2016

### vintageplayer

Power is consumed by a reactance half the time, and released half the time. Just because on the average the power consumed is 0 doesn't mean a reactance doesn't draw power for half the cycle. The peak power it draws is equal to Q.

That value is the "reactive power" Q agree?

21. Jan 24, 2016

### sophiecentaur

You can give it a name and you can calculate it but how does that value contribute to understanding the AC losses in power engineering? What counts is the wasted mean power in the resistive parts of the supply circuit. You could just as easily talk about the "peak"power being dissipated in a resistive load but who does? If there were no resistance in the supply chain, your "reactive power" would be irrelevant because there would be no supply losses.

22. Jan 24, 2016

### jim hardy

A picture is generally wort a thousand words.
It was helpful for me to look at sinewaves:
draw two of them one above the other
label one Volts and the other Amps
draw a pair in phase,
a pair 180 degrees out of phase,
a pair 90 degrees out of phase and
a pair 45 degrees out of phase

then consider instantaneous power, product volts X amps

For the in phase sinewaves , product is always positive , P = volts X amps, which represents power factor of 1.0 (cos 0 degrees)
for the pair 180 out, product is always negative, power = volts X amps and one of them is always negative , power factor = -1.0 which is cosine 180 degrees

for the others , power is sometimes positive and sometimes negative , over a complete cycle it will average to VIcosθ,
that's what reactive power does - shuttle back and forth between source and load during subcycle intervals.
At 90 degrees observe positive and negative intervals are equal and their average power is zero, e power factor of zero which is cos 90 degrees. Nothing gets hot or does any work.
At 45 degrees - would someone less awkward copy and paste some sinewaves into Paint and post ? That's been done someplace on PF (i remember doing it) but i was unable to find, and paint frustrates me terribly.

You'll find P = VICosθ and Q = VISinθ , and Pythagoras Rules !
I think that visual exercise is a good learning tool. It'll cement the concept.

old jim

23. Jan 24, 2016

### vintageplayer

You might want to know the reactive power if you ever want to control the voltage on a power grid. Or size your capacitors for power factor correction. Or size your power cables efficiently. Also, peak power is an important design spec for almost all electrical devices...

24. Jan 24, 2016

### sophiecentaur

To correct for the power factor of your load you need to know its reactance and then you tune it out as near as you can, using another reactance network. When would you calculate 'Peak Power'? True, the Peak Voltage could be very relevant - but that actually corresponds to the maximum Energy (E = CV2/2) and not to the rate of energy supplied (Power). Sizing your power cables will affect the Resistance and not the Reactance and I've already been into that.
I realise that these terms are all down to usage and it is not likely that they will change. It doesn't make them correct though. "Make it Work' type Engineering is full of slightly dodgy terms and statements - from Water Analogies where they don't apply, to the expression "dB Volts". The field is littered with pitfalls because people got taught these things and pass them on. But Engineers do 'make things work' very successfully and we mostly stuff get away with approximate thinking and that magic stuff called 'experience'.

25. Jan 24, 2016

### jim hardy

aha found it

single phase volts, amps, and power
when volts and amps are in phase ie power factor = 1
observe instantaneous power is always positive

single phase volts, amps and instantaneous power
when volts and amps are out of phase by 90 degrees
observe instantaneous power no longer unipolar, in fact averages zero; cos(90degrees) = 0....

now
no i dont agree.
do we have a definition of terms question here?

Where would you select peak power on that graph? Copy&paste into Paint and add an arrow....
I'd guess Q is average of the orange power wave .

those graphs came from www.electrical4u.com/electric-power-single-and-three-phase/11