Real and Reactive Power Analysis

[Dave Loucks]

Lots of good material on the web, but I've sometimes found it confusing to understand why reactive power flow to an inductor is considered "positive", but at the same time you hear that the phase angle of inductor current is lagging (which to my ear sounds "negative").  For capacitors this is reversed (e.g. negative reactive power flow for leading current).  Anyway, I thought I'd outline a quick overview of how all this fits together.

We begin with a diagram that might be familiar to many.  It shows real power flow on the x-axis and reactive power flow on the y-axis.  Since both forward and reverse (positive and negative) real and reactive power flow is possible, there are four categories (+W/+vars, +W/-vars, -W/+vars, -W/-vars) that the diagram places in separate quadrants.  I've shown it with ABC clockwise rotation (meaning that increasing angles are increasingly "lagging" which is explained below).

The circle shows the line etched by a constant value of S at different values of 

θ

(0 to 360 degrees).

The exact same value of S (VA or apparent power) can result in a variety of different P (real) and Q (reactive) values just by changing the phase angle!

So what is the phase angle and how do you calculate it?

Let's say you just define the voltage reference right now to be 0⁰ and you measure your current to be

θ

degrees away from that axis.  Depending on which way the angle points (above the x-axis or below) determines whether the power factor is lagging (consuming vars) or leading (producing vars), respectively.

Here's a diagram of system with where the current is lagging the voltage by 36.9 deg.

Here's where it gets somewhat tricky... the current is delayed by 36.9 degrees, which might sound like a "negative" angle, but you can see from the phasor diagram as shown that it actually is a more positive phase angle. 

sin (36.9⁰) = 0.6

By the way, in these waveforms the peak voltage and current were both 1.  When we calculate power (whether real or reactive), we use root mean square versions of the signals.  Since we have a two nice sine waves the rms value can be calculated easily from the peak:

In a single phase system:

• S = V * A
• P = S * cos(
  θ
  ) = V * A * cos(
  θ
  )
• Q = S * sin(
  θ
  ) = V * A * sin(
  θ
  )

(in a 3-phase system, each parameter is just multiplied by 1.732)

Plugging and chugging:

Note that sin (-36.9⁰) = - sin (+36.9⁰).  This means that moving the phase of the current  either ahead or behind the voltage causes leading (positive) var production or lagging (negative) var consumption, respectively.

That is not the case for the real power (cosine).  Since cos(-36.9⁰) = cos(+36.9⁰) = 0.8 this tells you that changing PF over this range doesn't affect real power flow over this small range.

The waveform and phasor diagrams show how the same value of S (in these cases it is assume to be 1) can result in very different values of P (real) or Q (reactive) power.

The phase angle of the current relative to the voltage would need to increase to more than 90 degrees (but less than 270 degrees) in order for the the sign of the real (W or P) term to turn negative.  This makes sense since consider a signal exactly 180 degrees out of phase.  180 degree out of phase current would look like this:

It is by convention among power systems engineers that capacitive circuits "produce" vars and have leading PF and inductive circuits "consume" vars and have lagging PF.   Here's an example where the current leads the voltage by 135 degrees resulting in negative real power, but positive reactive power.

In the scheme of things this is totally arbitrary because ideal versions of both components never "keep" the vars.  They store energy during a portion of the half cycle and then return it the next. 

So why do people talk about capacitors "producing vars"?

It is just convention to say that the reactive power is "consumed" when it is positive and "generated" when it is negative.  From the diagrams above, negative vars occur when you have leading power factor.

Maybe then someone asks "why do capacitor circuits result in leading phase angles and inductors result in lagging phase angles?"

Let's start with looking at a leading PF circuit.

This is a 3-phase diagram, but it works for 1-phase (just remove the B and C phases).  The way I look at it is to realize that the phasors are rotating in time.  At the instant this "snapshot" was taken, the Va phase was exactly aligned with the x-axis (0⁰), but that is totally arbitrary.  What is important is the angle between that voltage and its corresponding phase current.  While we said at this instant the voltage was measured and/or defined to be at 0 degrees, a millisecond later it may not be.

As these phasors rotate (I'll assume clock-wise for ABC rotation, meaning first A, the B, then C cross the 0⁰ axis as it rotates...), the Ia phase will always cross any arbitrary angle before the Va phasor "gets there". 

Since current "gets there" before the voltage we say the current is leading the voltage. 

Here's the "why" that this happens.  Capacitors look like short circuits when discharged.  For a moment in time after you apply a non-zero charging current, the voltage is 0 until it begins to charge up.  A voltage of 0, by definition, is a short-circuit.  In an ideal short circuit, infinite current flows but there is no voltage drop.  Now, in the physical world that isn't the case, but you will see current flowing in a capacitor before the voltage changes.  How fast that happens depends on the size of the capacitor and the voltage applied.  If you have current flowing before voltage changes, then that is another way of saying current changes precede (or lead) voltage changes -- or more simply, current leads the voltage.  This can also be described by looking at the mathematical relationship between current and voltage in a capacitor.  Danger Will Robinson - Differential Equations ahead  :

What this says is that for voltage to change instantaneously (dt = 0) you would have to apply infinite current.  Guess that isn't going to happen!  So for capacitors, any nominal current will result in a delayed voltage change, or in the vernacular of speaking about current in relation to the voltage we say the current leads the voltage.

If the phase angle was reversed, the voltage would cross any arbitrary angle before the current and we'd say the current lagged the voltage.  Here's that phasor diagram:

This is known to be an inductive circuit since with inductors no current flows immediately after a voltage change is applied across the inductor.  Mathematically we write:

According to this equation,  for the current in an inductor to change instantaneously (dt =  0) you would have to apply infinite voltage.  That ain't gonna happen either, so the result is that current can't change instantaneously with a voltage change and we say the current lags the voltage. 

Armed with this fabulous knowledge, we can then attack the problem using our standard trig equations that show how to solve for unknown values of a right triangle. 

If, for example, you know its hypotenuse (S or VA or apparent power) and you have been given either one other side (kW) or the angle (PF) you're good to go.

You can derive everything else using these equations:

W = Wh / h
Q = varh / h

S² = W² + Q² 
cos^-1(W/S) =

θ

W = S cos

θ

cos

θ

= W/S
sin^-1(Q/S) =

θ

Q = S sin

θ

sin

θ

= Q/S
tan

θ

= sin

θ

/ cos

θ

  = (Q/S)/(W/S) = Q/W

To help visualize these phase angles, I've attached two Excel spreadsheets that can be used to create a voltage waveform, then superimpose on the same graph the current waveform.  I went ahead and included the ability to add harmonic currents (which is a topic for a future discussion).

• 1-phase_harmonics.xls
  Allows entering the phase shift as power factor.  The spreadsheet will perform the math to shift the current waveform.
• 1-phase_harmonics_Phase_shift.xls
  Allows entering the phase shift in degrees.
• X over R2.xls
  Converts PF to X/R and reverse.  Calculates Z, %Z, X/R, R, X and L for a given voltage, desired  and current.  Useful when creating a simulation and you want to choose an R and X to limit current to a particular short circuit value at a particular X/R ratio.
• phase_angle_diagram_V1_V2.xls
  Allows entering the phase shift in degrees and seeing corresponding changes to real, reactive and apparent power.