Power Factor: Calculation and Correction

The power factor is the ratio between active power and apparent power of a load or installation.

In the previous article we saw the Power Triangle. We saw that Apparent Power (S) is the vector sum of Active (P, useful) and Reactive (Q, oscillating).

The Power Factor (PF) is simply the ratio that tells us what percentage of the apparent power () is actually being converted into useful work ().

In pure sinusoidal waves, this value mathematically coincides with the cosine of the phase angle () between voltage and current. That is why it is commonly called “cosine phi”.

Value Interpretation

• PF = 1 (Ideal): All apparent energy is transformed into active energy (S = P). Occurs in heaters and incandescent bulbs. The angle 𝜑 is 0.
• PF < 1 (Real): Part of the energy is reactive.
• A value of 0.95 is excellent.
• A value of 0.80 starts to be problematic.
• A value of 0.50 is disastrous (typical of motors running idle).

Why is a Low PF a Problem?

Suppose a single-phase load draws 10 kW of active power at 230 V RMS.

1. Case A (Efficient, PF = 1):

2. Case B (Inefficient, PF = 0.5): To achieve the same 10 kW of work, because the PF is 0.5, the apparent power doubles (S = 20 kVA).

The physical result: The motor performs the same work in both cases, but in Case B we are forcing the electrical grid to transport TWICE the current.

This causes:

• Cable overheating: Heat losses (I²·R) are multiplied by 4.
• Voltage drops: The voltage arrives lower at the end of the line.
• Less available capacity: The transformer reaches its current and temperature limits sooner, even though the active power is less than its rated power in kVA.

Reactive Energy Billing

Since transporting this extra current costs the electric company money (they need thicker cables and larger generators), they will not forgive you.

On the industrial bill, if your Power Factor drops below a certain limit (usually 0.95 in Europe), they apply a surcharge for reactive energy. It is a very painful fine. I have seen factories paying thousands of euros per month just for this concept.

Power Factor Correction

Most industrial loads are Inductive (Motors, transformers, welders). This means they consume Inductive Reactive Power (), causing the current to lag.

To fix this, we cannot remove the motors (we need them to work). The strategy is: Generate the reactive energy locally so as not to have to ask the electrical grid for it.

Who generates reactive energy opposite to the coil? The Capacitor.

Capacitor Bank

If we connect capacitors in parallel with the motor:

1. The motor needs inductive reactive power () to magnetize.
2. The capacitor provides capacitive reactive power ().
3. If we calculate correctly, cancels out .

Physically, what happens is fascinating: the reactive energy stops traveling from the power plant. It now bounces locally between the capacitor and the motor.

• The motor “asks” for energy → The capacitor gives it.
• The motor “returns” energy → The capacitor stores it.

The electric company, which is upstream, only sees the active power consumption (). We have “cleaned” the signal.

Capacitor Calculation ()

To go from a bad initial angle (𝜑₁) to a good final angle (𝜑₂), the reactive power the capacitor must provide is:

With this value ( in VARs), we can buy the appropriate capacitor bank.