Power factor and correction

In a DC circuit, power is simply voltage times current. In an AC circuit with inductance or capacitance, it is not: a motor drawing 50 A at 600 V or 480 V may be doing much less useful work than the product of those numbers suggests, because part of its current is spent building and collapsing magnetic fields rather than delivering energy. That part of the current still heats conductors, loads transformers and causes voltage drop, and utilities often charge for it. Power factor measures how effectively a load uses current. This lesson explains real, reactive and apparent power, the power triangle and power factor, why low power factor costs money, how to calculate and install power factor correction capacitors, and the code rules and pitfalls that go with them.

Power in AC circuits

Instantaneous power is the product of instantaneous voltage and current, \(p = vi\). In a resistor, voltage and current are in phase and \(p\) is always positive. In a pure inductor or capacitor, current is 90° out of phase with voltage, so \(p\) is positive for a quarter cycle (energy flowing into the magnetic or electric field) and negative for the next quarter (energy flowing back to the source); the average is zero. Energy sloshes back and forth but none is consumed. In a real load with both resistance and reactance, the average power is

where \(V\) and \(I\) are RMS values and \(\theta\) is the phase angle between them.

Real, reactive and apparent power

Three quantities describe power in AC circuits:

• Real (active) power \(P\), in watts (W): the average power actually converted to heat, light or mechanical work, \(P = VI\cos\theta\). Energy meters measure real energy in kWh.
• Reactive power \(Q\), in volt-amperes reactive (var): the power that oscillates between the source and the magnetic or electric fields, \(Q = VI\sin\theta\). Inductive loads absorb reactive power (lagging); capacitors supply it (leading).
• Apparent power \(S\), in volt-amperes (VA): the product of RMS voltage and current, \(S = VI\). It determines the current, and therefore the heating, voltage drop and size of conductors, transformers, generators and switchgear, which is why transformers and UPSs are rated in kVA.

They form the power triangle:

Power factor (PF) is the ratio of real to apparent power. It is 1.0 (unity) for a resistive load, and less than 1 when reactive power flows. It is described as lagging for inductive loads and leading for capacitive loads.

Typical power factors

Lightly loaded induction motors are the most common cause of poor power factor in industry: the magnetizing current needed to establish the motor's field is nearly the same at all loads, so at light load it forms most of the current. Oversized motors running well below their rating therefore have poor power factor as well as reduced efficiency.

Why low power factor matters

For a given real power, current is inversely proportional to power factor: \(I = P/(V \times \mathrm{PF})\). Low power factor therefore means:

• Higher current, requiring larger conductors, switchgear and transformers for the same useful load.
• Higher \(I^2R\) losses in conductors and transformers, which grow with the square of current.
• Greater voltage drop along feeders, which can cause equipment to malfunction.
• Less spare capacity: a transformer rated 500 kVA can supply 500 kW at unity power factor but only 350 kW at 0.7.
• Utility charges for poor power factor.

Canadian utilities recover the cost of supplying reactive power in several ways. Many bill demand in kVA rather than kW for larger customers, or bill demand at the greater of the measured kW and a percentage (often 90%) of the measured kVA, so that customers with a power factor below about 0.9 pay for more demand than they use. Others apply explicit power factor penalties. The details are set out in each utility's rate schedules.

Power factor correction

Since inductive loads absorb reactive power and capacitors supply it, capacitors connected in parallel with an inductive load can supply its reactive power locally, so that it no longer flows from the utility through the service, feeders and transformers. The real power is unchanged; the apparent power and current fall. The capacitor rating needed to raise the power factor from \(\mathrm{PF}_1\) to \(\mathrm{PF}_2\) at real power \(P\) is

Where to install capacitors

• At individual motors: the capacitor is connected at the motor terminals and switched with it, so reactive power is supplied right where it is consumed, relieving the branch circuit and feeder as well as the service. Simple, but many small capacitors are needed.
• Group correction: a capacitor bank on a motor control centre or distribution panel serving several motors.
• Central automatic correction: a bank at the main switchboard with a controller that switches capacitor steps in and out to track the load, keeping power factor near a target without overcorrecting at light load. This is the most common approach in larger facilities, though it does not relieve the feeders downstream.

Overcorrection, making the load leading, is undesirable: it raises voltage, can still incur utility charges, and wastes capacitors. Fixed capacitors sized for full load can overcorrect when the load is light, which is why automatic banks are used where load varies.

Code requirements for capacitors

The Canadian Electrical Code (Section 26) sets requirements for capacitor installations, including:

• Conductors supplying a capacitor must have an ampacity of at least 135% of the capacitor's rated current, because capacitor current rises with voltage and harmonics and capacitors are manufactured with a positive tolerance.
• Overcurrent protection and a means of disconnection for each capacitor bank, unless the capacitor is connected on the load side of a motor's overload device and switched with the motor.
• Means to discharge the stored energy automatically to a safe voltage within a short time after disconnection.
• Where capacitors are connected at a motor on the load side of its overload device, the overload setting must be based on the reduced line current, and the capacitor must not exceed the value that would cause the motor to self-excite when disconnected.

Pitfalls

• Self-excitation: a capacitor at a motor's terminals that supplies more than the motor's no-load magnetizing reactive power can keep the motor generating voltage after it is disconnected while it coasts, producing overvoltages and dangerous reclosing transients. Motor manufacturers publish maximum capacitor ratings for each motor.
• Harmonic resonance: capacitors and the system inductance form a resonant circuit (Lesson 2). Where variable frequency drives and other nonlinear loads produce harmonics, a capacitor bank can amplify them, overheating the capacitors and distorting the voltage. Detuned banks with series reactors, or active harmonic filters, are used instead.
• Switching transients: energizing a capacitor bank causes a brief inrush and voltage transient that can trip drives. Automatic banks use soft-switching contactors or thyristor switching to reduce this.
• True versus displacement power factor: with harmonics present, the true power factor (\(P/S\) with all currents included) is lower than the displacement power factor (the cosine of the angle between the fundamental voltage and current). Capacitors correct only the displacement part; harmonic currents need filters. A power quality analyzer measures both.
• Capacitor failure: capacitors age, especially when hot or subjected to harmonics, and lose capacitance. Automatic banks should be inspected and their step currents measured periodically.

Lesson 5 shows how power, energy, demand and power factor are measured in the field, and Lesson 3 explains why capacitor banks in buildings with drives often need reactors.