True power factor versus displacement factor (cos φ) — why a 'perfect' cos φ can still hide a poor power factor

The guide on power-factor correction covers how a capacitor bank compensates for the displacement between the fundamental voltage and current waveforms. The harmonics/THD guide covers the distortion of the current waveform itself caused by non-linear loads. This article connects both topics through one central concept: the difference between displacement factor and true (total) power factor, and why these two can diverge sharply for a modern electronic load.

Two different quantities that happen to share the same symbol

• Displacement factor (cos φ, displacement power factor, DPF): the cosine of the phase-shift angle between the fundamental (50 Hz component) of voltage and current. This is the classic power factor that is corrected with a capacitor bank, and the one most simple or older power meters actually display.
• True (total) power factor (true/total power factor, PF): the ratio between real power and apparent power, P/S, calculated over the full, non-sinusoidal current and voltage waveform — so including all harmonics, not just the fundamental.

For a purely sinusoidal current (no harmonics), both quantities are identical. As soon as the current becomes distorted — as with a variable-frequency drive, switch-mode power supply or LED driver — they diverge, and always in the same direction: the true power factor is never higher, and usually noticeably lower, than the displacement factor.

The mathematical relationship: distortion factor as an extra multiplier

As an approximation (for a purely sinusoidal supply voltage):

True power factor ≈ displacement factor (cos φ) × distortion factor

where the distortion factor is determined by the total harmonic distortion of the current (THDi):

Distortion factor ≈ 1 / √(1 + THDi²)

At a THDi of, say, 100% (not an unusual value for a single-phase switch-mode load without active power-factor correction built into the supply itself), the distortion factor already drops to roughly 0.71 — even though the displacement factor itself is nearly 1. The true power factor of that load then sits around 0.71, despite a cos φ that looks perfect on a simple meter.

Why this gets overlooked in practice

Many simple or older power meters, and some cheaper energy meters, measure only the phase angle between the fundamentals of voltage and current, and display that as "cos φ" or "PF" — without accounting for the distortion component. On such a meter, a modern electronic load (variable-frequency drive, LED lighting, switch-mode power supply) can show a cos φ close to 1, while the actual true power factor — as it would be determined by the utility or a proper true-RMS power meter — turns out considerably lower. This explains why an installation full of electronic loads sometimes still draws noticeably higher current than the cos φ meter would suggest for the same delivered real power.

Note: a higher current at the same real power means a greater thermal load on cables and transformers, and — if the utility measures true power factor rather than just cos φ — can lead to a power-factor penalty that a simple cos φ correction via a capacitor bank does not resolve.

Why capacitor correction doesn't fix the distortion component

A capacitor bank sized based on the power-factor correction guide compensates only the displacement component (the inductive or capacitive phase shift on the fundamental). It does nothing about the distortion component caused by harmonics — and, without the right precautions, can even introduce a resonance problem with the harmonics present (see the [detuned reactor/detuned filter guide](/guides/nen-1010/afgestemde-reactor-detuned-filter-p-factor-condensatorbank-resonantie) for that pitfall). To actually improve the true power factor of an installation with many non-linear loads, an approach is needed that reduces the harmonics themselves — for example active harmonic filtering or loads with built-in active power-factor correction (PFC) circuitry — rather than a capacitor bank against displacement alone.

Practical relevance

When assessing the power factor of an installation with many electronic loads, it is important to check whether the measuring instrument used measures the true, total power factor (this requires a true-RMS power meter, see the true RMS vs. average-value guide) or only the displacement factor. An installation with an apparently excellent cos φ can still show a noticeably lower value under a proper true power factor measurement, with all the resulting consequences for cable and transformer sizing and possible utility power-factor penalties.

Common mistakes

1. Treating cos φ (displacement factor) and true power factor as synonyms, while they diverge clearly for a distorted current.
2. Sizing a capacitor bank based on a cos-φ-based measurement, while the installation's actual true power factor is already lower than that measurement suggests due to harmonics.
3. Assuming that capacitor correction also fixes the distortion component from harmonics — capacitors only correct the displacement on the fundamental, not the distortion itself.
4. Using a simple, non-true-RMS power meter to assess the power factor of an installation with many non-linear loads, resulting in an overly optimistic picture of the actual true power factor.

Related

• Power-factor correction
• Harmonics & THD
• Detuned reactor / detuned filter — capacitor bank and resonance
• True RMS vs. average-value multimeter