# Total Harmonic Distortion (THD) and Power Factor Calculation

## In this article, we will discuss how to measure total harmonic distortion and the power factor calculations utilized.

Total harmonic distortion (THD) is the amount of harmonics on a line compared to the line fundamental frequency, e.g., 60Hz. The THD considers all of the harmonic frequencies on a line. THD can be related to either current harmonics or voltage harmonics, The following equation can be used to calculate the distortion of the line voltage:

*Figure 1. Total harmonic distortion (THD) should be measured at the transformer, not at the load.*

where Vn_rms is the RMS voltage of the nth harmonic and Vfund_rms is the RMS voltage of the fundamental frequency. The THD of a pure sine waveform with no higher harmonics, such as the ideal voltage supply, is 0%. A value of THD greater than zero means the sine waveform has become distorted. THD is often given as a percentage, such as 5% or 50%. THD can be measured for current and voltage.

Current harmonics are caused by non-linear loads for example those that draw pulses of current. Voltage harmonics are caused by the harmonic currents flowing through different system impedances. The current flowing through a transformer causes a voltage drop across the coil. When current flows in pulses, the voltage will also be in pulses. High voltage distortion is a problem because voltage distortion becomes a carrier of harmonics to linear loads such as motors. Voltage harmonics cause problems (extra heat) in the power distribution system and to the loads connected to the system.

## Measuring THD

When troubleshooting a circuit for harmonics, the voltage THD and the current THD should be measured. For best results, the voltage THD should not exceed 5%, and the current THD should not exceed 20% of the fundamental frequency. THD should be calculated at the transformer rather than at the harmonic-generating loads for an accurate calculation of THD in a system (see Figure 1). Measuring THD at the loads provides the highest THD reading because THD cancellation has not occurred along the system.

Figure 1. Total harmonic distortion (THD) should be measured at the transformer, not at the load.

When THD current is measured during full load, the THD is approximately equal to the total demand distortion (TDD). Total demand distortion (TDD) is the ratio of the current harmonics to the maximum load current. A THD measurement is taken when testing or troubleshooting a system. The TDD is different from the THD because TDD is referenced to the maximum current measurement taken over time. The THD is a measurement of current on a power line only at the specific time of the measurement. The purpose of the TDD measurement is to account for situations where the THD is relatively high, but the total load is fairly low. In this type of situation, the TDD is relatively low, and overheating is minimized.

## Power Factor

Power factor is the ratio of true power to apparent power in a circuit or distribution system. Any AC circuit consists of real, reactive, harmonic, and apparent (total) power. True power is the power, in W or kW, used by motors, lights, and other devices to produce useful work. Reactive power is the power, in VAR or kVAR, stored and released by inductors and capacitors. Reactive power shows up as a phase displacement between the current and voltage waveforms. Harmonic power is power, in VA or kVA, lost to harmonic distortion. Apparent power is the power, in VA or kVA, that is the vector sum of true power, reactive power, and harmonic power. Apparent power is not a simple summation but a vector summation.

The displacement power factor is the ratio of true power to apparent power due to the phase displacement between the current and voltage (see Figure 2). Capacitors can usually be added to a circuit or distribution system to correct the displacement power factor. The displacement power factor is calculated as follows:

PF = cos(θ)

where

PF = displacement power factor

θ = Difference between the phase of the voltage and the phase of the current (phase displacement) in degrees.

Note: DPF or PFD are sometimes used instead of PF to describe displacement power factor.

*Figure 2. The displacement power factor can be used to calculate the amount of power that is actually available for a load.*

The presence of harmonics complicates the discussion of the power factor. The distortion power factor is the ratio of true power to apparent power due to THD. Capacitors cannot be added to a circuit to compensate for the distortion power factor. The impedance of capacitors decreases with frequency. Therefore, a capacitor can become a sink for high-frequency harmonics. Special types of transformers or tuned harmonic filters consisting of capacitors and inductors are used to correct distortion power factor. The distortion power factor is calculated as follows:

$$P{{F}_{THD}}=\sqrt{\frac{1}{1+{{\left( THD \right)}^{2}}}}$$

where

PF_{THD }= distortion power factor

THD = total harmonic distortion

The total power factor is the product of the displacement power factor and the distortion power factor and is calculated as follows:

PF_{Tot} = PF × PF_{THD}

where

PFT_{ot }= total power factor

PF = displacement power factor

PF_{THD }= distortion power factor

For example, what is the total power factor when the displacement between voltage and current is 25°, and the THD is 49% (0.49)? The displacement power factor is calculated as follows:

PF = cos(θ)

PF = cos (25°)

PF = 0.906

The distortion power factor is calculated as follows:

$$P{{F}_{THD}}=\sqrt{\frac{1}{1+{{\left( THD \right)}^{2}}}}=\sqrt{\frac{1}{1+{{\left( 0.49 \right)}^{2}}}}=\sqrt{0.8064}=0.898$$

The total power factor is calculated as follows:

PF_{Tot} = PF × PF_{THD}

PF_{Tot} = 0.906 × 0.898

PF_{Tot }= 0.814

It is important to know the total power factor because it relates to apparent power. Apparent power is used to size the elements of a power distribution system.

## Current Crest Factor

The current crest factor is the peak value of a waveform divided by the rms value of the waveform. The purpose of a current crest factor is to give an idea of how much distortion is occurring in a waveform. The current crest factor is calculated as follows:

$$CCF=\frac{{{I}_{peak}}}{{{I}_{rms}}}$$

where

CCF = current crest factor

I_{peak} = peak value (in A)

I_{rms} = root mean square value (in A)

For example, what is the current crest value of a perfect sine waveform? In a perfect sine waveform with a peak value of 1, the rms value is 0.707.

$$CCF=\frac{{{I}_{peak}}}{{{I}_{rms}}}=\frac{1}{0.707}=1.414$$

A high current crest factor can cause overheating of circuits and devices. A typical distorted current waveform on a 120 V circuit supplying digital devices like computers may have a current crest factor of about 2 to 6 (see Figure 3). In general, a circuit with a higher current crest factor has more energy contained in the higher harmonics.

A power source must be able to supply the maximum power required by the circuit at the required voltage and current. A typical backup power system, such as a computer uninterruptible power source, has the capability of supplying a current crest factor of 3 at full load but can exhibit higher crest factors at lower loads.

*Figure 3. The current crest factor comparison*

## Source Impedance

Source impedance has an effect on the crest factor created by a non-linear load. Once the voltage rises to a predetermined point, the power supply starts charging a smoothing capacitor. The current drawn by the capacitor is high when the source impedance is low, and the charging cycle is short. Higher impedance limits the amount of current that can be drawn, extending the time it takes to charge the capacitor. The extended charge time has the effect of reducing the crest factor. The source impedance can be increased by adding line reactors or drive isolation transformers.

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