Harmonics & PF formulas

Line-current harmonics, power factor and distortion factor. 16 formulas for GATE EE, IN Power Electronics, each with when to use it. Each is explained in the topic notes.

All 16 Harmonics & PF formulas

AVERAGE OUTPUT VOLTAGE, SINGLE-PHASE FULLY CONTROLLED BRIDGE

V_{d} = \frac{2 2}{π} V_{s} cos α = 0.9 V_{s} cos α

Continuous conduction, zero source inductance.

AVERAGE OUTPUT VOLTAGE, THREE-PHASE FULLY CONTROLLED BRIDGE

V_{d} = \frac{3 2}{π} V_{LL} cos α = 1.35 V_{LL} cos α

Continuous conduction, zero source inductance. At $α = 0$ this is the diode bridge value $V_{d 0} = 1.35 V_{LL}$ .

RMS LINE CURRENT, THREE-PHASE BRIDGE

I_{s} = \frac{2}{3} I_{d} = 0.816 I_{d}

In each cycle a line carries $+ I_{d}$ for $12 0^{\circ}$ , nothing for $6 0^{\circ}$ , $- I_{d}$ for $12 0^{\circ}$ , and nothing for $6 0^{\circ}$ . With an ideal ripple-free $I_{d}$ this does not depend on $α$ .

LINE CURRENT OF PHASE A (SIX-PULSE BRIDGE)

i_{a} (t) = \frac{2 3}{π} I_{d} [cos (ω t - α) - \frac{1}{5} cos 5 (ω t - α) + \frac{1}{7} cos 7 (ω t - α) - \frac{1}{11} cos 11 (ω t - α) + \frac{1}{13} cos 13 (ω t - α) - \dots]

Reference $v_{an} = V_{m} cos ω t$ . At $α = 0$ the positive $12 0^{\circ}$ block is centred on the positive peak of $v_{an}$ . For $α > 0$ the whole waveform is the $α = 0$ waveform delayed by $α / ω$ . The peak of the $n$ -th term is $\frac{4 I _{d}}{nπ} sin \frac{nπ}{3}$ . Zero overlap, ripple-free $I_{d}$ .

RMS FUNDAMENTAL LINE CURRENT, THREE-PHASE BRIDGE

I_{s 1} = \frac{6}{π} I_{d} = 0.78 I_{d}

The peak of the fundamental is $\frac{2 3}{π} I_{d}$ . Divide by $2$ for the rms value.

RMS OF THE N-TH HARMONIC, THREE-PHASE BRIDGE

I_{s n} = \frac{I _{s 1}}{n}, n = 6 k \pm 1

Six-pulse bridge, zero overlap, ripple-free $I_{d}$ . Relative to the fundamental the magnitude falls as $1/ n$ , and it does not depend on $α$ .

LINE CURRENT, SINGLE-PHASE FULLY CONTROLLED BRIDGE

i_{s} (t) = \frac{4 I _{d}}{π} [sin (ω t - α) + \frac{1}{3} sin 3 (ω t - α) + \frac{1}{5} sin 5 (ω t - α) + \dots], I_{s n} = \frac{2 2 I _{d}}{nπ}

Reference $v_{s} = V_{m} sin ω t$ . The current is a square wave of $\pm I_{d}$ delayed by $α$ , so every odd order $n$ is present. Ripple-free $I_{d}$ , zero overlap. The magnitudes do not depend on $α$ .

LINE CURRENT, SINGLE-PHASE HALF-CONTROLLED BRIDGE

I_{s n} = \frac{2 2 I _{d}}{nπ} cos \frac{n α}{2} (n odd), I_{s} = I_{d} \frac{π - α}{π}

Half-controlled bridge (semi-converter), or a full bridge with a freewheeling diode, with ripple-free $I_{d}$ . The line current is $+ I_{d}$ from $α$ to $π$ , zero while the load freewheels, then $- I_{d}$ from $π + α$ to $2 π$ . Here the harmonic magnitudes do depend on $α$ , and the fundamental lags the supply voltage by $α /2$ .

DISPLACEMENT FACTOR

DPF = cos ϕ_{1}

$ϕ_{1}$ is the phase angle between the supply voltage and the fundamental line current. With zero overlap, $ϕ_{1} = α$ for a fully controlled bridge and $ϕ_{1} = α /2$ for a single-phase half-controlled bridge.

DISTORTION FACTOR

DF = \frac{I _{s 1}}{I _{s}}

The ratio of the fundamental rms current to the total rms line current. DF $\leq 1$ .

INPUT POWER FACTOR

PF = \frac{P}{S} = \frac{I _{s 1}}{I _{s}} cos ϕ_{1} = DF \times DPF

Sinusoidal supply voltage. For the three-phase case $P = 3 V_{LL} I_{s 1} cos ϕ_{1}$ and $S = 3 V_{LL} I_{s}$ . For the single-phase case $P = V_{s} I_{s 1} cos ϕ_{1}$ and $S = V_{s} I_{s}$ .

TOTAL HARMONIC DISTORTION OF THE LINE CURRENT

THD = \frac{I _{s}^{2} - I _{s 1}^{2}}{I _{s 1}} = \frac{1}{DF ^{2}} - 1

Harmonics are measured relative to the fundamental. Equivalently, $DF = 1/ 1 + THD^{2}$ .

SIX-PULSE BRIDGE, IDEAL INDUCTIVE LOAD

DF = \frac{3}{π} = 0.955, PF = \frac{3}{π} cos α, THD = \frac{π ^{2}}{9} - 1 \approx 31%

Fully controlled bridge, ripple-free $I_{d}$ , zero overlap. DF and THD do not change with $α$ . Only DPF does. A diode bridge ( $α = 0$ ) has PF $= 0.955$ .

SINGLE-PHASE HALF-CONTROLLED BRIDGE, IDEAL INDUCTIVE LOAD

DPF = cos \frac{α}{2}, PF = \frac{2 2}{π} \frac{cos ^{2} ( α /2 )}{( π - α ) / π} = \frac{2 ( 1 + cos α )}{π ( π - α )}

Ripple-free $I_{d}$ , zero overlap. At $α = 0$ this gives PF $= 0.900$ , the same as the full bridge. For $α > 0$ its PF is higher than the full bridge's $0.9 cos α$ .

POWER AND REACTIVE POWER, THREE-PHASE FULLY CONTROLLED BRIDGE

P = V_{d} I_{d} = V_{d 0} I_{d} cos α, Q_{1} = 3 V_{LL} I_{s 1} sin α = V_{d 0} I_{d} sin α

Zero overlap, lossless converter. $Q_{1}$ is the fundamental reactive power. With $V_{d 0} = \frac{3 2}{π} V_{LL}$ , the fundamental apparent power is $P^{2} + Q_{1}^{2} = V_{d 0} I_{d}$ .

OUTPUT VOLTAGE WITH OVERLAP, THREE-PHASE FULLY CONTROLLED BRIDGE

V_{d} = V_{d 0} cos α - \frac{3 ω L _{s}}{π} I_{d} = V_{d 0} \frac{cos α + cos ( α + μ )}{2}, cos α - cos (α + μ) = \frac{2 ω L _{s} I _{d}}{V _{LL}}

$L_{s}$ is the source inductance per phase, with ripple-free $I_{d}$ . For the single-phase fully controlled bridge the matching result is $V_{d} = 0.9 V_{s} cos α - \frac{2 ω L _{s}}{π} I_{d}$ .

Where this was asked: 2018 (1), 2020 (1), 2022 (2), 2024 (1), 2025 (1). See the 6 previous-year questions on Harmonics & PF. This shows what was asked, not what will be asked.

KEEP GOING

Harmonics & PF: the notes and 6 solved previous-year questions