Rectifiers formulas

Uncontrolled and controlled rectifiers, AC-DC converters. 24 formulas for GATE EE, IN Power Electronics, each with when to use it. Each is explained in the topic notes.

All 24 Rectifiers formulas

RIPPLE FACTOR FROM FORM FACTOR

R F = (\frac{V _{r m s}}{V _{d c}})^{2} - 1 = F F^{2} - 1

Holds for any output waveform; $V_{r m s}$ is the total rms including the DC part.

HALF-WAVE RECTIFIER, R LOAD

V_{d c} = \frac{V _{m}}{π}, V_{r m s} = \frac{V _{m}}{2}, F F = \frac{π}{2} = 1.57, R F = 1.21, P I V = V_{m}

One diode conducts for 180 degrees of each cycle.

FULL-WAVE RECTIFIER (CENTRE-TAP OR BRIDGE), R LOAD

V_{d c} = \frac{2 V _{m}}{π}, V_{r m s} = \frac{V _{m}}{2}, F F = \frac{π}{2 2} = 1.11, R F = 0.483

For the centre-tap type $V_{m}$ is the peak of each half of the secondary. PIV is $V_{m}$ for the bridge and $2 V_{m}$ for the centre-tap.

HALF-WAVE RECTIFIER, RL LOAD (NO FREEWHEELING DIODE)

V_{d c} = \frac{V _{m}}{2 π} (1 - cos β), sin (β - ϕ) + sin ϕ e^{- β / t a n ϕ} = 0

$ϕ = tan^{- 1} (ω L / R)$ . The current is $i = \frac{V _{m}}{Z} [sin (ω t - ϕ) + sin ϕ e^{- ω t / t a n ϕ}]$ with $Z = R^{2} + (ω L)^{2}$ . The current continues after the supply goes negative, up to the extinction angle $β > π$ where $i = 0$ , so the output voltage has a negative portion from $π$ to $β$ .

HALF-WAVE DIODE CHARGING A BATTERY $E$ THROUGH $R$ (NO INDUCTANCE)

θ_{1} = sin^{- 1} \frac{E}{V _{m}}, I_{d c} = \frac{1}{2 π R} [2 V_{m} cos θ_{1} - E (π - 2 θ_{1})]

The diode conducts only while the instantaneous supply voltage exceeds $E$ , from $θ_{1}$ to $π - θ_{1}$ . In a full-wave bridge the same interval repeats in every half cycle, so the average current doubles.

THREE-PHASE HALF-WAVE (3-PULSE) DIODE RECTIFIER

V_{d c} = \frac{3 3}{2 π} V_{m} = 0.827 V_{m}, P I V = 3 V_{m}

Each diode conducts for 120 degrees; its average current is $I_{d c} /3$ and its rms current is $I_{d c} / 3$ . The output ripple frequency is $3 f$ .

THREE-PHASE BRIDGE (6-PULSE) DIODE RECTIFIER

V_{d c} = \frac{3 V _{LL, m}}{π} = \frac{3 3}{π} V_{m} = 1.35 V_{LL}

The output is made of six line-voltage segments per cycle, so the ripple frequency is $6 f$ . PIV $= V_{LL, m}$ .

BRIDGE CURRENTS WITH CONSTANT $I_{DC}$

I_{D, a v g} = \frac{I _{d c}}{3}, I_{D, r m s} = \frac{I _{d c}}{3}, I_{s, r m s} = \frac{2}{3} I_{d c}, I_{s 1, r m s} = \frac{6}{π} I_{d c}, P F = \frac{3}{π} = 0.955

Each diode conducts 120 degrees. The line current has one positive and one negative 120-degree block per cycle, and its fundamental is $0.78 I_{d c}$ .

HALF-WAVE CONTROLLED RECTIFIER, R LOAD

V_{d c} = \frac{V _{m}}{2 π} (1 + cos α)

Applies to an R load and also to an RL load with a freewheeling diode. Range $0 \leq α \leq π$ .

CONTROLLED RECTIFIERS WITH R LOAD: RMS OUTPUT

V_{r m s} = \frac{V _{m}}{2} \frac{1}{π} (π - α + \frac{sin 2 α}{2}) (half-wave), V_{r m s} = \frac{V _{m}}{2} \frac{1}{π} (π - α + \frac{sin 2 α}{2}) (bridge)

Pure R load, so the output follows the supply from $α$ to $π$ in each conducting half cycle. The bridge result holds for both the fully controlled and the half-controlled bridge.

FULLY CONTROLLED BRIDGE, CONTINUOUS CONDUCTION (LARGE L)

V_{d c} = \frac{2 V _{m}}{π} cos α

Positive for $α < 9 0^{\circ}$ (rectification) and negative for $α > 9 0^{\circ}$ (inversion, which needs an active load such as a reversed battery or a regenerating machine). With an RLE load and continuous current, $I_{d c} = (V_{d c} - E) / R$ .

HALF-CONTROLLED BRIDGE, OR FULLY CONTROLLED BRIDGE WITH R LOAD OR FREEWHEELING DIODE

V_{d c} = \frac{V _{m}}{π} (1 + cos α)

The output cannot go negative, so only the rectifying quadrant is available.

FULLY CONTROLLED BRIDGE WITH LARGE L: SOURCE CURRENT

I_{s, r m s} = I_{d c}, I_{s 1, r m s} = \frac{2 2}{π} I_{d c} = 0.9 I_{d c}, D P F = cos α, P F = \frac{2 2}{π} cos α

Square-wave source current; each thyristor conducts for 180 degrees, with average current $I_{d c} /2$ and rms current $I_{d c} / 2$ . The source current has odd harmonics $n = 3, 5, 7, \dots$ with $I_{n} = I_{s 1} / n$ , and THD $= 48.3%$ . Input powers: $P = \frac{2 V _{m}}{π} I_{d c} cos α = V_{d c} I_{d c}$ and $Q = \frac{2 V _{m}}{π} I_{d c} sin α$ .

HALF-CONTROLLED BRIDGE WITH LARGE L: SOURCE CURRENT

I_{s, r m s} = I_{d c} \frac{π - α}{π}, I_{s 1, r m s} = \frac{2 2}{π} I_{d c} cos \frac{α}{2}, D P F = cos \frac{α}{2}, P F = \frac{2 ( 1 + cos α )}{π ( π - α )}

The source current is $+ I_{d c}$ from $α$ to $π$ , $- I_{d c}$ from $π + α$ to $2 π$ , and zero while the load freewheels. Its displacement factor $cos (α /2)$ is better than the fully controlled bridge's $cos α$ .

THREE-PHASE FULLY CONTROLLED BRIDGE, CONTINUOUS CONDUCTION

V_{d c} = \frac{3 V _{LL, m}}{π} cos α = \frac{3 3}{π} V_{m} cos α = 1.35 V_{LL} cos α

Negative for $α > 9 0^{\circ}$ , which is inversion. The load current stays continuous for all $α$ when $L$ is large enough.

THREE-PHASE HALF-CONTROLLED BRIDGE

V_{d c} = \frac{3 V _{LL, m}}{2 π} (1 + cos α)

The output never goes negative, so $0 \leq α \leq 18 0^{\circ}$ gives rectification only.

FULLY CONTROLLED BRIDGE, R LOAD, DISCONTINUOUS ($\ALPHA>60^\CIRC$)

V_{d c} = \frac{3 V _{LL, m}}{π} [1 + cos (α + 6 0^{\circ})]

Valid for $6 0^{\circ} \leq α \leq 12 0^{\circ}$ ; the output reaches zero at $α = 12 0^{\circ}$ . For $α \leq 6 0^{\circ}$ with an R load the output stays positive and the cosine law $\frac{3 V _{LL, m}}{π} cos α$ still holds.

THREE-PHASE FULLY CONTROLLED BRIDGE WITH LARGE L: SOURCE CURRENT

I_{s, r m s} = \frac{2}{3} I_{d c}, I_{s 1, r m s} = \frac{6}{π} I_{d c}, D P F = cos α, P F = \frac{3}{π} cos α

The line current has one positive and one negative 120-degree block per cycle. Its harmonics are of order $n = 6 k \pm 1$ ( $5, 7, 11, 13, \dots$ ) with $I_{n} = I_{s 1} / n$ , and THD $= 31.1%$ . Input powers: $P = \frac{3 V _{LL, m}}{π} I_{d c} cos α$ and $Q = \frac{3 V _{LL, m}}{π} I_{d c} sin α$ .

SINGLE-PHASE FULLY CONTROLLED BRIDGE WITH $L_S$

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

Large L, continuous conduction. The drop equals $4 f L_{s} I_{d c}$ and does not depend on $α$ .

THREE-PHASE FULLY CONTROLLED BRIDGE WITH $L_S$

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

Constant $I_{d c}$ ; $L_{s}$ is per phase. The drop equals $6 f L_{s} I_{d c}$ .

VOLTAGE (IMPULSE) COMMUTATION

t_{c} = \frac{C V _{C}}{I _{0}}

An auxiliary thyristor switches a capacitor $C$ , charged to $V_{C}$ , across the conducting thyristor, reverse-biasing it at once. With the load current $I_{0}$ constant during commutation, the capacitor voltage changes linearly from $- V_{C}$ to zero in time $t_{c}$ .

CURRENT (RESONANT-PULSE) COMMUTATION

I_{c p} = V_{C} \frac{C}{L} > I_{0}, t_{c} = L C [π - 2 sin^{- 1} \frac{I _{0}}{I _{c p}}]

An LC circuit drives a resonant current pulse of peak $I_{c p}$ through the thyristor, opposite to its load current $I_{0}$ , assumed constant during commutation. The thyristor current reaches zero when the pulse reaches $I_{0}$ . A diode across the thyristor then carries the excess, and its forward drop reverse-biases the thyristor until the pulse falls back to $I_{0}$ . Commutation needs $I_{c p} > I_{0}$ .

AC VOLTAGE CONTROLLER, R LOAD (SINGLE-PHASE, PHASE CONTROL)

V_{o, r m s} = V_{s} \frac{1}{π} (π - α + \frac{sin 2 α}{2})

Two antiparallel thyristors (or a triac), fired symmetrically. $V_{s}$ is the rms supply voltage and $α$ is measured from the voltage zero crossing. Control range $0 \leq α \leq 18 0^{\circ}$ . The output has no DC component. Input power factor $= V_{o, r m s} / V_{s}$ .

RECTIFIER-FED SEPARATELY EXCITED DC MOTOR

V_{d c} = E_{b} + I_{a} R_{a}, E_{b} = K ϕ ω_{m}

Steady state with continuous armature current. Speed is raised by increasing $V_{d c}$ (reducing $α$ ).

Where this was asked: 2018 (2), 2020 (2), 2022 (1), 2023 (1), 2024 (2), 2025 (2), 2026 (2). See the 12 previous-year questions on Rectifiers. This shows what was asked, not what will be asked.

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Rectifiers: the notes and 12 solved previous-year questions