DC-DC converters formulas

DC-DC converters: buck, boost and buck-boost. 21 formulas for GATE EE, IN Power Electronics, each with when to use it. Each is explained in the topic notes.

All 21 DC-DC converters formulas

VOLT-SECOND BALANCE (INDUCTOR)

\int_{0}^{T} v_{L} d t = 0 \Rightarrow V_{L, o n} D T + V_{L, o f f} (1 - D) T = 0

In steady state the average inductor voltage over one period is zero. This gives the voltage gain of every converter.

CHARGE BALANCE (CAPACITOR)

\overline{i_{C}} = 0 \Rightarrow \overline{i_{L}} = I_{o} (buck), \overline{i_{D}} = I_{o} (boost, buck-boost)

In steady state the average capacitor current is zero, so the average current delivered to the load equals the average current of the element feeding the output node: the inductor in the buck, the diode in the boost and buck-boost.

RMS INDUCTOR CURRENT (TRIANGULAR RIPPLE)

I_{L, r m s} = I_{L}^{2} + \frac{( Δ i _{L} ) ^{2}}{12}

CCM, $I_{L}$ = average inductor current, $Δ i_{L}$ = peak-to-peak ripple. Used for rms ratings and copper loss; with negligible ripple $I_{L, r m s} \approx I_{L}$ .

BUCK: VOLTAGE GAIN

V_{o} = D V_{s}

CCM. Output is positive, same polarity as the input, and $V_{o} \leq V_{s}$ .

BUCK: INDUCTOR CURRENT RIPPLE

Δ i_{L} = \frac{( V _{s} - V _{o} ) D T}{L} = \frac{V _{o} ( 1 - D )}{L f}

Peak-to-peak ripple, CCM. Average inductor current equals $I_{o}$ ; average input current is $I_{s} = D I_{o}$ .

BUCK: MAXIMUM RIPPLE

Δ i_{L} = \frac{V _{s} D ( 1 - D )}{L f} \leq Δ i_{L, ma x} = \frac{V _{s}}{4 L f} at D = 0.5

For fixed $V_{s}$ , $L$ and $f$ in CCM. The same result holds for a step-down chopper feeding an inductive load (with or without back emf) when the load resistance is neglected.

BUCK: CAPACITOR VOLTAGE RIPPLE

Δ v_{C} = \frac{Δ i _{L}}{8 f C} = \frac{V _{o} ( 1 - D )}{8 L C f ^{2}}

Assumes the whole ripple component of the inductor current flows into the capacitor (the load takes only the average).

BUCK: CRITICAL INDUCTANCE

L_{cr i t} = \frac{( 1 - D ) R}{2 f}

CCM if $L > L_{cr i t}$ (boundary where $I_{min} = 0$ , i.e. $I_{o} = Δ i_{L} /2$ ).

BOOST: VOLTAGE GAIN

V_{o} = \frac{V _{s}}{1 - D}

CCM. Same polarity as input, $V_{o} \geq V_{s}$ . In the ideal circuit the gain rises without limit as $D \to 1$ ; in a real converter the inductor and switch resistances make the gain reach a maximum and then fall as $D \to 1$ .

BOOST: AVERAGE INDUCTOR (INPUT) CURRENT

I_{L} = I_{s} = \frac{I _{o}}{1 - D} = \frac{V _{s}}{( 1 - D ) ^{2} R}

The diode conducts only for $(1 - D) T$ and carries $I_{L}$ , so its average current is $(1 - D) I_{L} = I_{o}$ ; the switch average current is $D I_{L}$ .

BOOST: INDUCTOR CURRENT RIPPLE

Δ i_{L} = \frac{V _{s} D T}{L} = \frac{V _{s} D}{L f}

CCM, peak-to-peak.

BOOST: CAPACITOR VOLTAGE RIPPLE

Δ v_{C} = \frac{I _{o} D T}{C} = \frac{V _{o} D}{R C f}

During the ON interval the capacitor alone supplies $I_{o}$ for $D T$ .

BOOST: CRITICAL INDUCTANCE

L_{cr i t} = \frac{D ( 1 - D ) ^{2} R}{2 f}

CCM if $L > L_{cr i t}$ (from $I_{L} = Δ i_{L} /2$ ).

BUCK-BOOST: VOLTAGE GAIN

V_{o} = - \frac{D}{1 - D} V_{s}

CCM. Output is negative with respect to the input common; $∣ V_{o} ∣ < V_{s}$ for $D < 0.5$ , $∣ V_{o} ∣ > V_{s}$ for $D > 0.5$ .

BUCK-BOOST: AVERAGE INDUCTOR CURRENT

I_{L} = \frac{I _{o}}{1 - D} = \frac{V _{s} D}{( 1 - D ) ^{2} R}

Here $I_{o} = ∣ V_{o} ∣/ R$ . Diode average current is $(1 - D) I_{L} = I_{o}$ ; average input (switch) current is $I_{s} = D I_{L} = D I_{o} / (1 - D)$ .

BUCK-BOOST: RIPPLES

Δ i_{L} = \frac{V _{s} D T}{L}, Δ v_{C} = \frac{∣ V _{o} ∣ D}{R C f}

CCM, peak-to-peak. The capacitor alone feeds the load during the ON interval.

BUCK-BOOST: CRITICAL INDUCTANCE

L_{cr i t} = \frac{( 1 - D ) ^{2} R}{2 f}

CCM if $L > L_{cr i t}$ .

DCM: GAIN IN TERMS OF THE DIODE INTERVAL $D_2$

Buck: \frac{V _{o}}{V _{s}} = \frac{D}{D + D _{2}}; Boost: \frac{V _{o}}{V _{s}} = \frac{D + D _{2}}{D _{2}}; Buck-boost: \frac{V _{o}}{V _{s}} = \frac{D}{D _{2}}

From volt-second balance with the inductor voltage zero during $D_{3} T = (1 - D - D_{2}) T$ . $D_{2}$ is the fraction of the period for which the diode conducts.

DCM: GAIN IN TERMS OF $D$ AND $K$

Buck: \frac{V _{o}}{V _{s}} = \frac{2}{1 + 1 + 4 K / D ^{2}}; Boost: \frac{V _{o}}{V _{s}} = \frac{1 + 1 + 4 D ^{2} / K}{2}; Buck-boost: \frac{V _{o}}{V _{s}} = \frac{D}{K}

Valid for $K < K_{cr i t}$ (DCM) only. At $K = K_{cr i t}$ they agree with the CCM gains. For the buck-boost the diode interval is $D_{2} = K$ ; in every case a DCM answer must satisfy $D + D_{2} < 1$ .

ARMATURE-VOLTAGE SPEED CONTROL

ω = \frac{D V _{s} - I _{a} R _{a}}{K ϕ}

Separately excited DC motor fed by a step-down chopper, continuous armature current, constant flux; average values.

REGENERATIVE BRAKING (CLASS B CHOPPER)

E_{b} = (1 - D) V_{s} + I_{a} R_{a}, P_{returned} = (1 - D) V_{s} I_{a}

Step-up (Class B) chopper with the machine generating: $D$ is the duty ratio of the switch that shorts the armature, $I_{a}$ is the magnitude of the continuous armature current flowing from the machine to the supply, and the average voltage across the machine terminals is $(1 - D) V_{s}$ . Average values, ripple neglected.

Where this was asked: 2018 (2), 2020 (1), 2022 (1), 2023 (1), 2024 (1), 2025 (2), 2026 (1). See the 9 previous-year questions on DC-DC converters. This shows what was asked, not what will be asked.

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DC-DC converters: the notes and 9 solved previous-year questions