Buck Boost Duty⚠ unverified

Understanding the Parameters

• $V_{in}$ — positive DC input. Larger input at fixed $|V_{out}|$ lowers the required $D$.

• $V_{out}$ — target output; the registry formula uses absolute values so polarity of the inverting topology does not flip the sign of $D$.

• $D$ — fraction of each cycle the main switch is on. $D < 0.5$ steps down in magnitude; $D > 0.5$ steps up; $D = 0.5$ gives $|V_{out}| = V_{in}$.

Derivation (Approaching a Proof)

Inductor volt-second balance in steady state: average voltage across the inductor over one period must be zero (else current ramps unboundedly).

• Switch on (duration $DT$): inductor charged from the input, $V_L = V_{in}$.
• Switch off (duration $(1-D)T$): inductor dumps into the output, $V_L = V_{out}$ with $V_{out} < 0$ in the classic inverting topology.

$$V_{in}\,D + V_{out}(1-D) = 0 \;\Rightarrow\; V_{out} = -V_{in}\,\frac{D}{1-D}.$$

Taking magnitudes, $|V_{out}| = V_{in}\,D/(1-D)$. Solving for $D$:

$$D = \frac{|V_{out}|}{|V_{in}| + |V_{out}|}.$$

Because the inductor is never a direct input-to-output path, the converter can both step up and step down and inverts polarity — unlike a pure buck or boost.

History

The inverting buck-boost and its isolated cousin (flyback) became standard low-power supply topologies as power MOSFETs matured. The duty-cycle formula is still the first-order design equation before including switch/diode drops and discontinuous conduction.

Related Concepts: Buck Boost Duty Cycle, Buck Converter Duty Cycle, Boost Converter Duty Cycle, Switching Loss, Conduction Loss

Notes: Registry calculator buck-boost-duty (unverified). Ideal continuous-conduction model. Near- duplicate of Buck Boost Duty Cycle (buck-boost-duty-cycle) — same formula, separate cards; Track B may dedupe. Real $D$ is slightly higher to cover conduction drops and losses.