Buck Converter Duty Cycle⚠ unverified

Understanding the Parameters

• $V_{in}$, $V_{out}$ — with $V_{out} \le V_{in}$; the ratio is the duty cycle.
• $D$ — the on‑time fraction; a controller adjusts it to regulate the output as input or load varies.

Derivation (Approaching a Proof)

The key principle is inductor volt‑second balance: in steady state the average voltage across the inductor over one switching period is zero (otherwise its current would ramp away). In a buck converter:

• Switch on (time $DT$): the inductor sees $V_L = V_{in} - V_{out}$.
• Switch off (time $(1-D)T$): the diode conducts and $V_L = -V_{out}$.

Setting the average to zero,

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

Hence $D = V_{out}/V_{in}$. The result is independent of load current (in continuous conduction) — the converter regulates purely by timing.

History

The switch‑mode buck converter (and volt‑second balance analysis) became practical with fast power semiconductors from the 1960s–70s, displacing lossy linear regulators. It is now the workhorse point‑of‑load regulator in virtually every computer and phone.

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

Notes: Registry calculator buck-converter-duty-cycle (unverified). Ideal, continuous‑conduction model; real duty cycle is slightly higher to cover switch/diode drops and losses.