• In a Buck Converter, the derivation of the Duty Cycle (𝐷) is fundamentally based on the Inductor Volt-Second Balance principle

• The principle states that: In a steady-state switching regulator, the net change in inductor current over a complete switching cycle must be zero

• In other words, the average voltage across the inductor over one full period is equal to zero

• When the switch (MOSFET) is turned on, the freewheeling diode becomes reverse-biased and turns off. Current flows from the input through the inductor to the output

The left side of the inductor is connected to 𝑉_𝑖𝑛, and the right side is at 𝑉_𝑜𝑢𝑡

The voltage across the inductor is:

· 𝑉_{𝐿(𝑜𝑛)} = 𝑉_𝑖𝑛 − 𝑉_𝑜𝑢𝑡 = 𝐿 𝑑𝑖/𝑑𝑡

The inductor current 𝑖_𝐿 rises linearly. According to the inductor voltage formula 𝑉_𝐿 = 𝐿 𝑑𝑖/𝑑𝑡 , the current ripple Δ𝑖_𝑜𝑛 is:

· Δ𝑖_𝑜𝑛 = [ (𝑉_𝑖𝑛 − 𝑉_𝑜𝑢𝑡)/𝐿 ] ⋅ 𝑡_𝑜𝑛 = [ (𝑉_𝑖𝑛 − 𝑉_𝑜𝑢𝑡)/𝐿 ] ⋅ 𝐷𝑇

• When the switch is turned off, the inductor opposes the sudden drop in current by reversing its voltage polarity, forcing the freewheeling diode to turn on

The left side of the inductor is now connected to ground (0V) via the diode, while the right side remains at 𝑉_𝑜𝑢𝑡

The voltage across the inductor is:

· 𝑉_{𝐿(𝑜𝑓𝑓)} = 0 − 𝑉_𝑜𝑢𝑡 = −𝑉_𝑜𝑢𝑡 = 𝐿 𝑑𝑖/𝑑𝑡

The inductor current 𝑖_𝐿 falls linearly. The current ripple Δ𝑖_𝑜𝑓𝑓 is:

· Δ𝑖_𝑜𝑓𝑓 = [ (−𝑉_𝑜𝑢𝑡) / 𝐿 ] ⋅ 𝑡_𝑜𝑓𝑓 = [ (−𝑉_𝑜𝑢𝑡) / 𝐿 ] ⋅ (1−𝐷)𝑇

Substituting the voltages and time intervals for both states:

· 𝑉_{𝐿(𝑜𝑛)} ⋅ 𝑡_𝑜𝑛 + 𝑉_{𝐿(𝑜𝑓𝑓)} ⋅ 𝑡_𝑜𝑓𝑓 = 0

·( 𝑉_𝑖𝑛 − 𝑉_𝑜𝑢𝑡 )⋅𝐷𝑇 + ( −𝑉_𝑜𝑢𝑡 )⋅(1−𝐷)𝑇 = 0

Divide both sides by the switching period 𝑇:

· ( 𝑉_𝑖𝑛 − 𝑉_𝑜𝑢𝑡 )⋅𝐷 − 𝑉_𝑜𝑢𝑡⋅(1−𝐷) = 0

Expand the terms:

· 𝑉_𝑖𝑛 ⋅ 𝐷 − 𝑉_𝑜𝑢𝑡 ⋅ 𝐷 − 𝑉_𝑜𝑢𝑡 + 𝑉_𝑜𝑢𝑡 ⋅ 𝐷 = 0

Cancel out the −𝑉_𝑜𝑢𝑡⋅𝐷 and +𝑉_𝑜𝑢𝑡⋅𝐷 terms:

· 𝑉_𝑖𝑛⋅𝐷 − 𝑉_𝑜𝑢𝑡 = 0

· 𝑉_𝑜𝑢𝑡 = 𝑉_𝑖𝑛 ⋅ 𝐷