• In a Boost 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

Substituting the voltages and time intervals for both states:

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

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

Divide both sides by the switching period 𝑇:

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

Expand the terms:

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

Cancel out the 𝑉_𝑖𝑛 ⋅ 𝐷 and − 𝑉_𝑖𝑛 ⋅ 𝐷 terms:

· 𝑉_𝑖𝑛 − ( 1−𝐷 ) ⋅ 𝑉_𝑜𝑢𝑡 = 0

· 𝑉_𝑖𝑛 = 𝑉_𝑜𝑢𝑡⋅( 1−𝐷 )