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Mutual Induction

In the figure...

Question

In the figure shown below, duty ratio of buck-boost converter is $D_{1}$ and duty ratio of buck converter is $D_{2}$ . If the input voltage of 20 V is given to first converter and output voltage of 10 V is obtained from the second converter, then value of $D_{1}$ in terms of $D_{2}$ is

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Solution

For Buck-boost converter, $V_{2} = \frac{D_{2}}{(1 - D_{1})} V_{1}$
For buck converter, $V_{3} = D_{2} V_{2}$
$V_{3} = D_{2} \times \frac{D_{1}}{(1 - D_{1})} V_{1}$
$10 = \frac{D_{2} D_{1}}{(1 - D_{1})} 20$
$1 - D_{1} = 2 D_{1} D_{2}$
$D_{1} (1 + 2 D_{2}) = 1$
$D_{1} = \frac{1}{(1 + 2 D_{2})}$

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