EE$331$ Lesson $43$: DC$-$DC Conversion

Additional Exercise: An idealized buck converter circuit is depicted at right. This buck converter produces an output voltage of $5$ V from a $9$ V battery. Derive the transfer function of an ideal buck converter by following the steps below. Assume the capacitor is 'large enough' that $V_O$ is constant.

Sketch the buck converter effective circuit when the switch is $\text{'}$ON$\text{'}$ and the diode is 'OFF.$\text{'}$

Replace the switch and diode with their idealized models.

Based on the given input voltage and desired output voltage, calculate $V_{L,\text{s}\text{w}\text{o}\text{n}}$ when this circuit is in effect. Is inductor current increasing or decreasing?

Sketch the buck converter effective circuit when the switch is 'OFF' and the diode is $\text{'}$ON$.\text{'}$

Replace the switch and diode with their idealized models.

Based on the components attached to the inductor at this time, calculate $V_{L_L,S{W}_{0ff}}$ when this circuit is in effect. Is inductor current increasing or decreasing?

Given that $($:$V_L$:$)=0V,$ we know the integral of $V_L$ over a period must be zero. That is: $DT{V}_{L,S{W}_{0n}}=-(1-D)T{V}_{L,S{W}_{0ff}}$ determine the value of $D$ required to achieve the $9$ V to $5$ V conversion. Does this agree with the equation presented in the lesson?

The average capacitor current $=0$ in periodic steady state. Why?

Given $R_{LD}=100,$ apply KCL at the right side inductor node using average currents and assuming $($:$V_O$:$)=5V$ to solve for average inductor current $($:$I_L$:$).$

$5.$ Sketch duty cycle, $V_L,$ and $I_L$ in time, qualitatively.