Convention Confusion

• Use uppercase $V$ and $I$ to represent steady-state DC quantities

  • Constant or mean values

• Use lowercase $v$ and $i$ to represent instantaneous or time-varying quantities

  • These represent variations due to switching ripples or AC components in the circuit

    $$ i_L(t) = I_L + \Delta i_L(t) $$

• Inductors resist changes in current, leading to current ripple described using $i_L(t)$, and average current described using $I_L$

• Capacitors resist changes in voltage, leading to voltage ripple described using $v_C(t)$, and average voltage described using $V_C$

Volt-Second Balance and Small Ripple Approximation

• Volt-Second Balance: in a steady state condition, average voltage across an inductor over one switching period must be zero, since inductors resist changes in current

  $$ \int V_L(t) dt = 0 $$

• Small Ripple Approximation: the ripple variations in the inductor current $(\Delta I_L)$ and capacitor voltage $(\Delta V_{out})$ during each switching cycle are small compared to their average (DC) values

  • Assumes converter operates in continuous conduction mode (CCM), meaning the inductor current never goes to 0 during a switching cycle
  • The voltage across the inductor and the current through the capacitor can be approximated by their average values rather than accounting for the detailed ripple variations in every switching cycle

Buck Converter

Boost Converter

Capacitor Charge Balance and Ripple Calculations

• Capacitor Charge Balance: same as volt-second charge balance but for boost converters, in steady-state operation, net charge entering and leaving a capacitor over one switching period must be zero

  $$ \int i_C(t)dt = 0 $$

• Ripple analysis in the frequency domain is harder for a buck converter because of the nonlinear duty cycle relationship, inductor current shape, and the dual pathway for ripple

  • Any perturbations in the duty cycle affect both the inductor current and output voltage

Boost Converter

DC Ripple:

$$ \Delta I_L = \frac{V_{in}}{L} \times D \times T_s \newline \Delta V_{out} = \frac{I_{out} \times D}{C \times f_s} $$

Buck Converter

DC Ripple:

$$ \Delta I_L = \frac{V_{in} - V_{out}}{L} \times D \times T_s \newline \Delta V_{out} = \frac{\Delta I_L}{8 \times f_s \times C} $$