Answer :

Final answer:

The critical resistance expressions for Buck, Boost, and Buck-Boost converters are derived based on their circuit dynamics under continuous conduction mode, considering the duty cycle, inductance, input voltage, and switching frequency.

Explanation:

To derive the expressions for the critical resistance for Buck, Boost, and Buck-Boost converters, we need to analyze the circuit dynamics under the continuous conduction mode (CCM). These expressions take into account the transfer of energy from the input to the output through the inductor while considering the converter's operating principles.

Buck Converter

For a Buck converter, the critical resistance (Rcrit) can be found by equating the inductor ripple current at the boundary of CCM and Discontinuous Conduction Mode (DCM) by considering the duty cycle (D), input voltage (Vin), inductance (L), and switching frequency (fsw). The expression for Rcrit is derived as:

Rcrit = (2×L×Vin) / (D2fsw)

Boost Converter

For a Boost converter, the critical resistance is dependent on similar parameters as the Buck converter but is derived differently due to the inversion in voltage gain. The expression for Rcrit is:

Rcrit = (1 / (1−D)2) × (2×L) / (fsw)

Buck-Boost Converter

The Buck-Boost converter, combining the principles of both Buck and Boost converters, has its own Rcrit expression, derived as:

Rcrit = (D / (1−D)2) × (2×L×Vin) / fsw

Each of these expressions enables the design and optimization of the respective converters for maximum efficiency and operational stability.

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