Boost Converter Efficiency Through Accurate Calculations

Sep 1, 2008 12:00 PM
By Travis Eichhorn, Senior Applications Engineer, National Semiconductor, Grass Valley, Calif.

Higher-Order Model

Until now, the duty-cycle calculations assumed no ripple current, which isn't a problem because an average inductor current is used instead of root-mean-square (rms) inductor current. And assuming the converter operates in the CCM only changes the previous answer a slight amount. However, with boost circuits operating at such low-output power levels, it is likely that a large portion of the load current will cause the converter to operate in a discontinuous conduction mode (DCM).

Unfortunately, the previous calculations only applied to the CCM. If the designer wants to model the DCM efficiency, then the ripple current is no longer trivial. Furthermore, to calculate efficiency over the entire load-current range, ignoring the ripple current in the CCM and then including it in the DCM would lead to a discontinuity in efficiency as the load current transitions through the modes.

The solution is to include the ripple current in the CCM duty-cycle equation and also solve for the duty cycle using a separate equation for the DCM. The DCM equation is developed using a power-balance equation in much the same way as for the CCM case.

As a starting point, the rms value of a triangle waveform, which is the true inductor current, is given by:

I_LDC is the dc or average value of the inductor current, which is I_OUT/1 - D for the CCM and zero for the DCM. ΔI_L is the peak-to-peak inductor current ripple and is given by (ΔI_L = V_IN ´ D/(f_SW ´ L)) for both the CCM and the DCM. Finally, d is the portion of the switching period that the rms current is conducting.

For example, d = D for the components conducting during the switch-on time in both the CCM and the DCM; d = (1 - D) for the components conducting during the off time in the CCM and d = (2 ´ I_OUT)/ ΔI_L for the components conducting during the off time in the DCM; and lastly, d = 1 for the components conducting during both portions of the period in the CCM and d = D + (2 ´ I_OUT)/ ΔI_L for the components conducting during the first two portions of the period in the DCM.

Adding the ripple component to the F(D²) equation adds a third-, fourth- and fifth-order term for the resistive power-loss components. On the other hand, the DCM equation becomes a third-order polynomial. To avoid the complexity of drawing out these equations, the table categorizes the power-loss components in the asynchronous boost converter with the factored D terms to the right. In the table, the first column lists the CCM components and the second column lists the DCM components.

Each term in the CCM column is normalized to I_OUT (i.e., it is divided by (1 - D)² to eliminate the (1 - D)² term in the denominator). The I_AC term is the inductor current slope over the entire switching period (or the inductor current ripple divided by D). Collecting the terms of the power-loss components in the CCM column results in the fifth-order polynomial, F_CCM (D⁵) = a ´ D⁵ + b ´ D⁴ + c ´ D³ + d ´ D² + e ´ D¹ + f ´ D⁰, and collecting the terms of the power-loss components in the DCM column gives a third-order polynomial, F_DCM (D³) = g ´ D³ + h ´ D² + j ´ D¹ + k ´ D⁰, where the a through k coeffecients are functions of the circuit parameters V_IN, V_OUT, I_OUT, R_N, R_L, etc.

This higher-order model using both the F_CCM(D⁵) and F_DCM(D³) equations has no simple solution like the F(D2) equation. Fortunately, as with F(D²), the useful solutions to the fifth- and third-order polynomials are those occurring at the first crossing of zero with increasing D.

Finding this solution is possible, using a spreadsheet like Excel, by setting F(D⁵) and F(D³) to zero and plotting F(D) versus D and varying D from zero to one. Alternatively, implementing the functions F_CCM(D⁵) and F_DCM(D³) in C and using a “while” loop to test for the first F(D) = 0 crossing versus increasing D works, also.

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