Model Current-Mode Control With Ease and Accuracy

Nov 1, 2008 12:00 PM
By Rendon Holloway, Principal Engineer, Fairchild Semiconductor, San Jose, Calif., and Gabriel Eirea, Ph.D., Instituto de Ingenieria Electrica, Universidad de la Republica, Uruguay

Proposed New Model

The new model proposed here incorporates most of the assumptions of Ridley's continuous-time model with one small but important difference in the computation of the average inductor current.^[8] From Fig. 2, the sensed peak current (I_P) can be expressed in terms of V_C as:

and in terms of the average sensed inductor current (I_L) as:

(Note that the average inductor current is computed as in steady-state operation, under the assumption that the inductor current returns to the same valley value at the end of the cycle.)

From Eqs. 6 and 7:

Then, under the assumption that slope S_N is constant, the modulator gain can be expressed as:

This expression is different from the one used in Ridley's continuous-time model.^[8] Actually, this modulator gain was reported even earlier (Middlebrook, Caltech), but in the context of a different derivation method.^[5-6]

Following Ridley's derivation, a perturbation in the command voltage is introduced and the variation in the inductor current is computed (Fig. 4).^[8] The variation in the inductor current is approximated by the sampled waveform i_s(k). Then, the closed-loop-transfer function between the command voltage V_c and the sampled inductor current (i_s) can be computed in the discrete-time domain, resulting in:

The equivalent continuous-time transfer function can be obtained as:

where a = (S_F - S_E)/(S_N + S_E).

In Fig. 3:

The new sampling gain H_E(s) is computed as:

A rational approximation can be obtained applying the same Padé approximation^[8], resulting in:

where ω_N = π/T_S and Q_Z = -(S_N/S_F + 1)2/π.

This expression is almost equal to the one in Eq. 5 with a slight difference.^[8] Although both the dc gain and the frequency of the double zero are the same, the quality factor is different.

Comparison of Models

The loop-transfer function of both the proposed new model and Ridley's model are compared with that obtained from a switched model simulated with the SIMetrix/SIMPLIS software tool.^{[8, 17]} The advantage of the SIMPLIS simulator is that it can compute a periodic operation point and introduce perturbations to obtain a small-signal transfer function around said operation point, thus computing numerically the transfer function. The former models were simulated using SIMetrix SPICE and performing a traditional ac analysis.

For this example, a buck converter with Fairchild's FAN2013 controller was used. The FAN2013 is a 2-A low-voltage current-mode synchronous pulse-width-modulated buck regulator designed for applications like hard-disk drives, set-top boxes, notebook computers and communications equipment. The circuit parameters used in these simulations are listed in the table.

The results are shown in Fig. 5. The new model proposed here shows a more accurate dc gain and improved mid-frequency phase characteristics than Ridley's model. References to prior work mentioned in this article are listed in the online version at www.powerelectronics.com.

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