4 C_ff Choose Method

This method is implemented by letting the gain cross 0dB with -20dB/decade slope [7]. To be noted, -20dB/decade gain at 0dB is not a necessary condition for stability. So this method just provides an allowable range for C_ff and it doesn’t mean the converter will be unstable if C_ff exceeds this range.

For the case as Figure 4-1(a) when the ripple injection zero frequency ω_RI is larger than bandwidth, we can add the C_ff zero ω_z inside bandwidth, then the loop gain can cross 0dB with -20dB/decade, as shown in Figure 4-1(b).

In the case as Figure 4-2, the zero and pole introduced by C_ff are both added inside bandwidth and the system stability can still be ensured. Although the slope of loop gain becomes -40dB/decade again after the pole ω_p, the loop gain increasement with C_ff makes the bandwidth increased and the ω_RI becomes smaller than bandwidth. A -20dB/decade crossing is achieved and system will have enough phase margin.

In the case as Figure 4-3, both the zero and pole introduced by C_ff are inside bandwidth, but the ripple injection zero frequency ω_RI is still larger than the bandwidth. At this condition, the loop gain will still cross 0dB with -40dB/decade, which can’t ensure the system phase margin.

Above all, we can conclude two restrictions to achieve a stable state with -20dB/decade crossing after adding feedforward capacitor:

A. ω_z<ω_cross; B. avoiding the condition as Figure 4-3.

First we could deduct the limit for restriction A. Based on Figure 4-4, we can know that ω_z<ω_cross can be achieved by ensuring ω_z<ω_c. The expression of ω_c is as equation (4). With equations (2) and (4), the lower limit of C_ff is derived as equation (5).

To get the limits for restriction B, we could know that Equation 6 corresponds the condition as Figure 4-3. Then the restriction B to avoid that condition corresponds to Equation 7.

where A_p is the amplitude of gain at ω_p.

To get the expressions of A_p and ω_cross, we can first get the relation between gain and frequency as Equation 8 through Equation 10.

Substituting Equation 2 through Equation 3 into Equation 8 through Equation 10, the expressions of A_p and ω_cross can be received as Equation 11 and Equation 12.

Substituting Equation 11 and Equation 12 into Equation 7, we can get Equation 13 as equation for restriction B.

Combine the Equation 5 for restriction A and Equation 13 for restriction B, Equation 14 and Equation 15 are the limits for C_ff.

At the meantime, the bandwidth after adding C_ff should also be limited under 1/3*f_sw. Since it is hard to give a unified method for the estimation of bandwidth and phase margin after adding C_ff, it is suggested to verify with the help of simulation model or bench loop test results.