4 Method for Selecting C_ff

This method is implemented by letting the loop gain cross 0 dB with –20 dB/dec slope. To be noted, –20 dB/dec gain at 0 dB is not a necessary condition for stability. So this method just provides an allowable range for C_ff and it does not mean the converter will be unstable if C_ff exceeds this range.

For the case as Figure 4-1(a) shows, when the EA zero frequency f_{Z_EA} is larger than bandwidth, the C_ff zero f_z can be added inside bandwidth, then the loop gain can cross 0 dB with –20 dB/dec, as Figure 4-1(b) shows.

As the case in Figure 4-2 shows, 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 –40 dB/dec again after the pole f_p, the increase in loop gain with C_ff makes the bandwidth increase and the f_{Z_EA} becomes smaller than the bandwidth. A –20-dB/dec crossing is achieved and the system has enough phase margin.

As the case in Figure 4-3 shows, both the zero and pole introduced by C_ff are inside bandwidth, but the EA zero frequency f_{Z_EA} is still larger than the bandwidth. At this condition, the loop gain will still cross 0 dB with –40 dB/dec, which cannot ensure the system phase margin.

Therefore, the two restrictions deduced to achieve a stable state with –20 dB/dec crossing after adding feedforward capacitor are (1) f_z < f_c, and (2) avoiding the condition as exhibited in Figure 4-3. f_c is the gain crossover frequency when not adding C_ff.

The expression of f_c has already been derived in Part I, as shown in Equation 7.

With the restriction f_z < f_c, the low limit of C_ff is obtained based on Equation 5 and Equation 7:

This equation corresponds to the limit for restriction 1. Since Equation 9 corresponds to the condition as Figure 4-3(b), restriction 2 to avoid that condition corresponds to Equation 10.

where A_p is the amplitude of gain at f_p.

To get the expressions of A_p and f_cross, first determine the relation between gain and frequency as Equation 11, Equation 12, and Equation 13.

Next, the expressions of A_p and f_cross as Equation 14 and Equation 15 are determined.

Substituting expressions for A_{P_OUT}, f_{P_OUT}, f_z, and f_p into Equation 14 and Equation 15, then Equation 16 is calculated as the equation for restriction 2 based on Equation 10, Equation 14, and Equation 15.

Combining Equation 8 for restriction 1 and Equation 16 for restriction 2, Equation 17 and Equation 18 are the limits for C_ff.