4 Loop Stability Analysis With Hybrid Output Capacitor Network

The loop of the D-CAP3 converter is stable with the MLCC output capacitors network and the bandwidth is less than 1/3 switching frequency to make sure the loop is stable. When the hybrid output capacitor network is used, the loop stability is discussed in more detail.

Equation 9 shows the zero frequency minus the pole frequency, C1<C2, r1<r2, the result is less than 0. So the zero frequency is always less than the pole frequency, which can simplify the subsequent analysis and calculations. Since the ESR and capacitance of the MLCC is low, the zero ω_{z_C1} is typically located at a very high frequency and does not affect loop stability. The discussion focuses on the zero ω_{z_C2} and pole ω_{p_C2,}, which is divided into two cases.

Equation 9.

ω_{z_C 2} - ω_{p_C 2} = \frac{1}{C_{2} r_{2}} - \frac{1}{(r_{1} + r_{2}) \frac{C_{1} C_{2}}{C_{1} + C_{2}}} = \frac{C_{1} r_{1} - C_{2} r_{2}}{(r_{1} + r_{2}) C_{1} C_{2} r_{2}} < 0

Case 1: When the ESR and capacitance of C2 is not large enough, and the zero ω_{z_C2} produced by the C2 is greater than ω_cross and outside the bandwidth as shown in Figure 4-1. The ω_{z_C2} and ω_{p_C2} have little effect on the bandwidth and the bandwidth is less than 1/3×fsw.

Figure 4-1 Loop Gain of a D-CAP3 Converter With ω_{z_C2} > ω_cross

The loop stable condition is ω_{z_C2} > ω_cross, the crossover frequency can be calculated by:

Equation 10.

\begin{array}{l} \frac{20 \lg (Acp \frac{Vref}{V_{0}}) - 20 \lg (G_{RI})}{\lg (ω_{0}) - \lg (ω_{RI})} = - 40 dB/decade \\ \frac{20 \lg (G_{RI}) - 0}{\lg (ω_{RI}) - \lg (ω_{cross})} = - 20 d B/decade \end{array}

The ω_cross can be given by:

Equation 11.

ω_{c r o s s} = \frac{A c p V r e f ω_{0}^{2}}{V_{0} ω_{R I}}

At the same time, the introduced C2 makes the double pole frequency ω₀ become small and causes the crossing frequency to decrease as well. If the ω_cross ≤ ω_RI, the loop gain crosses 0 dB with -40-dB/decade slope, which results in the loop becoming unstable without enough phase margin. So, the loop stable condition is ω_{z_C2} > ω_cross and ω_cross > ω_RI, which can be given by:

Equation 12.

C_{2} r_{2} < \frac{V_{0} ω_{RI}}{AcpVref ω_{0}^{2}}

and

Equation 13.

C_{2} < \frac{{AcpVrefω}_{0}^{2}}{L V_{0} ω_{RI}} - C_{1}

Case 2: when the ESR and capacitance is large enough, and ω_{z_C2} is pushed into the bandwidth as shown in Figure 4-2, ω_{z_C2} < ω_cross. Since the gain curve passes through the zero ω_{z_C2}, the slope of the gain curve becomes 0, which changes to -20dB/decade only when the slope encounters the pole ω_{p_C2}. So, when the zero ω_{z_C2} enters within the bandwidth, the poles ω_{p_C2} must be inside the bandwidth, and the crossing frequency occurs after the ω_{p_C2}. The zero and pole inside crossover frequency increases the bandwidth and can cause the loop to become unstable when the crossover frequency exceeds 1/3×fsw. To provide loop stabilization, the conditions that ω_cross < 1/3×fsw and ω_{z_C2} < ω_cross need to be met.

Figure 4-2 Loop Gain of a D-CAP3™ Converter With ω_{z_C2} < ω_cross

The crossover frequency can be calculated by:

Equation 14.

\begin{array}{l} \frac{20 l g (A c p \frac{V r e f}{V_{0}}) - 20 l g (G_{R I})}{l g (ω_{0}) - l g (ω_{R I})} = - 40 d B / d e c a d e \\ \frac{20 \lg (G_{R I}) - 20 l g (G_{z_C 2_{}})}{l g (ω_{R I}) - l g (ω_{z_C 2_{}})} = - 20 d B / d e c a d e \\ \frac{20 l g (G_{p_C 2_{}}) - 0}{l g (ω_{p_C 2_{}}) - l g (ω_{c r o s s})} = - 20 d B / d e c a d e \\ \lg (G_{p_C 2_{}}) = \lg (G_{z_C 2_{}}) \end{array}

The ω_cross can be given by:

Equation 15.

ω_{c r o s s} = \frac{A c p V r e f ω_{0}^{2} ω_{p_C 2}}{V_{0} ω_{R I} ω_{z_C 2}}

Summarizing the two previous conditions, the loop stable conditions can be given by:

Equation 16.

ω_{z_C 2} > \frac{A c p V r e f ω_{0}^{2}}{V_{0} ω_{R I}}

Equation 17.

\frac{A c p V r e f ω_{0}^{2} ω_{p_C 2}}{V_{0} ω_{R I} ω_{z_C 2}} < \frac{1}{3} f s w

If the ESR and capacitance of C2 is too large and the ω_{z_C2} is less than ω_RI or ω₀, the bandwidth increases more than case 2. This situation rarely occurs in the application design and the calculation of crossing frequency is more complex. Getting the crossover frequency through bench test is recommended because this method is simple and accurate. The ω_cross calculation is not discussed more in-depth in this section. The calculation results are ideal and have deviation with the bench test results. Verifying the loops stability of the hybrid output capacitor network is recommended with the help of the bench loop test or simulation model. The suitable C2 can be selected based on the above principles and analysis method.