SLUAAJ0 February 2024 TPS51397A , TPS54308 , TPS54320 , TPS54350 , TPS54620 , TPS54622 , TPS54821 , TPS54824 , TPS563300 , TPS566231 , TPS566235 , TPS566238 , TPS568230 , TPS56C215 , TPS62933 , TPS62933F , TPS62933O

5 Zero and Pole Analysis

The analysis about poles and zeros is introduced in the following. In the open loop transfer function Equation 15, the poles and zeros generated by G_EA(s) and G_ci(s) are the same as those in general purpose PCM buck converters without second stage filter.

G_EA(s) with type II compensation consists of an initial pole with frequency close to zero f_P1-EA, a middle frequency zero f_Z-EA, and a high frequency pole f_P2-EA:

Equation 16.

f_{Z-EA} = \frac{1}{2π R_{COMP} C_{COMP}}

Equation 17.

f_{P2-EA} = \frac{1}{2π R_{COMP} C_{O_EA}}

A pole is generated by G_ci(s) and the frequency is:

Equation 18.

f_{P - ci} = \frac{V_{IN} R_{i} f_{sw}}{2π [V_{Se} f_{sw} L+ ({0.5V}_{IN} - V_{O}) R_{i}]}

For the rest of part in Equation 15, calculate Z_O(s) x G_FB-TOTAL(s) and we can get:

Equation 19.

Z_{O} (s)× G_{FB-TOTAL} (s) = \frac{R_{2} (R_{L} + C_{ff} R_{1} R_{L} s+ C_{ff} L_{2} R_{1} s^{2} + C_{2} C_{ff} L_{2} R_{1} R_{L} s^{3})}{(R_{1} + R_{2} + C_{ff} R_{1} R_{2} s) [C_{2} C_{O} L_{2} R_{L} s^{3} + C_{O} L_{2} s^{2} + (C_{2} + C_{O}) R_{L} s+1]}

As shown from the previous transfer function has four poles and three zeros. Among all the parameters, output load resistance R_L varies during operation when load current changes. Figure 5-1 reflects the frequency response with different R_L.

Figure 5-1 Frequency Response of Equation (19) with Different R_L

As shown in Figure 5-1, the middle frequency and high frequency responses with different R_L are almost same. While in the low frequency range, the response with larger R_L has smaller phase. For the power solution design with low bandwidth, that can cause less phase margin. Thus the larger R_L corresponds to worse case for stability.

To simplify Equation 19, consider the worst case when the output resistance is infinite R_L->+∞:

Equation 20.

Z_{O} (s)× G_{FB-TOTAL} (s) |_{R_{L} →+ ∞} ≈ \frac{R_{2} (1+ C_{ff} R_{1} s+ C_{2} C_{ff} L_{2} R_{1} s^{3})}{s (R_{1} + R_{2} + C_{ff} R_{1} R_{2} s) [C_{2} C_{O} L_{2} s^{2} + (C_{2} + C_{O})]}

Figure 5-2 shows the map of zeros and poles of above transfer function.

Figure 5-2 Location of Zeros and Poles in Equation (20)

The four poles in Equation 20 include an initial pole P_out whose frequency is 0Hz, a pole P_ff at frequency f_Pff and a pair of conjugate poles P_2nd at frequency f_P-2nd. Expression of f_Pff and f_P-2nd are shown as Equation 21 and Equation 22:

Equation 21.

f_{Pff} = \frac{1}{2π C_{ff}} (\frac{1}{R_{1}} + \frac{1}{R_{2}})

Equation 22.

f_{P-2nd} = \frac{1}{2π \sqrt{\frac{C_{2} C_{O}}{C_{2} + C_{O}} L_{2}}}

The three zeros in Equation 20 include a zero Z_ff at frequency f_Zff and a pair of conjugate zeros Z_2nd at frequency f_Z-2nd.

The expression of f_Zff is complicated as shown in Equation 23. And it can be proven that f_Zff≤f_Pff.

Equation 23.

f_{Zff} =- \frac{1}{2π} \frac{5.67× 10^{15} {(\sqrt{3} \sqrt{{C_{2}}^{3} {C_{ff}}^{4} {L_{2}}^{3} {R_{1}}^{4} (4 {C_{ff}}^{2} {R_{1}}^{2} +27 C_{2} L_{2})} -9 {C_{2}}^{2} {C_{ff}}^{2} {L_{2}}^{2} {R_{1}}^{2})}^{\frac{2}{3}} -1.3× 10^{16} C_{2} {C_{ff}}^{2} L_{2} {R_{1}}^{2}}{1.49× 10^{16} C_{2} C_{ff} L_{2} R_{1} \sqrt[3]{\sqrt{3} \sqrt{{C_{2}}^{3} {C_{ff}}^{4} {L_{2}}^{3} {R_{1}}^{4} (4 {C_{ff}}^{2} {R_{1}}^{2} +27 C_{2} L_{2})} -9 {C_{2}}^{2} {C_{ff}}^{2} {L_{2}}^{2} {R_{1}}^{2}}}

The conjugate zeros Z_2nd are located in the right half plane if considering L₂ as a designed for inductor with no DCR. But for the real inductor or ferrite bead with DCR of several mΩ or larger, the conjugate zeros normally move to left half plane with positive damping effects.

The expression of the conjugate zeros frequency f_Z-2nd is too complicated. As an article to guide application design, the detailed mathematical analysis can not be applied here, since the conjugate zeros Z_2nd are not utilized for any important target like compensating phase margin in the following design guide.

Normally, the frequency f_Z-2nd is smaller than f_P-2nd, and the they are close. With larger C_ff, the frequency f_Z-2nd can be closer f_P-2nd. Figure 5-3 shows a typical gain response of PCM converter with second stage filter and hybrid sensing. The poles and zeros marked in red are newly introduced after adding the second stage filter.

Figure 5-3 PCM Converter Loop Response with Second Stage Filter and Hybrid Sensing