The open-loop response of a boost regulator is defined as the product of modulator transfer function and feedback transfer function. When plotted on a dB scale, the open loop gain is shown as the sum of modulator gain and feedback gain. The modulator transfer function of a current mode boost regulator includes a power stage transfer function with an embedded current loop. The transfer function is simplified as one pole, one zero, and one right-half-plane zero (RHPZ) system.

The modulator transfer function is defined as follows:

where

• Modulator DC gain, ${\mathrm{A}}_{\mathrm{M}}=\frac{{\mathrm{R}}_{\mathrm{o}\mathrm{u}\mathrm{t}}\times \mathrm{D}\mathrm{\text{'}}}{{2\times \mathrm{A}}_{\mathrm{c}\mathrm{s}}\times {\mathrm{R}}_{\mathrm{c}\mathrm{s}\_\mathrm{e}\mathrm{q}}}$
• Load pole, ${\mathrm{\omega }}_{\mathrm{P}\_\mathrm{L}\mathrm{F}}=\frac{2}{{\mathrm{R}}_{\mathrm{o}\mathrm{u}\mathrm{t}}\times {\mathrm{C}}_{\mathrm{o}\mathrm{u}\mathrm{t}}}$
• ESR zero, ${\mathrm{\omega }}_{\mathrm{Z}\_\mathrm{E}\mathrm{S}\mathrm{R}}=\frac{1}{{\mathrm{R}}_{\mathrm{E}\mathrm{S}\mathrm{R}}\times {\mathrm{C}}_{\mathrm{o}\mathrm{u}\mathrm{t}}}$
• RHPZ, ${\mathrm{\omega }}_{\mathrm{R}\mathrm{H}\mathrm{P}\mathrm{Z}}=\frac{{\mathrm{R}}_{\mathrm{o}\mathrm{u}\mathrm{t}}\times {\mathrm{D}\mathrm{\text{'}}}^2}{{\mathrm{L}}_{\mathrm{m}\_\mathrm{e}\mathrm{q}}}$
• The equivalent load resistance, ${\mathrm{R}}_{\mathrm{o}\mathrm{u}\mathrm{t}}=\frac{{\mathrm{V}}_{\mathrm{o}\mathrm{u}\mathrm{t}}^2}{{\mathrm{P}}_{\mathrm{o}\mathrm{u}\mathrm{t}\_\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}$
• The equivalent inductance, ${\mathrm{L}}_{\mathrm{m}\_\mathrm{e}\mathrm{q}}=\frac{{\mathrm{L}}_{\mathrm{m}}}{{\mathrm{N}}_{\mathrm{p}}}$
• The equivalent current sense resistor, ${\mathrm{R}}_{\mathrm{c}\mathrm{s}\_\mathrm{e}\mathrm{q}}=\frac{{\mathrm{R}}_{\mathrm{c}\mathrm{s}}}{{\mathrm{N}}_{\mathrm{p}}}$
• N_p is the number of the phases.
• Active current balancing circuit gain, ${\mathrm{G}}_{\mathrm{A}\mathrm{C}\mathrm{B}}\left(\mathrm{s}\right)=\frac{1}{2}\times \frac{\mathrm{s}\times 4\times {10}^{-6}+1}{\mathrm{s}\times 2\times {10}^{-6}+1}$. An active current balancing circuit is employed in LM5126A-Q1 to reduce the average current error caused by the difference of the two inductors.

If the equivalent series resistance (ESR) of C_out (R_ESR) is small enough and the RHPZ frequency is far away from the target crossover frequency, the modulator transfer function is further simplified to a one pole system and the voltage loop is closed with only two loop compensation components, R_COMP and C_COMP, leaving a single pole response at the crossover frequency. A single pole response at the crossover frequency yields a very stable loop with 90 degrees of phase margin.

As shown in Figure 7-1, a g_m amplifier is utilized as the output voltage error amplifier. The feedback transfer function includes the feedback resistor divider gain and loop compensation of the error amplifier. R_COMP, C_COMP, and C_HF configure the error amplifier gain and phase characteristics, create a pole at origin, a low frequency zero and a high frequency pole.

Figure 7-1 Type II g_m Amplifier Compensation

Feedback transfer function is defined as follows:

where

• The middle-band voltage gain, ${\mathrm{A}}_{\mathrm{V}\mathrm{M}}={\mathrm{K}}_{\mathrm{F}\mathrm{B}}\times {\mathrm{g}}_{\mathrm{m}}\times {\mathrm{R}}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{P}}$
• g_m = 1mA/V.
• The feedback resistor divider gain ${\mathrm{K}}_{\mathrm{F}\mathrm{B}}=\frac{{\mathrm{R}}_{\mathrm{F}\mathrm{B}\mathrm{B}}}{{\mathrm{R}}_{\mathrm{F}\mathrm{B}\mathrm{T}}+{\mathrm{R}}_{\mathrm{F}\mathrm{B}\mathrm{B}}}$. ${\mathrm{K}}_{\mathrm{F}\mathrm{B}}=\frac{1}{30}$ for the internal feedback resistor divider.
• Low frequency zero, ${\mathrm{\omega }}_{\mathrm{Z}\_\mathrm{E}\mathrm{A}}=\frac{1}{{\mathrm{R}}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{P}}\times {\mathrm{C}}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{P}}}$
• High frequency pole, ${\mathrm{\omega }}_{\mathrm{P}\_\mathrm{E}\mathrm{A}}\approx \frac{1}{{\mathrm{R}}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{P}}\times {\mathrm{C}}_{\mathrm{H}\mathrm{F}}}$

The pole at the origin minimizes the output steady state error. Place the low frequency zero to cancel the load pole of the modulator. Use the high frequency pole to cancel the zero created by the output capacitor ESR or to decrease noise susceptibility of the error amplifier. By placing the low frequency zero an order of magnitude less than the crossover frequency, the maximum amount of phase boost is achieved at the crossover frequency. Place the high frequency pole beyond the crossover frequency because the addition of C_HF adds a pole in the feedback transfer function.

The crossover frequency (open loop bandwidth) is usually limited to one fifth of the RHPZ frequency.

Increase R_COMP and proportionally decreasing C_COMP for higher crossover frequency. Conversely, decreasing R_COMP while proportionally increasing C_COMP, results in lower bandwidth while keeping the same zero frequency in the feedback transfer function.