2 Current Transfer Function G_{ii_CL}(s) of Load Current to Inductor Current

From Figure 1, the summation of all inductor current is called I_sum, undershoot and overshoot occur when I_sum cannot track I_o quickly. To calculate undershoot and overshoot without DCLL, based on unbalanced electric discharges and charges of output capacitors, the relationship of I_sum and I_o needs to be clarified. Thus, it's important to know how inductor current I_L(I_sum in multiphase application) responds when load current I_o varies rapidly.

Based on small-signal model of a multiphase buck converter, and the D-CAP+ control modeling, we can derive small-signal transfer functions both in open loop and closed loop, of any input to any output within the system, shown in Figure 3.

All open-loop small-signal transfer functions based on D-CAP+ control mode are listed below:

For example, from input I_o to output v_out, the transfer function is named Z_{o_OP}(s), representing how I_o variation affects v_out in open loop, also called open-loop output impendence from physical meaning.

Inside the dot line at bottom below, is the common closed-loop control loop called loop gain L(s)=H_comp(s)*G_c2vo(s). All closed-loop transfer functions can be calculated based on open-loop transfer functions and loop gain together, with signal-flow diagram transformation in Figure 3.

Load current I_o to inductor current I_L (I_sum in multiphase application), as current transfer functions, are our interests here. For clarity, we define symbols as below:

• G_{ii_OP}(s) is open-loop transfer function of I_o to I_sum
• G_{ii_CL}(s) is closed-loop transfer function of I_o to I_sum

They describe how I_o will affect I_sum with specific expression, under open-loop and D-CAP+ closed-loop control in frequency domain.

Taking parameters in Table 5 as an example, the bode plot of loop gin L(s) is shown in Figure 4 with results of both simulation and bench test. We can find these two match well with cross frequency f_c=~100kHz.

The good matching result can direct us to extend the usage of simulation tool to estimate other transfer functions. By performing signal-flow diagram transformation in Figure 3, we get the closed-loop equation of G_{ii_CL}(s):

In Figure 5, simulation shows what G_{ii_CL}(s) looks like, when compared with G_{ii_OP}(s) and L(s) together.

In conclusion, by D-CAP+ closed-loop control, we can strongly increase the bandwidth of the G_{ii_OP}(s) to much higher, and improve dynamic response of I_sum, ensuring itself to follow I_o load step as quick as possible. From Figure 5, we can approximately take G_{ii_CL}(s) as a first-order system with corner frequency f_ci=1.5f_c=~150kHz.

Figure 6 is one simulated example showing that real response of I_sum, matches very well with emulated response of the approximate first-order system, under a given ramp-step input.

We will use the response of this approximate G_{ii_CL}(s) to emulate real response of I_sum, under a given ramp-step load step, in a linear system without saturation.