*4.1. Simplified Model and Feedforward*

Both synchronous buck and H-bridge converters can be represented by a simplified form as shown in Figure 7a. Modulation techniques are used to get the desired value of the midpoint voltage, *Vmid*. The direction of average output current, *Iout*, depends on value of *Vmid* and *Vbat*. *Iout*(*t*) determines the mode of operation for the controller. In the steady state equilibrium condition, *Iout*=0 and *Vmid*=*Vout*=*Vbat*. A feedforward duty can maintain the power converters in equilibrium. The feedforward duty, *df f* , in averaged control mode can be expressed by (23) and (24).

$$\overline{V\_{mid}} = V\_{in}d\_{ff} = V\_{bat} \tag{23}$$

$$d\_{ff} = \frac{V\_{\text{bat}}}{V\_{\text{in}}} \tag{24}$$

**Figure 7.** Small signal modeling: (**a**) simplified model of power converters, and (**b**) AR-ECM model of a battery.

*4.2. Transfer Function of Switching Power Pole*

The average midpoint voltage for a PWM cycle at any instant can be expressed by (25).

$$
\overline{V\_{mid}}(t) = V\_{in}d(t) \tag{25}
$$

where, *d* is average duty for that cycle and the value of *d* is updated for every cycle based on control requirement. A small perturbation to steady state duty causes perturbation to average midpoint voltage. The small perturbations can be expressed by (26) and (27).

$$
\overline{d}(t) = d\_{ff}(t) + \dot{d}(t) \tag{26}
$$

$$
\overline{V\_{mid}}(\mathbf{t}) = V\_{in} d\_{ff}(\mathbf{t}) + \overline{V\_{mid}}(\mathbf{t}) \tag{27}
$$

Considering the average mode control perturbation and steady input voltage, frequency domain interpretation of the midpoint voltage can be expressed by (28).

$$
\overline{V\_{mid}}(\mathbf{s}) = V\_{in}d(\mathbf{s})\tag{28}
$$

where, *s* = *jω*, and *ω* is angular frequency.
