*2.2. Ripple Genertation Mechanism*

The output voltage and current of the single-phase QZSI inverter can be expressed as (4), where *Vo* and *Io* are the amplitude of the output voltage and current, ω is the fundamental frequency, and ϕ is the impedance angle of the load or grid, respectively.

$$v\_o = V\_o \sin(\omega t), \ i\_o = I\_o \sin(\omega t - \varphi) \tag{4}$$

The DC link voltage *vPN* and the DC link current *iPN* can be expressed as (5), where *VPN*, *IPN* are the average value and *<sup>v</sup>*<sup>ˆ</sup>*PN*, ˆ *iPN* are the 2ω components of *vPN*, *iPN*, respectively.

$$
\omega\_{\rm PN} = V\_{\rm PN} + \mathfrak{d}\_{\rm PN}, \ i\_{\rm PN} = I\_{\rm PN} + \hat{I}\_{\rm PN} \tag{5}
$$

From (4) and (5), the input power *pPN* and output power *po* of the H-bridge can be calculated as (6) and (7), respectively.

$$p\_{\rm PN} = (1 - D)\upsilon\_{\rm PN} i\_{\rm PN} = (1 - D)(V\_{\rm PN} I\_{\rm PN} + V\_{\rm PN} \hat{i}\_{\rm PN} + I\_{\rm PN} \theta\_{\rm PN} + \hat{i}\_{\rm PN} \theta\_{\rm PN}) \tag{6}$$

$$p\_o = \frac{1}{2}V\_o I\_o \cos\varphi - \frac{1}{2}V\_o I\_o \cos(2\omega t - \varphi) \tag{7}$$

The 2ω component of the DC link current can be given as (8), where *I*2ω is its amplitude and α is the phase angle.

$$\hat{\mathbf{i}}p\_{\text{N}} = I\_{2\omega}\cos(2\omega t - \alpha) \tag{8}$$

From *pPN* = *po* and Reference [12], the amplitude and phase angle of ˆ *iPN* can be calculated as

$$I\_{2\omega} = -\frac{V\_o I\_o \left[4a^2 LC - (1 - 2D)^2\right]}{2(1 - D)\sqrt{\left[4\omega^2 LC - (1 - 2D)^2\right]^2 V\_{PN}^2 + \left[4a\omega I\_{PN}(1 - D)\right]^2}}\tag{9}$$

$$\alpha = \varphi + \arctan \frac{4\omega L I p N (1 - D)}{\left[4\omega^2 L C - (1 - 2D)^2\right] V\_{PN}} \tag{10}$$

In (9), <sup>4</sup>ω*LIPN*(<sup>1</sup> − *D*) is much smaller than <sup>4</sup>ω2*LC* − (1 − 2*D*)<sup>2</sup>*VPN* and can be ignored, therefore, the 2ω component of the DC link current can be expressed as

$$\hat{H}\_{\rm PN} = -\frac{V\_o I\_o}{2(1 - D)V\_{\rm PN}} \cos(2\omega t - \alpha) \tag{11}$$

From (6) and (7), *IPN* can be calculated as

$$I\_{\rm PN} = \frac{V\_o I\_o}{2(1 - D)V\_{\rm PN}} \cos \varphi \tag{12}$$
