*2.3. Ripple Transmission Mechanism*

When the shoot-through duty cycle is a variable value, from (1), the small-signal model of single-phase QZSI can be obtained as (13) using the small signal analysis method with a constant DC source voltage, where ˆ *iL*1 , ˆ *iL*2 , *<sup>v</sup>*<sup>ˆ</sup>*C*1 , *<sup>v</sup>*<sup>ˆ</sup>*C*2 are the 2ω components of the inductor current *iL*1 , *iL*2 , the capacitor voltage *vC*1 , *vC*2 , respectively.

*Energies* **2019**, *12*, 3344

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

$$\begin{cases} L\_1 \frac{d\hat{\boldsymbol{u}}\_1}{dt} = -(1-D)\boldsymbol{\hat{v}}\_{\mathcal{C}\_1} + D\boldsymbol{\hat{v}}\_{\mathcal{C}\_2} + (\boldsymbol{V}\_{\mathcal{C}\_1} + \boldsymbol{V}\_{\mathcal{C}\_2})\boldsymbol{\hat{d}} \\ L\_2 \frac{d\hat{\boldsymbol{u}}\_2}{dt} = -(1-D)\boldsymbol{\hat{v}}\_{\mathcal{C}\_2} + D\boldsymbol{\hat{v}}\_{\mathcal{C}\_1} + (\boldsymbol{V}\_{\mathcal{C}\_1} + \boldsymbol{V}\_{\mathcal{C}\_2})\boldsymbol{\hat{d}} \\ \mathcal{C}\_1 \frac{d\hat{\boldsymbol{v}}\_1}{dt} = (1-D)\boldsymbol{\hat{i}}\_{\mathcal{L}\_1} - D\boldsymbol{\hat{i}}\_{\mathcal{L}\_2} - (1-D)\boldsymbol{\hat{i}}\_{\mathcal{P}\mathcal{N}} + (\boldsymbol{I}\_{\mathcal{P}\mathcal{N}} - \boldsymbol{I}\_{\mathcal{L}\_1} - \boldsymbol{I}\_{\mathcal{L}\_2})\boldsymbol{\hat{d}} \\ \mathcal{C}\_2 \frac{d\hat{\boldsymbol{v}}\_2}{dt} = (1-D)\boldsymbol{\hat{i}}\_{\mathcal{L}\_2} - D\boldsymbol{\hat{i}}\_{\mathcal{L}\_1} - (1-D)\boldsymbol{\hat{i}}\_{\mathcal{P}\mathcal{N}} + (\boldsymbol{I}\_{\mathcal{P}\mathcal{N}} - \boldsymbol{I}\_{\mathcal{L}\_1} - \boldsymbol{I}\_{\mathcal{L}\_2})\boldsymbol{\hat{d}} \end{cases} \tag{13}$$

The small signal model can be simplified as (14) under *L*1 = *L*2 = *L* and *C*1 = *C*2 = *C*, which means ˆ*iL* = ˆ*iL*1 = ˆ*iL*2 and *<sup>v</sup>*<sup>ˆ</sup>*C* = *<sup>v</sup>*<sup>ˆ</sup>*C*1 = *<sup>v</sup>*<sup>ˆ</sup>*C*2 referring to [8].

$$\begin{cases} L\frac{d\hat{l}\_{\rm L}}{dt} = -(1 - 2D)\mathfrak{d}\_{\rm C} + (V\_{\rm C\_{1}} + V\_{\rm C\_{2}})\hat{d} \\\ C\frac{d\hat{l}\_{\rm C}}{dt} = (1 - 2D)\hat{l}\_{\rm L} - (1 - D)\hat{l}\_{\rm PN} + (I\_{\rm PN} - 2I\_{\rm L})\hat{d} \end{cases} \tag{14}$$

From (2), (3) and (14), the transmission mechanism model of the inductor current ripple (15) and capacitor voltage ripple (16) can be obtained with Laplace transforms.

$$\hat{\eta}\_{L}(s) = \frac{(1 - 2D)(1 - D)}{LCs^{2} + (1 - 2D)^{2}} \hat{\eta}\_{PN}(s) + \frac{CsV\_{DC} + (1 - 2D)I\_{PN}}{(1 - 2D)\left[LCs^{2} + (1 - 2D)^{2}\right]} d(s) \tag{15}$$

$$\psi\_{\mathbb{C}}(\mathbf{s}) = \frac{-(1-D)Ls}{LCs^2 + (1-2D)^2} \hat{l}\_{\text{PN}}(\mathbf{s}) + \frac{(1-2D)V\_{\text{DC}} - LsI\_{\text{PN}}}{(1-2D)\left[LCs^2 + (1-2D)^2\right]} \hat{d}(\mathbf{s}) \tag{16}$$

In (15), the transfer functions can be expressed as (17) and (18), where *G*ˆ*iL* ˆ*iPN* represents the transfer function of ˆ*iPN* to ˆ*iL*1 , and *G*ˆ*iL* ˆ *d* represents the transfer function of ˆ *d* to ˆ*iL*.

$$\left. \right\vert\_{\text{l}^{\text{l}\text{s}}}^{\text{d}}(s) \vert\_{\text{d}=0} = \frac{(1 - D)(1 - 2D)}{L\text{Cs}^2 + (1 - 2D)^2} \tag{17}$$

$$\left.G\_{\stackrel{\text{dil}}{d}}^{\stackrel{\text{dil}}{\text{s}}}(\text{s})\right|\_{\text{l}^{\text{p}}\text{N}=\text{0}} = \frac{\text{Cs}V\_{\text{DC}} + (1 - 2D)I\_{\text{PN}}}{(1 - 2D)\left[L\text{Cs}^{2} + (1 - 2D)^{2}\right]} \tag{18}$$

From (2), (11), (12), (15), and (16), in the steady state, assuming a constant shoot-through duty cycle, we can ge<sup>t</sup>

$$\begin{cases} i\_{L\_1} = I\_{L\_1} + \hat{i}\_{L\_1} = \frac{V\_o I\_o}{2(1 - 2D)V\_{PN}} \cos \varphi + \frac{(1 - 2D)V\_o I\_o}{2[4\omega^2 LC - (1 - 2D)^2]V\_{PN}} \cos(2\omega t - \alpha) \\\ i\_{L\_2} = I\_{L\_2} + \hat{i}\_{L\_2} = \frac{V\_o I\_o}{2(1 - 2D)V\_{PN}} \cos \varphi + \frac{(1 - 2D)V\_o I\_o}{2[4\omega^2 LC - (1 - 2D)^2]V\_{PN}} \cos(2\omega t - \alpha) \\\ v\_{\mathcal{C}\_1} = V\_{\mathcal{C}\_1} + \hat{v}\_{\mathcal{C}\_1} = \frac{1 - D}{1 - 2D} V\_{\mathcal{DC}} + \frac{\alpha \mathcal{U}\_o V\_{\mathcal{C}\_1}}{[4\omega^2 LC - (1 - 2D)^2]V\_{PN}} \sin(2\omega t - \alpha) \\\ v\_{\mathcal{C}\_2} = V\_{\mathcal{C}\_2} + \hat{v}\_{\mathcal{C}\_2} = \frac{D}{1 - 2D} V\_{\mathcal{DC}} + \frac{\alpha \mathcal{U}\_o V\_{\mathcal{O}}}{[4\omega^2 LC - (1 - 2D)^2]V\_{PN}} \sin(2\omega t - \alpha) \end{cases} \tag{19}$$

From (19), the 2 ω current ripple ratios of *iL*1 and *iL*2 , the 2 ω voltage ripple ratios of *vC*1 and *vC*2 can be defined as (20), (21), and (22), respectively.

$$a = \frac{|\hat{l}\_{L\_1}|}{I\_{L\_1}} \times 100\% = \frac{|\hat{l}\_{L\_2}|}{I\_{L\_2}} \times 100\% = \frac{(1 - 2D)^2}{\left[4LC\alpha^2 - (1 - 2D)^2\right] \cos \varphi} \times 100\% \tag{20}$$

$$b\_1 = \frac{|\mathbb{G}\_{\mathbb{C}\_1}|}{V\_{\mathbb{C}\_1}} \times 100\% = \frac{(1 - 2D)aLV\_oI\_o}{(1 - D)\left[4\omega^2 LC - (1 - 2D)^2\right]V\_{\text{PN}}V\_{\text{DC}}} \times 100\% \tag{21}$$

*Energies* **2019**, *12*, 3344

$$b\_{2} = \frac{\left| \vartheta\_{\text{C}\_{2}} \right|}{V\_{\text{C}\_{2}}} \times 100\% = \frac{(1 - 2D)\omega LV\_{o}I\_{o}}{D \left[4\omega^{2}LC - (1 - 2D)^{2}\right] V\_{\text{PN}}V\_{\text{DC}}} \times 100\% \tag{22}$$

#### **3. Modulation Strategy Based on Ripple Vector Cancellation**

The ripple transmission models (15) and (16) show that the 2 ω inductor current ripple and the capacitor voltage ripple in the DC side are related to the variation ˆ*iPN* and ˆ *d*. If the shoot-through duty cycle or the DC link current has a variation, the current of the inductors will contain variation ˆ*i* ∗ *L* and *v*<sup>ˆ</sup><sup>∗</sup> *C*. This paper mainly focuses on the reducing of the 2 ω current ripple in the DC side. Therefore, regarding ˆ*i* ∗ *L* caused by ˆ*iPN* as the disturbance, which caused by ˆ *d* as the compensation, theoretically, the 2 ω current ripple will be suppressed to 0 with an appropriate ˆ *d*.A2 ω shoot-through duty cycle ˆ *d* = ˆ *d*2ω with specific amplitude and phase angle, according to the magnitude of the 2 ω inductors current ripple, can then be calculated based on the thought of the ripple vector cancellation. Adding the 2 ω compensation variation ˆ *d*2ω to the constant shoot-though duty cycle *D*, at this time, the shoot-through duty cycle can be expressed as

$$d = D + \hat{d}\_{2\nu} \tag{23}$$

From (23), the shoot-through duty cycle consists of a constant value and a 2 ω component, where *D* is determined by the output voltage of the inverter and the DC source voltage and ˆ *d*2ω is determined by the actual 2 ω current ripple in the DC side. Di fferent from the CMS, as shown in Figure 3, the proposed ripple vector cancellation modulation strategy (RVCMS) uses two sine waveforms with 2 ω components to generate the shoot-through duty cycle, where *vp* = 1 − *d* and *vn* = −1 + *d*.

**Figure 3.** Proposed ripple vector cancellation modulation strategy (RVCMS).

The 2 ω compensated shoot-through duty cycle can be expressed as (24).

$$
\hat{d}\_{2\omega} = A \sin(2\omega t + \beta) \tag{24}
$$

From (17) and (18), the transfer function of *iPN* to *d* used to cancel the 2 ω current ripple of inductors can be calculated as

ˆ

ˆ

$$G(s) = \frac{G\_{\stackrel{i\_{l\_{l}}}{i\_{l\text{PN}}}}^{\stackrel{i\_{l\text{L}}}{}}(s)}{G\_{\stackrel{i\_{l}}{i}}(s)} = \frac{(1 - D)(1 - 2D)^2}{\text{Cs}V\_{\text{DC}} + (1 - 2D)I\_{\text{PN}}} \tag{25}$$

and ˆ *d*2ω can be calculated by (26), where ∗ represents the convolution.

$$
\hat{d}\_{2\omega} = -\hat{l}\_{\rm PN} \ast \mathrm{L}^{-1} \left[ G(s) \right] \tag{26}
$$

From (3), (10), (11), (24)–(26), the amplitude *A* and phase angle β of ˆ *d*2ω can be calculated as

$$A = \frac{V\_o l\_o (1 - 2D)^3}{2V\_{\rm DC} \sqrt{4\alpha^2 C^2 V\_{\rm DC}^2 + I\_{\rm PN}^2 (1 - 2D)^2}} \tag{27}$$

$$\beta = \arctan\left[\frac{(1 - 2D)I\_{\rm PN}}{2\omega CV\_{\rm DC}}\right] - \arctan\frac{(1 - 2D)(1 - D)4\omega LI\_{\rm PN}}{\left[4\omega^2 LC - (1 - 2D)^2\right]V\_{\rm DC}} - q\tag{28}$$

When the proposed *d*2ω is applied, the entire 2ω power is buffered by the capacitors and there is no 2ω power in the inductors and DC source. It should be noted that the 2ω voltage ripple of capacitors will reduce slightly when the 2ω current ripple of the inductors are limited to 0. The phasor diagram of the 2ω power flows with a different modulation strategy are shown in Figure 4, where *PL*−2<sup>ω</sup>, *PC*−2<sup>ω</sup>, *Pdc*−2<sup>ω</sup>, and *Po*−2<sup>ω</sup> express the 2ω power of two inductors, two capacitors, and DC source and loads, respectively; the derivation and demonstration are given in Reference [12].

**Figure 4.** Phasor diagram of double-frequency (2ω) power flows with the CMS (**a**) and RVCMS (**b**).

## **4. Simulation and Experimental Results**

ˆ

In order to verify the correctness and demonstrate the effectiveness of the proposed modulation strategy for the ripple suppression effect, the 2ω inductor current ripple must be large enough with the CMS, and in Reference [8] the simulation model was built, and its parameters are shown in Table 1. Figure 5 shows the simulation waveforms of *d* (a), *iL*1 (b), *io* (c), *iPN* (d) *vC*1 (e), and *vC*2 (f) with the CMS. The average values of *iL*1 , *iPN*, *vC*1 , and *vC*2 are 3.014 A, 3.022 A, 90.21 V, and 30.21 V, and the amplitude of *io* is 4.154 A, respectively. From Figure 5a, the shoot-through duty cycle is a constant; Figure 5b,d–f show the 2ω ripple of the inductor current and the DC link current, the capacitor voltage are quite large with the CMS.


**Table 1.** Parameters of the single-phase QZSI.

Figure 6 shows the simulation results of *d* (a), *iL*1 (b), *io* (c), *iPN* (d), *vC*1 (e), and *vC*2 (f) with the RVCMS. With the proposed modulation strategy, the shoot-through duty cycle is a sine wave with a 2ω component; the current of inductor and the DC link have a little 2ω ripple. Figure 6e,f show the 2ω voltage ripple of capacitors reduced slightly compared to Figure 5e,f. The average values of *iL*1, *iPN*,

*vC*1 , and *vC*2 are 2.993 A, 2.995 A, 88.98 V, and 28.98 V, and the amplitude of *io* is 4.145 A, respectively. The simulation results show that the proposed RVCMS can effectively suppress the 2ω current ripple of inductors and can suppress the capacitor voltage ripple slightly compared to the CMS.

**Figure 5.** Simulation results of *d* (**a**), *iL*1 (**b**), *io* (**c**), *iPN* (**d**), *vC*1 (**e**), and *vC*2 (**f**) with the CMS.

**Figure 6.** Simulation results of *d* (**a**), *iL*1 (**b**), *io* (**c**), *iPN* (**d**), *vC*1 (**e**), and *vC*2 (**f**) with the RVCMS.

**Table 2.** The 2ω ripple ratios with

RVCMS

Table 2 lists the 2ω ripple ratios with the CMS and RVCMS. It can be seen that the 2ω inductor current ripple ratios decreased from 40.15% to 1.69%; the 2ω voltage ripple ratios of *vC*1 decreased from 3.14% to 2.53%, and *vC*2 decreased from 9.40% to 7.75%. If low 2ω ripple ratios in the DC side with the CMS are required, the large inductance and capacitance are necessary. From (20) and (21), assuming that *a* = 1.69% and *b*1 = 2.53%, the inductance and capacitance of quasi-Z-source network will be *L* = 36.89 mH and *C* = 1.03 mF, respectively, and the inductance is much larger than the value with the RVCMS, achieving the same ripple suppression effect. Figure 7 shows the FFT spectrum of the AC output current with the CMS and the RVCMS, and the total harmonic distortion (THD) are 3.46% and 3.54%, respectively. It can be seen that the proposed RVCMS has little effect on the output power quality.

 **Strategy** *iL*1 *vC*1 *vC*2 CMS 40.15% 3.14% 9.40%

 1.69% a different modulation

> 2.53%

 strategy.

 7.75%

**Figure 7.** FFT spectrum of the AC output current with the CMS (**a**) and RVCMS (**b**).

A single-phase QZSI experimental prototype was built in the laboratory, as shown in Figure 8. The PWM control signals of the proposed RVCMS and CMS for switches were generated by a TMS320F28335 DSP. Figures 9 and 10 show the experimental results with the CMS and RVCMS, respectively. The experimental results show that the 2ω current ripple of inductors reduced greatly and the 2ω ripple of capacitors reduced slightly with the RVCMS compared to the CMS. Simulation and experimental results verify that the proposed RVCMS can realize the suppression of 2ω ripples in addition to the functions of the DC-AC and voltage boost for the single-phase QZSI.

**Figure 8.** Experimental prototype in the laboratory.

**Figure 9.** Experimental results with the CMS.

**Figure 10.** Experimental results with the RVCMS.
