Design and Implementation of Peak Current Mode with PI Controller for Coupled Inductor-Based High-Gain Z-Source Converter

Abstract

In this study, a peak current mode controller design and implementation is carried out for a high-gain integrated Z-Source DC–DC Converter. In this context, firstly, a small-signal dynamic modeling method for switching converters, based on volt–charge balance methodology, has been proposed and then applied to the Z-Source converter. With the proposed method, the modeling procedure of the integrated complex converters with multiple operating points can be simplified. To verify the obtained model, a small-signal simulation of the converter is also performed by using the Frequency Response Estimator tool of MATLAB/Simulink. The transfer function from the control current to the output was then obtained from the developed converter model to design the current mode controller. Then, a PI (proportional–integral) controller was designed according to the obtained transfer function, and a simulation study was performed with PLECS. Finally, an experimental study was carried out and compared with the simulation study. The results of the simulation and experimental studies show that the designed control method maintains the output voltage at desired levels regardless of the changes in load conditions and input voltage, successfully.

1. Introduction

For applications requiring DC voltage with small size and weight and high efficiency, switching mode power supplies (SMPSs) are extensively used. Basic converters like boost, buck, cuk, etc., are widely used for low power and low output voltage applications. However, for high voltage and/or power applications, the aforementioned basic converters become insufficient in terms of efficiency and reliability [1,2,3]. Recently, interest in high-gain integrated converters has been increasing for applications requiring high output voltages with low DC input voltages. Examples of such applications can be given as Photovoltaic or battery-fed converters, in which the source units are preferably connected in parallel to increase reliability [4,5,6,7,8]. In this context, Z-Source converters provide very high gains in efficient manners especially when used with coupled inductors [9,10,11,12,13,14,15].

In terms of control applications, on the other hand, linear control methods are frequently used to control switching converters. For this reason, a linear model of the converter is required to design a suitable controller. For non-linear systems with explicit non-linear system models, linearization around an operating point, called Jacobian linearization, is the most preferred linearization method, if the system is required to operate around a dedicated operating point. However, switching converters are non-linear systems, which are a combination of sublinear systems with a non-linear switching operation, and are difficult to describe with explicit non-linear system models. For this reason, instead of Jacobian linearization, the state-space averaging method is the commonly preferred approach used to linearize switching converters [16,17]. Additionally, the average PWM switch modeling approach is the other popular method, where average switching converter models are developed using average PWM switch models and then operating points and small-signal dynamic equations are obtained [18,19,20]. While the PWM switch model is preferred to model converters with Discontinuous Conduction Mode (DCM), the state-space averaging method is also preferred to analyze recently developed integrated converters, which are hard to model with PWM switches [21,22,23].

However, as the boost ratio requirements of the converters increase, accordingly developed recent integrated converters can be quite complicated to analyze and model. An example of such a converter is given in [13]. The converter operates in Continuous Conduction Mode (CCM) and is quite advantageous in many respects, as explained in [13]. However, it has seven sub-modes even in CCM operation, and dynamic modeling of the converter by taking into account all sub-modes would be quite challenging and confusing. Fortunately, as described in [13], among the seven sub-modes, two of the seven sub-modes constitute the main conversion action (corresponding to one ON and one OFF state of the switch) and the remaining sub-modes may be ignored to simplify the modeling procedure. However, in this case, because of the eliminated sub-modes, some dynamic variables may lose their continuity and have different average values for different switch states. In such a case, the state-space averaging method will not converge because of the absence of a single operating point of the variables. The state-space averaging method solves such continuity problems by eliminating the corresponding variable from being a state variable as in DCM [16], but, as shown in [19], this reduces the accuracy of the model. On the other hand, because of the mode elimination, the PWM switch can not be equivalently represented. To overcome this problem, a small-signal modeling method, based on volt–charge balance (VCB), is proposed in this paper which was briefly introduced in [24]. Contrary to the average PWM switch method, where the perturbation is applied only to the switches, in this approach, the perturbation inherently exists for all linear dynamic elements of the converter in the form of volt-balance and charge-balance for inductors and capacitors, respectively. Although the VCB method is well-known and widely used in the static analysis of converters, its use in dynamic modeling provides some advantages. First of all, volt/charge balance equations emerge in the form of non-linear explicit functions of state variables where Jacobian linearization can be applied. This eliminates the extra switch modeling procedures and simplifies the modeling procedure. Secondly and more importantly, differently from the state-space averaging method, in the VCB approach, it is possible to assign different average values for dynamic variables for different switch states, as will be seen in the following sections. In this way, the insignificant sub-modes can be ignored and the analysis of the complex, integrated converters can be simplified.

Once the dynamic model of the converter is obtained, control techniques, such as sliding mode control [25], fuzzy logic control [26], and the neural network control method [27], can be used to control DC–DC converters with coupled inductor topologies. Sliding mode control was proposed for the coupled-inductor boost converter by Niliana Carrero [25]. Although this control technique is robust to changes in system parameters and model uncertainties, the selection of a suitable sliding surface may not be an easy task for the calculation. In [26], Sharma presented a fuzzy logic controller for DC–DC converters. The disadvantages of implementing fuzzy logic controllers (FLCs) are related to the complexity of the controller implementation and the high computational complexity. A fuzzy logic-based control system is more computationally demanding compared to conventional linear control systems, which can place a large computational burden on digital signal processors. Besides these non-linear control methods, in [27], a neural network-based control method was presented. However, this method brings again complexity and also optimization problems.

In addition to nonlinear control methods, linear control methods are also widely used to control DC–DC converters. Generally, two common approaches to implementing linear control are voltage mode control and current mode control. Although voltage mode control (VMC) offers simpler implementation, current mode control is more advantageous due to its inherent current sharing and limiting possibility, fast dynamic response, and insensitiveness to input voltage variations. Therefore, current mode control for switching converters is preferable to voltage mode controllers. The current mode control technique primarily consists of two control loops: the first one is the inner current loop, which improves the response to dynamic changes, and the second one is the outer voltage loop, which provides the current reference for the inner loop. Different configurations such as hysteresis current control [28] and Valley mode current control [29] have been proposed in the literature. Although the hysteresis current control algorithm is straightforward to implement and offers good robustness, its major drawback is the instability in switching frequency [28]. Valley mode current control is another option as a current mode control technique. Faizan analyzed the comparison of different current mode control techniques. It is shown that Valley mode control circuitry is quite complex to implement, despite its faster transient response [29].

In addition to the current mode control methods mentioned above, average current mode and peak current mode control are commonly used in applications. Although average current mode control enhances current accuracy, its transient response is slower compared to peak current mode control [30]. Peak current control provides some advantages such as preventing inductor–transformer flux imbalance, providing cycle-by-cycle current limits, and offering short circuit and overcurrent protection [31]. Additionally, according to [31,32], the saturation effect of non-linear inductors can be beneficial in peak current mode control to protect overcurrent and overvoltage protections. Another benefit of saturable inductors is that they reduce inductor size, cost, and weight [33]. Additionally, peak current mode control simplifies the parallel operation of converters and includes feed-forward features that make the output voltage less sensitive to input voltage variations [34]. These benefits of the peak current mode control make it preferable compared to other available current control techniques for the inner loop.

For the outer loop of the controller, numerous compensation techniques have been analyzed. In this context, nonlinear model predictive controllers (MPCs) have been proposed to improve voltage stability [35,36]. However, it requires an accurate dynamical model that is quite difficult in practice, and it requires high computation times that slow down the dynamic performance of the converter under wide variations of the operating regime [37].

A PI controller is a common approach combined with a peak current mode controller. PI controllers for the outer loop with inner peak current mode control loop were presented for buck [38], boost [39], and flyback [40] converters, in the literature. This work will show that the use of a simple PI control in the peak current mode control can regulate the output voltage of the Z-Source converter with a coupled inductor without the need to use more complex methods such as MPC or artificial intelligence-based controllers.

In this paper, a peak current mode control with a PI controller is developed for the Z-Source converter in [13]. The rest of this paper is organized as follows: In Section 2, the transfer function of duty to output voltage is obtained using the proposed volt–charge balance-based small-signal modeling approach. The frequency response of the Z-Source converter is also obtained by using the Frequency Response Estimator tool of MATLAB-Simulink to verify the VCB-based small-signal modeling approach. Then, the transfer function of control current to output voltage is obtained by using the duty to output voltage and duty to control current transfer functions of the developed model. In Section 3, the PI tuning approach is explained for the closed voltage control loop of the given Z-Source converter. The comparisons of simulation and experimental results are given in Section 4. Finally, conclusions are given in Section 5.

2. Volt/Charge Balance Based Dynamic Modelling

The VCB small-signal modeling method can be applied in two steps, as described below:

• From the static analysis of the converter, write down volt–balance and charge–balance expressions for all inductors and capacitors existing in the converter, respectively. As is well known, through the static analysis of these equations the operating points, around which the system is to be linearized, are obtained.
• Using Jacobian Linearization, linearize the volt–charge balance expressions w.r.t. dynamic variables of the converter. Then, obtain dynamic equations describing inductor currents and capacitor voltages in differential form, each of which corresponds to the state equations of the system.

Modeling Z-Source Converter

The circuit diagram of the isolated Z-Source-based high step-up DC–DC converter is shown in Figure 1a. Its properties, operating modes, and detailed analysis are given in [13]. The converter is actually a quasi-Z-Source-based converter that provides continuous input current and reduced switch stress [41]. As explained in [13], the converter has seven sub-modes that can be represented with great accuracy by two dominating main operational modes, corresponding to the ON and OFF modes of the switch S. The remaining sub-modes have very short intervals, which are mainly accompanied by the parasitic components of the switch S, and diodes. In the main ON mode, the switch S and diode D_o are ON, whereas diodes D₁, D₂, and D₃ are OFF. In the main OFF mode, S and D₀ are OFF, and D₁, D₂, and D₃ are ON. Referred to as the secondary side, equivalent circuit diagrams of the converter corresponding to the main ON and OFF modes are shown in Figure 1b and Figure 1c, respectively. Some critical current waveforms of the converter, for the converter parameters given in

Table 1. Z-Source converter parameters.

Parameter	Value	Operating Points
Lin, rin	1.6 mH, 0.2 Ω	D = 0.3
Lm Llke, rlke	2.5 mH 25 µH, 0.2 Ω	Vo = 1000 V
C1	1.1 µF
C2	1.1 µF
C3	1.1 µF
C4	1.1 µF
C0	1.1 µF
n	1.12
k	0.99
Ro	1000 Ω
Vin	350 V
fs	50 kHz

, are shown in Figure 2. As seen, the coupled inductor primary current, corresponding also to the leakage inductor current $i_{lke}$, has sharp transitions during the neglected sub-intervals, and so the corresponding average values in the main ON and OFF periods have different values. Such existence of different average values for a state variable is the main challenge encountered in the state–space averaging method. As will be seen in the next part, the VCB method allows different average value assignments to the variables for the ON and OFF periods.

In the circuit diagram of the Z-Source converter given in Figure 1a,$L_{in}$ is the inductance of the input inductor and $L_m$ is the magnetizing inductance of the coupled inductor. $L_{lke}$ in Figure 1b,c is the total leakage inductance of the coupled inductor according to the simplified transformer equivalent circuit representation. Note that secondary capacitors C₃ and C₄ are connected in series in ON mode and in parallel in OFF mode, and, with equal capacitances, their dynamic characteristics will be equal to each other. Therefore, their common parameters will be denoted by a single parameter with a subscript ‘34’, as in the capacitor currents of Figure 2. In Figure 1b,c, and in the following equations, the parameters are marked with a prime ′, indicate their values referred to the secondary side, and are defined as $C_1^{\prime }=C_1/n^2$, $C_2^{\prime }=C_2/n^2$, $L_m^{\prime }=n^2{L}_m$, $L_{in}^{\prime }=n^2{L}_{in}$, and $L_{lke}^{\prime }=n^2{L}_{lke}.$ Note also that v′ = nv and i′ = i/n for all primary voltages and currents, respectively, where $n=n_2/n_1$.

From Figure 1b,c, the volt–balance equations of the input inductor $L_{in}^{\prime },$ magnetizing inductor $L_m^{\prime }$ and leakage inductor $L_{lke}^{\prime }$ can be expressed as follows:

(1)

(2)

(3)

where, are average voltages of the corresponding inductors, the voltages with uppercase on the right hand side are average voltages of the corresponding capacitors, and D is duty cycle. Operating points (average values) of the capacitor voltages in (1)–(3) can be obtained from static analysis of the converter as follows [16]:

$$V_{C1}^{\prime }=\frac{\left(1-D\right)n{V}_{in}}{\left(1-2D\right)}$$

(4)

$$V_{C2}^{\prime }=\frac{Dn{V}_{in}}{\left(1-2D\right)}$$

(5)

$$V_{C34}=V_{C3}=V_{C4}=k{V}_{C2}^{\prime }$$

(6)

$$V_{C0}=nk{V}_{in}\frac{\left(1+D\right)}{\left(1-2D\right)}$$

(7)

where, $k=\frac{L_m}{L_m+L_{lke}}$ is the coupling coefficient of the coupled inductor. Linearizing (1)–(3) around the operating points by taking $V_{in}$ constant, and rearranging obtained dynamic equations using (4)–(7), dynamic equations of the small signal-inductor currents can be obtained as follows:

$$\frac{d{i}_{in}^{\prime }}{dt}=\frac{-r_{in}}{L_{in}^{\prime }}{i}_{in}^{\prime }+\frac{\left(D-1\right)}{L_{in}^{\prime }}{v}_{C1}^{\prime }+\frac{D}{L_{in}^{\prime }}{v}_{C2}^{\prime }+\frac{\left(V_{C1}^{\prime }+V_{C2}^{\prime }\right)}{L_{in}^{\prime }}d$$

(8)

$$\frac{d{i}_{Lm}^{\prime }}{dt}=\frac{-\left(D+1\right)}{L_m^{\prime }}{v}_{C34}+\frac{D}{L_m^{\prime }}{v}_{C0}+\frac{\left(V_{C0}-V_{C34}\right)}{L_m^{\prime }}d$$

(9)

$$\begin{gathered} \frac{d{i}_{lke}^{\prime }}{dt}=\frac{-r_{lke}^{\prime }}{L_{lke}^{\prime }}{i}_{lke}^{\prime }+\frac{D}{L_{lke}^{\prime }}{v}_{C1}^{\prime }+\frac{\left(D-1\right)}{L_{lke}^{\prime }}{v}_{C2}^{\prime }+\frac{\left(D+1\right)}{L_{lke}^{\prime }}{v}_{C34}-\frac{D}{L_{lke}^{\prime }}{v}_{C0} \\ +\frac{\left(V_{C1}^{\prime }+V_{C2}^{\prime }+V_{C34}-V_{C0}\right)}{L_{lke}^{\prime }}d \end{gathered}$$

(10)

where, $r_{in}$ and $r_{lke}^{\prime }$ are internal resistances of the input inductor and referred coupled inductor, respectively. In (8)–(10), $i_{in}^{\prime }$, $i_{Lm}^{\prime }$, and $i_{lke}^{\prime }$ are small-signal inductor currents, and all lowercase symbols on the right-hand side are small-signal variables of the corresponding parameters. Similarly, from Figure 1b,c, the charge–balance equations of the capacitors $C_1^{\prime }$, $C_2^{\prime }$, C₃₄, and C₀ can be obtained as follows:

(11)

(12)

(13)

(14)

where, $I_{lke-on}$ and $I_{lke-off}$ are average values of the equivalent leakage inductor, $L_{lke}$ currents in ON and OFF states, respectively. As can be seen, in charge balance equations it is possible to assign different average values for different switch states. The operating points of the currents in (11)–(14) can be obtained from a charge–balance analysis of the converter, as follows:

$$I_{in}^{\prime }=\frac{V_{C0}{I}_0}{n{V}_{in}}$$

(15)

$$I_{lke-on}^{\prime }=\frac{\left(1-D\right)}{D}{I}_{in}^{\prime }$$

(16)

$$I_{lke-off}^{\prime }=\frac{D}{\left(1-D\right)}{I}_{in}^{\prime }$$

(17)

$$I_{Lm}^{\prime }=\frac{\left(2-D\right)}{\left(1+D\right)}{I}_{in}^{\prime }$$

(18)

where,

$$I_0=\frac{V_{C0}}{R_0}$$

(19)

Linearizing (11)–(14) around the operating points, dynamic equations of small-signal capacitor voltages can be obtained as follows:

$$\frac{d{v}_{C_1}^{\prime }}{dt}=\frac{\left(1-D\right)}{C_1^{\prime }}{i}_{in}^{\prime }-\frac{D}{C_1^{\prime }}{i}_{lke}-\frac{(I_{lke-on}+I_{in}^{\prime })}{C_1^{\prime }}d$$

(20)

$$\frac{d{v}_{C_2}^{\prime }}{dt}=\frac{-D}{C_2^{\prime }}{i}_{in}^{\prime }+\frac{\left(1-D\right)}{C_2^{\prime }}{i}_{lke}-\frac{\left(I_{lke-off}+I_{in}^{\prime }\right)}{C_2^{\prime }}d$$

(21)

$$\frac{d{v}_{C34}}{dt}=\frac{\left(1+D\right)}{2C_{34}}{i}_{Lm}^{\prime }-\frac{\left(1+D\right)}{2C_{34}}{i}_{lke}+\frac{\left(I_{Lm}^{\prime }+I_{lke-off}-2I_{lke-on}\right)}{2C_{34}}d$$

(22)

$$\frac{d{v}_{C0}}{dt}=\frac{-D}{C_0}{i}_{Lm}^{\prime }+\frac{D}{C_0}{i}_{lke}-\frac{1}{R_o{C}_0}{v}_{C0}+\frac{(I_{lke-on}+I_{Lm}^{\prime })}{C_0}d$$

(23)

where, $C_{34}$=$C_3=C_4$. Finally, combining (8)–(10) and (20)–(23) in state–space form, the following dynamic system equation is obtained:

$$\left\{\begin{array}{c}\dot{x}=Ax+Bd\\ y=Cx\end{array}\right.$$

(24)

where,

$$A=\left[\begin{array}{ccccccc}\frac{{-r}_{in}}{L_{in}^{\prime }}& 0& 0& \frac{(D-1)}{L_{in}^{\prime }}& \frac{D}{L_{in}^{\prime }}& 0& 0\\ 0& 0& 0& 0& 0& \frac{-(D+1)}{L_m^{\prime }}& \frac{D}{L_m^{\prime }}\\ 0& 0& \frac{{-r}_{lke}}{L_{lke}^{\prime }}& \frac{D}{L_{lke}^{\prime }}& \frac{(D-1)}{L_{lke}^{\prime }}& \frac{(D+1)}{L_{lke}^{\prime }}& \frac{-D}{L_{lke}^{\prime }}\\ \frac{\left(1-D\right)}{C_1^{\prime }}& 0& \frac{-D}{C_1^{\prime }}& 0& 0& 0& 0\\ \frac{-D}{C_2^{\prime }}& 0& \frac{\left(1-D\right)}{C_2^{\prime }}& 0& 0& 0& 0\\ 0& \frac{\left(1+D\right)}{2C_{34}}& \frac{-\left(1+D\right)}{2C_{34}}& 0& 0& 0& 0\\ 0& \frac{-D}{C_0}& \frac{D}{C_0}& 0& 0& 0& \frac{-1}{R_o{C}_0}\end{array}\right]$$

(25)

$$\left\{\begin{array}{c}x=\left[\begin{array}{c}\begin{array}{c}{i}_{in}^{\prime }\\ i_{Lm}^{\prime }\\ i_{lke}\\ v_{C_1}^{\prime }\\ v_{C_2}^{\prime }\end{array}\\ \begin{array}{c}{v}_{C34}\\ v_{C0}\end{array}\end{array}\right], y=v_{C0}, C=\left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 1\end{array}\right] \hfill \\ \hfill B=\left[\begin{array}{c}\frac{\left(V_{C1}^{\prime }+V_{C2}^{\prime }\right)}{L_{in}^{\prime }}\\ \frac{\left(V_{C0}-V_{C34}\right)}{L_m^{\prime }}\\ \frac{\left(V_{C1}^{\prime }+V_{C2}^{\prime }+V_{C34}-V_{C0}\right)}{L_{lke}^{\prime }}\\ \frac{-(I_{lke-on}+I_{in}^{\prime })}{C_1^{\prime }}\\ \frac{-\left(I_{lke-off}+I_{in}^{\prime }\right)}{C_2^{\prime }}\\ \frac{\left(I_{Lm}^{\prime }+I_{lke-off}-2I_{lke-on}\right)}{2C_{34}}\\ \frac{(I_{lke-on}+I_{Lm}^{\prime })}{C_0}\end{array}\right]\hfill \end{array}\right.$$

(26)

To verify the model, a simulation was performed with the Z-Source converter parameters given in Table 1. In the simulations, the dynamic response of the converter was obtained using the Frequency Response Estimation tool of MATLAB/Simulink. In this context, a sequence of sinus signals with 0.01 V amplitude and with 30 different frequencies in the frequency range of 10 to 10⁵ rad/s was applied in MATLAB as the input perturbation for the control signal, D. The comparison of the frequency responses of the mathematical model (24) and simulation model are shown in Figure 3.

The main mismatches between the model and simulation results are especially at the resonant frequency points, where the model has quite less damping. This result is expected, considering that the primary inductor current has a rapid state change in the eliminated sub-modes, as seen in Figure 2. Mode elimination in the model results in less damped response in the system and as a result, lack of damping is more apparent in the model. Of course, more accurate results could be obtained without mode elimination but with much more burden. On the other hand, the state-space averaging method could also be used by eliminating the leakage inductor current, which has multiple operating points, from state variables. However, in this case, the results would be more inconsistent than in the VCB case. However, the model and simulation results have similar average traces, and it is considered that the model can be used in designing a suitable controller for the Z-Source converter.

3. Closed Loop Control of the System

In the previous section, a dynamic model of the Z-Source converter was obtained in state–space form. The duty-to-output transfer function of the Z-Source converter can be derived from $G_{vd}\left(s\right)=\frac{V_o\left(s\right)}{d\left(s\right)}=C{\left(sI-A\right)}^{-1}B$. However, the peak current mode controlled DC–DC converter requires control current to output transfer function $G_{iv}\left(s\right)=\frac{V_o\left(s\right)}{i_L\left(s\right)}$ for tuning PI controller parameters. A block diagram of the closed-loop peak current mode controller is shown in Figure 4. It is divided into three parts: Z-Source converter (Power Level), peak current control stage (Inner Control Loop), and voltage control stage (Outer Control Loop).

$G_{iv}\left(s\right)$ can be obtained from the duty to output transfer function $G_{vd}\left(s\right)=\frac{V_o\left(s\right)}{d\left(s\right)}$ and the duty to control current $G_{id}\left(s\right)=\frac{i_L\left(s\right)}{d\left(s\right)}$ transfer functions. To determine $G_{id}\left(s\right)$, state equations in (24) are converted into transfer function form as the output as being control current $i_L\left(s\right)$ instead of output voltage $V_o\left(s\right)$. To obtain $G_{id}\left(s\right)=\frac{i_L\left(s\right)}{d\left(s\right)}=C{\left(sI-A\right)}^{-1}B$, C matrix given in (26) must be changed as in (27) below.

$$C=\left[\begin{array}{cc}\begin{array}{ccc}1& 0& 0\end{array}& \begin{array}{cccc}0& 0& 0& 0\end{array}\end{array}\right]$$

(27)

Multiplying the duty to output $\frac{V_o\left(s\right)}{d\left(s\right)}$ transfer function and the inverse of the duty to control current $\frac{i_L\left(s\right)}{d\left(s\right)}$ transfer function, $\frac{V_o\left(s\right)}{i_L\left(s\right)}$ can be defined as in the following expression. Note that it contains one pole less than $\frac{V_o\left(s\right)}{d\left(s\right)}$.

$$G_{vi}=\frac{V_o\left(s\right)}{i_L\left(s\right)}=\frac{1.173\times {10}^7{s}^6+4.759\times {10}^{11}{s}^5+5.387\times {10}^{17}{s}^4+6.044\times {10}^{20}{s}^3+2.025\times {10}^{26}{s}^2-2.979\times {10}^{29}s+8.728\times {10}^{33}}{4.922\times {10}^5{s}^6+7.61\times {10}^9{s}^5+3.05\times {10}^{16}{s}^4+1.977\times {10}^{20}{s}^3+8.205\times {10}^{24}{s}^2+4.814\times {10}^{28}s+4.116\times {10}^{31}}$$

(28)

PI controller parameters can be designed according to Bode diagrams obtained from $\frac{V_o\left(s\right)}{i_L\left(s\right)}$. The transfer function of a PI controller is expressed as follows [42]:

$$C\left(s\right)=K_p+\frac{K_i}{s}=K_p\left(1+\frac{1}{T_{i}s}\right)=\frac{K_p\left(T_{i}s+1\right)}{T_{i}s}=K_i\left(\frac{T_{i}s+1}{s}\right)$$

(29)

where, $K_i$ and $K_P$ are the integral and proportional gain, respectively, and $T_i$ is the integration time. Bode diagram of $\frac{V_o\left(s\right)}{i_L\left(s\right)}$ is shown in Figure 5. At the selected crossover frequency (${\omega }_G=2560 rad/s$) phase delay of the converter is ${\theta }_G=-{101}^{°}$. Therefore, PI controller should have a phase $\varphi =-{21.7}^{°}$ at ${\omega }_G$ to ensure a phase margin of $\gamma ={57}^{°}$. Then, accordingly, the $T_i$ parameter is calculated as follows:

$$\angle C\left(s\right){|}_{\omega ={\omega }_G}=\varphi =\mathit{arctan}\left({\omega }_G{T}_i\right)-{90}^{°}$$

(30)

$$T_i=\frac{tan\left(\varphi +90°\right)}{{\omega }_G}$$

(31)

By using Equations (30) and (31), $T_i$ can be calculated as $9.82\ast {10}^{-4}$. Subsequently, in order to calculate $K_i$, it is necessary to plot a Bode diagram of $C^{\prime }\left(s\right)G_{iv}\left(s\right)$:

$$C^{\prime }\left(s\right)=\left(\frac{T_{i}s+1}{s}\right)$$

(32)

A Bode diagram of $C^{\prime }\left(s\right)G_{iv}\left(s\right)$ is shown in Figure 6. It presents a gain which equals −23.7 dB at the crossover frequency. The controller should provide −20 dB slope/decade given by 23.7 dB at the crossover frequency. Then, accordingly, K_i is calculated as follows:

$$K_i={10}^{\left(23.7/20\right)}=15.031$$

(33)

Finally, the PI controller is given as follows:

$$C\left(s\right)=15.031\left(\frac{9.82\times {10}^{-4}s+1}{s}\right)$$

(34)

From expression (34), $K_i$ and $K_P$ are calculated as 15.031 and 0.015, respectively. A Bode plot of the compensated system is shown in Figure 7. The controller meets the design specifications such that the loop transfer function has unity gain with 57° PM at the cross-over frequency.

4. Results

For testing the output voltage regulation of the controller, two selected cases were implemented in PLECS. Firstly, load ($R_0$) was changed from 1000 Ω to 2000 Ω and then from 2000 Ω to 1000 Ω. Figure 8 shows the responses of output voltages and currents of the designed Z-Source converter with corresponding load changes. It can be seen from these figures that the overshoot of the output voltage is about 7% and the convergence time that the output voltage is settled within 7% is 0.01 s. For the second case, the input voltage is changed from 350 V to 300 V and from 300 V to 400 V, respectively. The results are shown in Figure 9. As expected from current mode control, input voltage variations are not reflected in the output.

To validate the simulation results, an experimental study of the Z-Source converter, with the parameters given in Table 1, was designed and controlled in a closed loop with peak current mode control. A picture of the experimental setup is given in Figure 10. The equipment used in this work is listed in

Table 2. Equipment list.

Components Used	Components Parameters
MOSFET	G3R40MT12J, 1200 V, 47 A, RDSON: 40 mohm
Diodes	IDM05G120C5, 1200 V, 5 A. Schottky
Capacitors	C4AQPBU, 1200 V, 1.1 μF
Input Inductor Core	79726A7, AL = 175, Kool Mu Max toroid
Coupled Inductor Core	79617A7, AL = 189, Kool Mu Max toroid
Microcontroller	STM32G474RE MCU
MOSFET Driver	IED3431MC12M

. Note that Silicon Carbide Mosfet and diodes were used in this work.

Firstly, the experiment was conducted to check the output voltage response of the controller during the load variations. By keeping the input voltage constant, the controller’s performance is examined by varying the output loads from 1000 ohms to 2000 ohms, and then from 2000 ohms to 1000 ohms, as illustrated in Figure 11a,b. An electronic switch was used to adjust the load between the half and full load values. According to Figure 11a,b, it can be observed that in both cases the output voltage variations are consistent with the simulation results. The output voltage makes an 8.5% over/undershoot and returns to the set value in 1 millisecond. It can be concluded that the designed controller performs well and regulates the output voltage successfully in case of load variations.

A second experiment was conducted to check controller performance during the input voltage variations. For this reason, firstly by keeping the load at a nominal value (1000 ohm), the input voltage is reduced from 350 V to 300 V and the output voltage is observed. The results are shown in Figure 12a. Then input voltage was increased from 300 V to 400 V while keeping the load constant and the output voltage was observed as shown in Figure 12b. As seen in both cases, the output voltage is insensitive to the input voltage variations, as in the simulations. This is an expected property of the current mode control where the inner current control loop responds immediately to the input voltage variations preventing reflections of input voltage variations to the output. The general results of the experimental study are consistent with the simulation results in the case of the input voltage variations, as well.

5. Conclusions

In this paper, a PI controller for the outer loop of the Peak Current Mode Control was designed and implemented for a high-gain Z-Source converter. Firstly, a volt–charge balance-based small-signal modeling method for the Z-Source converter has been proposed. The required transfer functions are obtained from the developed state–space model of the converter. Although some deviations were observed between the Z-Source converter model and simulation results due to the elimination of sub-modes, the model was accurate enough for the design of the controller. Then, PI control parameters are determined by using Bode diagrams derived from the input current to output voltage transfer function. Finally, simulation results are compared with experimental studies to verify the design. As a result, the Peak Current Mode with PI controller implementation shows that it is sufficient and reliable for the new generation Z-Source DC–DC converter.