# A Fast, Decentralized, Self-Aligned Carrier Method for Multicellular Converters +
## 2 Conventional Phase-Shifted Carrier Generation Method  
Figure 1(a) shows the conventional phase-shifted carrier (CPSC) generation scheme based on chaining cells of a multiphase converter (parallel structure). Each cell communicates with its two neighboring cells to exchange the information of carrier angles. Then, the carrier angle of each cell is self-corrected so that it can converge toward the expected interleaved state (Figure 1(b)). The basic updating rules are executed by using the Formula (1) [23]:  
\[\theta_{n}^{k+1}=mod(\theta_{n+1}^{k}+\frac{1}{2}mod(\theta_{n-1}^{k}-\theta_ {n+1}^{k},360),360) \tag{1}\]  
where \(\theta_{n}^{k+1}\): the phase-shifted angle of the \(n\)th carrier at step \([k+1]\).  
Figure 2: _Cont._  
However, Ref [10] shows that, in the case of using digital implementation for (1), the system can be unstable in certain cases. To solve this issue, a modified rule is proposed as follows [10]:  
\[\left\{\begin{array}{c}\theta_{n}^{k+1}=\theta_{n}^{k}+K(\widehat{\theta}_{n}^ {k+1}-\theta_{n}^{k}),with\;K\in[0\ldots 1]\\ \widehat{\theta}_{n}^{k+1}=mod(\theta_{n+1}^{k}+0.5 mod(\theta_{n-1}^{k}-\theta _{n+1}^{k},360),360))\end{array}\right. \tag{2}\]  
\(\widehat{\theta}_{n}^{k+1}\): ideal corrected angle; \(K\): attenuation factor.  
It should be noted that (Equation (1)) is a special case of (2) when \(K\) is equal to unity. However, as shown in [10], \(K\) closer to unity results in higher instability issues.  
The restrictions of using the above updating rules are: (1) the system must have at least three carriers (\(N\geq 3\)), and (2) at start up, one carrier must have a different angle from the others to ensure the proper operation of the algorithm. If the system is converged, the inter-carrier phase angle reaches the desired value after a number of iterations. In the case  
Figure 2: Conventional decentralized method for multiphase converters, (**a**) intercellular communication; (**b**) basic iterative updating rule principle; (**c**) multiple reconfiguration.  
of the basic updating rule in (1), the number of iterations to achieve the convergence is given by the equation below [29]:  
\[I_{configuration}=\left\{\begin{array}{rl}&\frac{N}{2}+\left(\frac{N}{2}\right)^{2} \;\;\;\mbox{when}\;N\;\mbox{is}\;\mbox{even}\\ &\frac{N+1}{2}+\left(\frac{N+1}{2}\right)^{2}\;\mbox{when}\;N\;\mbox{is}\; \mbox{odd}\end{array}\right. \tag{3}\]  
where \(N\) is the number of active cells.  
If the modified updating rule (2) is used with the same number of \(N\), the number of iterations needed for convergence is higher than that of the basic updating rule. Besides, the lower the value of \(K\) is, the higher the number of iterations is required to converge to the desired value of \(\theta_{n}\). Further, in digital control with a digital signal processor (DSP), the system operates in discrete time where each iteration is completed in one time step (or sampling time). Thus, the configuration time is also the number of time steps needed from startup to achieve the convergence of expected interleaved carriers with an acceptable error of inter-carrier phase-shifts.  
In case of reconfiguration (Figure 2c), it is then straightforward to implement a bypass circuit into the deactivated cell in order to maintain a closed-loop chain of intercellular communications.  
Figure 3 shows the simulation results of interleaving process of a six-carrier system using the CPSC method. It should be noted that, in case the converter has a large number of cells, the process requires an increased amount of iterations, and consequently the time of convergence to reach the steady-state increases.