## 2 The Spectrum Signature of SMPSs
### Spectral Superposition Characteristics of Diode-Based Parallel Redundant SMPS System  
The diode-based simple parallel redundant SMPS system utilizes the unidirectional conduction characteristic of diodes, which also make the output power of the branch power supply closely related to the output voltage of all branch power supplies. Since the power supply is affected by factors such as the use environment, working conditions, and component ageing, the output voltage of SMPSs will change slowly over time. The variation of the output voltage of the branch power supply leads to the current uneven state of redundant power supply systems. For an ideal SMPS, the output voltage is constant, and there is no ripple, even if the voltage difference of mv level will cause a severe uneven current state of the system. For actual SMPSs, this situation is improved due to ripple. For any SMPS, its output voltage signal can be expressed as:  
\[v(t)=A_{0}+\sum_{i=1}^{N}A_{i}\sin(w_{i}t+\varphi_{i}(t)) \tag{3}\]  
where \(v(t)\) is the output voltage, \(A_{0}\) is the DC component of the output voltage of the SMPS, and \(A_{i}\sin(w_{i}t+\varphi_{i}(t))\) is the \(i\)-th ripple component. Due to the existence of the ripple component, the output voltage of the power supply actually changes within the range of \([A_{0}+A_{min}(t),A_{0}+A_{max}(t)]\), where \(A_{min}(t)\) and \(A_{max}\) (\(t\)) are the synthesized ripple components Minimum and maximum values. When the variation range of the output voltage of each branch power supply has overlapping areas, the output voltage of the redundant system is related to these branches, and the output power of the outgoing power supply is also related to the degree of overlap between the sections. In order to simplify the analysis process, the dual power supply redundancy system is taken as an example. Assuming that there is only one sinusoidal ripple component whose initial phase  
Figure 3: (**a**) Voltage waves in a diode-based simple parallel redundant system with two SMPSs; (**b**) The spectrum of three voltage waves. is 0 in the output voltage of each branch power supply, the output voltages of branch 1 and branch 2 can be expressed as:  
\[v_{1}(t)=A_{01}+A_{11}\sin(w_{11}t+\varphi_{12}(t)) \tag{4}\]  
\[v_{2}(t)=A_{02}+A_{12}\sin(w_{12}t+\varphi_{12}(t)) \tag{5}\]  
Then the voltage variation range of branch 1 is \(|v_{1}(t)|=\left[A_{01}-A_{11},A_{01}+A_{11}\right],\) and the voltage variation range of branch 2 is \(|v_{1}(t)|=\left[A_{02}-A_{12},A_{02}+A_{12}\right].\) If \(|v_{1}(t)|\cap|v_{2}(t)|\
eq\varnothing\), the output voltage of the redundant power supply is determined by the maximum value of the instantaneous voltage of branch 1 and outgoing 2 voltages at time \(t\), which can be Expressed as:  
\[v_{o}(t)=max\{v_{1}(t),v_{2}(t))\} \tag{6}\]  
Since the output DC component \(A_{0i}\) of each branch power supply, the initial phase of the ripple \(\varphi_{i}(t)\) and the ripple frequency \(w_{1i}\) are different, the output voltage of the redundant power supply is formed by the competition between the two outgoing paths in the overlapping interval. Figure 4 shows that when the power outputs of branches 1 and 2 are \(v_{1}(t)=24+0.15\sin(243*2\pi t)\) and \(v_{2}(t)=23.9+0.15\sin(200*2\pi t)\), respectively, the double Simulation results of the output voltage of a parallel redundant system of power supplies. As can be seen from the figure, the output voltage of a redundant system consists of voltages of branch 1 and branch 2. Since the voltage of branch 2 is relatively small, it accounts for a relatively small proportion of the overall output signal. The above signal is subjected to spectrum analysis, and the result is shown in Figure 5. As can be seen from the figure, the frequency spectrum of the output voltage consists of two frequency points, branch 1 and branch 2, and the amplitude of the ripple frequency corresponding to branch 2 is quite different from that of branch 1, which is different from that in the time domain. The signal ratio is the same.