## 2 The Spectrum Signature of SMPSs  
### Spectral Characteristics of Single SMPS  
A typical AC/DC SMPS topology is shown in Figure 1, which is mainly composed of two parts: power factor correction circuits (PFC) and DC/DC modules. The electronic switch is a critical component in both of them. Therefore, the impulse generated by switching on and off is one of a power supply's primary sources of output ripple [15]. In the PFC part, the duty cycle of the control PWM signal changes within the same AC signal cycle, while in the DC/DC part, the duty cycle of the control system signal is constant. Since they are all periodic square wave signals, the spectral distribution is discrete at any static power, and there are only odd harmonics [12; 15]. According to the current continuity in inductor coils, the operating mode of a SMPS can be divided into two basic modes: continuous current mode (CCM) and discontinuous current mode (DCM) [12]. The advantage of CCM mode is that it is suitable for high current loads, and the output ripple is small, while the disadvantage is that the conversion efficiency is low. The DCM mode has relatively higher conversion efficiency, but the output ripple is large and is suitable for application scenarios with small current loads. In order to adapt to these two load states, the mainstream SMPS will automatically switch between two modes according to the load conditions to achieve the best balance state at each load power point. The switching frequency can theoretically be fixed in either mode, and the output power can be adjusted only by adjusting the PWM pulse width. However, this method will cause the power supply to send a large amount of electromagnetic radiation to  
Figure 1: The topology of a typical SMPS. the outside near the switching frequency point, causing serious EMI problems. Therefore, most SMPSs usually adopt the spread spectrum method. By changing the PWM signal's frequency, the switching signal's interference ripple is dispersed into a boarder spectrum area, reducing the average electromagnetic radiation energy at the frequency point of the switching signal [16]. In unit time, the actual on-time of the PWM signal with frequency f can be expressed as:  
\[t_{on}=\frac{1}{T_{f}}\Big{(}T_{f}*R_{duty}-T_{rise}-T_{fall}\Big{)}=\Big{(}R_{ duty}-\Big{(}T_{rise}+T_{fall}\Big{)}f\Big{)} \tag{1}\]  
where \(T_{f}\) is the period of the signal, \(T_{rise}\) and \(T_{fall}\) are the turn-on rise and fall times, respectively, and \(R_{duty}\) is the duty cycle. It can be seen from the above formula that when the duty cycle is constant, with the increase of the frequency \(f\), the actual on-time decreases, and the output power is also smaller. On the contrary, the smaller the frequency, the larger the output power. At the same time, each time the switch is switched, there will be a short-term pulse impact, so the higher the pulse frequency, the more times the switch is switched, and the greater the accumulated interference energy. Therefore, in the output ripple, the spectral change around the switching frequency is shown in Figure 2.  
As can be seen from the figure, the switching signal frequency and its amplitude in the output ripple signal are negatively correlated with power. Since the transmission line seriously affects the ripple amplitude in the redundant power supply circuit system, the frequency can remain unchanged. The following relationship can express the relationship between the frequency and the output power. \[P(t)=F(f(t)) \tag{2}\]  
where \(f(t)\) represents the frequency change of the ripple signal, \(P(t)\) represents the output power change corresponding to the power supply, and F represents the functional relationship between the output power and the ripple amplitude and frequency. On the other hand, because the power switch, input filter capacitor, inductor and other devices have subtle parameter differences in each power supply, the pulse interference signal generated at the moment when the switch is turned on and off will also be different.