## 2 Converter Modulation  
The converter Pulse Width Modulation (PWM) is categorized in most cases by a constant parameters like constant duty cycle and switching frequency, which leads to specific narrowband EMI at specific given frequency as shown in Figure 1. However, in the case of the spread spectrum (SS) modulation, one or more parameters could be changedin order to provide a mitigation in the generated EMI by distributing the power of the signal over a wide range of frequencies. Based on the literature [11], the SS-modulation techniques could be divided into four main types:  
* Random Carrier Frequency Modulation with Constant Duty cycle (RCFMFD) [21; 22; 23; 24; 25; 26].
* Random Carrier Frequency Modulation with Variable Duty cycle (RCFMVD) [18].
* Randomized Pulse Width Modulation (RPWM) [20].
* Randomized Pulse Position Modulation (RPPM) [16].  
In this paper, the RCFMFD is chosen as the most common SS-modulation technique used in a lot of applications. This technique works by changing the switching frequency within a certain bandwidth, following a certain pattern as shown later in this paper. The carrier frequency \(f_{c}\) varies for every nth impulse according to a modulating signal \(\varepsilon(t)\). The signal of the carrier signal could be represented in the time domain as [13]  
\[S(t)=A_{c}\cos(2\pi f_{c}t+2\pi\Delta f\int_{-\infty}^{t}\varepsilon(t)dt) \tag{1}\]  
\[\Delta f=\alpha f_{c} \tag{2}\]  
where \(A_{c}\) is the amplitude of the carrier frequency, \(\Delta f\) is the frequency deviation around the main switching frequency, \(\alpha\) is the spreading factor used to set the required frequency bandwidth, i.e., the Carson bandwidth of the switching signal \([f_{c}-\Delta f/2,f_{c}+\Delta f/2]\) as shown in Figure 1. The increase in the \(\alpha\) value increases signal bandwidth and decreases its spectral amplitude. The SSM signal can be expressed in terms of its Fourier series as  
\[S_{ss}(t)=\sum_{k=-\infty}^{\infty}A_{k}e^{i(2\pi f_{c}t+\beta 2\pi k\Delta f\int_{-\infty}^{t}\varepsilon(t)dt)}, \tag{3}\]  
Consequently, the power spectral density of the EMI of the signal \(S_{ss}(t)\) will be expressed as [14]  
\[S_{ss}(f)=|H(f)|^{2}\sum_{k=-\infty}^{\infty}|A_{k}|^{2}\gamma \bigg{(}\frac{f-kf_{c}}{k\Delta f}\bigg{)}, \tag{4}\]  
where \(H(f)\) is the transfer function of the EMI coupling, and \(\gamma(x)\) is the amplitude probability density function of the spread function. The power spectral density of the EMI in Equation (5) is limited by the EMI standards for CISPRA Range which is between 9 KHz and 150 KHz.  
Figure 1: Conventional (narrowband noise vs. spread spectrum modulation (wideband noise).  
In the same context, as \(\varepsilon(t)\) is the modulating signal, \(\varepsilon(t)\) is changing by a certain sampling frequency \(f_{s}\), which sets the instantaneous frequency of the PWM signal. The function of the modulating signal \(\varepsilon(t)\) could be sinusoidal, triangular, or a RandomPAM signal [13]. Thus, the modulating signal profile is formed by a punch of points that form the required shape, which form the shape with a certain modulating signal frequency \(f_{m}\). The shape of the spectrum could be controlled in terms of the modulation index \(\beta\), controlled by both \(\Delta f\) and \(f_{m}\) and equal to  
\[\beta=\frac{\Delta f}{f_{m}}, \tag{5}\]  
The resulting spectrum varies according to the modulating signal pattern is shown in Figure 2.