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60 DESIGN007 MAGAZINE I NOVEMBER 2019 The Fourier Theorem states that every func- tion can be completely expressed as the sum of sine and cosine waves of various amplitudes and frequencies. The Fourier series expansion of a square wave is made up of a sum of har- monics. If the waveform has an even mark to space ratio, then the even harmonics cancel. Also, as the frequency increases, the amplitude decreases. A square wave can be expressed as Equa- tion 1: Equation 1: Any waveform in the time domain can be completely and uniquely described by a com- bination of sine waves. If we switch to the fre- quency domain and use sine wave descrip- tions, we can solve problems faster than in the time domain. Impedance is defined in both the time and frequency domains. However, it is far easier to understand and apply the concepts of AC im- pedance in the frequency domain. Decoupling capacitors are spread throughout the PDN. Thinking in the time domain, you can say that decoupling capacitors store and supply charge on-demand to the loads. However, in the fre- quency domain, decoupling capacitors also lowers the impedance at different frequencies to help to meet the AC impedance target. So, there are two distinct functions of capacitors that work in unison but in different domains. Further, when dealing with electromagnetic compatibility (EMC) issues, both FCC/CISPR specifications and methods of measuring the emissions of the product are more readily per- formed in the frequency domain. The Fourier series expansion of a square wave is made up of a sum of harmonics. Fig- ure 3 shows the conversion of a square wave from the time to the frequency domain and the resultant amplitudes of the frequency compo- nents. If the waveform has an even mark to space ratio, then the even harmonics cancel. The Zeroth harmonic is the DC value, and the fundamental (first harmonic) frequency is the largest in amplitude tapering off as the odd harmonic frequency increases. If the AC wave- form is offset (from zero), then the DC voltage component will appear at DC (0 Hz). The high-frequency content of a square wave is significantly affected by the rise time of the wave- form. A fast rise time results in higher frequen- cy components. Also, as the frequency increases, the amplitude decreases. In the real world, one needs to consider the maximum bandwidth of a signal, including harmonics, rather than assume the perfect square wave fundament frequency model. For example, a 200-MHz clock may have harmonics up to the fifth, meaning we need to consider the bandwidth up to 1 GHz. Technology moves fast, and much has changed over the years since I have been in high-speed multilayer design, particularly ad- vances in lithography that enable IC manufac- turers to ship smaller and smaller dies on chips. In 1987 we thought that 0.5-micron technolo- gy was the ultimate, but today, 5-nm technol- ogy is at the cutting-edge. As of 2019, Samsung and TSMC have begun commercial production of 5-nm nodes. Figure 3: Harmonics of a square wave transformed from the time to the frequency domain.

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