 # Basic Concepts for Semiconductor Power Delivery

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## 1.9 PDN Modeling Methodology

The design process that captures the modeling methodology described in this book is shown in Figure 1-50. The entire methodology is centered around plane modeling because as packages and boards migrate toward gigahertz frequencies, planes play a very important role by reducing inductance and supporting the return current of signal lines. Moreover, since the planes are electrically large along the lateral dimensions, an efficient numerical method is required from the modeling standpoint that reduces complexity (number of unknowns). So, in the methodology shown in Figure 1-50, the PDN consisting of planes, vias, capacitors, and solder balls is first modeled in the frequency domain. The goal of this analysis is to ensure that the target impedance of the PDN is met at the frequencies of interest, as dictated by the system application. The modeling of planes with vias and decoupling capacitors in the frequency domain is the subject of Chapter 2.

Assuming the target impedance is met in the frequency domain, the next step is to incorporate the signal lines along with the PDN. Doing so allows a designer to evaluate the coupling between the signal lines through the PDN and between the signal lines and PDN due to the return currents. Details on the incorporation of the signal lines into the PDN and coupling analysis in the frequency domain are found in Chapter 3.

Assuming the coupling meets the specifications in the frequency domain, the next step is to convert the frequency information into a time-domain signal to compute SSN. Assuming no nonlinear circuits are connected to the signal lines, this process is straightforward, since inverse fast Fourier transform (IFFT) methods can be used for this purpose. However, this is not a true representation of SSN, since nonlinear circuits can have memory and feedback effect whereby excessive SSN can slow down the drivers, reducing or saturating the power supply noise . The interaction between the nonlinear drivers, signal lines, and PDN can only be captured through a time-domain simulation. Chapter 4 describes time-domain simulation methods that convert the frequency-domain response of the signal lines and PDN into a modified nodal analysis (MNA) formulation, which is used by most circuit simulators such as Spice. The frequency response can therefore be converted into a Spice subcircuit to which nonlinear transistor circuits can be connected. During time-domain simulation, various PRBS patterns can be simulated to ensure that SSN is within specifications. If the specifications are violated for any particular bit stream, the impedance of the PDN or the routing of the transmission lines can be re-optimized around the frequencies corresponding to the bit stream before final tape out.

In Chapter 5, various applications are discussed using methods described in chapters 2, 3, and 4 to analyze impedances and scattering parameters in the frequency domain and SSN in the time domain. Advanced technologies such as embedded decoupling capacitors and electromagnetic bandgap (EBG) structures for controlling SSN are also discussed in Chapter 5. In addition, more focused examples are provided in each chapter to capture specific effects that can be used to evaluate commercial tools.