# How Simulation Tools Perform Linear Network Extraction

Linear networks are a central part of electronics simulations involving interconnects, power delivery networks, and components in the same system. When a designer wants to simulate the behavior of an entire system, they need to have a mathematical description of each element in their system. When we connect these elements together to match a system topology, we have just created a linear network.

This article will outline what is a linear network, and some brief points regarding how the elements in a linear network are constructed. You can start from either measurements or field solver results, and in either case an appropriate simulation tool can build a model from your data for use in simulations.

## What is a Linear Network?

A linear network contains elements and devices used in a system-level simulation. For example, a linear network could represent a simple circuit, transmission line, vias in a PCB, filter, modulator, connector, or any other electrical element that would operate as a linear circuit.

Linear networks are generally drawn out as two-port block diagrams, as shown below. We input a voltage and current, and the internal components of the linear network transform these to an output voltage and current. The network has a set of network parameters (S, H, Y, etc.) that describe how the network takes the input values and transforms them to outputs. Each element has also its own set of network parameters or circuit parameters.

*A diagram showing a linear network with 3 elements.*

Linear networks operate in a similar way as components in a SPICE simulation, but they may use a different mathematical formulation to predict the electrical behavior of a system. To model the input and output signals in a linear network, a set of network parameters is needed. The three most common sets of network parameters used in linear network simulations are S-parameters, ABCD parameters, or H-parameters. For certain structures in a package or a PCB (namely the PDN), Z-parameters might be preferred. Transfer functions can also be used to model linear networks.

*Each element in a linear network has its own set of network parameters.*

If you have a component in your PCB, such as a transmission line or connector, how can you get its network parameters to use in a simulation? This is where measured data, simulation data, or both can be used with sophisticated algorithms to determine the network parameters for an element in a system-level simulation.

### Network Parameter Extraction Methods

A device under test (DUT), interconnect under test, or circuit under test can be used in a system-level simulation only after its network parameters are determined. If you want your device/circuit under test to be used in SPICE simulations, then it’s common to go a step further and extract distributed circuit parameters from the network parameters.

With direct measurement, network parameters (such as S-parameters) can generally be determined after de-embedding from a set of test structures, connectors, etc. From simulation data, some more sophisticated methods are used to determine network parameters, such as:

- Genetic algorithms
- Nonlinear gradient methods
- Regression to power series models
- AI-guided model extraction
- Lp-norm minimization

The operations research experts reading this will notice that these are all essentially generalized optimization methods based on random or guided search. This is because network parameter extraction from simulation data can be thought of as an error minimization technique. Once network parameters or equivalent circuit parameters are determined, the model for a device/circuit under test can be used in a system-level simulation.

### What Happens in System-Level Simulation

Once the network parameters for a specific element in a system are determined, that element can then be used in a system-level simulation with other elements that will appear in a system. SPICE is common for system-level simulations, including 2-port networks; the standard SPICE solution process can be used to predict system behavior in the time or frequency domain for many systems.

Unfortunately, SPICE simulations take very long to converge to useful results when a system becomes very large, or subcircuits that are used to represent elements become very complex. This is why network parameter simulations are preferred for higher-level systems that are more complex. Simulation with network parameters is simple matrix multiplication in the frequency domain. As long as the network parameters for each element in your system are known or can be determined individually from simulation/measurement, then you can use these in system-level simulations.

Once you have extracted your linear network models and you’re ready for high-level simulations, use the complete set of system analysis tools from Cadence to evaluate systems functionality. Only Cadence offers a comprehensive set of circuit, IC, and PCB design tools for any application and any level of complexity. Cadence PCB design products also integrate with a multiphysics field solver for thermal analysis, including verification of thermally sensitive chip and package designs.

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