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Linear Network Analysis

Linear network analysis is a tool for determining the reaction of networks to changes in the inputs. While most commonly applied to electrical power, data and communication networks, linear network analysis is useful for examining any interconnected entity that has inputs and outputs. When behavior is not linear, networks react in an almost linear fashion over particular time frames. Linear network analysis can provide good approximations over these time frames, and can analyze almost any network.

Basics
- The simplest network has one input, one function and one output. For a linear network, the function that changes the input to the output is linear. A linear function changes the output by numerical factors rather than by a complex expression. Linear network analysis specifies the initial conditions as initial inputs and examines how the network reacts. It changes the input conditions to study the network stability. For linear networks, the equations governing these changes are simpler than the corresponding equations for non-linear systems. Equations for non-linear networks often can't be solved.
Initial Conditions
- A network analysis starts with specifying initial conditions. These can be conditions that prevail as the analysis starts, when the network starts operating, or at some arbitrary time. Ideally, the initial conditions are simple ones. Analysts often make all initial conditions zero to start, and then look for the network outputs. This special case puts the network in its zero-state condition and is a good starting point for further analysis.
Stability
- A key network characteristic is stability. Network analysis determines what inputs result in stable operation and may be permitted. If an input drives an output beyond design limits, the network is not operating in a stable mode, and the input in question can't be permitted as part of normal operation. Typical inputs for linear network analysis are step functions, ramp functions and periodic functions. A step function is an input that is suddenly increased by a set amount. A ramp function is an input that increases steadily, and a periodic function is an input that cycles, as in a sine wave. If these inputs result in stable operation, linear network analysis studies the outputs to determine how the network is functioning.
Applications
- Most real-life networks behave in a non-linear fashion over much of the operating range. The challenge for linear network analysis is to identify the operating ranges that approximate linear behavior closely enough to allow the calculation of useful application parameters. If networks must operate in ranges that exhibit non-linear characteristics, linear analysis can yield results if the network behaves according to different linear approximations on a piece-wise basis. Using such a piece-wise approach, linear network analysis can be applied to complex networks.

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