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{{Main|Image impedance}}
Transmission lines and certain types of filter design use the image method to determine their transfer parameters. In this method, the behaviour of an infinitely long cascade connected chain of identical networks is considered. The input and output impedances and the forward and reverse transmission functions are then calculated for this infinitely long chain. Although the theoretical values so obtained can never be exactly realised in practice, in many cases they serve as a very good approximation for the behaviour of a finite chain as long as it is not too short.
== Time-based network analysis ==
Most analysis methods will solve static networks, which are circuits consisting of memoryless components only, independent of time. That is, the calculated voltages across and the currents through all network components are valid at all times.
Simple dynamic networks that have capacitors and inductors, can be solved with the [[Laplace transform]]. To do this, the [[Network analysis (electrical circuits)#Transfer function|transfer function]] of the network is found and multiplied with the Laplace ___domain representation of the input signal to find the Laplace ___domain representation of the output signal. The output signal in terms of time can be found by using the inverse Laplace transform.
=== Circuit simulation ===
{{main|Electronic circuit simulation}}
Laplace transform-based methods do not work for more complicated circuits or arbitrarily defined input signals. In these cases, numerical methods can be used to calculate the voltage and current values. Even though these methods can also be calculated by hand, they are mostly performed by computers, since it is usually necessary to do more than hundreds (in many cases much more) of calculations.
In general, the equations that describe the behaviour of a dynamic circuit are in the form of a [[differential-algebraic system of equations]] (DAEs). DAEs are challenging to solve and the methods for doing so are not yet fully understood and developed (as of 2010). Also, there is no general theorem that guarantees solutions to DAEs will exist and be unique. <ref name="Circuit Simulation, Najm">{{cite book |last=Najm |first=Farid N. |date=2010 |title=Circuit Simulation |publisher=John Wiley & Sons |isbn=9780470538715}}</ref>{{rp|pages=204-205}}
In special cases, the equations of the dynamic circuit will be in the form of an [[ordinary differential equation]] (ODE), which are easier to solve, since numerical methods for solving ODEs have a rich history, dating back to the late 1800s. One strategy for adapting ODE solution methods to DAEs is called direct discretization and is the method of choice in circuit simulation. {{r|"Circuit Simulation, Najm"|p=204-205}}
For the time-___domain solution of circuit equations, the problem is posed as an [[initial value problem]] (IVP). That is, the values of the components with memories (for example, the voltages on capacitors and currents through inductors) are given at an initial point of time {{math|t<sub>0</sub>}}, and the analysis is done for the time <math>t_0\leq t\leq t_f</math>. {{r|"Circuit Simulation, Najm"|p=206-207}} Since finding numerical results for the infinite number of time points from {{math|t<sub>0</sub>}} to {{math|t<sub>f</sub>}} is not possible, this time period is discretized into discrete time instances, and the numerical solution is found for every instance. The time between the time instances is called the time step.
The numerical algorithms solve the problem in a step-by-step fashion. First, the circuit is solved for time {{math|t<sub>0</sub>}}; then, this information is used to solve for time {{math|t<sub>1</sub>}}, and so on. To see how the solution for time {{math|t<sub>n+1</sub>}} can be found, suppose that the solution for time {{math|t<sub>n</sub>}} is known. Then, [[temporal discretization]] can be used to replace the derivatives with differences. For example, the [[backward Euler method]] can be used to replace <math>x'(t_{n+1})</math> approximately with <math>\frac{x_{n+1}-x_n}{h_{n+1}}</math> where {{math|h<sub>n+1</sub>}} is the time step. After replacing the derivatives with differences, the equation system stops being differential. {{r|"Circuit Simulation, Najm"|p=266}}
If all circuit components were linear, the equation system at this point becomes a [[system of linear equations]] and can be solved with [[numerical linear algebra]] methods. If there are nonlinear components that were not linearized up until this point, the resulting nonlinear algebraic equations can be solved with [[Equation solving#Numerical methods|nonlinear numerical methods]] such as [[Root-finding algorithms]].
==== Choice of time step ====
It is possible to use the above methods with a fixed time step (that is, a constant time step throughout the whole simulation). In this case, an arbitrary initial time step can be chosen for an initial simulation, and based on the coarseness of the results of the first simulation, a new simulation can be done with an adjusted time step. However, intuitively, small time steps must be taken for fast dynamics, whereas large time steps can be taken for slow dynamics. {{r|"Circuit Simulation, Najm"|p=296}} To achieve this, an [[adaptive step size]] may be used.
==Non-linear networks==
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