Editing Network analysis (electrical circuits) (section)

== Time-based network analysis with simulation ==
{{see also|Electronic circuit simulation}}
Most analysis methods calculate the voltage and current values for static networks, which are circuits consisting of memoryless components only but have difficulties with complex dynamic networks. 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|ordinary differential equations]] (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}}

Simulation-based methods for time-based network analysis solve a circuit that 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 and can be fixed throughout the whole simulation or may be [[adaptive step size|adaptive]].

In an IVP, when finding a solution for time {{math|t<sub>n+1</sub>}}, the solution for time {{math|t<sub>n</sub>}} is already known. Then, [[temporal discretization]] is used to replace the derivatives with differences, such as <math display="block">x'(t_{n+1}) \approx \frac{x_{n+1}-x_n}{h_{n+1}}</math> for the [[backward Euler method]], where {{math|h<sub>n+1</sub>}} is the time step. {{r|"Circuit Simulation, Najm"|p=266}}

If all circuit components were linear or the circuit was linearized beforehand, the equation system at this point is a [[system of linear equations]] and is solved with [[numerical linear algebra]] methods. Otherwise, it is a nonlinear algebraic equation system and is solved with [[Equation solving#Numerical methods|nonlinear numerical methods]] such as [[Root-finding algorithms]].

=== Comparison to other methods ===
Simulation methods are much more applicable than [[Laplace transform]] based methods, such as [[Network analysis (electrical circuits)#Transfer function|transfer functions]], which only work for simple dynamic networks with capacitors and inductors. Also, the input signals to the network cannot be arbitrarily defined for Laplace transform based methods.