Network analysis (electrical circuits)
A particular technique might directly reduce the number of components, for instance by combining impedances in series.
For equivalence, the impedances between any pair of terminals must be the same for both networks, resulting in a set of three simultaneous equations.
The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm.
[2]: 2-10 In principle, nodal analysis uses Kirchhoff's current law (KCL) at N-1 nodes to get N-1 independent equations.
There is an underlying assumption to this method that the total current or voltage is a linear superposition of its parts.
This approach is standard in control theory and is useful for determining stability of a system, for instance, in an amplifier with feedback.
For two terminal components the transfer function, or more generally for non-linear elements, the constitutive equation, is the relationship between the current input to the device and the resulting voltage across it.
Another example of this technique is modelling the carriers crossing the base region in a high frequency transistor.
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.
[5]: 204–205 In special cases, the equations of the dynamic circuit will be in the form of an 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.
[5]: 204-205 Simulation-based methods for time-based network analysis solve a circuit that is posed as an initial value problem (IVP).
Also, the input signals to the network cannot be arbitrarily defined for Laplace transform based methods.
There are several options for dealing with non-linearity depending on the type of circuit and the information the analyst wishes to obtain.
For a network composed of linear components there will always be one, and only one, unique solution for a given set of boundary conditions.
Device manufacturers will usually specify a range of values in their data sheets that are to be considered undefined (i.e. the result will be unpredictable).
Since resistors are linear components, it is particularly easy to determine the quiescent operating point of the non-linear device from a graph of its transfer function.
Starting from a plot provided in the manufacturers data sheet for the non-linear device, the designer would choose the desired operating point and then calculate the linear component values required to achieve it.
In this case however, the plot of the network transfer function onto the device being biased would no longer be a straight line and is consequently more tedious to do.
This method can be used where the deviation of the input and output signals in a network stay within a substantially linear portion of the non-linear devices transfer function, or else are so small that the curve of the transfer function can be considered linear.
Under a set of these specific conditions, the non-linear device can be represented by an equivalent linear network.
The most important parameter for transistors is usually the forward current gain, h21, in the common emitter configuration.
The small signal equivalent circuit in terms of two-port parameters leads to the concept of dependent generators.
These dependencies must be preserved when developing the equations in a larger linear network analysis.
The diode is modelled as an open circuit up to the knee of the exponential curve, then past this point as a resistor equal to the bulk resistance of the semiconducting material.
If the signal crosses a discontinuity point then the model is no longer valid for linear analysis purposes.
In linear analysis, the components of the network are assumed to be unchanging, but in some circuits this does not apply, such as sweep oscillators, voltage controlled amplifiers, and variable equalisers.
Sidney Darlington disclosed a method of analysing such periodic time varying circuits.
