Editing Network analysis (electrical circuits) (section)

===Methods===

====Boolean analysis of switching networks====
A switching device is one where the non-linearity is utilised to produce two opposite states. CMOS devices in digital circuits, for instance, have their output connected to either the positive or the negative supply rail and are never found at anything in between except during a transient period when the device is switching.  Here the non-linearity is designed to be extreme, and the analyst can take advantage of that fact.  These kinds of networks can be analysed using [[Boolean algebra (logic)|Boolean algebra]] by assigning the two states ("on"/"off", "positive"/"negative" or whatever states are being used) to the Boolean constants "0" and "1".

The transients are ignored in this analysis, along with any slight discrepancy between the state of the device and the nominal state assigned to a Boolean value. For instance, Boolean "1" may be assigned to the state of +5V.  The output of the device may be +4.5V but the analyst still considers this to be Boolean "1". 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).

The transients are not entirely uninteresting to the analyst. The maximum rate of switching is determined by the speed of transition from one state to the other. Happily for the analyst, for many devices most of the transition occurs in the linear portion of the devices transfer function and linear analysis can be applied to obtain at least an approximate answer.

It is mathematically possible to derive [[Boolean algebra (structure)|Boolean algebra]]s that have more than two states. There is not too much use found for these in electronics, although three-state devices are passingly common.

====Separation of bias and signal analyses====
This technique is used where the operation of the circuit is to be essentially linear, but the devices used to implement it are non-linear. A transistor amplifier is an example of this kind of network. The essence of this technique is to separate the analysis into two parts.  Firstly, the dc [[Biasing (electronics)|biases]] are analysed using some non-linear method. This establishes the [[Biasing|quiescent]] operating point of the circuit. Secondly, the [[small signal]] characteristics of the circuit are analysed using linear network analysis. Examples of methods that can be used for both these stages are given below.

====Graphical method of dc analysis====
In a great many circuit designs, the dc bias is fed to a non-linear component via a resistor (or possibly a network of resistors). 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. The method is as follows: from linear network analysis the output transfer function (that is output voltage against output current) is calculated for the network of resistor(s) and the generator driving them. This will be a straight line (called the [[load line (electronics)|load line]]) and can readily be superimposed on the transfer function plot of the non-linear device. The point where the lines cross is the quiescent operating point.

Perhaps the easiest practical method is to calculate the (linear) network open circuit voltage and short circuit current and plot these on the transfer function of the non-linear device. The straight line joining these two point is the transfer function of the network.

In reality, the designer of the circuit would proceed in the reverse direction to that described. 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.

It is still possible to use this method if the device being biased has its bias fed through another device which is itself non-linear, a diode for instance. 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.

====Small signal equivalent circuit====
{{main|Small-signal model}}
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.  It must be remembered that this equivalent circuit is entirely notional and only valid for the small signal deviations.  It is entirely inapplicable to the dc biasing of the device.

For a simple two-terminal device, the small signal equivalent circuit may be no more than two components.  A resistance equal to the slope of the v/i curve at the operating point (called the dynamic resistance), and tangent to the curve.  A generator, because this tangent will not, in general, pass through the origin.  With more terminals, more complicated equivalent circuits are required.

A popular form of specifying the small signal equivalent circuit amongst transistor manufacturers is to use the two-port network parameters known as [[two-port network#Hybrid parameters (h-parameters)|[h] parameters]].  These are a matrix of four parameters as with the [z] parameters but in the case of the [h] parameters they are a hybrid mixture of impedances, admittances, current gains and voltage gains.  In this model the three terminal transistor is considered to be a two port network, one of its terminals being common to both ports.  The [h] parameters are quite different depending on which terminal is chosen as the common one.  The most important parameter for transistors is usually the forward current gain, h<sub>21</sub>, in the common emitter configuration.  This is designated h<sub>fe</sub> on data sheets.

The small signal equivalent circuit in terms of two-port parameters leads to the concept of dependent generators.  That is, the value of a voltage or current generator depends linearly on a voltage or current elsewhere in the circuit.  For instance the [z] parameter model leads to dependent voltage generators as shown in this diagram;

[[Image:Z-equivalent two port.png|thumbnail|none|400px|[z] parameter equivalent circuit showing dependent voltage generators]]

There will always be dependent generators in a two-port parameter equivalent circuit.  This applies to the [h] parameters as well as to the [z] and any other kind.  These dependencies must be preserved when developing the equations in a larger linear network analysis.

====Piecewise linear method====
In this method, the transfer function of the non-linear device is broken up into regions.  Each of these regions is approximated by a straight line.  Thus, the transfer function will be linear up to a particular point where there will be a discontinuity.  Past this point the transfer function will again be linear but with a different slope.

A well known application of this method is the approximation of the transfer function of a pn junction diode.  The transfer function of an ideal diode has been given at the top of this (non-linear) section.  However, this formula is rarely used in network analysis, a piecewise approximation being used instead.  It can be seen that the diode current rapidly diminishes to -I<sub>o</sub> as the voltage falls.  This current, for most purposes, is so small it can be ignored.  With increasing voltage, the current increases exponentially.  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.

The commonly accepted values for the transition point voltage are 0.7V for silicon devices and 0.3V for germanium devices.  An even simpler model of the diode, sometimes used in switching applications, is short circuit for forward voltages and open circuit for reverse voltages.

The model of a forward biased pn junction having an approximately constant 0.7V is also a much used approximation for transistor base-emitter junction voltage in amplifier design.

The piecewise method is similar to the small signal method in that linear network analysis techniques can only be applied if the signal stays within certain bounds.  If the signal crosses a discontinuity point then the model is no longer valid for linear analysis purposes.  The model does have the advantage over small signal however, in that it is equally applicable to signal and dc bias. These can therefore both be analysed in the same operations and will be linearly superimposable.