Diode modelling
A very accurate, but complicated, physical model composes the I-V curve from three exponentials with a slightly different steepness (i.e. ideality factor), which correspond to different recombination mechanisms in the device;[1] at very large and very tiny currents the curve can be continued by linear segments (i.e. resistive behaviour).
This nonlinearity still complicates calculations in circuits involving diodes so even simpler models are often used.
the formula can be simplified to: This expression is, however, only an approximation of a more complex I-V characteristic.
Its applicability is particularly limited in case of ultra-shallow junctions, for which better analytical models exist.
[2] To illustrate the complications in using this law, consider the problem of finding the voltage across the diode in Figure 1.
An explicit expression for the diode current can be obtained in terms of the Lambert W-function (also called the Omega function).
-function becomes The final explicit solution being With the approximations (valid for the most common values of the parameters)
, this solution becomes Once the current is determined, the diode voltage can be found using either of the other equations.
The diode law becomes By taking natural logarithms of both sides the exponential is removed, and the equation becomes For any
is on both sides of the equation, which forces an iterative solution: a starting value for
Graphical analysis is a simple way to derive a numerical solution to the transcendental equations describing the diode.
By plotting the I-V curves, it is possible to obtain an approximate solution to any arbitrary degree of accuracy.
This process is the graphical equivalent of the two previous approaches, which are more amenable to computer implementation.
In practice, the graphical method is complicated and impractical for complex circuits.
In mathematics, this means taking a function and breaking it down into several linear segments.
This method is used to approximate the diode characteristic curve as a series of linear segments.
The figure shows a real diode I-V curve being approximated by a two-segment piecewise linear model.
Typically the sloped line segment would be chosen tangent to the diode curve at the Q-point.
Then the slope of this line is given by the reciprocal of the small-signal resistance of the diode at the Q-point.
In order to get the diode to conduct, the voltage at the anode will need to be taken to Vt.
This circuit approximates the cut-in voltage present in real diodes.
The combined I-V characteristic of this circuit is shown below: The Shockley diode model can be used to predict the approximate value of
: Typical values of the saturation current at room temperature are: As the variation of
[5] The last thing needed is a resistor to limit the current, as shown below: The I-V characteristic of the final circuit looks like this: The real diode now can be replaced with the combined ideal diode, voltage source and resistor and the circuit then is modelled using just linear elements.
of the diode can be derived about some operating point (Q-point) where the DC bias current is
It is the charge stored on the diode by virtue of simply having a voltage across it, regardless of any current it conducts.
In fact, this is generally not the case: as temperature rises, the saturation current
[8] In addition, there is a small change of the material parameter bandgap with temperature.
For LEDs, this bandgap change also shifts their colour: they move towards the blue end of the spectrum when cooled.
Since the diode forward-voltage drops as its temperature rises, this can lead to thermal runaway due to current hogging when paralleled in bipolar-transistor circuits (since the base-emitter junction of a BJT acts as a diode), where a reduction in the base-emitter forward voltage leads to an increase in collector power-dissipation, which in turn reduces the required base-emitter forward voltage even further.
