In
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, diode modelling refers to the mathematical models used to approximate the actual behaviour of real diodes to enable calculations and circuit analysis. A
diode's
I-
V curve is
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
.
A very accurate, but complicated, physical model composes the I-V curve from three
exponentials
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
with a slightly different steepness (i.e. ideality factor), which correspond to different
recombination mechanisms in the device; at very large and very tiny currents the curve can be continued by linear segments (i.e. resistive behaviour).
In a relatively good approximation a diode is modelled by the single-exponential
Shockley diode law. This nonlinearity still complicates calculations in circuits involving diodes
so even simpler models are often used.
This article discusses the modelling of
p-n junction diodes, but the techniques may be generalized to other
solid state diodes.
Large-signal modelling
Shockley diode model
The
Shockley diode equation
The ''Shockley diode equation'' or the ''diode law'', named after transistor co-inventor William Shockley of Bell Telephone Laboratories, gives the I–V (current-voltage) characteristic of an idealized diode in either forward or reverse bias (appl ...
relates the diode current
of a
p-n junction diode to the diode voltage
. This relationship is the diode ''I-V characteristic'':
::
,
where
is the ''saturation current'' or ''scale current'' of the diode (the magnitude of the current that flows for negative
in excess of a few
, typically 10
−12A). The scale current is proportional to the cross-sectional area of the diode. Continuing with the symbols:
is the
thermal voltage (
, about 26 mV at normal temperatures), and
is known as the diode ideality factor (for silicon diodes
is approximately 1 to 2).
When
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 ultrashallow junctions, for which better analytical models exist.
[.
]
Diode-resistor circuit example
To illustrate the complications in using this law, consider the problem of finding the voltage across the diode in Figure 1.
![Diode Modelling Image2](https://upload.wikimedia.org/wikipedia/commons/e/e1/Diode_Modelling_Image2.svg)
Because the current flowing through the diode is the same as the current throughout the entire circuit, we can lay down another equation. By
Kirchhoff's laws, the current flowing in the circuit is
::
.
These two equations determine the diode current and the diode voltage. To solve these two equations, we could substitute the current
from the second equation into the first equation, and then try to rearrange the resulting equation to get
in terms of
. A difficulty with this method is that the diode law is nonlinear. Nonetheless, a formula expressing
directly in terms of
without involving
can be obtained using the
Lambert ''W''-function , which is the
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
of
, that is,
. This solution is discussed next.
Explicit solution
An explicit expression for the diode current can be obtained in terms of the
Lambert ''W''-function (also called the Omega function). A guide to these manipulations follows. A new variable
is introduced as
::
.
Following the substitutions
:
::
and
:
::
rearrangement of the diode law in terms of ''w'' becomes:
::
,
which using the Lambert
-function becomes
::
.
With the approximations (valid for the most common values of the parameters)
and
, this solution becomes
::
.
Once the current is determined, the diode voltage can be found using either of the other equations.
For large x,
can be approximated by
. For common physical parameters and resistances,
will be on the order of 10
40.
Iterative solution
The diode voltage
can be found in terms of
for any particular set of values by an
iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pr ...
using a calculator or computer.
[.
] The diode law is rearranged by dividing by
, and adding 1. The diode law becomes
::
.
By taking natural logarithms of both sides the exponential is removed, and the equation becomes
::
.
For any
, this equation determines
. However,
also must satisfy the Kirchhoff's law equation, given above. This expression is substituted for
to obtain
::
,
or
::
.
The voltage of the source
is a known given value, but
is on both sides of the equation, which forces an iterative solution: a starting value for
is guessed and put into the right side of the equation. Carrying out the various operations on the right side, we come up with a new value for
. This new value now is substituted on the right side, and so forth. If this iteration ''converges'' the values of
become closer and closer together as the process continues, and we can stop iteration when the accuracy is sufficient. Once
is found,
can be found from the Kirchhoff's law equation.
Sometimes an iterative procedure depends critically on the first guess. In this example, almost any first guess will do, say
. Sometimes an iterative procedure does not converge at all: in this problem an iteration based on the exponential function does not converge, and that is why the equations were rearranged to use a logarithm. Finding a convergent iterative formulation is an art, and every problem is different.
Graphical solution
![Diode Modelling Image3](https://upload.wikimedia.org/wikipedia/commons/b/ba/Diode_Modelling_Image3.jpg)
Graphical analysis is a simple way to derive a numerical solution to the
transcendental equations describing the diode. As with most graphical methods, it has the advantage of easy visualization. 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.
This method plots the two current-voltage equations on a graph and the point of intersection of the two curves satisfies both equations, giving the value of the current flowing through the circuit and the voltage across the diode. The figure illustrates such method.
Piecewise linear model
![Diode Modelling Image10](https://upload.wikimedia.org/wikipedia/commons/8/8c/Diode_Modelling_Image10.jpg)
In practice, the graphical method is complicated and impractical for complex circuits. Another method of modelling a diode is called
piecewise linear (PWL) modelling. 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 real diode is modelled as 3 components in series: an ideal diode, a voltage source and a
resistor.
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
In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processin ...
. Then the slope of this line is given by the reciprocal of the
small-signal
Small-signal modeling is a common analysis technique in electronics engineering used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC ...
resistance of the diode at the Q-point.
Mathematically idealized diode
![Diode Modelling Image5](https://upload.wikimedia.org/wikipedia/commons/0/0f/Diode_Modelling_Image5.png)
Firstly, consider a mathematically idealized diode. In such an ideal diode, if the diode is reverse biased, the current flowing through it is zero. This ideal diode starts conducting at 0 V and for any positive voltage an infinite current flows and the diode acts like a short circuit. The I-V characteristics of an ideal diode are shown below:
Ideal diode in series with voltage source
Now consider the case when we add a voltage source in series with the diode in the form shown below:
![Diode Modelling Image6](https://upload.wikimedia.org/wikipedia/commons/b/b1/Diode_Modelling_Image6.jpg)
When forward biased, the ideal diode is simply a short circuit and when reverse biased, an open circuit.
If the
anode
An anode is an electrode of a polarized electrical device through which conventional current enters the device. This contrasts with a cathode, an electrode of the device through which conventional current leaves the device. A common mnemonic ...
of the diode is connected to 0V, the voltage at the
cathode
A cathode is the electrode from which a conventional current leaves a polarized electrical device. This definition can be recalled by using the mnemonic ''CCD'' for ''Cathode Current Departs''. A conventional current describes the direction in wh ...
will be at ''Vt'' and so the potential at the cathode will be greater than the potential at the anode and the diode will be reverse biased. 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:
![Diode Modelling Image8](https://upload.wikimedia.org/wikipedia/commons/8/84/Diode_Modelling_Image8.png)
The Shockley diode model can be used to predict the approximate value of
.
::
Using
and
:
::
Typical values of the
saturation current The saturation current (or scale current), more accurately the reverse saturation current, is the part of the reverse current in a semiconductor diode caused by diffusion of minority carriers from the neutral regions to the depletion region. This ...
at room temperature are:
*
for silicon diodes;
*
for germanium diodes.
As the variation of
goes with the logarithm of the ratio
, its value varies very little for a big variation of the ratio. The use of base 10 logarithms makes it easier to
think in orders of magnitude.
For a current of 1.0mA:
*
for silicon diodes (9 orders of magnitude);
*
for germanium diodes (3 orders of magnitude).
For a current of 100mA:
*
for silicon diodes (11 orders of magnitude);
*
for germanium diodes (5 orders of magnitude).
Values of 0.6 or 0.7 volts are commonly used for silicon diodes.
[
]
Diode with voltage source and current-limiting resistor
The last thing needed is a resistor to limit the current, as shown below:
![Diode Modelling Image9](https://upload.wikimedia.org/wikipedia/commons/5/5a/Diode_Modelling_Image9.jpg)
The ''I-V'' characteristic of the final circuit looks like this:
![Diode Modelling Image11](https://upload.wikimedia.org/wikipedia/commons/f/fe/Diode_Modelling_Image11.png)
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. If the sloped-line segment is tangent to the real diode curve at the
Q-point
In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processin ...
, this approximate circuit has the same
small-signal
Small-signal modeling is a common analysis technique in electronics engineering used to approximate the behavior of electronic circuits containing nonlinear devices with linear equations. It is applicable to electronic circuits in which the AC ...
circuit at the Q-point as the real diode.
Dual PWL-diodes or 3-Line PWL model
![PWL diode characteristic](https://upload.wikimedia.org/wikipedia/commons/d/d7/PWL_diode_characteristic.PNG)
When more accuracy is desired in modelling the diode's turn-on characteristic, the model can be enhanced by doubling-up the standard PWL-model. This model uses two piecewise-linear diodes in parallel, as a way to model a single diode more accurately.
Small-signal modelling
Resistance
Using the Shockley equation, the small-signal diode resistance
of the diode can be derived about some operating point (
Q-point
In electronics, biasing is the setting of DC (direct current) operating conditions (current and voltage) of an active device in an amplifier. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processin ...
) where the DC bias current is
and the Q-point applied voltage is
.
[ To begin, the diode ''small-signal conductance'' is found, that is, the change in current in the diode caused by a small change in voltage across the diode, divided by this voltage change, namely:
::.
The latter approximation assumes that the bias current is large enough so that the factor of 1 in the parentheses of the Shockley diode equation can be ignored. This approximation is accurate even at rather small voltages, because the thermal voltage at 300K, so tends to be large, meaning that the exponential is very large.
Noting that the small-signal resistance is the reciprocal of the small-signal conductance just found, the diode resistance is independent of the ac current, but depends on the dc current, and is given as
::.
]
Capacitance
The charge in the diode carrying current is known to be
::,
where is the forward transit time of charge carriers: The first term in the charge is the charge in transit across the diode when the current flows. The second term is the charge stored in the junction itself when it is viewed as a simple capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
; that is, as a pair of electrodes with opposite charges on them. It is the charge stored on the diode by virtue of simply having a voltage across it, regardless of any current it conducts.
In a similar fashion as before, the diode capacitance is the change in diode charge with diode voltage:
::,
where is the junction capacitance and the first term is called the diffusion capacitance Diffusion Capacitance is the capacitance that happens due to transport of charge carriers between two terminals of a device, for example, the diffusion of carriers from anode to cathode in a forward biased diode or from emitter to baseforward-biase ...
, because it is related to the current diffusing through the junction.
Variation of forward voltage with temperature
The Shockley diode equation has an exponential of , which would lead one to expect that the forward-voltage increases with temperature. In fact, this is generally not the case: as temperature rises, the saturation current rises, and this effect dominates. So as the diode becomes ''hotter'', the forward-voltage (for a given current) ''decreases''.
Here is some detailed experimental data, which shows this for a 1N4005 silicon diode. In fact, some silicon diodes are used as temperature sensors; for example, the CY7 series from OMEGA has a forward voltage of 1.02V in liquid nitrogen (77K), 0.54V at room temperature, and 0.29V at 100 °C.[http://www.omega.com/Temperature/pdf/CY7.pdf datasheet]
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
Thermal runaway describes a process that is accelerated by increased temperature, in turn releasing energy that further increases temperature. Thermal runaway occurs in situations where an increase in temperature changes the conditions in a way t ...
in bipolar-transistor circuits (base-emitter junction of a BJT
A bipolar junction transistor (BJT) is a type of transistor that uses both electrons and electron holes as charge carriers. In contrast, a unipolar transistor, such as a field-effect transistor, uses only one kind of charge carrier. A bipolar ...
acts as a diode), where a change in bias leads to an increase in power-dissipation, which in turn changes the bias even further.
See also
* Bipolar junction transistor
A bipolar junction transistor (BJT) is a type of transistor that uses both electrons and electron holes as charge carriers. In contrast, a unipolar transistor, such as a field-effect transistor, uses only one kind of charge carrier. A bipola ...
* Semiconductor device modelling
References
{{reflist
Electronic device modeling