In electronics, 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.
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).
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.
The Shockley diode equation relates the diode current
I
VD
I=
| ||||
I | ||||
S\left(e |
-1\right)
where
IS
VD
VT
VT
kT/q
n
n
When
VD\ggnVT
I ≈ IS ⋅
| ||||
e |
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.[2]
To illustrate the complications in using this law, consider the problem of finding the voltage across the diode in Figure 1.
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
I=
VS-VD | |
R |
I
VD
VS
I
VS
VD
f(w)=wew
w=W(f)
An explicit expression for the diode current can be obtained in terms of the Lambert W-function (also called the Omega function).[3] A guide to these manipulations follows. A new variable
w
w=
ISR | \left( | |
nVT |
I | |
IS |
+1\right)
Following the substitutions
I/IS=
VD/nVT | |
e |
-1
wew=
ISR | |
nVT |
| ||||
e |
| ||||||||
e |
and
VD=VS-IR
wew=
ISR | |
nVT |
| ||||
e |
| ||||
e |
| ||||
e |
| ||||
e |
rearrangement of the diode law in terms of w becomes:
wew=
ISR | |
nVT |
| ||||
e |
which using the Lambert
W
w=W\left(
ISR | |
nVT |
| ||||
e |
\right)
The final explicit solution being
I=
nVT | |
R |
W\left(
ISR | |
nVT |
| ||||
e |
\right)-IS
With the approximations (valid for the most common values of the parameters)
IsR\llVS
I/IS\gg1
I ≈
nVT | |
R |
W\left(
ISR | |
nVT |
| ||||
e |
\right)
Once the current is determined, the diode voltage can be found using either of the other equations.
For large x,
W(x)
W(x)=lnx-lnlnx+o(1)
ISR | |
nVT |
| ||||
e |
The diode voltage
VD
VS
IS
| ||||
e |
=
I | |
IS |
+1
By taking natural logarithms of both sides the exponential is removed, and the equation becomes
VD | |
nVT |
=ln\left(
I | |
IS |
+1\right)
For any
I
VD
I
I
VD | |
nVT |
=ln\left(
VS-VD | |
RIS |
+1\right)
or
VD=nVTln\left(
VS-VD | |
RIS |
+1\right)
The voltage of the source
VS
VD
VD
VD
VD
VD
I
Sometimes an iterative procedure depends critically on the first guess. In this example, almost any first guess will do, say
VD=600mV
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.
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. Then the slope of this line is given by the reciprocal of the small-signal resistance of the diode at the Q-point.
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:
Now consider the case when we add a voltage source in series with the diode in the form shown below:
When forward biased, the ideal diode is simply a short circuit and when reverse biased, an open circuit.
If the anode of the diode is connected to 0V, the voltage at the cathode 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:
The Shockley diode model can be used to predict the approximate value of
Vt
\begin{align} &I=IS\left(
| ||||
e |
-1\right)\\ \Leftrightarrow{}&ln\left(1+
I | |
IS |
\right)=
VD | |
n ⋅ VT |
\\ \Leftrightarrow{}&VD=n ⋅ VTln\left(1+
I | |
IS |
\right) ≈ n ⋅ VTln\left(
I | |
IS |
\right)\\ \Leftrightarrow{}&VD ≈ n ⋅ VT ⋅ ln{10} ⋅ log10{\left(
I | |
IS |
\right)} \end{align}
Using
n=1
T=25°C
VD ≈ 0.05916 ⋅ log10{\left(
I | |
IS |
\right)}
Typical values of the saturation current at room temperature are:
IS=10-12
IS=10-6
As the variation of
VD
I | |
IS |
For a current of 1.0mA:
VD ≈ 0.53V
VD ≈ 0.18V
For a current of 100mA:
VD ≈ 0.65V
VD ≈ 0.30V
Values of 0.6 or 0.7 volts are commonly used for silicon diodes.[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. If the sloped-line segment is tangent to the real diode curve at the Q-point, this approximate circuit has the same small-signal circuit at the Q-point as the real diode.
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.
Using the Shockley equation, the small-signal diode resistance
rD
IQ
VQ
gD
gD=\left.
dI | |
dV |
\right|Q=
Is | |
n ⋅ VT |
| ||||
e |
≈
IQ | |
n ⋅ VT |
The latter approximation assumes that the bias current
IQ
VT ≈ 25mV
VQ/VT
Noting that the small-signal resistance
rD
rD=
n ⋅ VT | |
IQ |
The charge in the diode carrying current
IQ
Q=IQ\tauF+QJ
where
\tauF
IQ
In a similar fashion as before, the diode capacitance is the change in diode charge with diode voltage:
CD=
dQ | |
dVQ |
=
dIQ | |
dVQ |
\tauF+
dQJ | |
dVQ |
≈
IQ | |
VT |
\tauF+CJ
where
CJ=
dQJ | |
dVQ |
The Shockley diode equation has an exponential of
VD/(kT/q)
IS
Here is some detailed experimental data,[7] 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.[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.