Diode modelling

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A diode's I-V curve is non-linear, which complicates the modelling of diodes in circuits. There are two popular methods which are used for modelling.

1 Graphical analysis
2 Piecewise linear modelling
3 See also

[edit] Graphical analysis

Graphical analysis is a simple way to derive a numerical solution 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 can be accelerated through the use of computers.

Let us consider the following circuit:

The diode is forward biased and, assuming that the supply voltage is greater than the threshold voltage of the diode, a current, I, flows through the circuit. By Kirchhoff’s laws, we can find an equation for the current flowing in the circuit as follows: $I=\frac{V_S-V_D}{R}$

The problem with this equation is that it has two unknowns, I and V_d. However, since the current flowing through the diode is the same as the current throughout the entire circuit, we can lay down another equation relating I and V_d, and this is the diode I-V characteristic:

$I=I_S(e^{{qV_D}\over{nkT}}-1)$

When I >> I_S, the formula can be simplified to:

$I \approx I_S \cdot e^{{qV_D}\over{nkT}}$

Where $I S$ is the saturation current of the diode (typically 10^-12 A), q is the elementary charge value, 1.6x10^-19 C, n is known as the fiddle factor (for silicon diodes, n = 2), k is the Boltzmann constant, 1.38x10^-23 and T is the temperature in kelvins.

If we were to go about solving this equation analytically, we would substitute I from the second equation into the first equation, and then try and rearrange the resulting equation to get V_d. However, it turns out that this is impossible to do and the equations cannot be solved algebraically. The other method is to plot the two curves on a graph and the point of intersection will give the value of the current flowing through the circuit and the voltage across the diode.

[edit] Piecewise linear modelling

In practice, the graphical method is complicated and long and is impractical for complex circuits. The second 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. The example below shows how a curve can be approximated by three linear segments, forming a three-segment PWL model:

The same method is used to approximate the diode characteristic curve into linear segments. This enables us to substitute the real diode for an ideal diode, a voltage source and a resistor. The figure below shows a real diode I-V curve being approximated by a two segment piecewise linear model.

[edit] Ideal Diode

Firstly, let us consider the ideal diode. In an ideal diode, if the diode is reverse biased, the current flowing through it is zero. The ideal diode starts conducting at 0 V and an infinite current can flow through and the diode acts like a short circuit. The I-V characteristics of an ideal diode are shown below:

[edit] Ideal Diode in Series with Voltage

Now let us consider the case when we add a voltage source in series with the diode in the form shown below:

If the anode of the diode is connected to 0 V, 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 models the threshold voltage present in real diodes. The combined I-V characteristic of this circuit is shown below:

[edit] Diode with Voltage and Limited Current

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 be replaced with the ideal diode, voltage source and resistor and the circuit can now be modelled using just linear elements. When forward biased, the ideal diode is simply a short circuit and when reverse biased, an open circuit.