Shockley diode model The
Shockley diode equation relates the diode current I of a
p-n junction diode to the diode voltage V_D. This relationship is the diode
I-V characteristic: ::I = I_S\left(e^\frac{V_D}{nV_\text{T}} - 1\right), where I_S is the
saturation current or
scale current of the diode (the magnitude of the current that flows for negative V_D in excess of a few V_\text{T}, typically 10−12A). The scale current is proportional to the cross-sectional area of the diode. Continuing with the symbols: V_\text{T} is the
thermal voltage (kT/q, about 26 mV at normal temperatures), and n is known as the diode ideality factor (for silicon diodes n is approximately 1 to 2). When V_D\gg nV_\text{T} the formula can be simplified to: ::I \approx I_S \cdot e^\frac{V_D}{nV_\text{T}}. 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.
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. 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 = \frac{V_S - V_D}{R}. These two equations determine the diode current and the diode voltage. To solve these two equations, we could substitute the current I from the second equation into the first equation, and then try to rearrange the resulting equation to get V_D in terms of V_S. A difficulty with this method is that the diode law is nonlinear. Nonetheless, a formula expressing I directly in terms of V_S without involving V_D can be obtained using the
Lambert W-function, which is the
inverse function of f(w) = we^w, that is, w = W(f). 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 w is introduced as ::w = \frac{I_SR}{nV_\text{T}} \left(\frac {I}{I_S} + 1\right). Following the substitutions I/I_S = e^{V_D/nV_\text{T}} - 1: ::w e^w = \frac{I_SR}{nV_\text{T}} e^\frac{V_D}{nV_\text{T}} e^{\frac{I_SR}{nV_\text{T}} \left(\frac{I}{I_S} + 1 \right)} and V_D = V_S - IR: ::w e^w = \frac{I_SR}{nV_\text{T}} e^\frac{V_S}{nV_\text{T}} e^{\frac{-IR}{nV_\text{T}}} e^{\frac{IR I_S}{nV_\text{T} I_S}} e^\frac{I_SR}{nV_\text{T}} rearrangement of the diode law in terms of
w becomes: ::w e^w =\frac{I_SR}{nV_\text{T}} e^\frac{V_s + I_sR}{nV_\text{T}}, which using the Lambert W-function becomes ::w = W\left( \frac{I_SR}{nV_\text{T}} e^\frac{V_s + I_sR}{nV_\text{T}}\right). The final explicit solution being ::I = \frac{n V_\text{T}}{R} W\left( \frac{I_SR}{nV_\text{T}} e^\frac{V_s + I_sR}{nV_\text{T}}\right) -I_S. With the approximations (valid for the most common values of the parameters) I_sR \ll V_S and I/I_S \gg 1, this solution becomes ::I \approx \frac{n V_\text{T}}{R} W \left(\frac{I_S R}{n V_\text{T}} e^\frac{V_s}{n V_\text{T}}\right). Once the current is determined, the diode voltage can be found using either of the other equations. For large x, W(x) can be approximated by W(x) = \ln x - \ln\ln x + o(1). For common physical parameters and resistances, \frac{I_S R}{n V_\text{T}} e^\frac{V_s}{n V_\text{T}} will be on the order of 1040.
Iterative solution The diode voltage V_D can be found in terms of V_S for any particular set of values by an
iterative method using a calculator or computer. The diode law is rearranged by dividing by I_S, and adding 1. The diode law becomes ::e^\frac{V_D}{nV_\text{T}} = \frac{I}{I_S} + 1. By taking natural logarithms of both sides the exponential is removed, and the equation becomes ::\frac{V_D}{nV_\text{T}} = \ln \left(\frac{I}{I_S} + 1\right). For any I, this equation determines V_D. However, I also must satisfy the Kirchhoff's law equation, given above. This expression is substituted for I to obtain ::\frac{V_D}{nV_\text{T}} = \ln \left(\frac {V_S - V_D}{R I_S} + 1\right), or ::V_D = nV_\text{T} \ln \left(\frac{V_S - V_D}{R I_S} + 1\right). The voltage of the source V_S is a known given value, but V_D is on both sides of the equation, which forces an iterative solution: a starting value for V_D 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 V_D. This new value now is substituted on the right side, and so forth. If this iteration
converges the values of V_D become closer and closer together as the process continues, and we can stop iteration when the accuracy is sufficient. Once V_D is found, I 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 V_D = 600\,\text{mV}. 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 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 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.
Mathematically idealized diode 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: 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 V_t. ::\begin{align} &I = I_S \left( e^\frac{V_D}{n \cdot V_\text{T}} - 1 \right) \\ \Leftrightarrow {} &\ln \left( 1 + \frac{I}{I_S} \right) = \frac{V_D}{n \cdot V_\text{T}} \\ \Leftrightarrow {} &V_D = n \cdot V_\text{T} \ln\left(1+\frac{I}{I_S}\right) \approx n \cdot V_\text{T} \ln \left( \frac{I}{I_S} \right) \\ \Leftrightarrow {} &V_D \approx n \cdot V_\text{T} \cdot \ln{10} \cdot \log_{10}{\left( \frac{I}{I_S} \right)} \end{align} Using n = 1 and T = 25\,\text{°C}: ::V_D \approx 0.05916 \cdot \log_{10}{\left( \frac{I}{I_S} \right)} Typical values of the
saturation current at
room temperature are: • I_S = 10^{-12} \text{A} for silicon diodes; • I_S = 10^{-6} \text{A} for germanium diodes. As the variation of V_D goes with the logarithm of the ratio \frac{I}{I_S}, 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: • V_D \approx 0.53\,\text{V} for silicon diodes (9 orders of magnitude); • V_D \approx 0.18\,\text{V} for germanium diodes (3 orders of magnitude). For a current of 100mA: • V_D \approx 0.65\,\text{V} for silicon diodes (11 orders of magnitude); • V_D \approx 0.30\,\text{V} 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: 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.
Dual PWL-diodes or 3-Line PWL model 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==