 Diode modelling

In electronics, diode modelling refers to the mathematical models used to approximate the actual behavior of real diodes to enable calculations and circuit analysis. A diode's IV curve is nonlinear (it is well described by the Shockley diode law). This nonlinearity complicates calculations in circuits involving diodes, so simpler models are often required.
This article discusses the modelling of pn junction diodes, but the techniques may be generalized to other solid state diodes.
Contents
Largesignal modelling
Shockley diode model
The Shockley diode equation relates the diode current I of a pn junction diode to the diode voltage V_{D}. This relationship is the diode IV characteristic:

 ,
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_{T}, typically 10^{−12} A). The scale current is proportional to the diode area. Continuing with the symbols: V_{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 the formula can be simplified to:

 .
This expression is, however, only an approximation of a more complex IV characteristic. Its applicability is particularly limited in case of ultrashallow junctions, for which better analytical models exist.^{[1]}
Dioderesistor 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, which boil down to simply Ohm's law in this case, 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 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 Wfunction, 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 Wfunction (also called the Omega function). A guide to these manipulations follows. A new variable w is introduced as

 .
Following the substitutions and V_{D} = V_{S} − IR, rearrangement of the diode law in terms of w becomes

 ,
which using the Lambert Wfunction 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.
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 ^{[2]} The diode law is rearranged by dividing by I_{S}, and adding 1. The diode law becomes

 .
By taking natural logarithms of both sides the exponential is removed, and the equation becomes

 .
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

 ,
or

 .
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 . 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 IV 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 currentvoltage 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 following 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 modeled as 3 components in series: an ideal diode, a voltage source and a resistor. The figure below shows a real diode IV curve being approximated by a twosegment piecewise linear model. Typically the sloped line segment would be chosen tangent to the diode curve at the Qpoint. Then the slope of this line is given by the reciprocal of the smallsignal resistance of the diode at the Qpoint.
Mathematically idealized diode
Firstly, let us 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 IV characteristics of an ideal diode are shown below:
Ideal diode in series with voltage source
Now let us 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 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 approximates the cutin voltage present in real diodes. The combined IV characteristic of this circuit is shown below:
The Shockley diode model can be used to predict the approximate value of V_{t}.
Using n = 1 and T = 25C:
Typical values of the saturation current are:
 I_{S} = 10 ^{− 12} for silicon diodes;
 I_{S} = 10 ^{− 6} for germanium diodes.
As the variation of V_{D} 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 1.0 mA:
 for silicon diodes (9 orders of magnitude);
 for germanium diodes (3 orders of magnitude).
For a current of 100 mA:
 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 ^{[3]}
Diode with voltage source and currentlimiting resistor
The last thing needed is a resistor to limit the current, as shown below:
The IV 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 slopedline segment is tangent to the real diode curve at the Qpoint, this approximate circuit has the same smallsignal circuit at the Qpoint as the real diode.
Dual PWLdiodes or 3Line PWL model
When more accuracy is desired in modeling the diode's turnon characteristic, the model can be enhanced by doublingup the standard PWLmodel. This model uses two piecewiselinear diodes in parallel, as a way to model a single diode more accurately.
Smallsignal modelling
Resistance
Using the Shockley equation, the smallsignal diode resistance r_{D} of the diode can be derived about some operating point (Qpoint) where the DC bias current is I_{Q} and the Qpoint applied voltage is V_{Q}.^{[4]} To begin, the diode smallsignal conductance g_{D} 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 I_{Q} 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 V_{Q} / V_{T} tends to be large, meaning that the exponential is very large.
Noting that the smallsignal resistance r_{D} is the reciprocal of the smallsignal 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 I_{Q} is known to be
 Q = I_{Q}τ_{F} + Q_{J},
where τ_{F} is the forward transit time of charge carriers:^{[4]} The first term in the charge is the charge in transit across the diode when the current I_{Q} flows. The second term is the charge stored in the junction itself when it is viewed as a simple capacitor; 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, because it is related to the current diffusing through the junction.
References
 ^ . Popadic, Milo¿; Lorito, Gianpaolo; Nanver, Lis K. (2009). "Analytical Model of I – V Characteristics of Arbitrarily Shallow pn Junctions". IEEE Transactions on Electron Devices 56: 116–125. doi:10.1109/TED.2008.2009028. http://ieeexplore.ieee.org/search/srchabstract.jsp?arnumber=4717238&isnumber=4723893&punumber=16&k2dockey=4717238@ieeejrns&query=%28+%28%28electron+devices%29%3Cin%3Emetadata+%29+%3Cand%3E+%28%28popadic%29%3Cin%3Emetadata+%29+%29&pos=0&access=no.
 ^ . A.S. Sedra and K.C. Smith (2004). Microelectronic Circuits (Fifth ed.). New York: Oxford. Example 3.4 p. 154. ISBN 0195142519. http://worldcat.org/isbn/0195142519.
 ^ . Kal, Santiram (2004). "Chapter 2". Basic Electronics: Devices, Circuits and IT Fundamentals (Section 2.5: Circuit Model of a PN Junction Diode ed.). PrenticeHall of India Pvt.Ltd. ISBN 8120319524.
 ^ ^{a} ^{b} R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (second ed.). McGrawHill. ISBN 0072320990. http://books.google.com/?id=u6vH4Gsrlf0C&pg=PA883&dq=%22Microelectronic+Circuit+Design%22+inauthor:Jaeger+smallsignal+diode.
See also
Categories: Electronic device modeling

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