# III. Ideal Diode Equation

The **ideal diode equation** is an equation that represents current flow through an ideal *p*-*n* junction diode as a function of applied voltage. In realistic settings, current will deviate slightly from this ideal case.

### Ideal Diode Equation

As seen in the previous sections, a *p*-*n* junction diode creates the following current: under reverse bias, there is a small, constant reverse current, and under forward bias, there is a forward current that increases with voltage. The current-voltage function (also called the "*i*-*v* characteristic") for an ideal diode is

*i*(*v*) = *I*_{S}[exp(*v /* η*V _{T}*) - 1],

*v*>

*V*

_{Z}where *I _{S}* is the reverse saturation current,

*v*is the applied voltage (reverse bias is negative),

*V*=

_{T}*T*/ 11,586 is the volt equivalent of temperature, and η is the

*emission coefficient*, which is 1 for germanium devices and 2 for silicon devices. Note that

*i*is defined as positive when flowing from

*p*to

*n*. This is also called the

**Shockley ideal diode equation**or the

**diode law**. Note also that for

*v*≤

*V*, the diode is in breakdown and the ideal diode equation no longer applies; for

_{Z}*v*≤

*V*,

_{Z}*i*= -∞. The ideal diode

*i*-

*v*characteristic curve is shown below:

Ideal diode equation

The ideal diode equation is very useful as a formula for current as a function of voltage. However, at times the inverse relation may be more useful; if the ideal diode equation is inverted and solved for voltage as a function of current, we find:

*v*(*i*) = η*V _{T}* ln[(

*i*/

*I*) + 1].

_{S}### Approximations

#### Infinite step function

A number of approximations of diode behavior can be made from the ideal diode equation. The simplest approximation to make is to represent the diode as a device that allows no current through -- that is, it acts as an open circuit -- under reverse bias, and allows an unlimited amount of current through -- a closed circuit -- under forward bias. In this simplified model, the current-voltage relation (also called the "*i*-*v* characterstic") is an infinite step function:

This characteristic is depicted below:

This approximation is used in circuit analysis, as we will see in the next section.

#### Forward current approximation

In the case of large forward bias, a good approximation of the ideal diode equation is to simply set the second term to zero. This approximation is valid because the ideal diode i-v curve increases very quickly, and because reverse saturation current IS is typically very small. This approximation is acceptable for v > 0.2 V. The forward current approximation, as we will call it, results in the following formula:

*i*(*v*) ≈ *I _{S }*exp(

*v*/ η

*V*); v > 0.2 V.

_{T}#### Reverse current approximation

Under reverse bias, the resulting current can be treated as simply the reverse saturation current, IS. In reality, the current under reverse bias will asymptotically approach IS, but the small magnitude of the reverse saturation current makes this discrepancy negligible. The reverse current approximation is valid over the range VZ > v > 0 (the diode enters breakdown for v ≤ VZ):

*v* ≈ *I _{S}*,

*V*>

_{Z}*v*> 0.

### References

- "Chapter 6: Diodes." Fundamentals of Electrical Engineering. 2nd ed. New York, New York: Oxford UP, 1996. 363-64. Print.