3: 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

\[i(v) = I_S \left[\exp \left(\dfrac{v}{ηV_T}\right) - 1\right], \quad v > V_Z \label{eq1}\]

• 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. Equation \ref{eq1} is also called the Shockley ideal diode equation or the diode law. Note also that for \(v ≤ V_Z\), the diode is in breakdown and the ideal diode equation no longer applies; for \(v ≤ V_Z, \quad i = -∞\). The ideal diode i-v characteristic curve is shown below:

Figure \(\PageIndex{1}\): 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:

\[v(i) = ηV_T \ln \left[\left(\dfrac{i}{I_S}\right) + 1\right].\]

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:

\[i=\left\{\begin{array}{l}
0, v \leq 0 \\
\infty, v>0
\end{array}\right.\]

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 of Equation \ref{eq1} 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:

\[i(v) ≈ I_S \exp \left(\dfrac{v}{ηV_T}\right) \quad v > 0.2 \,V.\]

Reverse current approximation

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

\[i(v) ≈ I_S, \quad V_Z < v < 0.\]

References

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