Exponential growth
- <math>U = U_D</math>
- <math>I = -I_{D,D} = - I_S \left( e^{\frac{U_D}{m U_T}}-1 \right)</math>
Die Parameter in (2.1) haben hierbei folgende Bedeutung:
IS Sättigungsstrom in Diodensperrichtung en:reverse saturation current m Diodenfaktor U, I Solarzellenklemmspannung bzw. -strom UD, ID Diodenspannung bzw. -strom UT Temperaturspannung, UT = 25,7mV bei einer Temperatur von 25°C.
en
Shockley diode equation
The Shockley ideal diode equation or the diode law (named after transistor co-inventor William Bradford Shockley, not to be confused with tetrode inventor Walter H. Schottky) gives the I–V characteristic of an ideal diode in either forward or reverse bias (or no bias). The equation is:
- <math>I=I_\mathrm{S} \left( e^{V_\mathrm{D}/(n V_\mathrm{T})}-1 \right),\,</math>
where
- I is the diode current,
- IS is the reverse bias en:saturation current (or scale current),
- VD is the voltage across the diode,
- VT is the thermal voltage, and
- n is the ideality factor, also known as the quality factor or sometimes emission coefficient. The ideality factor n varies from 1 to 2 depending on the fabrication process and semiconductor material and in many cases is assumed to be approximately equal to 1 (thus the notation n is omitted).
The thermal voltage VT is approximately 25.85 mV at 300 K, a temperature close to “room temperature” commonly used in device simulation software. At any temperature it is a known constant defined by:
- <math>V_\mathrm{T} = \frac{k T}{q} \, ,</math>
where k is the Boltzmann constant, T is the absolute temperature of the p-n junction, and q is the magnitude of charge on an electron (the elementary charge).
The Shockley ideal diode equation or the diode law is derived with the assumption that the only processes giving rise to the current in the diode are drift (due to electrical field), diffusion, and thermal recombination-generation. It also assumes that the recombination-generation (R-G) current in the depletion region is insignificant. This means that the Shockley equation doesn’t account for the processes involved in reverse breakdown and photon-assisted R-G. Additionally, it doesn’t describe the “leveling off” of the I–V curve at high forward bias due to internal resistance.
Under reverse bias voltages (see Figure 5) the exponential in the diode equation is negligible, and the current is a constant (negative) reverse current value of −IS. The reverse breakdown region is not modeled by the Shockley diode equation.
For even rather small forward bias voltages (see Figure 5) the exponential is very large because the thermal voltage is very small, so the subtracted ‘1’ in the diode equation is negligible and the forward diode current is often approximated as
- <math>I=I_\mathrm{S} e^{V_\mathrm{D}/(n V_\mathrm{T})}</math>
The use of the diode equation in circuit problems is illustrated in the article on diode modeling.
en:Compound interest
Simplified calculation
Formulae are presented in greater detail at time value of money.
In the formula below, i is the effective interest rate per period. FV and PV represent the future and present value of a sum. n represents the number of periods.
These are the most basic formulae:
- <math> FV = PV ( 1+i )^n\, </math>
The above calculates the future value (FV) of an investment's present value (PV) accruing at a fixed interest rate (i) for n periods.
- <math> PV = \frac {FV} {\left( 1+i \right)^n}\,</math>
The above calculates what present value (PV) would be needed to produce a certain future value (FV) if interest (i) accrues for n periods.
- <math> i = \left( \frac {FV} {PV} \right)^\frac {1} {n}- 1</math>
The above calculates the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods.
- <math> n = \frac {\log(FV) - \log(PV)} {\log(1 + i)}</math>
The above formula calculates the number of periods required to get FV given the PV and the interest rate (i). The log function can be in any base, e.g. natural log (ln), as long as consistent bases are used all throughout calculation.