imported>WikiSysop: /* en:Compound interest */

2010-10-31T08:38:03Z

en:Compound interest

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:<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 Shockley|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,
:''I''S is the reverse bias [[:en:saturation current]] (or scale current),
:''V''D is the voltage across the diode,
:''V''T 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]] ''V''T 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 modelling#Shockley diode model|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.

Exponential growth - Revision history

imported>WikiSysop: /* en:Compound interest */