082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

10.8 Differentiation as a linear operator 379
For a fixed temperature the only variables in this equation are I andV. We can
differentiate thisequation togive
dI
dV ≈I q
S
nkT e qV
nkT
Back substituting the approximate expressionI ≈I S
e qV
nkT into this equation yields
dI
dV ≈I q
S
nkT e qV q
nkT ≈
nkT I
Inverting this equation gives
dV
dI ≈ nkT
qI
Itisconstructiveatthisstagetocomparethisresultwiththestaticequivalent,V =IR,
from which
V
I =R
The quantity dV
dI can be thought of as a dynamic value of the diode resistance,r d ,
which varies depending on the value of the current in the diode. It is valid for a
positive voltageV that is sufficiently large to ensure the diode is in the exponential
portion of the electrical characteristic.
Engineeringapplication10.5
Operatingpointforanidealsemiconductordiode
In Engineering application 2.18 we examined the mathematical model of a semiconductordiodeandthenconsideredthespecialisedcaseofanidealdiode.Atroom
temperaturetheequationdescribingthebehaviourofitselectricalcharacteristicswas
I =I s
(e 40V −1)
where I is the diode current,V is the applied voltage and I s
is the constant reverse
saturation current. Sometimes when a diode is used in a circuit it may be biased
to operate in a certain region of its I--V characteristic. This means that its use is
restricted to a certain voltage range. The point around which it operates is known as
its operating point. This isillustratedinFigure 10.17.
Deviationsfromthisoperatingpointmaybesmallincertaincases.Ifso,theyare
known assmall-signalvariations andarecausedbysmalla.c.voltagesbeingsuperimposedonthemaind.c.biasvoltage.Incalculatinghowthediodewillreacttosuch
small-signalvoltages,theslopeofthediodecharacteristicaroundtheoperatingpoint
is more relevant than the overall ratio of current to voltage. Provided the deviations
from the operating point are not large, the tangent to theI--V curve at the operating
point provides an adequate model for how the diode will behave. The slope of the
curve can beobtained by differentiating the diodeequation.If
then
I =I s
(e 40V −1) =I s
e 40V −I s
dI
dV = 40I s e40V
➔