Small-signal model: Difference between revisions

Revision as of 02:50, 5 November 2007

Revision as of 03:51, 5 November 2007

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The large-signal I-V characteristic of the PN junction [[diode]] under forward bias is described by the Shockley Equation (also known as the ''diode law''):

::<math>I = I_0(e^{qV/kT}-1)</math>

where <math>I_O</math> is the reverse current that flows when the voltage is large and negative, making the exponential very small. The charge in the diode is known to be

The large-signal capacitance of the diode is known to be

::<math>Q=I\tau_s +Q_J</math>

where <math>\tau_s</math> is the recombination lifetime of charge carriers {{ref|Hu36}}.

where <math>\tau_s</math> is the recombination lifetime of charge carriers {{ref|Hu36}}. The first term in the charge is the charge in transit across the diode when the current <math>I</math> flows. The second term is the charge stored in the junction itself when it is viewed as a simple capacitor; that is, as a pair of electrodes with opposite charges on them. It is the charge stored on the diode by virtue of simply having a voltage across it, regardless of any current it conducts.

Given these two relations, the small-signal resistance and capacitance of the diode can be derived about some operating point P.

Given these two relations, the small-signal diode resistance <math>r_D</math> and capacitance <math>C_D</math> of the diode can be derived about some operating point, or [[Q-point]] where the DC bias current is <math>I_Q</math> and the Q_point applied voltage is P<math>V_Q</math>.

<math>\frac{dI}{dV} = I_0 \frac{q}{kT} e^{qV / kT} \approx \frac{q}{kT} I</math>

::<math>\frac{dI}{dV}\Big|_Q = I_0 \frac{q}{kT} e^{qVqV_Q / kT} \approx \frac{q}{kT} II_Q</math>

The latter approximation assumes that the bias current I is large enough so that the factor of 1 in the parentheses of the Shockley Equation can be ignored. This approximation is fairly common in nonlinear circuit analysis.

Noting that <math>\frac {dI} {dV}</math> corresponds to the instantaneous conductivity of the diode, the small-signal resistance <math>rr_D</math> is the reciprocal of this quantity:

::<math>rr_D=\frac {1kT/q}{II_Q}\frac{kT} {q} </math>.

In a similar fashion, the diode capacitance is the change in diode charge with diode voltage:

::<math> C_D = \frac{dQ}{dV_D} =\frac{dI_Q}{dV_D} \frac{1}{{\tau}_s} + \frac {dQ_J}{dV_D}=\frac{q}{kT} \frac{I_Q}{{\tau}_s} + C_J </math>,

where <math> C_J = \frac {dQ_J}{dV_D}</math> is the junction capacitance and the first term is called the [[diffusion capacitance]], because it is related to the current diffusing through the junction.

==See also==