Small Signal Model

In some of applications circuit has been supplied to a bias to operate in the forward region & also a small ac signal is superimpose on dc quantity let the circuit indicated          

            While there is no ac, diode voltage is V_D thus

                                                            I_D =I_S e ^V_D ^{/ nV}_T

            and while signal V_d (t) has been applied,  then immediate voltage V_d(t) will be

                                                            v_D (t) = V_D +v_d(t)

                                    Therefore that instant diode current is

                                                        i_D(t) =I_s e ^v_D^/nV_T

                                                Putting for v_D

                                                 i_D (t) =I_Se ^(V_D^{+ v}_d⁾ /nV_T

                                                            = Is e ^V_D ^/nV_T . e ^v_d ^/nV_T

                                                         i_D (t) = I _D e ^{vd / nVT} ----------------- (i)

If the amplitude of the signal is adequately smaller than 1 , i.e V_d/nV_T <<1 expanding the earlier equation in the form of a taylor series for example

                                                  e^x= 1+ x / 1!+ x²/ 2! .........

These expression is a taylor series expansion of exponential function e^x

In the eq (i)  x = V_d /nV_T ,therefore we solve out the eq (i) upto two terms by taylor series

                                           e ^v_d ^/nV_T = ( 1 + v_d/nV_T)

                                    substitute this value in equation (i) we obtain          

                                                i_D(t) = I_D ( 1 + v_d /nV_T)

                                                                     = I_D + I_D/nV_T v_d --------------- (ii)

            we can also write

                                              I_D/nV_T v_d = i_d substitute in equation (ii) we get

                                                                        i_D =I_D + i_d

From earlier equation we can see that the quantity I_D/nV_T contain the dimensions of conductance provided in mho's and is said diode small signal conductance .The inverse of this is said incremental resistance or small signal resistance  given by

                                                             r_d= nV_T/I_D

                                                              r_d α 1/I_D

            nV_T are constants and given by the manufacturer.