Franclim F. Ferreira, Pedro Guedes de Oliveira, Vítor G. Tavares

The output resistance of a V_BE multiplier

We want to evaluate the output resistance of the montage represented on the left hand side of the figure below. The equivalent circuit is represented on the right. (cf. fig. 53)

Using the circuit transformations method and changing variables, first    v_p = ( R₂ // r_p ) i    and then    i = v / ( R₁ + R₂ // r_p )    we will successively obtain:

we can then write:                  

The evaluation of the output resistance of the V_BE multiplier can also be seen as the computation of the output resistance of the amplifier represented on the right.

It is a feedback amplifier in a shunt-shunt configuration, the output resistance of which, according to the feedback theory, is equal to the open loop output resistance divided by the amount of feedback, i.e.,     R_of = R_o / (1 + b A).

The following sequence shows the way to evaluate the equivalent circuit to the feedback block, using the equivalent two-port network technique, in the framework of the method for analysing a feedback amplifier.

Replacing this in the previous circuit, we get the new circuit as depicted in the figure on the right, from where, the open loop gain (b = 0) is easily obtained:

as    R_o = R₁   we have                   

that may be rewritten as                  

This value is slightly different from the exact value obtained above, due to the small error inherent to the method for analysing a feedback amplifier. It is easy to see that the error is really small: taking the values of exercise 7, 1/R₁ is 36.8 smaller than g_m, which induces an error of 2.6% in the evaluation of the output resistance.