Answer2

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Problem 2—Prove that the following circuit works as a logarithmic amplifier:

Answer

We need to prove that the output voltage Vo varies as the logarithm of the input voltage Vi. The diode current is given by the equation

I_f = I_oe^{V_f/nV_t – 1}

which can be approximated by

I_f = I_oe^V_f/nV_t

if I_f >> I_o.

Taking the logarithm of both sides and solving for V_f, we get

V_f = nV_t × ln I_f
I_o

But I_f = I = V_i/R due to virtual ground, so

V_o = –V_f = –nV_t × ln V_i/R
I_o

V_o = –nV_t × ln V_i
R × I_o

>From this relation, it is clear that the output Vo varies as the logarithm of the input Vi. However, it is also clear that the output also depends on n, V_t and I_o, which are characteristics of the particular diode used. Some of these characteristics are temperature-dependent as well, making this circuit difficult to apply in practice.

Contributor: Naveen P N

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