Introduction

The electronic circuits which can perform the mathematical operations such as logarithm and anti-logarithm with an amplification are called as Logarithmic and Anti-Logarithmic amplifiers respectively. The applications for log/anti-log amplifiers include signal compression and process control.

1) Log Amplifier

A logarithmic amplifier is an electronic circuit that produces an output that is proportional to the logarithm of the applied input. An op-amp based logarithmic amplifier produces a voltage at the output, which is proportional to the logarithm of the voltage applied to the resistor connected to its inverting terminal. The circuit diagram of op-amp based logarithmic amplifier is shown in figure 1.

Fig 1: Op-amp based Logarithmic Amplifier
In the above circuit, the non-inverting input terminal of the op-amp is connected to ground. That means zero volts is applied at the non-inverting input terminal of the op-amp.
The nodal equation at the inverting input is as follows
0 - V_i/R₁ + I_f = 0 ..........(1)
from the equation 1,the feedback current is
I_f = V_i/R₁
When the diode is forward biased, the current flowing through the diode is
I_f = I_s𝑒^(V_f/nV_T)..........(2)
Where,
I_s is the saturation current of the diode,
V_f is the voltage drop across diode, when it is in forward bias,
V_T is the diode's thermal equivalent voltage.
The KVL equation around the feedback loop of the op amp will be
0 - V_f - V₀ = 0
V_f = -V₀
Substituting the value of V_f in Equation 2, we get
I_f = I_s𝑒^(-V₀/nV_T) ..........(3)
V_i/R₁ = I_s𝑒^(-V₀/nV_T)
V_i/R_iI_s = 𝑒^(-V₀/nV_T)
Applying natural logarithm on the both sides
ln(V_i/R₁I_s) = -V₀/nV_T
V₀ = -nV_Tln(V_i/R₁I_s)
In the above equation, the parameters n, V_T and I_s are constants. So, the output voltage V₀ will be proportional to the natural logarithm of the input voltage V_i for a fixed value of resistance R₁.
The output voltage V₀ has a negative sign, which indicates that there exists a 180° phase difference between the input and the output.

2) Antilog Amplifier

An op-amp based anti-logarithmic amplifier produces a voltage at the output, which is proportional to the anti-logarithm of the voltage that is applied to the diode connected to its inverting input. The circuit diagram of an op-amp based anti-logarithmic amplifier is shown in figure 2.

Fig 2:Op-amp based Anti-Logarithmic Amplifier
In the circuit shown above, the non-inverting input of the op-amp is connected to ground. According to the virtual ground concept, the voltage at the inverting input of op-amp will be equal to the voltage present at its non-inverting input. So, the voltage at its inverting input will be zero volts.
The nodal equation at the inverting input node is
-I_f+0-V₀/R_f = 0
V₀ = -I_f*R_f ..........(4)
When the diode is forward biased, the current flowing through the diode is given by
I_f = I_s𝑒^(V_f/nV_T)
Substituting the value of I_f in Equation 4, then
V₀ = -R_fI_s𝑒^(V_f/nV_T)..........(5)
The KVL equation at the inverting input of the op-amp will be
V_i - V_f = 0
So,
V_i = V_f
Substituting the value of V_f in equation 5, then
V₀ = -R_fI_s𝑒^(V_i/nV_T)
In the above equation the parameters n, V_T and I_s are constants. So, the output voltage V₀ will be proportional to the anti-natural logarithm of the input voltage V_i, for a fixed value of feedback resistance R_f.
The output voltage V₀ has a negative sign, which indicates that there exists a 180° phase difference between the input and the output.