Log And Anti Log Amppfiers

The electronic circuits which perform the mathematical operations such as logarithm and anti-logarithm (exponential) with an amppfication are called as Logarithmic amppfier and Anti-Logarithmic amppfier respectively.

This chapter discusses about the Logarithmic amppfier and Anti-Logarithmic amppfier in detail. Please note that these amppfiers fall under non-pnear apppcations.

Logarithmic Amppfier

A logarithmic amppfier, or a log amppfier, is an electronic circuit that produces an output that is proportional to the logarithm of the appped input. This section discusses about the op-amp based logarithmic amppfier in detail.

An op-amp based logarithmic amppfier produces a voltage at the output, which is proportional to the logarithm of the voltage appped to the resistor connected to its inverting terminal. The circuit diagram of an op-amp based logarithmic amppfier is shown in the following figure −

In the above circuit, the non-inverting input terminal of the op-amp is connected to ground. That means zero volts is appped at the non-inverting input terminal of the op-amp.

According to the virtual short concept, the voltage at the inverting input terminal of an op-amp will be equal to the voltage at its non-inverting input terminal. So, the voltage at the inverting input terminal will be zero volts.

The nodal equation at the inverting input terminal’s node is −

$$frac{0-V_i}{R_1}+I_{f}=0$$

$$=>I_{f}=frac{V_i}{R_1}......Equation 1$$

The following is the equation for current flowing through a diode, when it is in forward bias −

$$I_{f}=I_{s} e^{(frac{V_f}{nV_T})} ......Equation 2$$

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 −

$$0-V_{f}-V_{0}=0$$

$$=>V_{f}=-V_{0}$$

Substituting the value of $V_{f}$ in Equation 2, we get −

$$I_{f}=I_{s} e^{left(frac{-V_0}{nV_T} ight)} ......Equation 3$$

Observe that the left hand side terms of both equation 1 and equation 3 are same. Hence, equate the right hand side term of those two equations as shown below −

$$frac{V_i}{R_1}=I_{s}e^{left(frac{-V_0}{nV_T} ight)}$$

$$frac{V_i}{R_1I_s}= e^{left(frac{-V_0}{nV_T} ight)}$$

Applying natural logarithm on both sides, we get −

$$Inleft(frac{V_i}{R_1I_s} ight)= frac{-V_0}{nV_T}$$

$$V_{0}=-{nV_T}Inleft(frac{V_i}{R_1I_s} ight)$$

Note that in the above equation, the parameters n, ${V_T}$ and $I_{s}$ are constants. So, the output voltage $V_{0}$ will be proportional to the natural logarithm of the input voltage $V_{i}$ for a fixed value of resistance $R_{1}$.

Therefore, the op-amp based logarithmic amppfier circuit discussed above will produce an output, which is proportional to the natural logarithm of the input voltage ${V_T}$, when ${R_1I_s}=1V$.

Observe that the output voltage $V_{0}$ has a negative sign, which indicates that there exists a 180⁰ phase difference between the input and the output.

Anti-Logarithmic Amppfier

An anti-logarithmic amppfier, or an anti-log amppfier, is an electronic circuit that produces an output that is proportional to the anti-logarithm of the appped input. This section discusses about the op-amp based anti-logarithmic amppfier in detail.

An op-amp based anti-logarithmic amppfier produces a voltage at the output, which is proportional to the anti-logarithm of the voltage that is appped to the diode connected to its inverting terminal.

The circuit diagram of an op-amp based anti-logarithmic amppfier is shown in the following figure −

In the circuit shown above, the non-inverting input terminal of the op-amp is connected to ground. It means zero volts is appped to its non-inverting input terminal.

According to the virtual short concept, the voltage at the inverting input terminal of op-amp will be equal to the voltage present at its non-inverting input terminal. So, the voltage at its inverting input terminal will be zero volts.

The nodal equation at the inverting input terminal’s node is −

$$-I_{f}+frac{0-V_0}{R_f}=0$$

$$=>-frac{V_0}{R_f}=I_{f}$$

$$=>V_{0}=-R_{f}I_{f}.........Equation 4$$

We know that the equation for the current flowing through a diode, when it is in forward bias, is as given below −

$$I_{f}=I_{s} e^{left(frac{V_f}{nV_T} ight)}$$

Substituting the value of $I_{f}$ in Equation 4, we get

$$V_{0}=-R_{f}left {{I_{s} e^{left(frac{V_f}{nV_T} ight)}} ight }$$

$$V_{0}=-R_{f}{I_{s} e^{left(frac{V_f}{nV_T} ight)}}......Equation 5$$

The KVL equation at the input side of the inverting terminal of the op amp will be

$$V_{i}-V_{f}=0$$

$$V_{f}=V_{i}$$

Substituting, the value of ?? in the Equation 5, we get −

$$V_{0}=-R_{f}{I_{s} e^{left(frac{V_i}{nV_T} ight)}}$$

Note that, in the above equation the parameters n, ${V_T}$ and $I_{s}$ are constants. So, the output voltage ${V_0}$ will be proportional to the anti-natural logarithm (exponential) of the input voltage ${V_i}$, for a fixed value of feedback resistance ${R_f}$.

Therefore, the op-amp based anti-logarithmic amppfier circuit discussed above will produce an output, which is proportional to the anti-natural logarithm (exponential) of the input voltage ${V_i}$ when, ${R_fI_s}= 1V$. Observe that the output voltage ${V_0}$ is having a negative sign, which indicates that there exists a 180⁰ phase difference between the input and the output.