What is the operation of a logarithmic amplifier?

Answer 1

A logarithmic amplifier is an amplifier which provides the logarithmic function.

Linear amplifiers produce an output which is equal to the equation y=kx, where y is the output, x is the input, and k is a constant. If the amp is unity-gain, y=x because k=1. If the amplifier linearly increases the input signal, k will be greater than 1. If the amplifier reduces the input signal, but retains the linear relationship, k will be less than 1.

A logarithmic amplifier (log amp) produces an output with relation to the input of the logarithmic function. The equation would be y=KLog(x). The base of the logarithmic function used is usually e, so the equation could be written y=Kln(x). Again, K is a multiplying constant which scales the output. e^y uses e (the "natural number", about 2.71828) and raises it to the y power. y=ln(x) means that y is the exponent necessary to get e raised to y equal to x.

It is easier to understand the logarithmic function in terms of log-base-10, than natural-log, because with log-base-10, the log number is the number of the power required to raise 10 to the given value. That means that each whole-number log represents a power of 10:

log 0=1

log 1=10

log 2=100 and so on.

Likewise, negative logs represent powers of 1/10:

log 0=1/1 (so it still =1)

log -1=1/10

log -2=1/100

This makes the line of the function rise at an accelerating rate to the right of x=1, and decrease at a decelerating rate to the left of 1. If y is the output of the log amp, then, the input (x) can increase by powers of 10 and the output will increase by additive values of 1.

An op-amp which is configured as a log-base-10 amp with an output maximum of 5 volts and positive-only input maximum of 1 volt and scaled for 1volt-per-decade will produce:

5volts for 1volt input

4volts for .1volt input

3volts for .01volt input

2volts for .001volt input

1volt for .0001volt input

0volts for .00001volt input

To understand this, look at the inputs for 1volt and 5volts output, and compare:

for inputs, .0001volt to 1volt is a dynamic range of 10,000: the largest value is 10,000 times larger than the input.

for outputs, 1 volt to 5 volts is 4 volts, so the largest value is 5 times larger than the input.

This is compression: expressing very large differences as small values. One benefit is that you can multiply the large numbers by adding the small numbers (equivalent to adding exponents!)

The amp could be configured the other way, providing 1volt for an input of 5volts, and .00001volt for an input of 0volts, with the function being log-base-10. This is an expander, making a small dynamic range into a large one.

Log amps are useful where natural measured phenomena are related by logs or natural logs. (Examples: Volume is log-base-2 related: each equal amount of increase in volume to the ear is actually an increase of a factor of 2), as is pitch (each octave increase is a doubling of frequency). Photodiodes produce a logarithmic current, so a log amp can be used to make the output match the linear nature of applied light.

Log amps are also useful when companding: taking a very large-dynamic signal (in audio terms, that means very loud to very quiet) and encoding it in a signal with a very small dynamic range, transmitting it, then expanding it back to the original characteristic.