The logarithmic amplifier circuit configuration is shown on the Picture 1. As we can see from the picture, this circuit is basically the inverting amplifier with the difference that the resistor R2 is replaced with the diode D. Here, the output voltage is equal in the magnitude with the diode's voltage, but they are with opposite signs. Since, for the current that flows through the diode D we can write:
id ~= Is*exp(Vd/Vt)
where the Vd is the diode's voltage and Vt is the thermal voltage, which in general is equal to 25 mV at room temperature. Thus, it folows that:
Vo = -Vd = -Vt(log(Vi/R1) - log(Is))
So, according to the last relation, we can state that the output voltage Vo is proportional to the logarithm of the input voltage Vi. That's why this circuit configuration is called logarithmic amplifier.
Picture 1: Logarithmic Amplifier
The wave forms of the input and output voltages are shown on the Picture 2. The input voltage Vi is represented with the blue line plot on the graph and the output voltage Vo is represented with the green line plot. The input voltage signal is with sinusoidal waveform with amplitude of 1 V and at frequency of 500 Hz. The output voltage signal is about -670 mV for the positive half period of the input voltage signal and about 11 V for the negative half period of the the input voltage signal.
Picture 2: The input and output voltage wave forms of the logarithmic amplifier
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