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Some inveter have a resistor between its input and output? I wonder what is this resistor for? to stablize the DC level? In which case should we use this kind of inverter instead of the convertional one?
Fom is right, an inverter with resistor connected between input and output acts as a linear amplifier, where the gain of inverter and the value of resistance affects the final amplifier gain.
You will see this use in crystal oscillator or AC coupled inverter amplifier where a small signal is amplified thru this so-called "inverter amplifier" until it becomes a large signal to feed into logic for further manipulation. One of the input signals is the FM signal or AM signal in radio.
For those crystal oscillator, you can search and almost all circuit higher than 1MHz will use this simple inverter to do crystal signal amplification.
A larger feedback resistance will decrease the negative feeback for better oscillation
The resistor autobiases the inverter to a point at (or near)
maximum sensitivity. It's a way to gain up small signals that
would not cross logic threshold if ground-referenced, you
DC-block them and let the inverter chain amplify it to full
scale (if you have the bandwidth, etc.).
This approach needs a fairly balanced waveform well within
the cutoff band of the input C, feedback R.