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I am thoroughly confused on this topic. How does "common base" even make sense? If the base is grounded then the transistor is turned off, so how does the input reach the output?

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The fact that the base is common to the input and output signals does not imply that the base is at the lowest voltage (assuming npn). Compare with common collector (aka emitter follower): the collector is at the highest voltage, not at the lowest!

In the common base configuration the base is held at a fixed voltage, the input is applied to the emitter, the output is taken from the collector. The input signal will be loaded heavily (input impedance is very low), output impedance is high, current amplification is ~ 1, voltage amplification is very high.

This configuration is sometimes used in HF stages.

With some hand-waving the long-tailed pair (input stage of an opamp) with one input fixed can be seen as an emitter follower + a common base.

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    \$\begingroup\$ I would like to know why the output impedance of a common base amplifier is high (compared with the common emitter stage). \$\endgroup\$ Sep 10, 2015 at 11:20
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If the base is grounded then the transistor is turned off, so how does the input reach the output?

The collector, base, and emitter currents are identical in the two circuits below. If the transistor in the left circuit is biased on, the transistor in the right circuit is biased on.

Yet, in the left circuit, the emitter is grounded while, in the right circuit, the base is grounded.

So, I think you should carefully reconsider your assumptions. Note that, in the common-base circuit, the emitter is no longer grounded.

schematic

simulate this circuit – Schematic created using CircuitLab

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Transistor will turn on (current will run from collector to emitter) when there is some current running thru base-emitter junction.

Everything is a bit more complicated in reality, but to put it simply - it does not matter if you have emitter at a fixed point and wiggle the base (common collector) or you have base fixed and wiggle the emitter (common base).

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    \$\begingroup\$ Evidently, I down voted this answer accidentally unintentionally but it is locked in unless you edit your answer. \$\endgroup\$ Dec 8, 2013 at 23:49
  • \$\begingroup\$ @Alfred: you can do a dummy edit yourself. The edit doesn't have to be by the original author. \$\endgroup\$ Dec 14, 2013 at 7:53
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Common base = base terminal voltage is used as a reference voltage for both the emitter terminal and the collector terminal.

A transistor is turned on when the diode junction of the base-emitter is forward (or reverse) biased. For a silicon npn device, with the base at 0V, the emitter terminal will be near -0.6V when the transistor is forward active.

To tell if a transistor is on or off, a helpful voltage measurement is the collector to emitter voltage. That is, the voltage between those two terminals, without regard to ground. This will tell you if the transistor is on, off, or saturated. The base to emitter voltage will tell you why the transistor is in that state. Ground is a relative term that must be defined for each circuit.

I hope this helps you understand how transistors operate.

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