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In common emitter circuits an increase in Base voltage causes a decrease in Vout and a decrease in Base voltage produces an increase in Vout. In other words the output signal is 180o out-of-phase with the input signal. The Voltage Gain of the common emitter amplifier is equal to the ratio of the change in the input voltage to the change in the amplifiers output voltage. Then ΔVL is Vout and ΔVB is Vin. But voltage gain is also equal to the ratio of the signal resistance in the Collector to the signal resistance in the Emitter or:
Vout/Vin = ΔVL/ΔVB = -(RL/RE)
I`ve got the impression that Andrew would like to know WHY there is a phase inversion.
The answer is simple:
An input voltage increase causes an increase in the collector current. Thus, the voltage drop across the collector resistor Rc increases also. Because the upper end of Rc is connected to the fixed supply voltage the voltage at the lower end of Rc (at the collector node) decreases - and is just in opposite phase to the input.