Voltage Gain in BJT

This is not a good circuit for several reasons:

1. The bias point is unpredictable, since it's quite dependent on the gain of the transistor. The effective resistance at the base is roughly RE times the gain. That means that the voltage divider of RB and this apparent base resistance varies strongly with the transistor gain, which in turn makes the bias point unpredictable since the gain of bipolar transistors is unpredictable. All you usually know is that it will be above some maximum level, not how high it could be. It's not at all unusual for actual devices to have several times higher gain than their minimum guaranteed gain.

2. The gain adjustment also effects the bias point. Even worse, nothing is compensating for this change.

3. The output impedance, and therefore the output rise time, change with the gain setting.

To fix these things, first create a circuit that has the highest gain you ever want and bias it predictably so that the output is close to the middle of its range with no signal in. Then put a potentiometer on the input of that amplifier to adjust overall gain. If the input signal has a DC offset, then AC couple the potentiometer both from the input signal and from the input to the amplifier. The potentiometer will therefore always operate with a DC level of 0, so changing the gain won't change any DC level anywhere.

"My question is that is it possible for a BJT VCVS to have a gain > 1. Thanks."

I suppose, with your question you refer not only to the shown circuit (common collector configuration). Thus, my interpretation of your question is as follows: "Is it possible to realize a VCVS using one single transistor?".

My answer: A device which we call VCVS must have a very large (ideal infinite) input resistance and a very small (ideal: zero) output resistance. Both requirements cannot be fulfilled with a transistor. However, the shown circuit can be a rather good approach because we can realize an input resistance in the order of some hundreds of kohms and an output resistance of 50 ohms or so. However, as was mentioned already, the gain of this circuit cannot be larger than unity.

Of course, there are other circuit alternatives (common emitter) with gain values that are much larger - however, the required characteristics for a VCVS (input, output resistances) cannot be fulfilled as we could for the common-collector circuit.

As a consequence, if you need a "good" VCVS with gain>1 you need 3 transistor stages in series: : A gain stage (common emitter) and two other stages (common collector) at the input as well as at the output of the gain stage.