We all know energy cannot be formed from nothing.
So how does a bipolar junction transistor (BJT) — for example — amplify the base current by beta and outputs it to collector current?
Where is the “catch?” Is there somewhere else where were losing energy?
The base current in a transistor controls the collector current. The energy comes from the power supply. It is not generated within the transistor.
The 'catch' is that a transistor only controls the flow of current; it does not itself generate power. The power would come from some other part of the circuit, perhaps from the electric company via a power supply or from a battery.
Now, one possible point of confusion is that transistors can be represented as equivalent circuits that contain a so-called "current source". This sounds like it could generate power, yes? Well, it turns out that a "source" can actually either generate or absorb power, depending on the relationship between the current through it and the voltage across it. The main thing a "source" does is fix the current (in the case of a current source) or the voltage (in the case of a voltage source) to a specific value. For example, a lithium-ion battery acts like a voltage source. If you connect a resistor across the battery terminals, the battery will supply current to hold the voltage more-or-less constant. However, if you connect an external power supply, the battery will start to charge, absorbing energy while trying to hold the voltage constant.
Now, there are several different 'models' or 'equivalent circuits' of different types of transistors, all of which use dependent sources in some way. The trick is that these models are only valid under particular operating conditions, and it turns out that there isn't any set of conditions under which a transistor will ever generate power. This isn't a trick of the mathematics, the reason for this is that there is nothing inside of a transistor that's capable of generating power; the only thing a transistor can do is generate a voltage drop to oppose the flow of current. It usually turns out that transistors end up dissipating a lot of power and end up needing to be mounted on large heat sinks.
A BJT is generally used as a power amplifier and the the power gained by the output signal comes from the DC power source that it uses.
Amplifying current on its own can be done without power amplification using a transformer but, if you want a power amp (i.e. the product of volts and amperes increased) then you need a power source.
Energy is average power times time: \$E = P \times t\$. Energy is conserved. Since power is just instant moments of energy, it is also a fact that, in general, power is also conserved. So you will often see statements like "power in must be equal to or greater than power out." On average and on human scales of time and locality, that's pretty much also true. But it is the conservation of energy that really rules the universe as we understand it.
Power, itself, though is like a kind of coinage. Each coin has two faces: voltage and current. Power is the product of volts times current: \$P = V \times I\$. Note that neither of these are time. So there is no implication here that current (\$I\$) must be conserved, because voltage (\$V\$) can be adjusted. There is also no implication that voltage is conserved, because current can be adjusted. On average what is conserved is energy and power. Not voltage and current.
The energy per unit time (power) comes from the power supply. That energy and power provides heat and electrical and transducer activities and it does have to follow basic energy conservation ideas. But the BJT transistor consumes a small amount of "re-combination" current provided to the base-emitter region in order to activate a larger in-rush of collector current. What's missing here is any discussion of the voltages involved and the rest of the circuit, too, most especially including the energy sources. The BJT transistor draws from those sources and those sources lose energy well in excess of the very tiny local effect by the BJT as one tiny part of a much larger circuit and energy source system.
The conservation law applies to the closed system as a black box. But tiny local increases in energy are possible, so long as those increases come from somewhere else you are ignoring. Just as life itself on Earth may appear to be organization coming from disorganization and violate entropy laws, the fact is that the sun's entropy increases far, far more than any small local decreases in entropy represented by an isolated life form on Earth. You have to take an appropriately complete system when applying conservation laws.
It is possible to view such an amplifier as a circuit which transfers DC power (from the power supply) into signal power - available at the amplifiers output.
There are so-called "power amplifiers", which - however - do NOT amplify power. But it is their main task not to amplify a signal voltage but to PROVIDE as much signal power as possible at its output. And the output-to-input power ratio is the so-called "efficiency".
Lets consider a typical NPN transistor, and consider two electrical paths through it: collector to emitter and base to emitter. These are the only paths through which current will flow when the device is being used in a conventional manner as an amplifier.
When no forward bias voltage (relative to the emitter) is present on the base terminal, no significant current flows through any path in the device.
When voltage on the base terminal (relative to the emitter) is in the forward bias range (depends on the specifics of the device, but perhaps 0.6V), a small amount of current flows through the base to the emitter. This has the effect of altering the current-carrying ability of the device in the other path - from the collector to the emitter. The small amount of current flowing from base to emitter permits a much larger current to flow from collector to emitter. Base-emitter current doesn't drive collector-emitter current. Think of it more like operating a valve. In the forward bias range, tiny changes in base-emitter current causes big changes in the current-carrying ability of the collector-emitter path. Thus we have an amplifier. To flow through the collector, current still has to be provided by an external source (battery, power supply, etc.).
Pretty much anything described as an amplifier works in an analogous way, whether electrical or even mechanical. There is a power supply which provides the output power, an input signal, and the amplification device which simply controls the flow of power through the device in response to the input signal. No conservation laws need be violated.
