The big breakthrough for me was the realization that a transistor is similar to a 4 pin relay, but with a shared ground between the main circuit and the control, resulting in 3 pins.
The next step is to realize that solid-state devices usually control current instead of voltage, but that we can add voltages, currents and resistances to the circuit that let us work with voltage in a linear region of a device's response curve. So our normalized 0-1 control corresponds linearly to 0-1 on the main circuit. From there, it's straightforward to build amplifiers where the control voltage or current is thousands or even millions of times smaller than the main.
After that it gets.. complicated. It took me 4 years of math and physics to finally understand solid state theory and be able to analyze large circuits by subdividing them into simpler linear sub-circuits for my degree. Then the really interesting stuff happens when we abandon all of that and convert to the frequency domain using the Fourier, Laplace or Z transform. So discrete signals have periodic frequencies and periodic signals have discrete frequencies. Which lets us analyze the transient and steady state portions of a signal separately and gain valuable insights about what a circuit will do.
Of course I've mostly forgotten all of that. One word of advice: keep your college textbooks. I still find concise explanations there that haven't made it onto the internet over 20 years later.
Edit: IMHO, the above way of translating between abstraction and application is the primary advantage of getting a degree from a university. For anyone young reading this, I highly recommend applying to the best schools you can. If you just settle, then you might miss out on the underlying theory behind each discipline. You'll want the theory later when you've forgotten everything like everyone else, because you'll be able to re-derive everything you've learned from first principles when you need it.
The next step is to realize that solid-state devices usually control current instead of voltage, but that we can add voltages, currents and resistances to the circuit that let us work with voltage in a linear region of a device's response curve. So our normalized 0-1 control corresponds linearly to 0-1 on the main circuit. From there, it's straightforward to build amplifiers where the control voltage or current is thousands or even millions of times smaller than the main.
After that it gets.. complicated. It took me 4 years of math and physics to finally understand solid state theory and be able to analyze large circuits by subdividing them into simpler linear sub-circuits for my degree. Then the really interesting stuff happens when we abandon all of that and convert to the frequency domain using the Fourier, Laplace or Z transform. So discrete signals have periodic frequencies and periodic signals have discrete frequencies. Which lets us analyze the transient and steady state portions of a signal separately and gain valuable insights about what a circuit will do.
Of course I've mostly forgotten all of that. One word of advice: keep your college textbooks. I still find concise explanations there that haven't made it onto the internet over 20 years later.
Edit: IMHO, the above way of translating between abstraction and application is the primary advantage of getting a degree from a university. For anyone young reading this, I highly recommend applying to the best schools you can. If you just settle, then you might miss out on the underlying theory behind each discipline. You'll want the theory later when you've forgotten everything like everyone else, because you'll be able to re-derive everything you've learned from first principles when you need it.