Is there software out there that will take a schematic as input and simulate its functioning?
I don't have a lot of cash for components and tools, so this would be a cheap and easy way for me to learn more about electronics.
We usually call that simulation, not emulation. If we were to emulate the board, that means having another device perform similarly. There are plenty of circuit simulators, most famous of which is Spice (LTspice is one popular version). Qucs, ktechlab, klogic (limited to digital), and gnucap are a few others; for those without a graphical interface, there typically is one separately, such as oregano or gspiceui.
As @LoneTech said, there are plenty of circuit simulators. My two cents:
If you are a student I would HIGHLY recommend National Instruments Multisim. With a valid student ID, or student email, or even a offical transcript you can pick up a copy for $40 dollars. It also comes packaged with Utilboard which is a perfboard layout program.
Some of the higher-level programs for general analog simulation are Multisim and OrCad's suite (including PSpice). Other programs which offer limited capabilities are Simulink in MATLAB and National Instruments LabView. If you are in higher education, you may find that these are available in a computer lab.
The Altera Quartus II software for their FPGAs and CPLDs supports schematic entry, for digital simulation and synthesis. Most people use VHDL or Verilog, though. There is a free version.
SIMetrix is another great tool for circuit simulation and design.
It should be noted, that simulation tools should be used carefully, because they can give wrong results
Check out SystemVision. It's completely online, free, and has a ton of models and designs for you to use for learning.
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