Ryan, you need to think about both voltages and currents. For a quick approximate AC analysis, treat the transistors as having infinite beta and zero base-to-emitter voltage. Try thinking through how the transistors work, rather than seeking equations. If you can think it through, you can derive equations for yourself. This approach has worked for me for over fifty years.
Given AC Vin, what is AC signal voltage across R2? What is AC current through R2? What is AC current through Q1's collector?
Given that R1 is connected to Q2's emitter, what is AC voltage across R1? What is AC current through R1? Given this and Q1's AC collector current, what is AC current through Q2's emitter? What is AC current through R3? What is AC voltage across R3? What is AC voltage at Q3's emitter? What is Vout/Vin?
You are correct that the voltage gain of Q1 is small because of the cascoded connection to Q2's emitter. The purpose of the cascode is to make the AC voltage across R1 small, to reduce the capacitive coupling between Q2's collector and Q1's base. This was important to prevent oscillation in VHF tuners in vacuum-tube analog TV receivers in the 1950s and 1960s. 6BK7 and 6BQ7 and 6DJ8 are dual-triode tubes intended for VHF cascode amplifiers.I agree with David that this particular circuit with these particular resistor values is probably meant for teaching purposes. Changing some of the resistor values would make this circuit useful. Try answering my questions above, and let us know what you find.
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