Hello. I have obtained a copy of this above circuit which is built using a blinking LED. The capacitor will slowly charge and eventually become an open circuit equivalent as no more direct current can flow through it, which causes the transistor to turn "off" meaning no current flows from the collector junction to the emitter junction because no more base current reaches the transistor. While this is happening, the blinking LED receives a voltage from the DC source to motivate the IC contained therein to blink repeatedly. The brightness of the blinking diminishes gradually as the transistor cuts "off" and the circuit is broken.
My question is how can I derive the DC and AC analysis of this circuit mathematically? For DC analysis, the capacitor can be replaced by an equivalent open circuit after a long time, which means that transistor is "off" and there is no conduction through any circuit elements. The voltage at the top of the LED is 14V. For AC analysis, there is no AC signal source, so do I have to replace the battery with a cosinusoidal source? How can I derive the differential equation relating the base current as a function of time to known circuit parameters? I found that capacitor current equals current through the bottom resistor plus current through the base junction, but all of these are functions of time and cannot be related to each other in any way.
Thank you for reading and good day.