The Physics of LED Current Limiting
Light Emitting Diodes (LEDs) are non-linear current-driven semiconductors. Unlike light bulbs that exhibit positive resistance, an LED's internal resistance plummets exponentially once voltage exceeds its forward threshold ((V_f)). Connecting an LED directly to a fixed DC voltage source without current limiting results in instantaneous thermal runaway and component burnout.
The Series Resistor Formula
The ballast resistor (R) absorbs the voltage difference between your power rail (V_s) and the total forward voltage drop of your series LED string ((N imes V_f)):
The power dissipated in each resistor is (P_{ ext{resistor}} = I_f^2 imes R). To ensure component longevity and prevent PCB charring, always choose a physical resistor with at least 2× the calculated wattage rating (e.g. use a 1/2W resistor if calculating 0.20W).
Why You Must Never Wire LEDs in Parallel with a Single Shared Resistor
A classic beginner mistake is placing 5 parallel LEDs together connected to a single large resistor. Due to semiconductor manufacturing tolerances, the LED with the lowest forward voltage ((V_f)) will draw more current. As it gets hotter, its (V_f) drops further (negative temperature coefficient), hogging even more current until it burns out. Then the next LED hogs the current and dies in a domino cascade. Always give each parallel branch its own dedicated resistor!