The Multi-LED Current Limiting Resistor & Wattage Sizer calculates the required series resistance and power rating to operate LEDs safely from DC power sources.

### Governing Electrical Formulas (Ohm's Law) Light-emitting diodes have non-linear voltage-current curves. Once forward voltage ($V_f$) is reached, current increases exponentially; without a series ballast resistor to drop excess voltage, the LED undergoes thermal runaway and burns out: - **Total LED String Forward Voltage**: $$V_{f,\text{total}} = V_f \times N_{\text{LEDs}}$$ *(Total string forward voltage must remain less than the power supply voltage $V_s$)* - **Voltage Dropped Across Resistor**: $$V_R = \max(0, V_s - V_{f,\text{total}})$$ - **Required Ballast Resistance**: $$R = \frac{V_R}{I_{\text{target}}}$$ - **Resistor Power Dissipation**: $$P = V_R \times I_{\text{target}}$$ - **Safety Rating Rule**: Uses a $2\times$ safety margin factor ($P_{\text{safe}} = P \times 2$) to select practical commercial ratings ($1/4\text{W}$, $1/2\text{W}$, or $1\text{W}+$) to avoid overheating.

### Worked Circuit Calculation Consider driving **3 Red LEDs** in series from a **$12\text{V}$ DC** automotive or bench supply at a target current of **$20\text{ mA}$** ($0.020\text{ A}$): - **Supply Voltage ($V_s$)**: $12.0\text{ V}$ - **LED Forward Voltage ($V_f$)**: $2.2\text{ V}$ per diode - **Total String $V_f$**: $2.2 \times 3 = 6.6\text{ V}$ - **Voltage Dropped on Resistor ($V_R$)**: $12.0 - 6.6 = 5.4\text{ V}$ - **Required Resistance**: $$R = \frac{5.4\text{ V}}{0.020\text{ A}} = 270\,\Omega$$ - **Resistor Heat Dissipation**: $5.4\text{ V} \times 0.020\text{ A} = 0.108\text{ W}$ ($108\text{ mW}$) - **Safe Wattage Rating**: $0.108 \times 2 = 0.216\text{ W} \le 0.25\text{ W} \implies$ standard **$1/4$ Watt** carbon-film or metal-film resistor.

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