The Physics of Semiconductor Light Emission & Negative Resistance
A light emitting diode (LED) is a specialized compound semiconductor PN junction diode composed of materials such as Gallium Arsenide (GaAs), Gallium Nitride (GaN), or Indium Gallium Aluminum Phosphide (InGaAlP). When forward biased, electrons from the N-region and holes from the P-region are injected into the active junction zone where they undergo radiative recombination, releasing discrete packets of electromagnetic radiation (photons) whose wavelength ($\lambda$) corresponds to the material's semiconductor bandgap energy ($E_g = \frac{h \cdot c}{\lambda}$).
Unlike linear ohmic resistors, LEDs are non-linear electronic devices governed by the Shockley diode equation:
Below the threshold forward voltage ($V_f$), the diode is virtually non-conductive. Once applied voltage reaches $V_f$, the curve rises nearly vertically. A minute increase in voltage of just $0.1\text{ V}$ can cause forward current to jump by $300\%$ to $500\%$. Furthermore, as the junction heats up during conduction, its forward voltage drop decreases at approximately $-2\text{ mV/}^\circ\text{C}$ (negative thermal coefficient). If connected directly to a fixed voltage rail without a series ballast resistor, this positive thermal feedback loop induces thermal runaway, destroying the diode in fractions of a second.
Mathematical Current Limiting Formulations
1. Calculating Series Resistance ($R$)
By applying Kirchhoff's Voltage Law (KVL) around a single closed circuit containing a DC supply voltage $V_s$, a current limiting resistor $R$, and $N$ series connected LEDs:
Applying Ohm's Law ($R = \frac{V_R}{I_f}$), the exact required resistance is:
Where:
- $V_s$ = DC power supply voltage in Volts ($\text{V}$).
- $N$ = Total quantity of series LEDs wired in the string.
- $V_f$ = Forward voltage drop of each LED in Volts ($\text{V}$).
- $I_f$ = Target forward design current in Amperes ($\text{A}$). Note: convert milliamperes to amperes ($\text{mA} \div 1000$).
2. Resistor Thermal Power Dissipation & Derating
Electric current passing through the series resistor converts electrical potential energy into thermal heat energy via Joule heating:
Standard industrial component guidelines mandate a $50\%$ thermal derating rule ($2\times$ wattage factor). Operating a resistor continuously at $100\%$ of its nominal wattage rating elevates surface temperatures above $100^\circ\text{C}$, scorching PCB substrate materials, drifting resistance value, and inducing premature solder fatigue.
LED Color, Forward Voltage ($V_f$) & Semiconductor Bandgap Table
Because photon energy is inversely proportional to wavelength, shorter wavelength light (blue, ultraviolet) requires higher bandgap energy, directly dictating higher forward operating voltages:
| Color Spectrum | Semiconductor Material | Wavelength ($\lambda$) | Typical $V_f$ Drop | Typical $I_f$ Rating |
|---|---|---|---|---|
| Infrared (IR) | Gallium Arsenide (GaAs) | 850: 940 nm | 1.2: 1.6 V | 20: 50 mA |
| Red | AlGaAs / GaAsP | 620: 660 nm | 1.8: 2.2 V | 15: 20 mA |
| Orange / Amber | GaAsP / AlInGaP | 590: 610 nm | 2.0: 2.3 V | 20 mA |
| Yellow | GaAsP / AlInGaP | 570: 590 nm | 2.1: 2.4 V | 20 mA |
| Pure Green | Indium Gallium Nitride (InGaN) | 515: 535 nm | 3.0: 3.4 V | 20 mA |
| Blue | Indium Gallium Nitride (InGaN) | 450: 475 nm | 3.1: 3.5 V | 20 mA |
| White (Phosphor) | InGaN Blue + YAG:Ce Phosphor | Broad Spectrum | 3.1: 3.6 V | 20: 30 mA |
๐ Real-World Engineering Worked Example: 12V Automotive Dash LED String
Scenario: An automotive electronics engineer is designing an instrument cluster backlighting circuit powered by a nominal $12.0\text{V}$ vehicle accessory rail. The design calls for three series-connected blue indicator LEDs ($V_f = 3.3\text{ V}$ each), operating at a conservative $I_f = 15\text{ mA}$ ($0.015\text{ A}$) to maximize service life.
- Calculate Cumulative LED Forward Voltage Drop: $$V_{\text{LED, total}} = N \cdot V_f = 3 \times 3.3\text{ V} = 9.9\text{ V}$$
- Determine Voltage Drop Across the Series Resistor: $$V_R = V_s - V_{\text{LED, total}} = 12.0\text{ V} - 9.9\text{ V} = 2.1\text{ V}$$
- Calculate Exact Resistance ($R$): $$R = \frac{V_R}{I_f} = \frac{2.1\text{ V}}{0.015\text{ A}} = 140.0\text{ }\Omega$$
- Select Standard E24 Commercial Resistor: The nearest standard E24 resistor value is $150\text{ }\Omega$ (Brown ยท Green ยท Brown ยท Gold).
- Verify Actual Operating Current with 150 ฮฉ Resistor: $$I_{\text{actual}} = \frac{2.1\text{ V}}{150\text{ }\Omega} = 0.014\text{ A} = 14.0\text{ mA}$$
- Calculate Resistor Heat Dissipation & Specify Wattage: $$P = V_R \times I_{\text{actual}} = 2.1\text{ V} \times 0.014\text{ A} = 0.0294\text{ Watts } (29.4\text{ mW})$$ With a $2\times$ derating requirement ($2 \times 29.4\text{ mW} = 58.8\text{ mW}$), a standard miniature $1/4\text{ Watt}$ ($0.25\text{W}$ / $250\text{ mW}$) carbon film or SMD 0805 resistor will run cool and provide decades of continuous operation.
Frequently Asked Technical Questions
What happens if the supply voltage is lower than the LED forward voltage?
If DC supply voltage Vs is less than total string forward voltage (N ร Vf), the diode remains in its non-conductive region. No current flows through the circuit, and the LED will emit zero light. For instance, powering a 3.3V blue LED with a 3.0V battery requires an active inductive DC-DC boost converter or charge-pump driver (like a Joule Thief) to step voltage above the diode's bandgap threshold.
When should you use an active constant current driver instead of a resistor?
Resistors are simple and cost-effective for low-power indicator LEDs (< 50 mA). However, in high-power lighting (1W, 3W, 10W COBs drawing 350 mA to 3 A), or in automotive circuits where battery voltage swings between 11.5V and 14.8V, series resistors waste excessive battery power as heat and cause visible brightness flickering. High-power systems require switch-mode buck constant current regulators (e.g., based on PT4115 or LM3409) that maintain steady amperage with > 90% electrical efficiency.
What are the standard E-series resistor values (E12, E24, E96)?
Standardized by the International Electrotechnical Commission (IEC 60063), E-series divide each decade logarithmic decade (e.g., 10 to 100 ฮฉ) into mathematically equal geometric intervals based on component tolerance. E12 (10% tolerance) provides 12 values per decade (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82). E24 (5% tolerance) provides 24 values per decade, and E96 (1% tolerance) provides 96 precision values.
Can an AC power supply be used to power LEDs directly with a resistor?
LEDs only conduct during the forward half-cycle of an AC wave, but have very low peak reverse breakdown voltage ratings (typically V_R,max is only 5V). Applying unrectified 12V or 120V AC will reverse-bias the diode past breakdown, causing avalanche punch-through and destroying the chip. Always use a full-wave bridge rectifier, or wire an antiparallel 1N4148 diode across the LED to clamp reverse voltage safely below 0.7V.