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LED-based lighting is replacing incandescent and gas-discharge lamps in many situations. As practical lighting sources, and considering their directional nature, LED fixtures usually use multiple LEDs arranged in an array, with “m” LEDs in a serial string and “n” such strings in parallel *(Fig. 1)*.

For the array configuration and the exponential I-V characteristics model of a single LED,^{1} it’s also possible to have a similar model representing the loading of such an array.

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Since the single LED is a diode with a nonlinear forward-voltage/current, driving it with a fixed voltage source is generally not recommended. Rather, a serial (current-limiting) resistor is needed to equalize the power source and the load,*(Fig. 2)*.

However, sizing this resistor requires a significant analytical manipulation. Using the single-LED exponential model, a Kirchoff’s voltage law (KVL) equation around the circuit loop gives:

which can be rearranged to:

where a, b, and c are model parameters for a selected LED, as defined in the reference.

This is a challenge to solve symbolically, but it can be solved numerically with mathematical software tools. Alternatively, it can be solved graphically by separating the two key I-V relationship equations, v+ R_{x}I_{F} = V_{s} and I_{F} = ae^{bv} + c.

For example, using a Lumileds Luxeon Rebel ES LED with a 3.2-V dc source and a desired drive current of 0.5 A, the horizontal current line intercepts the LED curve at 2.92 V *(Fig. 3)*. Therefore, the series resistor will be R_{x} = (3.2 – 2.92)/0.5 = 0.56 Ω.

However, the graphical approach would not handle an LED array effectively, and the numerical method works better. To represent the array current i_{A}, the exponential model for the (m × n) array is modified, since voltage drops across current-limiting resistor R_{n} and sense-resistor R_{Sense} are usually small compared with driving-source V_{s}:

With the array model, the switch-mode buck-regulating current driver in a closed-loop configuration *(Fig. 4) *can be described by a set of equations starting at the feedback point, circling around the control loop, and ending at the driver output:

• Current sensing:

V_{f} = i_{A} • R_{Sense }(4)

• Error amplifier:

V_{er} = A(V_{Ref} – V_{f}) (5)

• Effective error:• PWM ramp:• Duty cycle, D:

V_{Ramp}(D • T_{s}) = V_{ef }(8)

• Power stage:With effort, all the equations can be consolidated into a single, closed-loop equation:This is a transcendental equation with an exponential term that prevents us from solving it analytically, but it can be solved using computational techniques and software tools.

**Reference **

1.“Generate Realistic Models for LED Current Versus Voltage” Keng C. Wu; *Electronic Design*, Vol. 61, No. 1, p. 52, Jan. 10, 2013, http://electronicdesign.com/power/generate-realistic-models-led-current-versus-voltage

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