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The real question here is finding a transistor array which has a gain near to 20.
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There isn't one. At a collector current of 20 mA, all commonly available transistor arrays have a DC forward gain much greater than 20.
I don't understand why a transistor with a gain of 20 is the real question. A schematic would help greatly in clarifying your question. Also, can you share how driving an LED with a high-gain darlington transistor would ruin it? I can explain things more clearly if I know what your thinking is. It is clear from your statement in post #11 that your understanding of how transistor gain affects external circuits is incorrect. I brought up Ohm's Law because that is what determines the current in an LED, not a driver transistor's gain.
An LED is a "current mode" device. That is, its major characteristics (luminous flux, forward voltage, etc.) are defined by the current going through it. Because it is a diode, it has no inherent current limiting mechanism, and will burn up if the current through it is not limited externally. Since most circuits are powered by a constant voltage source, the usual way to limit the current through an LED is to put a resistor in series with it. Maxim and others make constant-current drivers for LEDs, but the power equation is the same. Even if you vary the LED brightness with a pulse-width modulated waveform, there still needs to be something to limit the current pulses to a value that is safe for the device. I'm not sure what you mean by efficiency, but for an individual LED the power equation always is the same:
(source voltage - LED Vf) x LED current = power dissipated in the current limiter.
For multiple LEDs you can use the persistence of the human retina to reduce the total circuit power by multiplexing the LEDs so they are not all on all the time. There still have to be current limiters in the circuits, but not as many.
If your goal is to have the drive transistor limit the LED current by not going into saturation, that is what Maxim does internally in their constant current LED driver chips. It is possible to do this with discrete components but not simple, and difficult to stabilize over temperature. Also, note that the system power dissipation will be exactly the same as if the transistor switch were saturated with a fixed resistor as the current limiter. Other than multiplexing, there is no way to reduce the total power requirement of a multi-digit display system (compared to an all direct-drive system).
ak