For what you're doing, you're gonna have to rethink a bit. From the data sheet, it appears that each 2" segment looks like
simulate this circuit – Schematic created using CircuitLab
Since each LED should have a current-limiting resistor, you'll need 450 of them, about 500 ohms, 1/4 watt each. The resistors will need to be mounted somewhere where they have good airflow, since they will dissipate about 100 watts total. (10 amps times 12 volts is 120 watts, and it's got to go somewhere.) Just bundling them together and putting them in a hidden box is almost certain to cause Bad Problems.
Your final assembly should look something like
simulate this circuit
except extended to 150 units.
Each of the 3 drive lines needs to provide 12 volts at about 3 amps, or you can drive all 3 simultaneously with a total of 9 amps, so a 10-amp supply will probably do. There is a small chance, though, that driving all the LEDs on may draw slightly more than 10 amps, and the power supply will get unhappy. I don't have enough information to be sure. If you want to play it safe, cut back on the current a bit and use something like 560 or 620 ohm resistors. You may even want to mix values a bit to produce different drive currents, in order to get just the color you want. Experiment on a single segment before you commit to a final design. Be aware, though, that if you drive all of them the result will be (approximately) white. If you only want to produce amber light, don't turn on the blues. As a matter of fact, if you only want amber, forget about hooking up the blue leads entirely and cut your workload by 30%.
Your logic schematic is not ideal in this case, since your master drive could easily be folded into the 3 color drive signals, but I'll keep it anyways.
The master drive is relatively easy, and can be done with an NPN transistor, or actually with an NPN Darlington (you won't get 10 amps with a single stage driven by LSTTL logic). What you really want is an n-type MOSFET. Your drive circuit would look like
simulate this circuit
The MOSFET should have a current rating of 20 volts or better, and 10 amps or better. However, that is very modest as MOSFETs go.
The 3 color channel drives are a little trickier, and for these you need p-type MOSFETs. The circuit is complicated by the fact that LSTTL will not handle 12 volts on the output without Letting The Magic Smoke Out, so you need to make slightly more complicated switches.
simulate this circuit
I've shown the NPN buffer transistor as a 2N3904, but almost any small-signal NPN will do. 2N3904s are cheap and readily available. Try someplace like Jameco.
I hope this helps.
I specialize in the clicking of brains.
I remember trying to figure this circuit out when I was around eleven. (The one in my old book used light bulbs, put in Halloween-mask eyes.) Here's my version below. The main trick to these is to re-draw the schematic so it reveals familiar patterns.
With yours, first turn it rightside up, and you'll see that it's actually two amplifier stages connected by capacitors. And, the signals are connected in a loop.
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Below is the rightside-up circuit without that looped capacitor connection:
Each transistor is wired as a common-emitter amplifier stage. Each amp will both invert the signal and also make it larger. If we apply a pulse to point A above, a large upside-down pulse will appear at point B. And this inverted pulse goes through a coupling capacitor C1 to point C, which is the input of the next stage Q1. Q1 inverts it again, and the twice-amplified signal appears at the output at point D. All together it's a 2-stage amplifier. Probably you could hook up a microphone and loudspeaker, and use it that way.
So, what happens if we connect the output directly to the input? Positive feedback. Then any small pulse will go through the loop from point A to point D, getting bigger each time. Or in other words, it will break into oscillation.
With C2 restored, we'd expect to see high-frequency sine waves, where their frequency is caused by the time-delay through the whole loop of amplification. It's similar to when you hold a microphone a bit too close to an auditorium loudspeaker. But in reality, this whole system has way too much gain, so it gets overloaded and goes nonlinear. It won't make a high-frequency sine-wave oscillation, instead it clips into big slow square waves. (I guess it's more like holding your auditorium-microphone directly against the loudspeaker cone.) In that case the output slams all the way to 9V and to zero, and the speed of the oscillation is determined by charging and discharging of the two capacitors, which mostly happens through the 100K resistors.
Once you know how it works, you can sit down and plot out the various events. Start out with one capacitor zero volts, and the other charged to 9V, then figure out what happens next, then next after that. You might have to go through several cycles to see how it settles into constant blinks. All the components are symmetrical, so it blinks equally back and forth like a logic flip-flop. Its official name comes from that: non-stable or "a-stable" flip-flop blinker circuit.
That's just one tack on an explanation. Paraphrasing Feynman: if you don't have three or four separate approaches to explain something, you don't really understand it.
This circuit can use very, very low DC voltage yet still keep running. Besides LED blinkers, I've seen it used as various beepers, signal injectors, even analog-synthe instruments (with tiny capacitor values, like 0.01uF etc.) A similar circuit with a big iron transformer can generate 120VAC 60Hz, for an electric shocker or as an automotive "Power Inverter" for low-power appliances. Or use coils wrapped on a CRT ferrite HV flyback transformer and make a mini Tesla Coil or a 20KV power supply. The exact same circuit is in a solar-powered pendulum toy, where the LEDs are replaced by electromagnet coils, with a tiny ceramic magnet on a little pendulum getting kicked back and forth when light shines on the solar cells (with four 1cm solar cells in series, for about 2V power supply in sunlight, far less w/indoor fluorescents.) Or, use very large resistors on the base connection, small capacitor values, and add a few-inches pickup wire to one transistor base, to create the world's cheapest Theremin or touch-sensitive audio generator.
Best Answer
This is a fun circuit! You will enjoy figuring this out. A couple of clues for you...
Enjoy!