I disassembled the connection box and discovered a hardware hacker's dream! In this box, there is a simple low voltage DC supply, a relay to switch the 120VAC to the load, and (in some) a transistor driver to operate the relay. I deduced that the sensor head itself runs on DC and outputs a DC control signal. This leads to a redesign that could have all our PIR sensors running on 9V batteries and outputting logic levels to computer controllers or Event Control Timers (ECTs). Alternatively, we could have the PIRs running off a common 24 VAC sprinkler or doorbell transformer, and retain the relay. The relay can directly switch 5 Amps at 120VAC, 24 VAC or DC, or act as a contact closure for input to a computer or ECT. This latter plan is what I decided to pursue.
The first model I found is an ElectriPak BC8010K, sold by Orchard Supply Hardware for $9.99. I suspected the low-end PIRs at Target were similar in design, so I picked up one of them, too. The Target units are BC8008K, by Intelectron, and sell for only $5.10. Note the similar model number families!
Look inside the domed cover plate. There should be a flat plastic box, held in place by two small screws on the sides. Remove these screws and the long captive screw that passes through the box. Turn the module over, and you will find the circuit board, with the relay and a handful of discrete components. About 2/3 of the way up the sides or at the top of the case, there should be little bumps on the plastic housing, one on each side. Slide a small screwdriver blade between the circuit board and the case, pry gently, and the board will pop out.
Connected to the board will be black and white power supply wires with labels on them, and two more black/white pairs that lead to the lamp sockets. Cut off or unsolder the two white lamp wires. Unsolder the two black lamp wires, and clean out at least one of the holes they were soldered through. Remove and toss the lamp sockets in the junk box. Near the remaining (supply) black wire, there will be a small resistor with a molded capacitor running parallel to it. On mine, these were labeled R1 and C1. Next to them (in line) should be another resistor, 100 Ohms, labeled R2 on mine. Remove and discard R1 and C1, either by cutting the leads off near the board or by unsoldering them. Clean out one of the holes where R1 and C1 connect to R2. This is the end away from the black wire's tie point.
Solder a 6" length of black wire to one of the pads where the black lamp socket wires were wired. Solder a 6" length of white wire to the junction of R1, C1 and R2 that you cleaned out, above. I labeled this point (A) in the schematic.
The circuit for this unit is a little simpler, leaving out the filter capacitor and the transistor for driving the relay. Otherwise it's about the same. Proceed as above, for the BC8010K, except the components to remove are the capacitor (C1) and resistor R3, 330k, in parallel with it. The new black wire goes on exactly as above. The added white wire should go where the end of C1 farthest from the black feed wire is, in a little cluster of pads next to the diodes. I labeled this point (A) again.
That's all the wiring and modifications! Snap the board back into its housing, turn over, and reattach with the two small self-tapping screws. You now have the original PIR unit, with four wires (two black and two white) hanging out of it. The two white wires go to your 24 VAC sprinkler or doorbell transformer. The two black wires are relay contacts that are normally open, then close when the sensor detects motion in its field of view.
Set the Time knob or switch to the "TEST" position. This makes the sensor turn on and off as quickly as it can.
The Sensitivity knob needs to be set by trial and error, under the operating lighting conditions.
The BC8008K (and to a lesser extent, the BC8010K) needs a short time, say 20 to 30 seconds, to start working when power is applied. No problem in the usual working situation, but when you are testing it, don't be alarmed - it isn't broken!