In this case, you are being led astray by what you don't know. A speaker coil has a (more or less) constant impedance because it drives a cone against air resistance. In your case, with no speaker attached, it behaves as an inductor, not a resistor. From the numbers you provide, it will have an inductance of ~ 5 milliHenrys (.005 H). For an inductor, the current (and the magnetic field) will go down as frequency goes up. The relevant equation deals with impedance, not resistance, represented by Z.
Zl = 2 x pi x L x f
and this will equal 8 ohms at f = 250 Hz (about). So, for frequencies below about 250 Hz, the total impedance (sqrt(Zl^2 + R^2)) will increase with frequency, until at 250 Hz your current amplitude will be about 70% of its DC value. Further increases in frequency will continue to increase impedance and current will drop further as you increase the frequency. You probably don't need to hear about phase shifts.
The effect of all of this is to reduce power dissipation in your coil as frequency increases. But particularly at low frequencies you do need to keep a very close watch on your temperatures. Dissipating 800 watts in a coil the size you're considering stands a pretty good chance of melting the interior layers. It certainly nay cause the bobbin to melt or catch fire.
This is basically a classic window comparator, with stuff around it to make is actually useful in the particular application.
PIR sensors report changes in IR accross the sensor area. C2 removes the DC bias, and the circuit around IC1D amplifies the result and also does some frequency filtering. This is probably in part to reduce frequencies that aren't relevant and therefore just add noise, and in part to get the response of the overall sensor+filter that is useful for detecting motion.
IC1C and IC1B are the window comparator. R7, R8, R9, and R10 are a divider chain making voltages for the output of IC1D to be compared against. Just from the topology without looking at any numbers, you can see that the threshold for IC1C is higher than that for IC1B. Also see that the input signal into the window comparator (outout of IC1D) is fed in to the two comparators (IC1C and IC1B) at opposite polarity. In the "window" region, which is the voltage range between the - input of IC1C and the + input of IC1B, both amps will be driving low. Below the window region, IC1B will drive high and IC1C low. Above the window, IC1C will drive high and IC1B low.
The two comparator outputs are averaged by R11 and R12, then the result compared to a threshold by IC1A. This threshold is set so that IC1A drives high only when both comparator amps are driving low, meaning the voltage is in the window region.
The digital signal that indicates whether the sensor output is within the window region is capacitively coupled into this HT2812 thing. I didn't look that up, so I don't know what exactly it does, but from the transistor and speaker it is probably intended to produce a beep when motion is detected.
I'm not sure what the point of the switch in series with the KEY input is. When the switch is open, the HT2812 block won't receive the motion signal. If that is the intent, then powering everything down would be the more obvious approach, so there is probably some additional feature it supports. I don't know why you'd want to only sound a beep due to motion when a button is pressed, but that appears to be what what this circuit will do.
Best Answer
It's like an op-amp but it has the output DC set-point control mechanism already inside the chip: -
R7 connects the output to pin 2 which is regarded generally as the "inverting input". So, any external negative feedback you apply should avoid upsetting the internallt set dc bias, hence C1 and C5 in your circuit.
At high frequencies (maybe above 10kHz or 20kHz) I expect that C1 starts to become significant and this effectively places R3 in parallel with R1 and the gain reduces.