Hi again,
Well maybe this is even simpler than that. The reason i say this now is because when i calculate the time constant of the circuit and multiply that by 5, i see that the cap charges up in 125ms which is twice as fast as the longest time period in one of the pulse tests. This means the resistor will charge the cap, and after that it doesnt matter anymore what the switch does. This is because the cap charges up and it takes a full 1.8 seconds to discharge the cap by about 250 volts or so, so the next pulse becomes only 250v instead of 400v. So it looks like this particular circuit can take any pulse width or duration that you can throw at it. We're lucky here in that the discharge resistor is as high as 1.8k.
If you have a particular pulse duty cycle you would like to test on this circuit we can do that next. Just for a simple example, 100ms on, 500ms off.
In the attachment there is shown a test set. Various duty cycles are tested and the average power calculated at the end of 2000 seconds for each entry. Keep in mind this chart shows results from some 500 thousand calculations. How this test set is generated is as follows...
1. Start with a zero cap voltage Vc=0 and Vcc=420 volts, and an 'on' time of 100us and total period of 200ms.
2. Generate the first charge cycle using the equation for a charging cap with TWO resistors, calculating the new cap voltage Vc.
3. Generate the first discharge cycle using the equation for a discharging cap with ONE resistor, calculating the new cap voltage Vc.
4. Do steps 2 and 3 ten thousand times, noting the last charge voltage Vc1 and last discharge voltage Vc2.
5. Calculate the duty cycle D, calculate the power P from Vcc and the last Vc1 and Vc2, calculate the average power Pavg, tabulate.
6. Step the 'on' time by 100us and repeat steps 1 through 5, tabulating each Pavg result.
Finally, compare all the Pavg results looking for the highest value. Note that this is the entry for 2.7ms which generates an average power of 24.186 watts. This tells us that with the duty cycle that produces the highest power we will see less than 25 watts in the resistor. Since it is a 50 watt resistor the power rating for this resistor was a very good choice.
The other way to do it is to generate a series from the two equations for charge and discharge, then try to find a closed form for the series, then calculate the average voltage across the cap, then using the duty cycle calculate the average power in the resistor.
Table:
Code:
Ton Time Vc1 Vc2 Pavg D
-------- ------ ----- ----- ------ -------
0.000100 2000.0 15.4 13.8 3.287 0.000500
0.000200 2000.0 29.8 26.7 6.138 0.001000
0.000300 2000.0 43.3 38.8 8.617 0.001500
0.000400 2000.0 55.9 50.1 10.775 0.002000
0.000500 2000.0 67.8 60.7 12.656 0.002500
0.000600 2000.0 79.0 70.7 14.296 0.003000
0.000700 2000.0 89.5 80.1 15.728 0.003500
0.000800 2000.0 99.5 89.0 16.977 0.004000
0.000900 2000.0 108.9 97.5 18.067 0.004500
0.001000 2000.0 117.8 105.5 19.018 0.005000
0.001100 2000.0 126.3 113.1 19.845 0.005500
0.001200 2000.0 134.3 120.3 20.564 0.006000
0.001300 2000.0 141.9 127.1 21.188 0.006500
0.001400 2000.0 149.2 133.6 21.727 0.007000
0.001500 2000.0 156.2 139.9 22.192 0.007500
0.001600 2000.0 162.8 145.8 22.590 0.008000
0.001700 2000.0 169.1 151.5 22.929 0.008500
0.001800 2000.0 175.2 156.9 23.215 0.009000
0.001900 2000.0 181.0 162.1 23.454 0.009500
0.002000 2000.0 186.5 167.1 23.652 0.010000
0.002100 2000.0 191.9 171.9 23.813 0.010500
0.002200 2000.0 197.0 176.5 23.940 0.011000
0.002300 2000.0 201.9 180.9 24.037 0.011500
0.002400 2000.0 206.6 185.2 24.108 0.012000
0.002500 2000.0 211.2 189.2 24.154 0.012500
0.002600 2000.0 215.6 193.2 24.180 0.013000
0.002700 2000.0 219.8 197.0 24.186 0.013500
0.002800 2000.0 223.8 200.6 24.175 0.014000
0.002900 2000.0 227.8 204.1 24.148 0.014500
0.003000 2000.0 231.6 207.5 24.108 0.015000
0.003100 2000.0 235.2 210.8 24.056 0.015500
0.003200 2000.0 238.7 214.0 23.992 0.016000
0.003300 2000.0 242.2 217.1 23.919 0.016500
0.003400 2000.0 245.5 220.1 23.837 0.017000
0.003500 2000.0 248.7 223.0 23.747 0.017500
0.003600 2000.0 251.8 225.8 23.650 0.018000
0.003700 2000.0 254.8 228.5 23.546 0.018500
0.003800 2000.0 257.7 231.1 23.438 0.019000
0.003900 2000.0 260.5 233.6 23.324 0.019500
0.004000 2000.0 263.3 236.1 23.206 0.020000
0.004100 2000.0 265.9 238.5 23.084 0.020500
0.004200 2000.0 268.5 240.8 22.959 0.021000
0.004300 2000.0 271.0 243.1 22.832 0.021500
0.004400 2000.0 273.5 245.3 22.701 0.022000
0.004500 2000.0 275.8 247.5 22.569 0.022500
0.004600 2000.0 278.1 249.5 22.434 0.023000
0.004700 2000.0 280.4 251.6 22.299 0.023500
0.004800 2000.0 282.6 253.5 22.161 0.024000
0.004900 2000.0 284.7 255.5 22.023 0.024500
0.005000 2000.0 286.8 257.3 21.884 0.025000