Need better full wave RMS Voltage Formula?

If you have access to a computer while you're doing your calcs, I'd recommend using Spice to look at the problem. LTSpice/SwitcherCAD is free, and easily up to the task.

If you insist on calculating things "by hand," then consider that there are two time periods: the first is when the capacitor is charging from the source, and the second is when it is discharging through the resistive load. The first will be close to a segment of a sinewave, offset by the voltage drop in the rectifier diodes, which can be significant, and also modified by the drop in the resistance of the source (transformer winding resistance, mainly)--which should be fairly small and often is ignored. The second is an exponential decay, determined just by C and R. One problem is that it's common to assume the exponential decay starts at the peak voltage, but with small capacitance, that's not true, and not even a good approximation. The output voltage will follow the sinusoid beyond the peak voltage. The point at which the circuit transitions from the first region to the second is when the negative slope of the sinusoid becomes greater than the slope of the exponential decay: V/RC.

You can put all that together and come up with a pretty accurate estimate of the waveform, and find the average DC from that, but it's a whole lot easier to just let Spice do it for you. Then it's easy to add in transformer winding resistance, leakage inductance, the inductance of filter inductors you specifically add, etc.

Cheers, Tom

See....

Newsgroups: alt.binaries.schematics.electronic Subject: From S.E.D - Need better full wave RMS Voltage Formula - RectifierRipple.pdf Message-ID:

for how this was calculated prior to Spice.

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC\'s and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
|       http://www.analog-innovations.com           |    1962     |
             
         America: Land of the Free, Because of the Brave

There's no closed-form solution. It requires iteration (that listing was from an ancient magnetic-card-reader calculator). Spice is an iterative solver.

You can do pretty well graphically... sine wave and a straight edge ;-)

...Jim Thompson

--
|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
|  Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    |
|  Phoenix, Arizona            Voice:(480)460-2350  |             |
|  E-mail Address at Website     Fax:(480)460-2142  |  Brass Rat  |
|       http://www.analog-innovations.com           |    1962     |
             
         America: Land of the Free, Because of the Brave

"Hawker"

** You need to carry out a sanity check.

A filter electro should not be subjected to large, 120 Hz ripple voltages - ie 50% of the peak supply or more.

Bad for the cap as the max ripple current rating is likely to be exceeded.

Eg: for a 1000uf, 50 volt cap with 35 volts of peak ripple,

I = C dv/dt = 5 amps.

...... Phil

On 3/27/2007 1:41 PM, The digits of Jim Thompson's hands composed the following:

Wow thanx, but a bit more than I can sort though. I was not able to grok your notes. Sorry. Seem this should be able to be dispelled down to a single formula. Given Frequency*2, RMS input AC voltage, capacitance and load (in amps or resistance) what is the RMS DC voltage assuming full wave rectified. The reason SPICE won't cut it is I have a bunch of dependant formulas in several configurations. I need to put the formula into a spreadsheet with the rest of the data to come up with slope formula for a program given this and every other set of input data that is part of this monstrosity.

thanx Hawker

So...put the whole thing into SPICE. Or feed SPICE parametric data for just this part of it.

How much is a closed-form formula worth to you? Would a reasonably compact iterative solution with a few input parameters work? (e.g., frequency, diode drop, capacitance, load resistance-or-current, whether full or half-wave rectification. Assumptions: constant diode drop when forward biased, no transformer/mains resistance [a "stiff" voltage driving the rectifier], ...) What exactly do you need for output? Average DC voltage? RMS output voltage? Peak output voltage? Peak-to-peak ripple voltage? What accuracy?

Cheers, Tom

On 3/27/2007 7:33 PM, The digits of Tom Bruhns's hands composed the following:

Well in essence this client is giving me a rectified AC voltage and has a small filter cap on it. I need to PWM a large DC load for a constant VA drive with it so I need to compute, given the load, what the RMS DC will be so I can figure out duty cycle. I know the input voltage, load and capacitor. Given that the input voltage will be "universal" or

85-265VAC before the full wave rectifier the error in voltage drop will me minimal compared to the line voltage and can be ignored (or assumed .7V). We can assume in infinitely low impedance source for this calculation. An accuracy of a few percent (5%?) should be good enough. It would be nice to know ripple but only RMS is really required. the only parameter not in your request is line frequency which could be 50 or 60 hz (100 or 120 for full wave). The device will always be full wave.

In the end I may end up just figuring this out my measuring. The issue is the source could be 60 or 50hz and my 50Hz supply can't give me the current I need to test this. Hopefully his will.

Hmmm...well, it wasn't much of a request on my part, more on yours I think! I did say that frequency would be an input parameter. I can see my way clear to doing the calculation, assuming zero source impedance and zero drop in the diodes. You didn't say how much it was worth to you...

If you're thinking of measuring some scenarios, I'll again ask, why not just use Spice? LTSpice will give you the RMS of a waveform with a mouse click, and setting up to do a bunch of calcs like you need is very easy. There's always enough current available in a Spice source. ;-) Also, if you are thinking of measuring, there's no reason you can't scale things to low current; a given R*C value will give you the same answer, regardless of what R is, given the assumption of zero source resistance and zero or constant diode drop

-- at least for the case of an RC discharge. A constant current discharge will give the same answer if you scale I proportionally to C, so 10A and 10000uF will give the same answer as 10mA and 10uF -- because dV/dt is the same in both cases. The zero source impedance will always keep the cap at the value of the input sinusoid until the sinusoid drops faster than the load is discharging the cap, and then the cap goes into either an exponential or a linear discharge ramp. Because of diode drops and finite source impedance, that won't be perfectly true, but for the voltage range you mentioned, it should give you a result easily within 5%.

Cheers, Tom

Try 'calculus', at least that what it was called when I attended High School.

Have fun.

Stanislaw.