FAQ- those %@*&! scaling resistors - do the math

[Richard Hull]

I note that on this day, 1/4/24, that Electronic Goldmine is offering a 1%, 1 watt, 500 megohm resistor for $5.00. Typically such resistors are 5%.
Sounds like a good deal. We need to involve our selves with only two formulae and have enough algebra behind us to move the knowns about to find a single unknown.

1. E=IR
2. P=I^2R

As we are dealing with large resistances and large voltages and small milliamp currents, it will help if we have a scientific calculator and work with scientific notation, while remembering the units demanded in the equations

I am not here to teach how this is done but to talk about precision and the best resistors to have in an effort to scale your output voltage with the best accuracy.

1. the absolute best resistors to have on hand are 100 and 200 megohm 1 or 2 watt resistors of 1% or 2% tolerance.
2. if you are planning on using a d'Arsonval meter, use only a 50 microamp meter movement.
3. a digital meter is a matter you will have to deal with, but make its input impedance in the 10 megohm range.

How accurate can we make a scaler meter arrangement for measuring 0-50kv?
A great rule of thumb is use a 100megohm resistor for each 10Kv to be scaled. 100ua current will flow per 10kv if series strung
With a 50ua meter it would be best to use a string of 5, 200 meg resistors to make a 50ua current flow through our 50 ua meter to show 50kv full scale.

Using this example only, what power will the resistors have to handle at 50kv? (50x10e-6)^2 x 1x10e9 = 2.5 watts.
As all 5 200meg resistors are 1 watt that is 5 watt total dissipation We are OK here and will not send our resistors up in smoke.

Now how accurate will we be? 5 1% resistors in a string can, under worst case conditions be off 5% (Incorrect! The total error is 1% ... See posts below)
I have corrected by edit all that follows.

1% or 50,000 volts can be a max error of 500 volts. If we can only afford 2% resistors then the error could be 1,000 volts off. If far more normal 5% resistors are used then the error could be +/-2,500 volts in error.

Since we need a gig-ohm above, what about two of those Electronic Goldmine resistors?? 2,1% 500meg resistors gives only a max error at 50kv of
+/- 500volts! However, we have only 2 watts total resistor capability and at full voltage they will be really hot and probably shift value outside of there 1% tolerance. Be mindful of your power dissipation in any string of resistors to scale.

The idea is to use as few resistors in a string as possible of 1% or 2% tolerance with the appropriate wattage as possible!

In short, you need to figure this out 6 ways from Sunday.

All high voltage metering chains and even current measurement chains can be dangerous and kill you if you get to close to the resistor chain and in some instances the metering readout device!! Put your readout devices in an all metal, grounded panel!

edit 1/12/24 Make sure to read all responses in all FAQs. I get things wrong, now corrected above thanks to the first response from Rich Feldman below. I try to correct mistakes, as above, in the original post herein and admit to the correction. Again, read all responses. I'm not always correct on a first pass FAQ and rely on other watchdogs to point them out so that I might make this first post ever more correct.

Richard Hull

[Rich Feldman]

>>Now how accurate will we be? 5 1% resistors in a string can, under worst case conditions be off 5%

Friendly critique here:
If we put any number of 1% resistors in series, how can the total be off by more than 1%?

I agree with Richard, that one should leave generous margins of power dissipation per resistor
if accuracy (even linearity) are critical.
Will go farther and remind people to mind the voltage rating per resistor.
In a notorious example, I got some 10 MΩ 2 watt carbon composition resistors (new old stock) at a flea market.
Current was more than 5% nonlinear on the way up to 2500 volts per R (only 0.6 watts) and back. This is not a temperature coefficient thing, it's a voltage dependent R thing.

Also beware of using high-gigohm resistors in glass tubes for kV metering. They are meant for pico and nano amp metrology, and can be abused with kilovolts even if not greatly heated by the power dissipation.

[Richard Hull]

Rich is correct in retrospect. If you use all 1% resistors and the resistors are modern ceramic types, the entire string will be 1% and the current ought to track well. However, this is provided you follow the voltage recommendation by the manufacturer. The resistors offered by EG were rated at 15kv. Obey that rule for more linear current. They know their stuff. 25kv across each of the two resistors, above, would not only over-watt them as, I mentioned, but is far beyond the manufacturer's voltage rating. The longer a series string is, the less likely an arc over will occur across any single resistor. Figure on ohmic value needed versus resistor voltage rating and the string's ability to handle the wattage!

If you are using a d'arsonval meter movement and not a high impedance digital meter, the meter resistance is pitiably small and should point out any gross errors in its calibration or the dropping resistor string. However such a meter is incapable of hyperfine readings anyway, like a digital meter can. So for those who can accept a +/- 500v -1000v reading error, the old mechanical meter is just fine.

Thanks for the correction, Rich. I hope anyone reading my original will read this next post here.

I have over one hundred of the high gig-ohm resistors in glass. These range from a near dead short in electrometers of 1 gig-ohm to a virtual open circuit at one tera-ohm. Such glass encased resistors are designed for under a hundred volts each! They will only be accurate in the pico-amp to fempto-amp range, provided their glass bodies are absolutely free of grease, dust, and moisture. These are not meant to be used in voltage dividers, but as a single dropping resistor to develop voltage at a grid of an electrometer tube or the gate of a special mosfet. Such professional electrometers are usually ranged in voltages of no more than 10 volts, currents as small as 10e-13amps and in pico-coulombs. I own and use several professional Keithley electrometers. When one experiences that we live in a world of constant potential energy charge exchanges, moment to moment, using these instruments, you realize just how free and on the loose electrons are.

For those interested, accurately measured currents of 10e-15 amps are possible using a special sapphire input jeweled vibrating reed electrometer. Here, in precision work, 1 to 10 electron differential currents can be detected. I have a Keithley vibrating reed electrometer, but haven't the skill or competence to set up such a system in isolation that could boast such a resultant effort on my part.

Having been involved deeply at one time in measuring static potential energies, I could go on and on, but I demur.

Richard Hull