There are two methods of calculating the values of multiplier resistors for a multirange voltmeter. In the first method, each multiplier is calculated the same as for a single-range voltmeter. Assume that you wish to extend the range of a 1-mA movement to measure 0- 10, 0- 100, and 0- 1000 volts, and you also want a 0- 1-V range. Since full-scale deflection equals 1 Von the 0- 1-V range (V = IM RM = 0.001 A x 1000 Q = I volt), no multiplier is needed. The total resistance (RT OT) needed to limit meter current (IM) to 1 mA on the 0- 10-V range is RrnT = V w vJ IM = 10 V/ 0.001A= 10,000 Q

Since the resistance of the meter (RM) is 1000 ohms, then the multiplier resistance RMuLT is 9000 ohms. A second method of calculating the values of voltmeter multiplier resistors is the series-multiplier arrangement in which the multiplier resistors are connected in series. R1 is the multiplier resistor for the 0-10 volt range. For the 0- 100-V range, R1 is in series with R2. Therefore, the value of the multiplier resistance for the 0- 100-V range is equal to R1 plus R2. Similarly, the multiplier resistance for the 0-1000-V range is equal to R1 plus R2 plus R3.

Now, let’s calculate the values for a series multiplier voltmeter. We will use the same 1-mA, 1000-Q meter movement that we used previously. Since this movement indicates 1 volt for a full-scale deflection, no multiplier resistor is needed for the 0- 1-V range. Therefore, your first step is to calculate the multiplier resistance needed for the 0- 10-V range. Again, using Ohm’s law, find the total resistance (RToT) needed to limit meter current (IM) to 1 mA at this range:

RrnT = V w.xl lM = 10 V/ 0.001 A = 10,000 Q

Therefore, multiplier resistor R1 for the 0-10-V range equals 10,000n minus the 1000-Qmeter resistance, or 9000 n. Thus far, the procedure is the same as in the other method, and the value of the multiplier resistor is the same for the 0- 10-V range. Having found the series multipliers for the 0- 1- and 0- 10-V ranges, let’s calculate the total resistance needed for the 0- 100-Vrange:

RrnT = vM AXl’IM = 100 v; o.001 A = 100,000 n

Subtracting the meter resistance from the total resistance, you find that the multiplier resistance for the 0- 100-V range is 99,000 ohms. Thus far, this method is the same as the previous, but now the multiplier resistance is made up of R I plus R2 in series. Therefore, since you need 99,000 n for the multiplier resistance and R I equal s 9000 n, R2 must equal 90,000 n.

Similarly, for the 0-1000-V range:

RT oT = VMNJ’IM = 1000 v; o.001 A = 1,000,000 n

Thus, RMuLT = RToT – RM = 1,000,000 – 1000 = 999,000 n. But RMuLT = R1 + R2 + R3.

Thus, RMuLT = 999,000 n = 9000 + 90,000 + R3

And R3 = 999,000 – 99,000 = 900,000 n

No matter which method you use, the value of the multiplier resistance for each range remains the same. However, in the first method, the multiplier is a single resistor, while in the second method, on all but the first extended range, it is made up of resistors in series.

References

https://www.allaboutcircuits.com/textbook/direct-current/chpt-8/voltmeter-design/

http://sound.whsites.net/articles/meters.htm

http://www.rfcafe.com/references/radio-craft/meter-shunts-multipliers-may-1931-radio-craft.htm