Resolved Problems about Measuring Instruments

Problema + Hard 16-1. Source: Problem prepared by the author of the site.

Calculate the values of R₁, R₂ and R₃ so that the voltmeter shown in the figure below can measure up to 1 volt when the selector switch is in the x1 position, 10 volts in the x10 position, and 100 volts in position x100. The measuring instrument is a microammeter with full scale of 100 µA and internal resistance of 500 Ω conform shown in Figure 16-01.1

Solution of the Problem + Hard 16-1.

Initially, let's calculate the necessary voltage for the instrument to have the maximum deflection. We know that the full scale is 100 µA and the internal resistance is 500 Ω. So the voltage is:

V_g = R_i I_g = 500 x 0.0001 = 0.05 V

Solution for position x1

Therefore, in order to measure 1 volt we need an external resistance that should cause a voltage drop of V_r = 1 - 0.05 = 0.95 volts. So it's easy to calculate the value of R₁ when the selector key is positioned at x1. Therefore:

R₁ = (V - V_g) / I_g = (1 - 0.05) / 0.0001 = 9,500 Ω

Another way to calculate the value of R₁ would be to calculate the total resistance that the circuit should have and then subtract the value of R_i. Look:

R_total = V / I_g = 1 / 0.0001 = 10,000 Ω

Now we subtract from R_total the value of R_i, obtaining:

R₁ = R_total - R_i = 10,000 - 500 = 9,500 Ω

Solution for position x10

Looking closely at the circuit we see that when the selector switch is in position x10, resistor R₁ will be in series with R₂ as well as R_i. Soon we can calculate the value of R₂ by calculating the total resistance and then subtract the values of R_i and R₁. Therefore:

R_total = V / I_g = 10 / 0.0001 = 100,000 Ω

Subtraindo de R_total o valor de R_i + R₁, obtem-se:

R₂ = R_total - R_i - R₁ = 100,000 - 500 - 9,500 = 90,000 Ω

Solution for position x100

Following the same reasoning above and realizing that when the selector switch is in position x100, resistor R₁, R₂ as well as R_i will be in series with R₃. Then, you can calculate the value of R₃ by calculating the total resistance and then subtract the values of R_i, R₁ and R_{2< /sub>}. Therefore:

R_total = V / I_g = 100 / 0.0001 = 1,000,000 Ω

Subtracting from R_total the value of R_i + R₁ + R₂, we get:

R₃ = R_total - R_i - R₁ - R₂

Substituting numerical values, we find for R₃ the value of:

R₃ = 900,000 Ω

Note that in this type of circuit we will always have the sum of the previous resistors for each scale we choose. This is normally the circuit used in instruments professionals.

To finish, we can calculate the sensitivity of the instrument:

S = 1/ I_g = 1 / 0.0001 = 10,000 Ω/V

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