Calculating Parallel Resistor Tolerance

This problem involves finding the tolerance of an equivalent resistor ($R$) when two resistors, $R_1$ and $R_2$, are connected in parallel. We are given the nominal values of $R_1$ and $R_2$, and their individual percentage tolerances.

Given Information

• Nominal value of Resistor 1 ($R_1$): 300 $\Omega$
• Tolerance of Resistor 1 ($\Delta R_1 / R_1$): 1%
• Nominal value of Resistor 2 ($R_2$): 200 $\Omega$
• Tolerance of Resistor 2 ($\Delta R_2 / R_2$): 1%

Step 1: Calculate the Nominal Equivalent Resistance

The formula for two resistors connected in parallel is:

$$ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} $$

Substitute the nominal values of $R_1$ and $R_2$:

$$ \frac{1}{R} = \frac{1}{300 \, \Omega} + \frac{1}{200 \, \Omega} $$

Find a common denominator (600) to add the fractions:

$$ \frac{1}{R} = \frac{2}{600 \, \Omega} + \frac{3}{600 \, \Omega} = \frac{5}{600 \, \Omega} $$

Invert the fraction to find $R$:

$$ R = \frac{600 \, \Omega}{5} = 120 \, \Omega $$

The nominal equivalent resistance is $120 \, \Omega$.

Step 2: Calculate the Absolute Tolerances

The absolute tolerance for each resistor is the percentage tolerance multiplied by its nominal value:

• Absolute tolerance for $R_1$: $\Delta R_1 = 1\% \times 300 \, \Omega = 0.01 \times 300 \, \Omega = 3 \, \Omega$.
• Absolute tolerance for $R_2$: $\Delta R_2 = 1\% \times 200 \, \Omega = 0.01 \times 200 \, \Omega = 2 \, \Omega$.

Step 3: Determine the Range of Possible Resistance Values

The actual resistance value for each component can lie within its tolerance range:

• $R_1$ can range from $300 \, \Omega - 3 \, \Omega = 297 \, \Omega$ to $300 \, \Omega + 3 \, \Omega = 303 \, \Omega$.
• $R_2$ can range from $200 \, \Omega - 2 \, \Omega = 198 \, \Omega$ to $200 \, \Omega + 2 \, \Omega = 202 \, \Omega$.

To find the tolerance of the equivalent resistance $R$, we calculate the minimum and maximum possible values for $R$ using these ranges.

Step 4: Calculate Minimum and Maximum Equivalent Resistance

We calculate the equivalent resistance using the boundary values:

• Minimum Equivalent Resistance ($R_{min}$): Use the minimum values of $R_1$ and $R_2$. $$ R_{min} = \frac{R_{1,min} \times R_{2,min}}{R_{1,min} + R_{2,min}} = \frac{297 \, \Omega \times 198 \, \Omega}{297 \, \Omega + 198 \, \Omega} = \frac{58806 \, \Omega^2}{495 \, \Omega} \approx 118.8 \, \Omega $$
• Maximum Equivalent Resistance ($R_{max}$): Use the maximum values of $R_1$ and $R_2$. $$ R_{max} = \frac{R_{1,max} \times R_{2,max}}{R_{1,max} + R_{2,max}} = \frac{303 \, \Omega \times 202 \, \Omega}{303 \, \Omega + 202 \, \Omega} = \frac{61206 \, \Omega^2}{505 \, \Omega} \approx 121.2 \, \Omega $$

Step 5: Calculate the Tolerance of the Equivalent Resistance

The deviation from the nominal value ($120 \, \Omega$) is:

• Maximum deviation = $R_{max} - R = 121.2 \, \Omega - 120 \, \Omega = 1.2 \, \Omega$.
• Minimum deviation = $R_{min} - R = 118.8 \, \Omega - 120 \, \Omega = -1.2 \, \Omega$.

The absolute tolerance for the equivalent resistance is $\pm 1.2 \, \Omega$. To express this as a percentage tolerance:

$$ \text{Percentage Tolerance} = \frac{\text{Absolute Tolerance}}{\text{Nominal Equivalent Resistance}} \times 100\% $$

$$ \text{Percentage Tolerance} = \frac{1.2 \, \Omega}{120 \, \Omega} \times 100\% $$

$$ \text{Percentage Tolerance} = 0.01 \times 100\% = 1\% $$

Final Answer

The tolerance of the equivalent resistor $R$ is 1%.