Question

Calculate the maximum and minimum values of the following resistances defined by their tolerance bands: (a) $35\mathrm{\Omega }\pm 5\mathrm{\%}$ (b) $470\mathrm{\Omega }\pm 10\mathrm{\%}$

Short answer

The maximum and minimum resistance values for (a) are 36.75 Ω and 33.25 Ω, respectively, and for (b), they are 517 Ω and 423 Ω, respectively.

Step-by-step solution

Identify resistance value and tolerance

For each resistance, identify the nominal resistance value (R) and the tolerance (T). For (a), the nominal resistance is $35\mathrm{\Omega }$ and the tolerance is $5\mathrm{\%}$. For (b), the nominal resistance is $470\mathrm{\Omega }$ and the tolerance is $10\mathrm{\%}$.

Calculate the absolute tolerance range

Find the absolute tolerance range by multiplying the nominal resistance value (R) by the tolerance (T). For (a), this is $(35\mathrm{\Omega })\ast (0.05)=1.75\mathrm{\Omega }$. For (b), this is $(470\mathrm{\Omega })\ast (0.10)=47\mathrm{\Omega }$.

Calculate maximum and minimum resistance values

To find the maximum and minimum resistance values, add and subtract the absolute tolerance range from the nominal resistance value (R). For (a): - Maximum Resistance: $35\mathrm{\Omega }+1.75\mathrm{\Omega }=36.75\mathrm{\Omega }$ - Minimum Resistance: $35\mathrm{\Omega }-1.75\mathrm{\Omega }=33.25\mathrm{\Omega }$ For (b): - Maximum Resistance: $470\mathrm{\Omega }+47\mathrm{\Omega }=517\mathrm{\Omega }$ - Minimum Resistance: $470\mathrm{\Omega }-47\mathrm{\Omega }=423\mathrm{\Omega }$ So, the maximum and minimum resistance values for (a) are $36.75\mathrm{\Omega }$ and $33.25\mathrm{\Omega }$ respectively, and for (b), they are $517\mathrm{\Omega }$ and $423\mathrm{\Omega }$ respectively.

Key concepts

Electrical Resistance

Electrical resistance is a fundamental concept in the study of electronics and electrical circuits. Essentially, it is a measure of the opposition that a material offers to the flow of electric current. Imagine water flowing through a pipe: the resistance would be anything that makes it harder for the water to flow, such as a narrow passage. Similarly, in an electrical circuit, components like resistors restrict the flow of electrical current.

Resistance is measured in ohms ( $\mathrm{\Omega }$ ), named after the physicist Georg Simon Ohm who discovered the law that governs how current, voltage, and resistance interrelate. Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points, and inversely proportional to the resistance (R). The formula is $V=I\times R$.

Understanding this concept is crucial for solving problems involving resistance, such as calculating the maximum and minimum resistance values within a tolerance range, which can affect the functionality of electrical devices.

Tolerance Range

The tolerance range of a resistor tells us how much the actual resistance value can vary from the stated nominal value. Tolerance is expressed as a percentage and represents the precision of the resistor. A lower tolerance indicates a more accurate resistor that will more closely match its labeled value. For instance, a $100\mathrm{\Omega }$ resistor with a $\pm 5\mathrm{\%}$ tolerance could actually have a resistance anywhere between $95\mathrm{\Omega }$ and $105\mathrm{\Omega }$.

Manufacturers use tolerance bands, often colored stripes on the resistors, to indicate this variance. Knowing the tolerance is important when designing circuits where precise resistance values are crucial for the circuit's performance. It affects how a circuit regulates voltage, controls current, or shapes signal waveforms. For example, for precision timing circuits, a low-tolerance resistor would be essential to maintain the right operation.

Maximum and Minimum Resistance Values

To manage uncertainties in electrical circuit design, it's necessary to calculate the maximum and minimum resistance values that a resistor might realistically exhibit due to its tolerance. This ensures that the circuit can function correctly, even if the resistors vary within their tolerance ranges.

The calculation is rather straightforward once you have the nominal value and the tolerance percentage. As demonstrated in the step-by-step solution, you get the absolute tolerance by multiplying the nominal resistance value by the tolerance. This number is then added to (for the maximum resistance) and subtracted from (for the minimum resistance) the nominal value.

Understanding these calculations helps in assessing the reliability of a circuit under different conditions and can be crucial in high-precision applications where the resistances of components must be within a tight range to avoid malfunctions or degraded performance.