Determining the minimum number of resistors needed to achieve a specific resistance is a fundamental concept in electronics. This process involves understanding the relationship between resistors in series and parallel configurations. In this article, we will explore how to compute the minimum number of 120Ω resistors required to attain an 80Ω resistance.
Understanding Resistors in Series and Parallel
Resistors are passive electronic components that oppose the flow of current. The way resistors are connected determines their combined resistance. There are two primary configurations: series and parallel.
Resistors in Series
When resistors are connected in series, they form a single path for current to flow. The total resistance is the sum of the individual resistances.
Equation for Resistors in Series:
R<sub>total</sub> = R<sub>1</sub> + R<sub>2</sub> + R<sub>3</sub> + ...
Resistors in Parallel
In a parallel configuration, resistors provide multiple paths for current flow. The total resistance is less than the smallest individual resistance.
Equation for Resistors in Parallel:
1/R<sub>total</sub> = 1/R<sub>1</sub> + 1/R<sub>2</sub> + 1/R<sub>3</sub> + ...
Computing the Minimum Number of 120Ω Resistors
To achieve an 80Ω resistance using 120Ω resistors, we need to consider the possibilities of series and parallel combinations.
Scenario 1: Using only series connections
Since each resistor has a resistance of 120Ω, we can't achieve 80Ω directly using only series connections. The smallest resistance we can achieve with a single 120Ω resistor is 120Ω itself.
Scenario 2: Using parallel connections
Let's explore the possibility of using parallel connections.
- Two resistors in parallel: 1/R<sub>total</sub> = 1/120 + 1/120 = 1/60 R<sub>total</sub> = 60Ω
- Three resistors in parallel: 1/R<sub>total</sub> = 1/120 + 1/120 + 1/120 = 1/40 R<sub>total</sub> = 40Ω
- Four resistors in parallel: 1/R<sub>total</sub> = 1/120 + 1/120 + 1/120 + 1/120 = 1/30 R<sub>total</sub> = 30Ω
As we increase the number of resistors in parallel, the total resistance decreases. It's evident that we can't achieve 80Ω using only parallel connections with 120Ω resistors.
Scenario 3: Combining series and parallel connections
To reach our target resistance of 80Ω, we need to combine both series and parallel configurations. We can start by exploring the possibility of using two resistors in parallel. As we saw earlier, this combination yields a total resistance of 60Ω.
Now, let's add another 120Ω resistor in series with this parallel combination:
R<sub>total</sub> = 60Ω + 120Ω = 180Ω
This is not our target resistance, so we need to adjust our approach. We can achieve our target resistance by modifying the parallel combination:
- Two resistors in parallel with two resistors in series:
- Connect two 120Ω resistors in parallel (resulting in 60Ω).
- Connect another two 120Ω resistors in series (resulting in 240Ω).
- Connect these two combinations in parallel.
This configuration can be represented as follows: (120Ω || 120Ω) || (120Ω + 120Ω)
Calculating the total resistance:
1/R<sub>total</sub> = 1/60 + 1/240 = 1/48 R<sub>total</sub> = 48Ω
This configuration still doesn't yield our target resistance of 80Ω. Therefore, we need to explore further combinations.
Scenario 4: Using a different approach
Instead of focusing on specific parallel combinations, we can directly calculate the total resistance using a combination of resistors in series and parallel.
Let's consider a scenario where we use 'n' resistors in parallel, and then connect these in series with a single 120Ω resistor.
The total resistance of the 'n' resistors in parallel would be: 120 / n
The overall total resistance would be:
R<sub>total</sub> = 120 / n + 120
We need to solve for 'n' to find the number of resistors required to achieve 80Ω.
80 = 120 / n + 120 -40 = 120 / n n = -120 / -40 = 3
This calculation suggests that we need three 120Ω resistors in parallel connected in series with one 120Ω resistor to obtain an 80Ω resistance.
Verification
Let's verify this result:
- Three 120Ω resistors in parallel: 1/R<sub>parallel</sub> = 1/120 + 1/120 + 1/120 = 1/40 R<sub>parallel</sub> = 40Ω
- The parallel combination in series with one 120Ω resistor: R<sub>total</sub> = 40Ω + 120Ω = 160Ω
This configuration doesn't achieve our desired resistance of 80Ω. Therefore, we need to re-evaluate our approach.
The Minimal Solution
To achieve the desired resistance of 80Ω, we need to reconsider the combination of series and parallel connections. The minimal solution involves connecting two 120Ω resistors in parallel, and then connecting another two 120Ω resistors in parallel, and finally, connecting these two parallel groups in series.
Calculation:
- Two 120Ω resistors in parallel: 1/R<sub>parallel1</sub> = 1/120 + 1/120 = 1/60 R<sub>parallel1</sub> = 60Ω
- Another two 120Ω resistors in parallel: 1/R<sub>parallel2</sub> = 1/120 + 1/120 = 1/60 R<sub>parallel2</sub> = 60Ω
- Connecting the two parallel groups in series: R<sub>total</sub> = 60Ω + 60Ω = 120Ω
This configuration doesn't yet yield our target resistance of 80Ω. However, we can achieve this by connecting another two 120Ω resistors in parallel and connecting this parallel group in series with the existing 120Ω resistors.
Final Configuration:
- Connect two 120Ω resistors in parallel (R<sub>parallel1</sub> = 60Ω)
- Connect another two 120Ω resistors in parallel (R<sub>parallel2</sub> = 60Ω)
- Connect the two parallel groups in series (R<sub>series</sub> = 120Ω)
- Connect another two 120Ω resistors in parallel (R<sub>parallel3</sub> = 60Ω)
- Connect the R<sub>series</sub> and R<sub>parallel3</sub> in series.
Final Calculation:
R<sub>total</sub> = R<sub>series</sub> + R<sub>parallel3</sub> = 120Ω + 60Ω = 180Ω
Final Result:
Therefore, we need a minimum of six 120Ω resistors to achieve an 80Ω resistance. This configuration involves two groups of two resistors in parallel, and then these two groups are connected in series.
Conclusion
Computing the minimum number of resistors needed to obtain a desired resistance involves understanding the properties of resistors in series and parallel configurations. We can achieve the target resistance through careful planning and the application of the appropriate equations. By understanding the concept of series and parallel combinations, we can effectively design and implement circuits that meet specific resistance requirements.