Determine the Thevenin equivalent resistance RTh for the given circuit.
Understand the Problem
The question is about calculating the equivalent resistance in a circuit where resistors are arranged in a combination of series and parallel. We need to find the Thevenin equivalent resistance, RTh, for the given circuit.
Answer
The Thevenin equivalent resistance is $R_{Th} \approx 4.67 \, \Omega$.
Answer for screen readers
The Thevenin equivalent resistance, $R_{Th}$, is approximately $4.67 , \Omega$.
Steps to Solve
- Identify Series and Parallel Combinations
In the circuit, we can see that $R_1$ (3 Ω) is in series with a combination of $R_2$ (10 Ω) and $R_3$ (2 Ω) in parallel.
- Calculate the Resistance of Resistors in Parallel
To find the equivalent resistance for resistors in parallel, use the formula:
$$ \frac{1}{R_{parallel}} = \frac{1}{R_2} + \frac{1}{R_3} $$
Substituting the values:
$$ \frac{1}{R_{parallel}} = \frac{1}{10} + \frac{1}{2} $$
- Compute the Equivalent Resistance
First, calculate the reciprocal:
$$ \frac{1}{R_{parallel}} = \frac{1}{10} + \frac{1}{2} = \frac{1 + 5}{10} = \frac{6}{10} = \frac{3}{5} $$
Now, take the reciprocal to find $R_{parallel}$:
$$ R_{parallel} = \frac{5}{3} , \Omega \approx 1.67 , \Omega $$
- Add Series Resistance
Now add $R_1$ to $R_{parallel}$ to find the total equivalent resistance $R_{Th}$:
$$ R_{Th} = R_1 + R_{parallel} $$
Substituting the values:
$$ R_{Th} = 3 + \frac{5}{3} $$
- Simplify the Result
Convert 3 into a fraction with a common denominator:
$$ 3 = \frac{9}{3} $$
Therefore,
$$ R_{Th} = \frac{9}{3} + \frac{5}{3} = \frac{14}{3} , \Omega \approx 4.67 , \Omega $$
The Thevenin equivalent resistance, $R_{Th}$, is approximately $4.67 , \Omega$.
More Information
The Thevenin equivalent resistance is a simplification that allows complex circuits to be analyzed easily. By finding the equivalent resistance, we can better understand how the circuit will behave when connected to a load.
Tips
- Confusing series and parallel connections. Always identify the configuration before calculating.
- Failing to convert values to a common denominator when adding fractions.
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