The condition of minimum impedance is:

Options

A. `X_(C ) ne X_(L)`
B. `X_(C ) = X_(R )`
C. `X_(C ) ne X_(R )`
D. `X_(C ) = X_(L)`

Explanation

To solve the question regarding the condition of minimum impedance in an RLC circuit, we can follow these steps:

### Step 1: Understand the Impedance Formula
The impedance \( Z \) in an RLC circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( R \) is the resistance,
- \( X_L \) is the inductive reactance,
- \( X_C \) is the capacitive reactance.

### Step 2: Identify the Condition for Minimum Impedance
To find the condition for minimum impedance, we need to minimize the expression for \( Z \). The term \( (X_L - X_C) \) is crucial here.

### Step 3: Set the Condition for Minimum Impedance
The impedance \( Z \) will be at its minimum when the difference between the inductive reactance and the capacitive reactance is zero:
\[ X_L - X_C = 0 \]
This implies:
\[ X_L = X_C \]

### Step 4: Relate Reactance to Frequency
The inductive reactance \( X_L \) is given by:
\[ X_L = 2\pi f L \]
and the capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{2\pi f C} \]
Setting these equal gives us the resonance condition:
\[ 2\pi f L = \frac{1}{2\pi f C} \]

### Step 5: Conclusion
Thus, the condition for minimum impedance in an RLC circuit occurs when:
\[ X_L = X_C \]
This is known as the resonance condition.

### Final Answer
The condition of minimum impedance is when the inductive reactance equals the capacitive reactance, i.e., \( X_L = X_C \).

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