The impedasnce of sereis `L-C-R` circuit in an `AC` circuit is

Options

A. `sqrt(R+(X_L-X_C))`
B. `sqrt(R^2+(X_L^2+X_C^2))`
C. `R`
D. none of these

Explanation

To find the impedance of a series L-C-R circuit in an AC circuit, we can follow these steps:

### Step 1: Understand the components of the circuit
In a series L-C-R circuit, we have:
- R: Resistance
- L: Inductance
- C: Capacitance

### Step 2: Define the reactances
The reactance due to the inductor (X_L) and the capacitor (X_C) are given by:
- \( X_L = \omega L = 2\pi f L \)
- \( X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \)

### Step 3: Calculate the total reactance
The total reactance (X) in the circuit is the difference between the inductive reactance and capacitive reactance:
\[ X = X_L - X_C \]

### Step 4: Write the formula for impedance
The impedance (Z) of the series L-C-R circuit is given by the formula:
\[ Z = \sqrt{R^2 + X^2} \]
Substituting the expression for total reactance, we have:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

### Step 5: Final expression for impedance
Thus, the impedance of the series L-C-R circuit can be expressed as:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

### Conclusion
From the options provided, the correct expression for the impedance of a series L-C-R circuit is:
**Option B: \( \sqrt{R^2 + (X_L - X_C)^2} \)**