Value of impedance of series LCR circuit is given by:

Options

A. `Z=sqrt(R^(2)+(X_(C)+X_(L))^(2))`
B. `Z=sqrt(R^(2)+(X_(L)-X_(C))^(2))`
C. `Z=sqrt(R^(2)+(X_(L)+X_(C))^(2))`
D. `Z=sqrt(R+(X_(L)+X_(C))^(2))`

Explanation

To find the impedance of a series LCR (Inductor-Capacitor-Resistor) circuit, we can follow these steps:

### Step 1: Understand the Components
In a series LCR circuit, we have three components:
- Resistor (R)
- Inductor (L) with inductive reactance \(X_L\)
- Capacitor (C) with capacitive reactance \(X_C\)

### Step 2: Identify Reactances
The inductive reactance \(X_L\) is given by the formula:
\[ X_L = 2\pi f L \]
where \(f\) is the frequency of the AC source and \(L\) is the inductance.

The capacitive reactance \(X_C\) is given by:
\[ X_C = \frac{1}{2\pi f C} \]
where \(C\) is the capacitance.

### Step 3: Determine the Phase Relationship
In a series LCR circuit, the inductive reactance and capacitive reactance are out of phase with each other. This means that:
- If \(X_L > X_C\), the circuit behaves more like an inductor.
- If \(X_C > X_L\), the circuit behaves more like a capacitor.

### Step 4: Calculate the Net Reactance
The net reactance \(X\) of the circuit can be calculated as:
\[ X = X_L - X_C \quad \text{(if } X_L > X_C\text{)} \]
or
\[ X = X_C - X_L \quad \text{(if } X_C > X_L\text{)} \]

### Step 5: Write the Impedance Formula
The total impedance \(Z\) of the circuit is given by:
\[ Z = \sqrt{R^2 + X^2} \]
where \(X\) is the net reactance calculated in the previous step.

### Step 6: Final Expression
Thus, we can express the impedance in terms of the reactances:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \quad \text{(if } X_L > X_C\text{)} \]
or
\[ Z = \sqrt{R^2 + (X_C - X_L)^2} \quad \text{(if } X_C > X_L\text{)} \]

### Conclusion
The impedance of a series LCR circuit can be summarized as:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]