Which of the shown graphs may represent the reactance of a series L-C combination?

Options

A.
B.
C.
D.

Answer

To determine which graph may represent the reactance of a series L-C combination, we can follow these steps:

### Step 1: Understand the Reactance of L-C Combination
The total reactance \( X \) of a series L-C circuit is given by:
\[ X = X_L - X_C \]
where \( X_L = \omega L \) (inductive reactance) and \( X_C = \frac{1}{\omega C} \) (capacitive reactance). Here, \( \omega = 2\pi f \) is the angular frequency.

### Step 2: Express Reactance in Terms of Frequency
Substituting \( \omega \) into the reactance equation, we get:
\[ X = 2\pi f L - \frac{1}{2\pi f C} \]
This equation shows how the reactance changes with frequency \( f \).

### Step 3: Identify Resonance Frequency
The resonance frequency \( f_r \) occurs when \( X_L = X_C \), or:
\[ 2\pi f_r L = \frac{1}{2\pi f_r C} \]
Solving for \( f_r \):
\[ f_r = \frac{1}{2\pi \sqrt{LC}} \]
At this frequency, the total reactance \( X \) becomes zero.

### Step 4: Analyze the Behavior of Reactance
- For frequencies \( f < f_r \), \( X_L > X_C \) which means \( X > 0 \) (inductive behavior).
- For frequencies \( f > f_r \), \( X_L < X_C \) which means \( X < 0 \) (capacitive behavior).

### Step 5: Graph Characteristics
- The graph of reactance \( X \) vs. frequency \( f \) will cross the x-axis at the resonance frequency \( f_r \), indicating where \( X = 0 \).
- For \( f < f_r \), the graph will be above the x-axis (positive reactance).
- For \( f > f_r \), the graph will be below the x-axis (negative reactance).

### Step 6: Evaluate Given Graphs
- **Graph A**: Starts at zero frequency, which is not possible.
- **Graph B**: Similar issues with zero frequency.
- **Graph C**: Does not cross the x-axis, indicating no resonance point.
- **Graph D**: Crosses the x-axis at a certain frequency, indicating resonance, and shows positive reactance for lower frequencies and negative for higher frequencies.

### Conclusion
The graph that represents the reactance of a series L-C combination is **Graph D**.

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