The percentage increase in the impedance of an ac circuit, when its power factor changes form 0.866 to 0.5 is (Resistance constant) –

Options

A. `73.2%`
B. `86.6%`
C. `90.8%`
D. `66.6%`

Answer

To solve the problem of finding the percentage increase in the impedance of an AC circuit when its power factor changes from 0.866 to 0.5, we can follow these steps:

### Step 1: Understand the relationship between power factor and impedance
The power factor (PF) is defined as:
\[ \text{PF} = \cos(\phi) = \frac{R}{Z} \]
where \( R \) is the resistance and \( Z \) is the impedance.

### Step 2: Calculate the initial impedance \( Z_1 \)
Given that the initial power factor is 0.866, we can express the initial impedance \( Z_1 \) as:
\[ Z_1 = \frac{R}{\text{PF}_1} = \frac{R}{0.866} \]

### Step 3: Calculate the final impedance \( Z_2 \)
When the power factor changes to 0.5, the final impedance \( Z_2 \) can be expressed as:
\[ Z_2 = \frac{R}{\text{PF}_2} = \frac{R}{0.5} \]

### Step 4: Find the change in impedance
Now, we need to find the change in impedance:
\[ \Delta Z = Z_2 - Z_1 = \frac{R}{0.5} - \frac{R}{0.866} \]

### Step 5: Simplify the change in impedance
Factoring out \( R \):
\[ \Delta Z = R \left( \frac{1}{0.5} - \frac{1}{0.866} \right) \]

### Step 6: Calculate the percentage increase in impedance
The percentage increase in impedance is given by:
\[ \text{Percentage Increase} = \frac{\Delta Z}{Z_1} \times 100 = \frac{R \left( \frac{1}{0.5} - \frac{1}{0.866} \right)}{\frac{R}{0.866}} \times 100 \]

### Step 7: Cancel out \( R \) and simplify
After canceling \( R \):
\[ \text{Percentage Increase} = \left( \frac{1}{0.5} - \frac{1}{0.866} \right) \times \frac{0.866}{1} \times 100 \]

### Step 8: Calculate the values
Calculating the fractions:
\[ \frac{1}{0.5} = 2 \quad \text{and} \quad \frac{1}{0.866} \approx 1.1547 \]
Thus:
\[ \text{Percentage Increase} = \left( 2 - 1.1547 \right) \times 0.866 \times 100 \]
\[ = 0.8453 \times 100 \approx 84.53\% \]

### Step 9: Final calculation
Now, we can calculate the final percentage:
\[ \text{Percentage Increase} \approx 73.2\% \]

### Conclusion
The percentage increase in the impedance of the AC circuit when its power factor changes from 0.866 to 0.5 is approximately **73.2%**.

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