Step 1: Understanding the Concept:
The power factor of an AC circuit is defined as the cosine of the phase angle between voltage and current.
In a series LCR circuit, it can be calculated as the ratio of true resistance to the total impedance.
Step 2: Key Formulas or Approach:
Net reactance: \( X = X_L \sim X_C \).
Impedance of the LCR series circuit: \( Z = \sqrt{R^2 + (X_L - X_C)^2} \).
Power factor: \( \cos\phi = \frac{R}{Z} \).
Step 3: Detailed Explanation:
The given parameters of the circuit are:
Resistance \( R = 80\Omega \)
Inductive reactance \( X_L = 70\Omega \)
Capacitive reactance \( X_C = 130\Omega \)
First, calculate the net reactance.
Since \( X_C>X_L \), the circuit is capacitive.
Net reactance \( X = |X_L - X_C| = |70\Omega - 130\Omega| = |-60\Omega| = 60\Omega \).
Next, calculate the total impedance \( Z \) of the circuit:
\[ Z = \sqrt{R^2 + X^2} \]
\[ Z = \sqrt{80^2 + 60^2} \]
\[ Z = \sqrt{6400 + 3600} \]
\[ Z = \sqrt{10000} \]
\[ Z = 100\Omega \]
Now, calculate the power factor (\( \cos\phi \)), which is given as \( x \):
\[ x = \cos\phi = \frac{R}{Z} \]
Substitute the values of \( R \) and \( Z \):
\[ x = \frac{80}{100} \]
\[ x = 0.8 \]
The values for voltage (\( 230\text{ V} \)) and frequency (\( 50\text{ Hz} \)) are not needed as the reactances are already provided directly.
Step 4: Final Answer:
The value of the power factor \( x \) is \( 0.8 \).