The Q of the coil using series connection measurement method is given by which of the following equations?
■ \( Q_x = \frac{(C_1 - C_2)(Q_1 Q_2)}{(C_1 Q_1 - C_2 Q_2)} \) [Option ID = 2982]
Conceptual Solution for Question 38
Question 38 asks for the Quality Factor ($Q_x$) of a coil measured using the Series Substitution Method on a Q-Meter. Here is the fundamental breakdown:
1. The Fundamental Principle: Series Resonance
A Q-meter operates on the principle of series resonance. At the resonant frequency ($f$):
- Inductive reactance ($X_L$) exactly cancels out Capacitive reactance ($X_C$).
- The voltage across the capacitor is magnified by the Quality Factor: \( Q = \frac{X_L}{R} = \frac{1}{\omega C R} \).
2. Measurement Steps
The circuit is tuned to resonance using only the internal work coil.
- Resonance capacitance: \( C_1 \)
- Measured Quality Factor: \( Q_1 \)
- Circuit resistance is given by: \( R_1 = \frac{1}{\omega C_1 Q_1} \)
The unknown coil is added in series, and the capacitor is retuned to find a new resonance.
- New resonance capacitance: \( C_2 \)
- New Quality Factor: \( Q_2 \)
- Total resistance is now: \( R_{total} = R_1 + R_x = \frac{1}{\omega C_2 Q_2} \)
- The unknown resistance is: \( R_x = \frac{1}{\omega C_2 Q_2} - \frac{1}{\omega C_1 Q_1} \)
- The unknown reactance is: \( X_x = \frac{1}{\omega C_2} - \frac{1}{\omega C_1} \)
3. Deriving the Final Formula
The \( Q \) of the unknown coil is the ratio of its reactance to its resistance: \( Q_x = \frac{X_x}{R_x} \).
Substituting the values derived in Step 2:
By simplifying the common denominators and cancelling the \( \omega \) terms, we arrive at the final equation:
Summary of Logic: The formula represents the ratio of the "Reactance Shift" to the "Resistance Shift" caused by adding the unknown component into the series resonant circuit.