EMI BYU 38

38. Series LC Circuit Analysis

Part 1: Capacitance for V_C = 220 V

The voltage across the capacitor in a series LC circuit is given by the voltage divider rule (magnitude):

$$ V_C = V_{source} \left| \frac{X_C}{X_L – X_C} \right| $$

We require $V_C = V_{source} = 220 \text{ V}$. Therefore:

$$ \left| \frac{X_C}{X_L – X_C} \right| = 1 \implies X_C = |X_L – X_C| $$

Case A (Trivial): $C \to 0 \implies X_C \to \infty$. The circuit is effectively open, so the voltmeter reads the full source voltage. ($C=0$ is a mathematical solution).

Case B (Non-Trivial): For finite $C$, we must have:

$$ X_C = X_L – X_C \implies 2X_C = X_L $$ $$ \frac{2}{\omega C} = \omega L $$ $$ C = \frac{2}{\omega^2 L} $$

Substitute values:

$$ C = \frac{2}{(10^5)(1)} = 2 \times 10^{-5} \text{ F} = 20 \, \mu\text{F} $$

Part 2: Forbidden Capacitance

A value is “forbidden” if it causes the current or voltage to become undefined or infinitely large (dangerous). This happens at Resonance in an ideal LC circuit ($Z = 0$).

$$ X_L = X_C $$ $$ \omega L = \frac{1}{\omega C} \implies C = \frac{1}{\omega^2 L} $$ $$ C = \frac{1}{(10^5)(1)} = 1 \times 10^{-5} \text{ F} = 10 \, \mu\text{F} $$