In the given problem, we have an RLC parallel circuit where the supply voltage is 100 V, and the resistance $R = 120 \, \Omega$. The inductive reactance $X_L$ and capacitive reactance $X_C$ at resonance are both given as $63 \, \Omega$. The question asks for the calculation of the supply current and the currents in the inductor and capacitor at resonance.

Given:

• Supply Voltage $V_s = 100 \, V$
• Resistance $R = 120 \, \Omega$
• Inductive Reactance $X_L = 63 \, \Omega$
• Capacitive Reactance $X_C = 63 \, \Omega$
• Resonant Frequency = 25 cycles (implied resonance condition)

To Calculate:

1. Supply Current at Resonance ($I_s$)
2. Currents in Inductor and Capacitor ($I_L$ and $I_C$)

Explanation:

• At resonance, in a parallel RLC circuit, the inductive reactance and capacitive reactance cancel each other out because $X_L = X_C$, and the net reactance of the circuit becomes zero.
• The current through the inductor $I_L$ and capacitor $I_C$ will be equal in magnitude but opposite in phase. Thus, they cancel each other, and the total current supplied by the source is primarily determined by the resistive branch.

(a) Supply Current at Resonance:

At resonance, the total impedance $Z_{\text{total}}$ is dominated by the resistance $R$, and the supply current is determined by Ohm's Law: $I_s = \frac{V_s}{R}$ $I_s = \frac{100 \, V}{120 \, \Omega} = 0.833 \, \text{A}$ Therefore, the supply current at resonance $I_s = 0.833 \, \text{A}$.

(b) Currents in the Inductor and Capacitor:

The currents through the inductor and capacitor can be calculated using the reactance values: $I_L = \frac{V_s}{X_L}$ $I_L = \frac{100 \, V}{63 \, \Omega} \approx 1.59 \, \text{A}$ Similarly, for the capacitor: $I_C = \frac{V_s}{X_C}$ $I_C = \frac{100 \, V}{63 \, \Omega} \approx 1.59 \, \text{A}$ Thus, $I_L = 1.59 \, \text{A}$ and $I_C = 1.59 \, \text{A}$.

Summary:

• The supply current at resonance $I_s = 0.833 \, A$.
• The current through the inductor $I_L = 1.59 \, A$.
• The current through the capacitor $I_C = 1.59 \, A$.

Let me know if you'd like more details, or if you have any questions! Here are some follow-up questions to deepen your understanding:

1. What would happen to the supply current if the resistance was decreased?
2. How would the resonant frequency change if the inductance or capacitance were altered?
3. What happens to the phase relationship between voltage and current in the circuit at resonance?
4. What are the implications of resonance in practical electrical circuits like power transmission systems?
5. How would you calculate the impedance of the circuit if it were not at resonance?

Tip:

In parallel resonance, while the voltage across the inductor and capacitor is high, their currents cancel each other, leaving only the current through the resistor as the net supply current.