I have a series resonant circuit driven with a voltage of 5V. The capacitor is 402nF and the inductor is 6.3uH. What is the resonant frequency and what is the maximum inductor current?

To determine the resonant frequency and the maximum inductor current in a series resonant circuit, we can follow these steps:

### 1. Resonant Frequency Calculation

The resonant frequency (\( f_0 \)) of a series LC circuit is given by the formula:

\[ f_0 = \frac{1}{2 \pi \sqrt{LC}} \]

Where:
- \( L \) is the inductance in henrys (H)
- \( C \) is the capacitance in farads (F)

Given:
- \( L = 6.3 \, \mu\text{H} = 6.3 \times 10^{-6} \, \text{H} \)
- \( C = 402 \, \text{nF} = 402 \times 10^{-9} \, \text{F} \)

Let's calculate the resonant frequency.

\[ f_0 = \frac{1}{2 \pi \sqrt{(6.3 \times 10^{-6}) (402 \times 10^{-9})}} \]

### 2. Maximum Inductor Current Calculation

At resonance, the impedance of the LC circuit is purely resistive and minimal (theoretically zero if there is no resistance in the circuit). The current is then limited by the source voltage and the series resistance (if any), but for an ideal LC circuit, the current can be very high.

The peak current (\( I_{max} \)) in the inductor at resonance can be found using:

\[ I_{max} = \frac{V_{max}}{X_L} \]

Where:
- \( V_{max} \) is the peak voltage across the circuit
- \( X_L \) is the inductive reactance at resonance

At resonance, \( X_L = \frac{V}{I} \)

For a sinusoidal AC supply, \( V_{max} = \sqrt{2} \times V_{rms} \).

Given \( V_{rms} = 5 \, \text{V} \):

\[ V_{max} = \sqrt{2} \times 5 = 5 \sqrt{2} \]

Using Ohm's Law and the fact that \( X_L = X_C \) at resonance, we use:

\[ I_{max} = \frac{5 \sqrt{2}}{R} \]

If we consider an ideal circuit (no resistance):

\[ I_{max} \text{ would theoretically be infinite.} \]

However, practically the resistance \( R \) would limit the current.

### Calculation

First, let's calculate the resonant frequency:

\[ f_0 = \frac{1}{2 \pi \sqrt{(6.3 \times 10^{-6} \text{ H}) (402 \times 10^{-9} \text{ F})}} \]

Let's compute this:

\[ f_0 = \frac{1}{2 \pi \sqrt{(6.3 \times 10^{-6} \times 402 \times 10^{-9})}} \]
\[ f_0 = \frac{1}{2 \pi \sqrt{2.5326 \times 10^{-15}}} \]
\[ f_0 = \frac{1}{2 \pi \sqrt{2.5326} \times 10^{-7.5}} \]
\[ f_0 = \frac{1}{2 \pi \times 5.0324 \times 10^{-8}} \]
\[ f_0 \approx 10 \text{ MHz} \]

Now for the maximum inductor current:

\[ I_{max} = \frac{5 \sqrt{2}}{R} \]

If there is negligible resistance:

\[ I_{max} \approx \text{very large (theoretically infinite)} \]

In practical circuits, the internal resistance and other factors limit the current.

### Summary

1. **Resonant Frequency**: \( f_0 \approx 10 \text{ MHz} \)
2. **Maximum Inductor Current**: Theoretically very high, practically limited by the circuit resistance.

Do you want more details or have any questions?

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### Follow-up Questions

1. What is the significance of the resonant frequency in a series LC circuit?
2. How does the presence of resistance in the circuit affect the resonant frequency?
3. Can you explain the concept of inductive reactance?
4. What is the role of a capacitor in a resonant circuit?
5. How would changing the inductance affect the resonant frequency?
6. What are some practical applications of resonant circuits?
7. How does the quality factor (Q-factor) relate to resonant circuits?
8. What are the potential consequences of having a high current in the inductor?

### Tip

Always double-check units and conversion factors when calculating electrical parameters to ensure accuracy.

Learn how to calculate resonant frequency and maximum inductor current in a series LC circuit driven by 5V voltage with a 402nF capacitor and 6.3uH inductor. Detailed step-by-step solution provided.

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