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Resonance Calculator

Calculate resonant frequency, bandwidth, and quality factor in RLC circuits with interactive visualization

Parameters

Hⓘ
Inductance in the circuit
Fⓘ
Capacitance in the circuit
Ωⓘ
Resistance in the circuit
Resonance Analysis
Enter values to calculate resonant frequency, bandwidth, and quality factor

Calculated Values

Resonant Frequency:
503.2921;Hz503.2921;Hz
Quality Factor:
31.6228;31.6228;
Bandwidth:
15.9155;Hz15.9155;Hz
Period:
0.0020;s0.0020;s

Examples

Example 1: Audio Circuit

A 0.1H inductor and 1μF capacitor with 10Ω resistance.

  • Resonant Frequency: 503.3503.3
  • Quality Factor: 31.631.6
  • Bandwidth: 15.915.9

Example 2: Radio Circuit

A 1mH inductor and 100pF capacitor with 1Ω resistance.

  • Resonant Frequency: 503292.4503292.4
  • Quality Factor: 3162.33162.3
  • Bandwidth: 159.2159.2

Example 3: Power Circuit

A 10H inductor and 1μF capacitor with 100Ω resistance.

  • Resonant Frequency: 50.350.3
  • Quality Factor: 31.631.6
  • Bandwidth: 1.591.59

Resonance in RLC Circuits

Resonance occurs in an RLC circuit when the inductive reactance equals the capacitive reactance, causing the circuit to oscillate at its natural frequency. At resonance, the impedance is purely resistive and the current is maximum.

The resonant frequency (f₀) is the frequency at which the circuit naturally oscillates. It is calculated using f₀ = 1/(2π√(LC)), where L is the inductance and C is the capacitance. This is also known as the natural frequency of the circuit.

The quality factor (Q) is a measure of how sharp the resonance peak is. It is calculated as Q = 2πf₀L/R = 1/(2πf₀CR). Higher Q values indicate a sharper, more selective resonance peak.

Bandwidth (BW) is the range of frequencies over which the circuit response is significant. It is calculated as BW = f₀/Q. A higher quality factor results in a narrower bandwidth, making the circuit more selective.

At resonance, the voltage across the inductor and capacitor can be much larger than the applied voltage due to the Q factor. This voltage magnification is useful in applications like radio tuning and signal filtering.

Key Concepts

  • Resonant Frequency: f₀ = 1/(2π√(LC)) (natural oscillation frequency)
  • Quality Factor: Q = 2πf₀L/R (sharpness of resonance)
  • Bandwidth: BW = f₀/Q (frequency range of significant response)
  • At Resonance: Xₗ = Xc, Z = R (minimum impedance)
  • Voltage Magnification: Vₗ = Vc = Q × Vₛ (at resonance)
  • Power Factor: PF = 1 (unity at resonance)

Real-World Applications

  • Radio Tuning: Selecting specific frequencies
  • Audio Filters: Crossover networks and equalizers
  • Power Factor Correction: Improving efficiency
  • Signal Processing: Frequency selective circuits
  • Oscillators: Generating stable frequencies

Physics Equations

Resonant Frequency:
f0=12πLCf_0 = \frac{1}{2\pi\sqrt{LC}}
Quality Factor:
Q=2πf0LR=12πf0CRQ = \frac{2\pi f_0 L}{R} = \frac{1}{2\pi f_0 CR}
Bandwidth:
BW=f0QBW = \frac{f_0}{Q}
At Resonance:
XL=XC,Z=RX_L = X_C, \quad Z = R
Voltage Magnification:
VL=VC=Q×VSV_L = V_C = Q \times V_S

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Identify Known Values

List the given values from the problem:

Equation:

\text{Given: } L = ${L} \text{ H}, C = ${C} \text{ F}, R = ${R} \text{ }\Omega

Calculation:

L=0.1 HC=0.000001 FR=10 ΩL = 0.1 \text{ H} \\ C = 0.000001 \text{ F} \\ R = 10 \text{ }\Omega

Explanation:

We start by identifying what values we know and what we need to find.

2

Step 2: Calculate Resonant Frequency

Use the resonant frequency formula:

Equation:

f0=12πLCf_0 = \frac{1}{2\pi\sqrt{LC}}

Calculation:

f0=12π0.1×0.000001=503.29 Hzf_0 = \frac{1}{2\pi\sqrt{0.1 \times 0.000001}} = 503.29 \text{ Hz}

Explanation:

Resonant frequency is calculated using the inductance and capacitance values.

3

Step 3: Calculate Quality Factor

Use the quality factor formula:

Equation:

Q=2πf0LRQ = \frac{2\pi f_0 L}{R}

Calculation:

Q=2π×503.29×0.110=31.62Q = \frac{2\pi \times 503.29 \times 0.1}{10} = 31.62

Explanation:

Quality factor measures the sharpness of the resonance peak.

4

Step 4: Calculate Bandwidth

Use the bandwidth formula:

Equation:

BW=f0QBW = \frac{f_0}{Q}

Calculation:

BW=503.2931.62=15.92 HzBW = \frac{503.29}{31.62} = 15.92 \text{ Hz}

Explanation:

Bandwidth is the range of frequencies over which the circuit response is significant.

Frequently Asked Questions (FAQ)

What is resonance in an RLC circuit?

Resonance occurs when the inductive reactance equals the capacitive reactance, causing the circuit to oscillate at its natural frequency. At resonance, the impedance is purely resistive.

How do I calculate resonant frequency?

Resonant frequency is calculated using f₀ = 1/(2π√(LC)), where L is the inductance and C is the capacitance. This is the frequency at which the circuit naturally oscillates.

What is the quality factor?

The quality factor (Q) measures how sharp the resonance peak is. It is calculated as Q = 2πf₀L/R. Higher Q values indicate a sharper, more selective resonance.

What is bandwidth in resonance?

Bandwidth is the range of frequencies over which the circuit response is significant. It is calculated as BW = f₀/Q. Higher quality factors result in narrower bandwidths.

What happens at resonance?

At resonance, the inductive and capacitive reactances cancel each other, making the impedance purely resistive and minimum. The current is maximum, and the power factor is unity.

How does resistance affect resonance?

Resistance affects the quality factor and bandwidth. Higher resistance reduces the quality factor, making the resonance peak broader and less selective.

Practice MCQs

  1. If inductance is doubled, the resonant frequency becomes:
  2. What is the resonant frequency of a 1mH inductor and 1μF capacitor?
  3. At resonance in an RLC circuit:
  4. Quality factor is directly proportional to:
  5. Which circuit has the highest quality factor?