What is the resonant frequency of a series RLC circuit?

fr = 1 / (2π√(LC)), where L is in henries and C is in farads. At this frequency XL = XC, the reactances cancel, impedance is at its minimum (= R), and current is at its maximum.

How do you calculate impedance in a series RLC circuit?

Z = √(R² + (XL − XC)²), where XL = 2πfL and XC = 1/(2πfC). At resonance XL = XC so Z = R. Away from resonance Z > R.

What is the Q-factor of a series RLC circuit?

Q = (1/R) × √(L/C). A higher Q gives a sharper, more selective resonance peak. In a series circuit, smaller R gives higher Q — the opposite of a parallel circuit.

What happens to current at series resonance?

At resonance the impedance is minimum (= R), so for a given source voltage the current I = V/R is at its maximum. The voltage across L and C can both be Q times larger than the source voltage.

What is bandwidth in an RLC circuit?

BW = fr / Q (in hertz). The bandwidth is the frequency range between the two half-power (−3 dB) points where impedance equals R√2 and current drops to 1/√2 of its peak value.

Does the resistor affect the resonant frequency?

No. The resonant frequency fr = 1/(2π√LC) depends only on L and C. The resistor affects the Q-factor, bandwidth, and the maximum current, but not the frequency at which resonance occurs.

What is inductive reactance XL?

XL = 2πfL (ohms). Inductive reactance increases with frequency. It is the inductor's opposition to AC current. At DC (f = 0), XL = 0 (short circuit); at high frequency, XL is very large (open circuit).

What is capacitive reactance XC?

XC = 1/(2πfC) (ohms). Capacitive reactance decreases with frequency — the opposite of XL. At DC (f = 0), XC is infinite (open circuit); at high frequency, XC approaches zero (short circuit).

What is the difference between series and parallel RLC resonance?

Series RLC: impedance is minimum, current is maximum, Q = (1/R)√(L/C) — smaller R gives higher Q. Parallel RLC: impedance is maximum, current is minimum, Q = R√(C/L) — larger R gives higher Q. Both resonate at fr = 1/(2π√LC).

⚡ AC Circuit Laboratory

RLC Series Resonance Simulation

Analyze Impedance, Reactance, Phase Angles, and Resonant Peaks

Resistance (R)

1 Ω500 Ω

Inductance (L)

0.5 mH150 mH

Capacitance (C)

µF

0.1 µF50 µF

Source Voltage (Vrms)

1 V120 V

AC Frequency (f)

10 Hz6 kHz

Reactance Status

Inductive Mode

Current lags Voltage (inductive reactance dominates)

⚙️ Series RLC Schematic & Live Probes

📈 Resonance Peak Analysis

Current (mA) vs Frequency Log Scale

Resonant Peak

240.00 mA

Q-Factor

1.34

📐 Phasor Vectors

Reactance Complex Plane (+j)

Phase Angle (θ)

0.0°

IN-PHASE

Total Impedance (Z)

50.00 Ω

Total opposition to current

Resonant Frequency (fr)

1068.72 Hz

Reactance balance point

Net Reactance (X)

0.00 Ω

Inductive/Capacitive load

Active Power (P)

2.88 W

Real energy consumed

1. Reactance & Resonance

X_L = 2πfL | X_C = 1/(2πfC) | f_r = 1/(2π√LC)

2. Impedance

Z = √[R² + (X_L − X_C)²]

3. Current & Phase

I = V / Z | θ = tan⁻¹((X_L − X_C)/R)

4. Power & Quality Factor

P = I² R | Q = ωL/R | BW = f_r/Q

Step 1: Calculate Reactances

X_L = 2π × f × L
X_C = 1 / (2π × f × C)

X_L = 314.16Ω | X_C = 318.31Ω

Step 2: Calculate Impedance

Z = √[R² + (X_L − X_C)²]

Z = √[100² + (314.16 − 318.31)²] = 100.09Ω

Step 3: Calculate Current

I = V / Z
P = I² × R

I = 12 / 100.09 = 0.1199A | P = 1.438W

Step 4: Calculate Phase Angle

φ = tan⁻¹((X_L − X_C)/R)

φ = tan⁻¹((−4.15)/100) = −2.38° (Capacitive)

Step 5: Calculate Resonant Frequency

f_r = 1 / (2π√(LC))

f_r = 1 / (2π√(0.010000 × 0.0000022000)) = 1.07 kHz

Step 6: Calculate Quality Factor

Q = (ωL) / R = f_r / BW

Q = 62.83 / 50 = 1.34 | BW = 1068.72 / 1.34 = 797.55 Hz

RLC Series Resonance Calculator — Complete Guide to Series RLC Circuit Analysis

This RLC series resonance calculator analyses a series RLC circuit at any frequency. Enter R, L, C, source voltage V, and frequency f to instantly find the resonant frequency (f_r = 1/2π√LC), inductive reactance X_L = 2πfL, capacitive reactance X_C = 1/(2πfC), total impedance Z = √(R² + (X_L−X_C)²), circuit current I = V/Z, phase angle φ, quality factor Q = (1/R)√(L/C), and bandwidth BW = f_r/Q — with a live circuit diagram, resonance curve, and phasor diagram. Use it for band-pass filter design, LC oscillator analysis, RF tuner modelling, and power factor correction.

Quick Reference: Series RLC Formulas

Quantity	Formula	At Resonance (f = fr)
Resonant Frequency	f_r = 1 / (2π√(LC))	—
Inductive Reactance	X_L = 2πfL	X_L = X_C
Capacitive Reactance	X_C = 1 / (2πfC)	X_C = X_L
Total Impedance	Z = √(R² + (X_L−X_C)²)	Z = R (minimum)
Circuit Current	I = V / Z	I = V/R (maximum)
Phase Angle	φ = arctan((X_L−X_C) / R)	φ = 0° (unity PF)
Quality Factor	Q = (1/R) × √(L/C)	—
Bandwidth	BW = f_r / Q	—
Voltage across R	V_R = I × R	V_R = V (all voltage)
Voltage across L	V_L = I × X_L	V_L = Q × V
Voltage across C	V_C = I × X_C	V_C = Q × V

What Is Series RLC Resonance?

A series RLC circuit places R, L, and C end-to-end in a single loop driven by an AC source. The same current flows through every component. At the resonant frequency, the inductive reactance X_L = 2πf_rL equals the capacitive reactance X_C = 1/(2πf_rC), so they cancel exactly. The total impedance drops to its minimum value (= R), the current rises to its maximum (= V/R), and the voltage and current are perfectly in phase (unity power factor).

Resonant Frequency: f_r = 1 / (2π × √(L × C))

At resonance: X_L = X_C, Z = R, I = V/R (maximum), φ = 0°

Reactance, Impedance, and Phase Angle

Away from resonance, one reactance dominates the other:

Below f_r: X_C > X_L → circuit is capacitive → I leads V (negative φ)

Above f_r: X_L > X_C → circuit is inductive → I lags V (positive φ)

Z = √(R² + (X_L − X_C)²) φ = arctan((X_L − X_C) / R)

Voltage Magnification at Resonance

One of the most remarkable features of a series RLC circuit is that the voltages across L and C can both be Q times larger than the source voltage at resonance. This is called voltage magnification or resonance rise:

V_L = V_C = Q × V_source (at resonance)

Example: Q = 20, V = 12 V → V_L = V_C = 240 V
Always rate L and C for their peak operating voltages!

Q-Factor and Bandwidth

Q = (1/R) × √(L/C)

In a series circuit, smaller R → higher Q → sharper resonance.
This is the opposite of a parallel RLC where larger R → higher Q.

Bandwidth: BW = f_r / Q
Half-power (−3 dB) frequencies: f₁ = f_r − BW/2, f₂ = f_r + BW/2
At f₁ and f₂: Z = R√2, I = I_max/√2

Worked Examples

📐 Example 1 — Resonant Frequency & Full Analysis at Resonance

GivenR = 50 Ω | L = 10 mH | C = 2.2 µF | V = 10 V | f = f_r

Step 1f_r = 1 / (2π × √(0.01 × 2.2×10⁻⁶)) = 1,073 Hz

Step 2X_L = 2π × 1073 × 0.01 = 67.4 Ω | X_C = 1 / (2π × 1073 × 2.2×10⁻⁶) = 67.4 Ω ✓ (equal at resonance)

Step 3Z = √(50² + (67.4 − 67.4)²) = √(2500 + 0) = 50 Ω (minimum — equals R)

Step 4I = V / Z = 10 / 50 = 200 mA (maximum) | Phase φ = 0° (unity power factor)

Resultf_r = 1,073 Hz | Z = 50 Ω | I = 200 mA | φ = 0°

🔍 Example 2 — Q-Factor, Bandwidth & Voltage Magnification

GivenR = 10 Ω | L = 10 mH | C = 2.2 µF | V = 10 V | f_r = 1,073 Hz

Step 1Q = (1/R) × √(L/C) = (1/10) × √(0.01 / 2.2×10⁻⁶) = (1/10) × 67.4 = 6.74

Step 2BW = f_r / Q = 1073 / 6.74 = 159 Hz | f₁ = 994 Hz | f₂ = 1,153 Hz

Step 3⚠️ V_L = V_C = Q × V = 6.74 × 10 = 67.4 V (across each component at resonance!)

ResultQ = 6.74 | BW = 159 Hz | V_L = V_C = 67.4 V — rate components accordingly

📊 Example 3 — Off-Resonance: Impedance & Phase Angle

GivenR = 50 Ω | f > f_r | X_L = 75 Ω | X_C = 40 Ω | V = 10 V

Step 1Net reactance = X_L − X_C = 75 − 40 = +35 Ω (positive → inductive, I lags V)

Step 2Z = √(50² + 35²) = √(2500 + 1225) = √3725 = 61.0 Ω

Step 3φ = arctan(35 / 50) = arctan(0.70) = 35.0° (inductive — current lags voltage)

Step 4I = V / Z = 10 / 61.0 = 163.9 mA (less than the 200 mA at resonance)

ResultZ = 61.0 Ω | φ = +35° (inductive) | I = 163.9 mA

Series vs Parallel RLC Resonance — Complete Comparison

	Series RLC	Parallel RLC (Tank)
Impedance at resonance	Minimum (= R)	Maximum (= R)
Current at resonance	Maximum (= V/R)	Minimum (= V/R)
Q-factor formula	Q = (1/R)√(L/C) ↓R → ↑Q	Q = R√(C/L) ↑R → ↑Q
Resonant frequency	f_r = 1/(2π√LC)	f_r = 1/(2π√LC)
Below resonance	Capacitive (I leads V)	Inductive (I lags V)
Above resonance	Inductive (I lags V)	Capacitive (I leads V)
Voltage magnification	V_L = V_C = Q × V_s	I_tank = Q × I_source
Main use	Band-pass filters, tuned amplifiers	Oscillators, notch filters, tuners

Practical Applications of Series RLC Circuits

Band-pass filters: Series RLC passes frequencies near f_r and attenuates the rest — used in audio equalisers, IF amplifiers, and signal processing.
Radio and TV tuning: The resonance condition selects one broadcast frequency while rejecting all others.
LC oscillators: The resonance frequency sets the oscillation frequency of series-mode crystal and ceramic resonators.
Impedance matching: Series RLC networks transform impedance between source and load for maximum power transfer.
Power factor correction: Adding capacitance in series with an inductive load brings the circuit toward resonance, reducing reactive power draw.
Notch / band-reject filters: Combining a series and parallel tank achieves precise frequency notches.
Voltage magnification: At resonance, V_L and V_C can both be Q times the source voltage — useful in Tesla coils and some power converters.

Common Mistakes to Avoid

Component voltage ratings: At resonance V_L = V_C = Q × V_source. For Q = 50, a 10 V source drives 500 V across L and C — always check component ratings.
Units mismatch: Convert L to henries (not mH) and C to farads (not µF) before substituting into formulas.
Resistor location: Real inductors have winding resistance r_L. The effective R in the Q formula is R + r_L, not just the series resistor.
Bandwidth vs selectivity: A very high Q (narrow BW) makes the circuit sensitive to component drift and temperature. Real designs often target Q of 10–100.

Frequency Response and Reactance Chart

Frequency vs f_r	X_L vs X_C	Circuit Type	Phase Angle	Impedance Z
f ≪ f_r	X_L ≪ X_C	Capacitive	φ ≈ −90°	Very high (≈ X_C)
f < f_r	X_L < X_C	Capacitive	φ < 0°	> R
f = f_r	X_L = X_C	Resistive	φ = 0°	Z = R (minimum)
f > f_r	X_L > X_C	Inductive	φ > 0°	> R
f ≫ f_r	X_L ≫ X_C	Inductive	φ ≈ +90°	Very high (≈ X_L)

Frequently Asked Questions

What is the resonant frequency of an RLC circuit?

f_r = 1 / (2π√(LC)). This is the frequency at which X_L = X_C, impedance is minimum, and current is maximum. It depends only on L and C, not R.

Does the resistor change the resonant frequency?

No. The resonant frequency f_r = 1/(2π√LC) is determined entirely by L and C. R affects the Q-factor, bandwidth, and peak current, but not the frequency at which resonance occurs.

Why can the voltage across L and C exceed the source voltage at resonance?

At resonance V_L = V_C = Q × V_source. The reactive voltages are equal and opposite, so they cancel in the total phasor, leaving only V_R = V_source. But each individual reactive component sees Q times the source voltage.

What is the difference between XL and XC?

X_L = 2πfL increases with frequency. X_C = 1/(2πfC) decreases with frequency. At resonance they are equal. The net reactance X = X_L − X_C determines the circuit's phase and how far it is from resonance.

Related Calculators

RLC Parallel Resonance Calculator — impedance maximum, current minimum
Inductors in Series Calculator
Capacitors in Series Calculator
AC Power Calculator — reactive power, power factor
Ohm's Law Calculator