RLC AC Impedance Calculator
Enter resistance, inductance, capacitance and frequency to compute inductive reactance X_L=2πfL, capacitive reactance X_C=1/(2πfC), impedance Z, phase angle and resonance frequency. R=100Ω, L=0.1H, C=10μF, f=50Hz → Z≈304.5Ω.
Input Data
Results
At a glance:AC impedance of a series RLC circuit: in AC, the opposition of R, L, C to current is expressed as impedance |Z|. Inductive reactance X_L=2πfL (blocks high frequencies, 0 at DC); capacitive reactance X_C=1/(2πfC) (blocks low frequencies, ∞ at DC); reactance X=X_L−X_C; impedance |Z|=√(R²+X²); phase angle φ=atan(X/R); power factor cos(φ)=R/|Z|. When X_L=X_C, reactance is zero, |Z|=R is minimum and current is maximum — this is series resonance, at f_0=1/(2π·√(LC)). Example: R=100Ω, L=0.1H, C=10μF, f=50Hz → X_L=2π×50×0.1≈31.4Ω, X_C≈318.3Ω, X≈−286.9Ω, |Z|≈304.5Ω, φ≈−70.7° (capacitive), cos(φ)≈0.328; resonance f_0≈159.2 Hz. Applications: filters (low/high/band-pass, audio crossovers, PSU EMI), resonant circuits (radio tuning, oscillators), radio tuning (f=1/(2π·√(LC))), impedance matching (max power transfer when Z_load=Z_source*), power-factor correction (shunt capacitor raises cos(φ)), audio crossovers. History: AC impedance by Oliver Heaviside (1880s), phasor method by Charles Steinmetz (1893).
Formula
Inductive reactance: X_L = 2πfL
Capacitive reactance: X_C = 1 / (2πfC)
Reactance: X = X_L − X_C
Impedance: |Z| = √(R² + X²)
Phase angle: φ = atan(X / R)
Power factor: cos(φ) = R / |Z|
Resonance: f₀ = 1 / (2π·√(LC))
$$X_L = 2\pi f L, \quad X_C = \frac{1}{2\pi f C}$$$$|Z| = \sqrt{R^2 + (X_L - X_C)^2}, \quad \varphi = \arctan\frac{X_L - X_C}{R}$$$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$How to Use
- Enter resistance R (Ω), inductance L (H), capacitance C (F), frequency f (Hz).
- The calculator gives X_L, X_C, X, |Z|, phase angle, power factor and resonance frequency.
- Capacitive circuit (X_C>X_L) has negative phase; inductive (X_L>X_C) positive; at resonance 0.
Impedance vs frequency for R=100Ω, L=0.1H, C=10μF
| Frequency f (Hz) | X_L (Ω) | X_C (Ω) | X (Ω) | |Z| (Ω) | φ (°) | cos(φ) |
|---|---|---|---|---|---|---|
| 10 | 6.28 | 1591.55 | -1585.27 | 1588.42 | -86.39 | 0.063 |
| 50 | 31.42 | 318.31 | -286.89 | 304.49 | -70.74 | 0.328 |
| 100 | 62.83 | 159.15 | -96.32 | 125.00 | -43.93 | 0.721 |
| 159.2 | 100.00 | 100.00 | 0.00 | 100.00 | 0.00 | 1.000 |
| 500 | 314.16 | 31.83 | 282.33 | 299.40 | 70.51 | 0.334 |
| 1000 | 628.32 | 15.92 | 612.40 | 620.45 | 80.73 | 0.161 |
At f=159.2 Hz (resonance) X=0, |Z|=R=100Ω, cos(φ)=1; capacitive below, inductive above.
Case Studies
Radio tuning circuit
AM radio tuning LC circuit selects 1000 kHz with C=100 pF.
From f=1/(2π·√(LC)), L=1/(4π²f²C)≈253 μH.
Adjusting C selects different station frequencies.
Power-factor correction
Factory motor is inductive: P=10 kW, cos(φ)=0.7, V=220V, f=50Hz.
Uncorrected apparent power S=P/cos(φ)=10/0.7≈14.3 kVA, line current I≈65 A.
Shunt capacitor raises cos(φ) to 0.95 → S≈10.5 kVA, I≈48 A, line loss down ~46%.
FAQ
What is the difference between impedance and resistance?
Resistance R opposes current in both DC and AC and dissipates energy as heat. Impedance |Z| includes resistance and reactance; reactance (X_L, X_C) matters only in AC, stores and returns energy rather than dissipating it (inductors store magnetic, capacitors store electric). R is real, Z is complex (Z=R+jX); at DC X_L=0 and X_C=∞, so a capacitor is open and an inductor is short.
What is series resonance?
When X_L=X_C (f=f₀=1/(2π·√(LC))), reactance is zero, impedance is minimum (|Z|=R), current is maximum and phase is 0 (voltage and current in phase). Energy sloshes between L and C while the source only replenishes resistive loss. The inductor voltage V_L=I·X_L can far exceed the source (Q times) — voltage resonance. Used in frequency-selective filters, oscillators, tuned amplifiers.
Why does an inductor block high frequencies?
Inductor voltage V=L·di/dt; higher frequency means larger di/dt, so a given current needs higher voltage — larger effective opposition (X_L=2πfL). At DC (f=0) X_L=0 (short); at high frequency X_L→∞ (open). Used in RF chokes, EMI filters, RF isolation.
Why does a capacitor block low frequencies?
Capacitor current I=C·dv/dt; lower frequency means smaller dv/dt, so a given voltage yields less current — larger effective opposition (X_C=1/(2πfC)). At DC (f=0) X_C=∞ (open); at high frequency X_C→0 (short). Used in coupling capacitors (block DC, pass AC), bypass capacitors, integrator/differentiator circuits.
Why correct the power factor?
Inductive loads (motors, transformers) with cos(φ)<1 make apparent power S>P, raising line current I=S/V, line loss I²R and voltage drop. A shunt capacitor raises cos(φ)→1, lowering S and I. In Hong Kong, CLP and HK Electric charge reactive-power penalties when industrial users' cos(φ)<0.85; correction can cut the bill by 5–15%.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.