Formula & Calculator
RLC Series Impedance
The total impedance of a series RLC circuit combining resistance and net reactance.
Interpretation
RLC series impedance is the sum of resistance R and the net reactance j(X_L − X_C).
The circuit is inductive if X_L > X_C, capacitive if X_C > X_L, and resistive at resonance.
Example: R=10Ω, X_L=20Ω, X_C=5Ω → Z = 10 + j15Ω.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Z | RLC series impedance | Ω |
| R | Resistance | Ω |
| j | Imaginary unit (√−1) | — |
| XL | Inductive reactance (ωL) | Ω |
| XC | Capacitive reactance (1/ωC) | Ω |
What it means
The total impedance of a series RLC circuit is the sum of the resistance and the reactance of the inductor and capacitor: Z = R + j(X_L − X_C), where X_L = ωL and X_C = 1/(ωC). The net reactance determines the circuit behaviour: if X_L > X_C, the circuit is inductive; if X_C > X_L, it is capacitive; at resonance (X_L = X_C), the impedance is purely resistive and equal to R. This impedance is used to calculate the current and voltage distribution in the circuit. The magnitude of Z is √(R² + (X_L−X_C)²). Example: R=10Ω, X_L=20Ω, X_C=5Ω → Z = 10 + j(20−5) = 10 + j15Ω. The magnitude is √(100+225) = √325 = 18.03Ω, and the phase angle is arctan(15/10) = 56.3°.
Worked example
RLC Series Impedance – Practical Example
Real‑World| Parameter | Value |
|---|---|
| R | 100 Ω |
| XL | 80 Ω |
| XC | 40 Ω |
| Formula | Z = R + j·(XL − XC) |
Common mistakes
- X_L: Inductive reactance = ωL.
- X_C: Capacitive reactance = 1/(ωC).
- Net reactance: X = X_L − X_C – if positive, inductive; if negative, capacitive.
- Resonance: When X_L = X_C, Z = R (purely resistive).
- Units: Ohms.
Applications
RLC series impedance is the sum of resistance and net reactance, Z = R + j(X_L − X_C). This determines the total opposition to AC current. Engineers use it to calculate the current and voltage distribution in series circuits, to design resonant circuits, and to match impedances. The net reactance determines whether the circuit is inductive, capacitive, or resistive (at resonance). By analysing the impedance, professionals can optimise power factor and filter characteristics. This formula is fundamental to all AC circuit analysis and is used in many applications, from audio to RF.
- Series resonant circuit design (band‑pass, notch filters)
- Impedance matching in communication systems
- Power factor correction and analysis
- Current and voltage phasor calculations
- Educational understanding of series RLC behaviour
Frequently Asked Questions
The total impedance is Z = R + j(X_L − X_C), where X_L = ωL and X_C = 1/(ωC). It is a complex number combining resistance and net reactance.
Resonance occurs when X_L = X_C, i.e., ωL = 1/(ωC). At resonance, the impedance is purely resistive (Z = R) and the current is maximum.
|Z| = √(R² + (X_L − X_C)²). This is the ratio of voltage magnitude to current magnitude.
φ = arctan((X_L − X_C)/R). Positive φ means inductive (voltage leads current), negative means capacitive (voltage lags).
The current is I = V / Z, with magnitude I = V/|Z| and phase shift equal to -φ (current lags voltage for inductive).
At low frequencies, X_C dominates, so |Z| is large (capacitive). At high frequencies, X_L dominates (inductive). At resonance, |Z| is minimum (=R).
At resonance, V_R = V_source, while V_L and V_C are equal and opposite, each being Q times V_source. They cancel out.
Filters (bandpass, bandstop), tuned amplifiers, impedance matching, and oscillator circuits.
Common errors include: 1) using the wrong sign for X_L or X_C, 2) forgetting the j factor, 3) applying the formula to parallel RLC, 4) using the wrong resonance frequency, and 5) ignoring the effect of component tolerances.