Series RLC circuit — components share the same current
Figure 1: Toggle between series and parallel to see how R, L and C connect.
Enter what you know and what impedance you need — the calculator finds the missing component.
Table of Contents
- Phase Angle and Power Factor
- Inductive vs Capacitive Reactance
- Resonance and Q Factor
- Complex Impedance Notation
What Is Impedance?
Impedance is the total opposition that a circuit presents to alternating current. Unlike pure resistance, which opposes both DC and AC equally, impedance accounts for the frequency-dependent behaviour of inductors and capacitors as well. The symbol is Z and the unit is the ohm (Ω).
In a DC circuit you only need Ohm’s law: V = IR. In an AC circuit the voltage and current may be out of phase, so you need a quantity that captures both magnitude and timing. That quantity is impedance: Z = R + jX, where R is resistance and X is reactance. The Ohm’s Law Calculator handles the DC case; this calculator extends it into the AC domain.
Series and Parallel Formulas
Series RLC
When R, L and C are connected in series they share the same current. The total impedance is the vector sum:
Where XL = 2πfL and XC = 1/(2πfC).
Magnitude: |Z| = √(R² + (XL − XC)²)
Phase: θ = arctan((XL − XC) / R)
Parallel RLC
When R, L and C are connected in parallel they share the same voltage. It is easier to work with admittance (Y = 1/Z):
Where G = 1/R, BL = 1/XL, BC = 1/XC.
Then |Z| = 1/|Y| and θZ = −θY
Worked Example — Series RLC at 1 kHz
Step 1 — Inductive reactance: XL = 2π × 1000 × 0.05 = 314.16 Ω
Step 2 — Capacitive reactance: XC = 1/(2π × 1000 × 1×10−6) = 159.15 Ω
Step 3 — Net reactance: X = 314.16 − 159.15 = 154.99 Ω (inductive)
Step 4 — |Z| = √(100² + 154.99²) = 184.5 Ω
Step 5 — Phase angle: θ = arctan(154.99/100) = 57.2° (voltage leads current)
The positive phase tells us the circuit is inductive at this frequency — the inductor’s reactance dominates. To explore how this phase angle affects real power delivery, see the Phase Angle Calculator.
Worked Example — Parallel RLC at 10 kHz
Step 1 — XL = 2π × 10000 × 0.01 = 628.3 Ω
Step 2 — XC = 1/(2π × 10000 × 10×10−9) = 1591.5 Ω
Step 3 — Admittance: G = 1/1000 = 1 mS, BL = 1/628.3 = 1.592 mS, BC = 1/1591.5 = 0.628 mS
Step 4 — Net susceptance: B = 0.628 − 1.592 = −0.964 mS
Step 5 — |Y| = √(1² + 0.964²) = 1.389 mS → |Z| = 720.0 Ω
In parallel circuits the impedance peaks at resonance rather than dipping to a minimum. If you need to find that resonant peak precisely, use the Resonant Frequency Calculator.
Worked Example — Speaker Crossover Impedance
Step 1 — XL = 0 (no inductor)
Step 2 — XC = 1/(2π × 3000 × 22×10−6) = 2.41 Ω
Step 3 — |Z| = √(8² + 2.41²) = 8.36 Ω
Step 4 — θ = arctan(−2.41/8) = −16.8° (capacitive)
A capacitor in series with a speaker forms a simple high-pass crossover. The impedance stays close to the nominal 8 Ω at the crossover frequency, but the capacitive phase shift means lower frequencies are attenuated. For the full crossover design with component values, check the Crossover Calculator.
Worked Example — At Resonance
Step 1 — f0 = 1/(2π√(0.1 × 100×10−9)) = 1591.5 Hz
Step 2 — At resonance XL = XC = 1000 Ω
Step 3 — Series: |Z| = R = 47 Ω (minimum), θ = 0°
Step 4 — Q = XL/R = 1000/47 = 21.3
A Q factor of 21.3 means this is a sharply tuned circuit — the voltage across the inductor or capacitor at resonance is 21 times the source voltage. This principle is fundamental to RLC circuit analysis, bandpass filter design, and radio tuning.
Phase Angle and Power Factor
The phase angle θ of the impedance tells you how much the voltage leads or lags the current. When θ = 0° the circuit is purely resistive and all power is real. When θ approaches ±90° most of the power is reactive — it sloshes back and forth without doing useful work.
Power factor is simply the cosine of the phase angle: PF = cos(θ). An industrial motor with a lagging power factor of 0.7 (inductive) draws 43% more current than necessary. Power factor correction adds capacitance to cancel the inductive reactance and bring θ closer to zero. The Power Factor Calculator can size the correction capacitor for you.
Inductive vs Capacitive Reactance
Reactance is the frequency-dependent part of impedance. Inductive reactance (XL = 2πfL) increases with frequency — an inductor passes DC freely but increasingly blocks higher frequencies. Capacitive reactance (XC = 1/(2πfC)) decreases with frequency — a capacitor blocks DC but passes higher frequencies.
This complementary behaviour is what makes filters, tuned circuits, and impedance matching possible. For standalone reactance calculations, use the dedicated Inductive Reactance Calculator or the Capacitive Reactance Calculator.
Resonance and Q Factor
Resonance occurs when XL = XC, at the frequency f0 = 1/(2π√(LC)). In a series circuit the impedance drops to its minimum (just R), and in a parallel circuit the impedance rises to its maximum.
The quality factor Q measures the sharpness of the resonance. For a series circuit Q = XL/R at resonance. Higher Q means a narrower bandwidth: BW = f0/Q. A radio tuner with Q = 50 at 1 MHz has a bandwidth of only 20 kHz — enough to select a single AM station.
The Resonant Frequency Calculator (linked above) provides detailed resonance analysis including bandwidth and damping ratio.
Complex Impedance Notation
Impedance is a complex number: Z = R + jX. The real part R is resistance, and the imaginary part X is reactance. Engineers use j (not i) for the imaginary unit to avoid confusion with current.
There are two equivalent representations. Rectangular form (R + jX) is best for adding impedances in series. Polar form (|Z| ∠ θ) is best for multiplying, dividing, and quick mental estimates. The calculator above shows both.
For circuits with multiple impedances in complex combinations, the Complex Impedance Calculator can handle arbitrary series-parallel networks.
Frequently Asked Questions
What is the difference between impedance and resistance?
Why does impedance use complex numbers?
What happens to impedance at resonance?
Can I use this calculator for a circuit with only R and C?
How is impedance measured in practice?
What is admittance and when is it useful?
Does impedance affect power consumption?
How do I convert between impedance forms?
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