RLC Impedance Calculator

X_L = 2πfL, X_C = 1/(2πfC), Z = √(R² + (X_L − X_C)²), φ = arctan((X_L − X_C)/R), f₀ = 1/(2π√(LC))

R

The resistance, in ohms, which is the one part of the impedance that does not change with frequency. It is what sets the floor under Z: however the reactances move, the impedance can never fall below R, and it is exactly R at resonance. Its other job is to set the scale of the phase angle — a large R next to a small net reactance means the circuit behaves almost like a plain resistor (Ω)

L

The inductance, entered in millihenries because that is what is printed on the part. Its reactance rises in proportion to frequency, so an inductor blocks high frequencies and passes low ones. The field is in millihenries, not henries: an entered 10 is 10 mH and is converted to 0.01 H before the formula sees it (mH)

C

The capacitance, entered in microfarads for the same reason. Its reactance falls as frequency rises, the opposite of the inductor, so a capacitor passes high frequencies and blocks low ones. Also converted before use: an entered 100 is 100 µF and becomes 0.0001 F. The two reactances are always in opposition, which is why the formula subtracts one from the other rather than adding them (µF)

f

The frequency of the driving voltage, in hertz. It is the input that makes this page different from Ohm's law: it appears twice in the arithmetic and the two appearances push in opposite directions, so the impedance falls at first and then rises again. Changing it changes the answer with no component touched, and there is no frequency at which the reactances are both zero (Hz)

X_L − X_C

The net reactance, which is what actually sets the phase angle. When the inductive term is larger the current lags the voltage and φ is positive; when the capacitive term is larger the current leads and φ is negative; when the two are equal the net reactance is zero, φ is zero, and the circuit is at resonance. Neither reactance is subtracted from the resistance — they are combined at right angles, which is why the formula uses a square root of squares and not a sum (Ω)

f₀

The resonant frequency, 1/(2π√(LC)), the frequency at which the two reactances cancel. It depends only on L and C — not on R, and not on the frequency currently entered — so it holds still while the rest of the page moves, which is easy to mistake for a bug. At f₀ a series circuit presents its smallest impedance, and that impedance is exactly R (Hz)

Use it whenever the circuit has an inductor or a capacitor in it and the voltage is alternating, because that is where Ohm's law stops being true rather than merely approximate. It answers what a component or a branch looks like to a particular frequency: the input impedance of a filter, the reactance of a choke at the mains frequency, the point at which an LC pair resonates. The phase angle is the reason to use it even when you already know the magnitude, because a load that draws current out of step with the voltage is not the same load as a resistor of the same ohmic value. Reach for Ohm's law instead when the load really is resistive — heating elements, filament lamps, a long run of wire — and treat this page as unnecessary there rather than as a more precise version of it.

1. The defaults: 50 Hz on 10 Ω, 10 mH and 100 µF

   1. Convert first: L = 10 mH = 0.01 H, C = 100 µF = 0.0001 F
   2. X_L = 2π × 50 × 0.01 = 3.142 Ω
   3. X_C = 1 / (2π × 50 × 0.0001) = 31.831 Ω
   4. Net reactance is 3.142 − 31.831 = −28.689 Ω, so the circuit is capacitive
   5. Z = √(10² + 28.689²) = 30.382 Ω, φ = arctan(−28.689 / 10) = −70.783°
   6. f₀ = 1 / (2π√(0.01 × 0.0001)) = 159.155 Hz

   The third and fourth results are the ones to read together: the capacitive reactance is more than ten times the inductive one, so at 50 Hz this circuit is a capacitor that happens to have a resistor in series with it, and the current leads the voltage by nearly 71 degrees. Note also how little the 10 Ω resistance matters here — it is only a third of the impedance, and the phase angle is close to −90°, which is what a pure capacitor would give.

2. The same components driven at 1 kHz

   1. X_L = 2π × 1000 × 0.01 = 62.832 Ω — twenty times its 50 Hz value
   2. X_C = 1 / (2π × 1000 × 0.0001) = 1.592 Ω — one twentieth of its 50 Hz value
   3. Net reactance is 62.832 − 1.592 = +61.240 Ω, so the circuit is now inductive
   4. Z = √(10² + 61.240²) = 62.051 Ω, φ = arctan(61.240 / 10) = +80.726°
   5. f₀ is still 159.155 Hz, unchanged

   Put this beside the first example: not one component was touched, and the phase angle has swung from −70.783° to +80.726°. The circuit that looked like a capacitor at the mains frequency looks like an inductor at 1 kHz, which is the single most important thing this page has to say and the thing a fixed resistance cannot express. The resonant frequency in the last row did not move, because it never depends on the frequency you are driving at — if it appears to change, something else was edited.

3. At resonance, 159.155 Hz

   1. Enter the resonant frequency itself: 159.155 Hz
   2. X_L = 2π × 159.155 × 0.01 = 10 Ω
   3. X_C = 1 / (2π × 159.155 × 0.0001) = 10 Ω
   4. The two cancel exactly: Z = √(10² + 0²) = 10 Ω, which is R itself
   5. φ = arctan(0 / 10) = 0°, so voltage and current are in step

   This is the row that gives the page its shape: at resonance a series circuit presents its minimum impedance, and that minimum is exactly the resistance. It is also the only frequency at which the current is in step with the voltage. One detail worth knowing before you check this by hand: 159.155 is f₀ rounded to three decimals, so each reactance is about four millionths of an ohm away from 10 (and the two differ from each other by about seven millionths), which leaves the phase angle at 0.000041° rather than exactly zero — the panel rounds both away. The exact resonance is at 159.1549431… Hz.

4. A 1 kΩ resistor in the same circuit

   1. The reactances are unchanged from the first example: 3.142 Ω and 31.831 Ω
   2. Net reactance is still −28.689 Ω, but R is now 1000 Ω
   3. Z = √(1000² + 28.689²) = 1000.411 Ω
   4. φ = arctan(−28.689 / 1000) = −1.643°
   5. f₀ is still 159.155 Hz: R has no effect on it

   This is where the page meets Ohm's law. Raising R from 10 Ω to 1 kΩ leaves the reactances untouched — they are properties of L, C and the frequency, not of R — but it drops the phase angle from −70.783° to −1.643°, and the impedance from 30.382 Ω to 1000.411 Ω, of which 1000 is just R. At this ratio the circuit is a resistor with a barely measurable trace of capacitance, and V = I × R is a good approximation rather than a mistake. That is the boundary between this page and the ohms law page, and it is a smooth one: there is no threshold, just an error that shrinks as R grows.

This page models a series circuit only — one resistor, one inductor and one capacitor in a single loop. There is deliberately no selector for a parallel circuit, because a parallel RLC network is not this formula with a flag flipped: it uses a different expression and it does the opposite thing at resonance, where a series circuit reaches its minimum impedance and a parallel one reaches its maximum. Filling the same four numbers in and getting a different answer would be reading a fact about a circuit you did not describe, so that variant belongs on its own page rather than behind a switch here. Ideal components are also assumed throughout: no winding resistance in the inductor, no leakage or equivalent series resistance in the capacitor, and no frequency at which the parts stop behaving as lumped elements. The input frequency is fixed rather than swept, so the page gives you one point on the impedance curve and the resonant frequency, not the curve itself.

Why can't I just use Ohm's law for this?

Because with an inductor or a capacitor in the circuit, the ratio of voltage to current is not a constant — it depends on the frequency, and the current stops staying in step with the voltage. Ohm's law gives a number here and the number is wrong, which is worse than giving no number at all. The ohms law calculator says so itself and sends you to this page; V = I × R remains a good approximation only while the resistance is large compared with the net reactance, as the fourth example shows, where 1 kΩ next to 28.689 Ω leaves the phase angle at less than 2 degrees.

What about a parallel RLC circuit?

It is a different calculation and a different page, which is why there is no switch for it here. A parallel network uses its own expression, and at resonance it behaves in the opposite way: a series circuit reaches its minimum impedance there, while a parallel one reaches its maximum. Worse, this page would happily accept those four numbers and return an impedance figure that is wrong by a large factor without any visible sign of trouble — which is exactly the sort of silent error a selector would have to prevent, not cause.

Why does the resonant frequency not change when I change the frequency?

Because the resonant frequency is a property of the inductor and the capacitor, while the frequency field is the frequency you are driving the circuit at — two different things that happen to share a name. f₀ = 1/(2π√(LC)) contains no R and no f, so it holds still while you sweep the drive frequency past it. It moves only when you change L or C. That independence is what makes it useful: it tells you where the circuit will resonate before you go looking for it.

What does a negative phase angle mean?

It means the current is leading the voltage, which happens whenever the capacitive reactance is the larger of the two. Positive means the current lags, with inductance in charge, and zero means they are equal and the circuit is at resonance. The sign is therefore a summary of the whole page in one character: −70.783° says the 100 µF capacitor is running this circuit at 50 Hz, and +80.726° at 1 kHz says the 10 mH inductor has taken over.

Why is the impedance only 10 ohms at resonance?

Because the two reactances cancel and leave only the resistance. X_L and X_C are equal and opposite in effect, so the net reactance is zero and Z = √(R² + 0²) = R. That is the minimum impedance for a series circuit, and it is also the frequency at which the current is largest and in phase with the voltage. It is the reason a series LC pair is used to select a frequency: everything else is attenuated, and the chosen one passes with only R in the way.

Are the reactances added to the resistance?

No, they are combined at right angles, which is why the formula takes the square root of the sum of the squares rather than a plain sum. At the defaults that difference is large: adding arithmetically would give 10 + 3.142 + 31.831 = 44.973 ohms, whereas the correct impedance is 30.382 ohms. The physical reason is the phase shift — the voltage across a reactance is a quarter cycle out of step with the voltage across a resistance, and quantities that are out of step do not simply add.

• NIST Guide for the Use of the International System of Units (SI), §6.5 — the hertz, the henry and the farad, and the derived unit for reactance — NIST
• Series RLC circuit — impedance as a vector sum of resistance and net reactance, the phase angle, and resonance as the point where the two reactances cancel — Wikipedia
• Electrical reactance — why an inductor and a capacitor react in opposite directions as frequency rises, and how each behaves at the two extremes — Wikipedia