Formula of Impedance: Z = √(R² + (X_L − X

The formula of impedance for a resistor, inductor and capacitor in series is $Z = \sqrt{R^2 + (X_L - X_C)^2}$ , where $X_L = \omega L$ and $X_C = \frac{1}{\omega C}$ . Impedance is the total opposition an AC circuit offers to current, and like resistance it is measured in ohms.

Why not just add R, X_L and X_C? Because the voltages across them are not in step. The voltage across the resistor is in phase with the current, the inductor voltage leads by 90° and the capacitor voltage lags by 90°. They add like perpendicular vectors (phasors), which is why a square root appears. Once you know Z, the AC version of Ohm's law gives the current: $I_{rms} = \frac{V_{rms}}{Z}$ . For the DC starting point, see the resistance formula and the electricity formulas section.

Quick facts

Formula
$Z = \sqrt{R^2 + (X_L - X_C)^2}$
Reactances
$X_L = 2\pi f L$ , $X_C = \frac{1}{2\pi f C}$
SI unit
ohm (Ω)
Dimensional formula
[M¹ L² T⁻³ I⁻²], same as resistance
Phase angle
$\tan\phi = \frac{X_L - X_C}{R}$
At resonance
$X_L = X_C$ , so $Z = R$
AC Ohm's law
$V = IZ$ for rms or peak values

Terms in the formula of impedance

Series LCR quantities
Symbol	Name	How to find it	Unit
$R$	Resistance	given, does not depend on frequency	Ω
$X_L$	Inductive reactance	$\omega L = 2\pi f L$	Ω
$X_C$	Capacitive reactance	$\frac{1}{\omega C} = \frac{1}{2\pi f C}$	Ω
$Z$	Impedance	$\sqrt{R^2 + (X_L - X_C)^2}$	Ω

Here f is the supply frequency in hertz and $\omega = 2\pi f$ is the angular frequency in rad/s. Raise the frequency and $X_L$ climbs while $X_C$ falls. The inductance L comes from the coil, as described on the self inductance formula page, and C from the capacitance formula.

How to find impedance in an exam question

Convert units: mH to H, μF to F, and note f in Hz.
Work out $X_L = 2\pi f L$ and $X_C = 1/(2\pi f C)$ to at least three figures.
Subtract to get the net reactance $X_L - X_C$ . A positive value means the circuit is inductive; negative means capacitive.
Square, add $R^2$ , and take the square root.
Use $I = V/Z$ and $\tan\phi = (X_L - X_C)/R$ if current or phase is asked.

The calculator above takes R and the net reactance (enter its size, without sign) and returns Z, or works back from Z to either of the others.

Solved impedance examples

Example 1: reactances given

A series circuit has R = 30 Ω, X_L = 70 Ω and X_C = 30 Ω. Find Z.

Net reactance: 70 − 30 = 40 Ω.

$Z = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50\ \Omega$

Example 2: mains supply with unit conversions

A 40 Ω resistor, a 200 mH inductor and a 100 μF capacitor are in series across a 230 V, 50 Hz supply. Find Z, the current and the phase angle.

Convert: L = 0.200 H, C = 1.00 × 10⁻⁴ F.

$X_L = 2\pi (50)(0.200) = 62.8\ \Omega$

$X_C = \frac{1}{2\pi (50)(1.00 \times 10^{-4})} = 31.8\ \Omega$

$Z = \sqrt{40^2 + (62.8 - 31.8)^2} = \sqrt{1600 + 961} = 50.6\ \Omega$

Current: $I = \frac{230}{50.6} = 4.5\ \text{A}$ (rms). Phase: $\tan\phi = \frac{31.0}{40} = 0.775$ , so $\phi = 37.8^\circ$ , with the current lagging the voltage because the circuit is inductive.

Special cases and resonance

Impedance of simpler AC circuits
Circuit	Impedance	Current versus voltage
R only	$Z = R$	in phase
L only	$Z = \omega L$	current lags by 90°
C only	$Z = \frac{1}{\omega C}$	current leads by 90°
R and L	$\sqrt{R^2 + X_L^2}$	lags by less than 90°
R and C	$\sqrt{R^2 + X_C^2}$	leads by less than 90°

When $X_L = X_C$ the reactances cancel, Z falls to its least value R, and the current is largest. This happens at the resonant frequency $f_0 = \frac{1}{2\pi\sqrt{LC}}$ . Radio tuning circuits use this to pick one station out of many.

Units and dimensions of impedance

Every term under the root is a squared resistance, so Z comes out in ohms. You can check $X_L$ : henry times 1/s is (V s/A)(1/s) = V/A = Ω. For $X_C$ : 1/(F × 1/s) = s/F = s V/C = V/A = Ω.

Impedance has the same dimensional formula as resistance, [M¹ L² T⁻³ I⁻²], set out in the dimensional formula of resistance guide. Its reciprocal, admittance, is measured in siemens.

Errors that lose marks

Adding ohms straight. Z is never R + X_L + X_C in a series AC circuit.
Using f where ω is needed. $X_L = \omega L$ , and ω = 2πf, so 50 Hz gives ω ≈ 314 rad/s.
Unconverted μF. 100 μF is 1.00 × 10⁻⁴ F; leaving it as 100 gives a tiny X_C.
Mixing peak and rms. V = IZ works for both, but use the same kind on each side.

Check your Z against R: impedance is always at least as large as the resistance, and equal only at resonance.

Frequently asked questions

What is the formula of impedance?

For a series LCR circuit, Z = √(R² + (X_L − X_C)²), with X_L = ωL and X_C = 1/(ωC). For R and L only, Z = √(R² + X_L²).

Which unit is used for impedance?

Ohms, Ω, just like plain resistance. Reduced to base quantities an ohm carries [M¹ L² T⁻³ I⁻²], so Z and R also share dimensions.

What is the difference between impedance and resistance?

Resistance opposes current by turning energy into heat and does not depend on frequency. Impedance also includes reactance from inductors and capacitors, which changes with frequency and shifts the phase of the current.

When is impedance equal to resistance?

At resonance, when X_L equals X_C. The reactances cancel, the current is in phase with the voltage and Z = R.

Does impedance depend on frequency?

Yes. Inductive reactance rises with frequency and capacitive reactance falls, so Z changes and has its smallest value at the resonant frequency.