Quick Answer
Impedance is the total opposition a circuit presents to alternating current at one frequency, written Z and measured in ohms. It combines resistance with reactance, but as the two perpendicular sides of a right triangle instead of a sum, so stating an impedance takes two numbers: a magnitude and a phase angle.
Intuition
Opposition with a size and a timing
A resistor obstructs a current by the same amount whatever the current happens to be doing. A capacitor and a coil do not. Capacitive reactance shrinks as the signal speeds up, inductive reactance grows, and both are quoted in ohms even though neither of them turns any energy into heat. Put all three parts in one circuit and the useful question is what the source has to push against overall.
Adding the ohms together gives the wrong answer, and not by a little. A resistance and a reactance hold the current back at different moments of the cycle. The voltage across the resistor is at its largest at the instant the current is; the voltage across a reactance peaks a quarter of a cycle away from that instant. Adding the two as plain numbers counts opposition that is never present at the same time, and the total always comes out too high.
Impedance is the quantity that puts them together properly. It is still measured in ohms and it still answers the resistance question: how much current a given voltage will drive. What it holds beyond a resistance is the timing. Two networks can each limit a source to the same number of amperes and still disagree about when that current arrives relative to the voltage, and the disagreement settles how much of the energy leaving the source is consumed and how much of it comes back.
An impedance is therefore quoted as a pair of figures. One says how large the opposition is. The other says how far the current runs ahead of or behind the voltage. Quote only the first and the more interesting half has gone missing.
Practitioner
From three component values to one impedance
Everything begins with the two reactances, each worked out at the frequency the circuit is running at.
The two are entered with opposite signs, inductive positive and capacitive negative, and what survives is the difference between them. Layer 3 sets out where the minus sign comes from; for now it is enough that the coil and the capacitor pull the current in opposite directions and partly cancel. What is left over, sign and all, is the net reactance, and it goes into the right triangle with the resistance:
Squaring removes the sign, so the magnitude alone will not say which way the network leans. The angle keeps that information, and it is the arctangent of the net reactance divided by the resistance: positive when the coil dominates, negative when the capacitor does, zero when the two cancel.
Worked example — One series chain at one frequency
A resistance of 100 Ω sits in series with an inductance of 100 mH and a capacitance of 10 µF, all driven at 50 Hz.
The coil contributes 31.42 Ω and the capacitor 318.3 Ω, which leaves a net reactance of -286.9 Ω, negative because the capacitor is much the larger of the two.
Against the resistance, that gives a magnitude of 303.8 Ω at an angle of -70.78°.
Drawn to scale, the triangle puts both numbers into one picture. The resistance barely reaches a third of the way along the hypotenuse, so this network is mostly reactive and the angle is a long way from zero. A resistance of a few kilohms with the same two parts would produce a stubby triangle and a small angle instead.
Reactance takes the place of resistance in Ohm's law at a single frequency, provided only magnitudes are being asked for:
Worked example — The current, and what the magnitude will not tell you
Drive the same chain from a source of 12 V RMS at the same frequency and it draws 39.50 mA.
Of the three parts only the resistance turns anything into heat, and the current through it is that same figure, so the power the network consumes is 156.0 mW. The source voltage multiplied by the current gives 0.474 VA instead, which is the burden the supply and the cable have to carry either way.
The ratio between them is 0.329, and it is the cosine of the impedance angle.
Those last two figures are the reason the angle is worth carrying. Treat the magnitude as though it were a resistance and the power calculation returns the larger number, several times too large here, and always in that direction. AC power gives the two quantities their proper names and power factor turns the ratio between them into a design problem.
Unit discipline matters more here than in a DC calculation. Convert to plain ohms, henries, farads and hertz before anything is squared, since a millihenry entered as a henry survives the square root looking entirely reasonable. And write the frequency down beside every impedance figure, because the same three components give a different answer at every point on the axis.
Engineer
Why the two reactances subtract
At one frequency, and once any switch-on transient has died away, every voltage and current in a linear circuit is a sinusoid of that same frequency. One such sinusoid differs from another in only two ways: how big it is, and where its peak falls relative to everything else. Both of those go into a phasor.
Series parts share a current, so take that current as the reference. The resistor's voltage rises and falls in step with it. The inductor's voltage arrives a quarter of a cycle early, since a coil responds to how fast the current is changing and the current changes fastest as it passes through zero. The capacitor's voltage arrives a quarter of a cycle late, for the mirror-image reason. A quarter turn one way and a quarter turn the other leaves the coil's voltage and the capacitor's voltage half a cycle apart, which is to say pointing in opposite directions. Two opposed quantities combine by subtraction. The minus sign in the formula above is nothing more than that.
Worked example — Three voltages on one shared current
The chain from Layer 2 carries 39.50 mA through all three parts.
Across the resistor that current develops 3.950 V. Across the coil it develops 1.241 V and across the capacitor 12.57 V, and because those two oppose each other the pair between them contributes 11.33 V.
Set the resistor's share and that remainder at right angles and the length of the pair is 12.00 V, which is the source voltage recovered.
The capacitor voltage there, 12.57 V, is larger than the source driving the whole chain. The coil holds an opposing voltage at the same instant, and only what is left of the pair ever reaches the terminals. Series RLC circuits pushes that effect much further.
Divide all four of those voltages by the shared current and the triangle keeps its shape while its sides turn into ohms. The impedance triangle is that same picture rescaled, and it shows the series expression for what it is, a length:
Worked example — The same magnitude by a different route
Put the resistance 100 Ω on one axis and the net reactance -286.9 Ω on the other, with no mention of coils or capacitors at all.
The length of that pair is 303.8 Ω, matching the 303.8 Ω the series expression produced.
Written the way most texts write it, an impedance is a resistance plus j times a reactance, with j marking the quarter turn between them. The magnitude is the length of that pair and the angle is its direction. Complex numbers for electronics supplies the notation, and once impedances are written that way the DC network theorems all carry across unchanged, which is the subject of AC circuit analysis.
The picture rests on conditions that are easy to leave unsaid, and each of them gives way somewhere real. One frequency at a time: a signal carrying harmonics has a different impedance presented to each component, and no single figure covers the lot. Steady state: during the first few cycles after a switch closes the circuit is still in the territory of the RC time constant and its relatives, where impedance says nothing useful. Linear parts: a saturating core or a rectifier changes the shape of the waveform, and once the current is not a sinusoid there is no phasor to draw. And a series connection: the right triangle above adds reactances that share a current. Parts in parallel share a voltage instead, so their reciprocals add, and the reciprocal of impedance gets its own name, admittance, to keep the two apart.
Professional
Reading impedance off real parts and real meters
An impedance figure describes a network at one point on the frequency axis and makes no promise anywhere else. Move far enough and even the sign of the angle changes.
Worked example — The same three parts three octaves up
Leave the components alone and raise the drive to 400 Hz.
The coil now contributes 251.3 Ω while the capacitor has dropped to 39.79 Ω, so the net reactance is 211.5 Ω and points the other way.
The magnitude falls to 234.0 Ω and the angle becomes 64.70°: the same box, inductive now, and the current lagging where it used to lead.
Somewhere between those two frequencies the reactances cancel and the network is briefly indistinguishable from its resistor alone. Resonance is that crossing, and one impedance measurement of an unknown box tells you far less about it than a sweep does.
An LCR meter reports either form. Ask for polar and it gives a magnitude and an angle; ask for rectangular and it gives a resistance and a reactance, which are the magnitude multiplied by the cosine and the sine of that angle. Resolving the reading above that way returns 100.0 Ω of resistance, the resistor's own value, unchanged by the move up the axis while the reactance moved a long way. Sweep an unknown part and the term that stays put is the resistive one.
The choice the meter puts in front of you next is the equivalent-circuit model. A single measurement gives one magnitude and one angle, and any number of two-element circuits reproduce them: a resistance in series with a reactance, or a different resistance in parallel with a different reactance, fit the same reading equally well. Meters offer both and the numbers differ, sometimes by a lot. Pick the one that matches how the part behaves. A low-loss capacitor is conventionally reported in series terms and a leaky one in parallel terms, and quoting the wrong model is a common way to make two datasheets look as though they disagree.
Real components make the calculation approximate before the meter is even attached. A capacitor carries a small series resistance and a small series inductance, so its impedance stops falling at some frequency and climbs again above it; ESR and capacitor parasitics works through the consequences. A wirewound resistor is a small inductor. A coil has winding resistance that no reactance calculation includes and self-capacitance that eventually overwhelms it. The conductors change too as the frequency climbs, since current crowds towards their surfaces and the effective resistance rises with it. The three-component arithmetic in Layer 2 remains true to the ideal parts it names, and drifts steadily away from the parts on the bench.
One more thing shares the ohm and the word without sharing the meaning. A cable's characteristic impedance, the familiar fifty or seventy-five ohms, is a ratio of voltage to current for a wave travelling along the line, fixed by the geometry and the insulation. It is not a resistance an ohmmeter can find between the ends of a short piece, and it does not vary with frequency the way the impedance of a lumped network does. Impedance matching is where the two ideas meet and where getting them confused costs the most.
The habit worth building is to treat a bare figure in ohms as incomplete. Which frequency was it taken at? Is it a magnitude or a resistance? Does the meter's series reading or its parallel reading describe this part? A number that cannot answer those three is not yet a measurement.
Common mistakes
- Adding the resistance and the reactance arithmetically — equal amounts of each do not make twice as much; they make the diagonal of a square. Perpendicular sides combine through Pythagoras.
- Adding the two reactances instead of subtracting them — the coil and the capacitor push in opposite directions, so what counts is the difference. Adding them can turn a nearly cancelling pair into a large fictitious opposition.
- Using the magnitude to work out dissipated power — only the resistive part heats anything. The power follows the current squared times the resistance, never the current squared times the impedance magnitude.
- Dropping the sign of the angle — the magnitude is the same whether the network is capacitive or inductive, and the sign is the only thing that says which. Lose it and a correction fitted afterwards will push the wrong way.
- Quoting an impedance with no frequency attached — the number belongs to one point on the axis. The same three parts here move from capacitive to inductive over three octaves.
- Reading a series meter value into a parallel model — both models fit the same measurement while giving different component values. State which one a figure came from before comparing it with anything.
Frequently asked questions
What is impedance?
The total opposition a circuit offers to alternating current at a stated frequency, measured in ohms. It has a magnitude, which fixes how much current a given voltage drives, and an angle, which says how far the current runs ahead of or behind that voltage.
How is impedance different from resistance?
Resistance is one part of an impedance, the part that consumes energy and holds its value at every frequency. Impedance adds the reactive part, which stores energy and returns it, changes with frequency, and shifts the current in time relative to the voltage.
Why can I not simply add the resistance and the reactance?
They act a quarter of a cycle apart, so their peaks never coincide. Treating them as two perpendicular sides of a right triangle gives the correct total, while adding them as ordinary numbers always overstates it.
What does a negative impedance angle mean?
That the capacitive reactance is the larger of the two, so the network behaves capacitively and its current reaches its peak before the voltage does. A positive angle means the opposite, with the inductance winning and the current lagging.
Can a multimeter measure impedance?
No. An ohmmeter applies a steady voltage, and at zero frequency a capacitor is an open circuit and a coil is only its winding resistance, so the reading describes neither. Impedance needs an instrument that excites the part at a chosen frequency and measures the phase as well as the amplitude.