Impedance & Reactance: X_L, X_C, Complex Z & AC Networks

1 Introduction to AC Circuits and Impedance

Beginner

In a DC circuit, opposition to current is just resistance R, a single number in ohms. In an AC circuit the story is richer: inductors and capacitors also oppose current, but they do so in a way that depends on the signal's frequency and that changes the timing between voltage and current. Engineers bundle all of that opposition into one quantity called impedance, written Z and also measured in ohms.

Impedance is what lets us use Ohm's law for AC: instead of V = IR we write V = IZ, where Z is a complex number that carries both a size (how much opposition) and an angle (how far out of step voltage and current are). That single generalization is the foundation of almost everything in power electronics, filtering, matching and radio work, which is why it deserves its own module.

Use the level filter above to jump to a level, or the jump list to go straight to any lesson. The live playground at the top is your reference throughout — keep an eye on it as the ideas click into place.

2 Resistance vs. Reactance

Beginner

Resistance (R) converts electrical energy into heat. Its opposition is the same at every frequency, and the voltage across a resistor is always in phase with the current through it. Reactance (X), by contrast, is the opposition offered by energy‑storing elements — inductors and capacitors. It doesn't dissipate power; it stores energy in a magnetic or electric field and gives it back. Crucially, reactance depends on frequency, and it shifts voltage and current by 90°.

Both R and X are measured in ohms, so they're "compatible," but they act at right angles to each other in the math. A resistor contributes a real part to impedance; reactance contributes an imaginary part. Inductive reactance is taken as positive, capacitive reactance as negative — that sign is how the math remembers which way the phase shift goes.

3 Inductive Reactance X_L

Beginner

An inductor resists changes in current (that's Faraday's law in action). The faster the current tries to change — i.e. the higher the frequency — the stronger that opposition. That opposition is the inductive reactance, X_L = 2πfL = ωL, where f is frequency in hertz, L is inductance in henries, and ω = 2πf is angular frequency. The unit is ohms. Notice the direct proportionality: double the frequency or the inductance and X_L doubles.

A useful limiting case: at DC, f = 0, so X_L = 0 — an ideal inductor is a short circuit to steady current. At high frequency X_L is large, so inductors "block" AC and pass DC, which is exactly how many filters and chokes work.

4 Capacitive Reactance X_C

Beginner

A capacitor does the opposite of an inductor with respect to frequency: it resists changes in voltage, and a rapidly changing (high‑frequency) voltage is easy for it to follow. So its opposition, the capacitive reactance, X_C = 1/(2πfC) = 1/(ωC), falls as frequency rises. Larger capacitance also means less opposition. Again the unit is ohms, but notice X_C is in the denominator of frequency — the exact mirror image of X_L.

The limiting cases are just as handy as for the inductor: at DC, f → 0 and X_C → ∞, so an ideal capacitor is an open circuit to steady current (it "blocks" DC). At high frequency X_C → 0, so capacitors "pass" AC — the basis of coupling and bypass capacitors.

5 Frequency and Its Effect on Reactance

Beginner

Put the two formulas side by side and a clear picture emerges: X_L climbs in a straight line with frequency, while X_C falls off like a hyperbola. They are mirror images across the frequency axis, and they cross at exactly one frequency — the point where X_L = X_C. That crossing is resonance, and it's one of the most useful phenomena in electronics (you'll analyze it fully in lesson 18).

The explorer below plots both curves and a marker at the frequency you choose. Drag the slider and watch the marker move from the capacitive region, through resonance, into the inductive region.

6 Phase Relationships in R, L and C

Beginner

Because inductors and capacitors store and return energy, their voltage and current are never in step. In a resistor they're in phase (0° apart). In an inductor the voltage leads the current by 90° — or, equivalently, the current lags the voltage. In a capacitor it's the reverse: the current leads the voltage by 90°. These three fixed relationships are the reason impedance has a phase angle at all.

A mnemonic engineers actually use: "ELI the ICE man" — in ELM (voltage, current, inductor) E comes before I (voltage leads); in ICE (current, voltage, capacitor) I comes before E (current leads). The sign convention in the table is the algebraic form of the same idea.

7 Complex Numbers for AC Circuit Analysis

Beginner

The cleanest way to handle "a size plus a 90° timing shift" is a complex number, written a + jb. Engineers use the letter j for the imaginary unit (where j = √(−1)) instead of the mathematicians' i, precisely to avoid clashing with i for current. The one rule you must remember is j² = −1, which makes the powers of j cycle: j, −1, −j, 1, and repeat every four steps.

Think geometrically: a + jb is an arrow on a 2‑D plane, a units right and b units up. Multiplying by j rotates any arrow by 90° without changing its length — which is exactly what a 90° phase shift is. That's the bridge: once voltage and current are arrows, circuit analysis becomes geometry. The next level makes that concrete.

8 Complex Impedance Z

Intermediate

Complex impedance generalizes resistance to AC: Z = R + jX, where R is the resistive (real) part and X is the net reactance (imaginary part). A single Z now tells you both how much a branch opposes current (its magnitude |Z|) and how far its voltage and current are out of step (its angle θ). For a branch that's just a resistor, X = 0 and Z = R — the DC case is a special case of AC.

The magnitude and angle come straight from the right‑triangle relationship in lesson 2: |Z| is the hypotenuse, θ the angle at R. The calculator below takes a resistance and the inductive and capacitive reactances, forms the net X = X_L − X_C, and draws the impedance triangle for you.

9 Impedance in Rectangular Form

Intermediate

Rectangular (Cartesian) form writes impedance as Z = R + jX. The real part is the resistance; the imaginary part is the net reactance. Its superpower is addition: to add impedances you simply add the real parts and the imaginary parts separately, (R₁ + jX₁) + (R₂ + jX₂) = (R₁+R₂) + j(X₁+X₂). That's why series analysis (next‑ish) is so clean — series impedances add component‑wise, just like resistors.

Rectangular form is the "bookkeeping" form: it keeps resistance and reactance in separate columns so you never lose track of which is which. When a problem gives you components or asks you to combine branches, rectangular is usually where you do the arithmetic.

10 Impedance in Polar Form

Intermediate

Polar form writes the same impedance as Z = |Z| ∠ θ — a magnitude and an angle. This form shines for multiplication and division: to multiply, multiply the magnitudes and add the angles; to divide, divide the magnitudes and subtract the angles. Ohm's law for AC, I = V/Z, is a division, so polar makes it almost trivial: the current's size is V/|Z| and its angle is the voltage's angle minus θ.

So the two forms are complementary tools: rectangular for building networks (add/subtract), polar for applying Ohm's law and power (multiply/divide). The next lesson is the conversion between them, which is the only real skill you need to switch freely.

11 Converting Between Rectangular and Polar

Intermediate

The conversions are just right‑triangle trigonometry. From rectangular to polar: magnitude M = √(a² + b²) and angle θ = atan2(b, a). From polar to rectangular: a = M cos θ and b = M sin θ. Use atan2 (available on every calculator as "atan2" or by checking the signs) rather than plain arctan, because it returns the angle in the correct quadrant based on the signs of a and b.

12 Resistors, Inductors & Capacitors as Impedance

Intermediate

Each basic element has a clean impedance. A resistor is just Z_R = R (purely real). An inductor is Z_L = jX_L = jωL (purely imaginary, +90°). A capacitor is Z_C = 1/(jωC) = −jX_C (purely imaginary, −90°). A real inductor or capacitor also has some resistance, so in practice you often see a series combination like Z = R + jωL or Z = R − jX_C.

13 Series Impedance Calculations

Intermediate

Impedances in series add exactly like resistors, because the same current flows through each: Z = Z₁ + Z₂ + …. In rectangular form that's component‑wise: add all the real parts and all the imaginary parts. For a series RLC this reduces to the familiar Z = R + j(X_L − X_C). The magnitude and phase then follow from the triangle: |Z| = √(R² + X²), θ = atan2(X, R).

The calculator adds two arbitrary series impedances (enter each in rectangular form) and draws them head‑to‑tail, which is the geometric meaning of vector addition. Notice the result is always a bit "longer" than either input — series opposition builds up.

14 Parallel Impedance Calculations

Intermediate

Impedances in parallel do not add directly. Instead their admittances (Y = 1/Z, in siemens) add, just as parallel conductances do: 1/Z = 1/Z₁ + 1/Z₂ + …, so Z = 1/(Y₁ + Y₂ + …). For exactly two, this simplifies to Z = (Z₁·Z₂)/(Z₁ + Z₂). The result is always smaller in magnitude than either branch — parallel paths ease opposition, just as with resistors.

The calculator uses the reciprocal (admittance) method and shows each step — Y₁, Y₂, their sum, and the final Z — so you can see the "reciprocal method" the task refers to. It's the same arithmetic you'll use for any number of parallel branches.

15 Mixed Series-Parallel Impedance Networks

Advanced

Real circuits mix series and parallel pieces. The method is always the same reduce step by step: find the impedance of each parallel sub‑branch (reciprocal method), replace that branch with a single equivalent Z, then add the remaining impedances in series. You never need a new rule — just the series and parallel tools applied in the right order, usually from the inside of the network out.

The discipline is bookkeeping: label each sub‑result, keep rectangular form while adding, and only convert to polar at the end when you need |Z| and θ. The playground's readout chain (X_L → X_C → X → |Z| → θ → I) is the same step‑by‑step discipline for the series case.

16 Voltage, Current & Impedance Relationships

Advanced

Ohm's law works unchanged in phasor form: V = IZ. Because Z is complex, dividing a voltage phasor by Z gives a current phasor whose size is V/|Z| and whose angle is the voltage's angle minus θ. KVL and KCL also hold with complex quantities: voltages around a loop still sum to zero, and currents into a node still sum to zero — you just add them as phasors.

The AC voltage divider generalizes the DC one directly: for two series impedances, V₂ = V·(Z₂/(Z₁+Z₂)). This is how filter networks set their output, and it's where the frequency dependence of X_L and X_C produces actual filtering.

17 Phasors and Impedance Diagrams

Advanced

A phasor diagram draws the impedances (or the voltages) as arrows so the geometry of the circuit is visible. For a branch with R and X, the impedance diagram has R along the horizontal axis, X vertically, and Z as the hypotenuse — the same right triangle as the voltage diagram, just scaled. The current phasor points along R (in phase with V_R); the source voltage leads the current by the angle θ.

Reading a phasor diagram is a skill: the length of Z is |Z|, the angle from the R‑axis is θ, and the direction tells you whether current leads (X below the axis, capacitive) or lags (X above, inductive). The interactive diagram below builds the I, V_R, V_X and V phasors from your R, X and a chosen current.

18 Resonance: X_L and X_C Cancel

Advanced

At one special frequency the inductive and capacitive reactances are exactly equal, X_L = X_C, so the net reactance X = X_L − X_C is zero. The series impedance collapses to Z = R (purely real), the phase angle is 0° (voltage and current in phase), and the current reaches its maximum for a given source. That frequency is the resonant frequency f₀ = 1/(2π√(LC)). It's the same frequency where the X_L and X_C curves crossed in lesson 5.

The quality factor Q = X_L/R (at f₀) measures how "sharp" the resonance is and how much the individual voltages are magnified: even when the total impedance is just R, the inductor and capacitor each develop a voltage Q times the source. Set the playground's frequency to its displayed f₀ and you'll see V_L and V_C swing large while |Z| = R and θ = 0 — resonance made visible.

19 Advanced Analysis & Troubleshooting

Advanced

In practice, impedance is how you design filters (an RC low‑pass passes low frequencies because X_C is large at low f), matching networks (choosing R and L/C so a source sees its preferred load), and power systems (the phase angle behind power factor correction). You'll measure impedance with an LCR meter or an impedance analyzer, and you'll reason about it with the tools from this module. The habits that pay off: keep R and X in separate columns, track the sign of X, and convert to polar only when you multiply or divide.

Troubleshooting is impedance reasoning in reverse — a symptom points to a change in Z. A current that drops to near zero suggests a large increase in |Z| (an open, e.g. a failed capacitor or broken wire). A current that rises and the phase swings suggest X has changed (a shorted inductor, a leaking capacitor). The table below collects the most common slip‑ups and their fixes.

20 Comprehensive Review & Final Challenge

Advanced

You now have the complete impedance toolkit: reactances that depend on frequency (X_L = 2πfL, X_C = 1/2πfC), complex impedance Z = R + jX with |Z| and θ, rectangular and polar forms and their conversions, series addition, parallel (admittance) combination, phasor diagrams, and resonance. The final challenge uses every one of them on a single circuit. Work it yourself first, then reveal the full solution.

Mark module complete