Reactance and Impedance

Why resistance isn't the whole story

A resistor is gloriously boring: by Ohm's law its opposition to flow is just R, the same value whether you push steady DC through it or a sine wave wiggling a million times a second. It does not care how fast the voltage changes. But the two energy-storing parts you met earlier in this rung — the capacitor and the inductor — care about almost nothing else. Their whole behavior is tied to the rate of change, through I = C times dV/dt for the capacitor and V = L times dI/dt for the inductor.

So when you drive these parts with AC — a voltage that is forever changing — how strongly they oppose the flow must depend on how fast the wave wiggles, that is, on its frequency. Lean on the analogies from the first guides. A capacitor is a tiny bucket that resists sudden voltage change: a fast sine barely starts filling it before reversing, so charge sloshes in and out easily and it lets a lot through; a slow or DC voltage fills it once and then it blocks. An inductor is a flywheel resisting current change: it coasts a steady DC current along happily but fights a fast wiggle hard. Both, in other words, present an opposition that changes with frequency — and that needs a name beyond "resistance."

Reactance: opposition that depends on frequency

That frequency-dependent opposition is called reactance, written X and measured in ohms, just like resistance. For a capacitor the capacitive reactance is Xc = 1 / (2 times pi times f times C), and for an inductor the inductive reactance is XL = 2 times pi times f times L. The factor 2 times pi times f is just the frequency in radians per second — the natural "how fast it spins" version of the wave you met in the AC guide. Plug in numbers: a 100 nF capacitor at f = 1 kHz has Xc = 1 / (2 times pi times 1000 times 100 times 10^-6 times 10^-3)... cleaner, 1 / (2 times pi times 1000 times 100 times 10^-9) ≈ 1.6 kΩ. A 10 mH inductor at the same 1 kHz has XL = 2 times pi times 1000 times 0.01 ≈ 63 ohm.

Stare at those two formulas and the opposite personalities jump out. Capacitive reactance shrinks as frequency rises (the f is on the bottom): at DC, f = 0 makes Xc infinite, so a capacitor blocks DC; at very high frequency Xc collapses toward zero, so it passes AC like a piece of wire. The inductor is the exact mirror (the f is on top): at DC its XL is zero — a plain wire — and the higher the frequency, the harder it pushes back. The one-line summary every engineer keeps in their head: a capacitor blocks DC and passes high frequencies; an inductor passes DC and blocks high frequencies. That single fact is the seed of every filter you will build in the next guide.

Phase: current and voltage out of step

Reactance carries a second secret that resistance lacks: it shifts the timing between voltage and current. In a resistor, voltage and current rise and fall together, perfectly in step. In a capacitor, the current is busiest exactly when the voltage is changing fastest — which is as the voltage crosses zero, a quarter-cycle ahead of the voltage peak. So the capacitor's current leads its voltage by 90 degrees, a quarter of a cycle. The inductor does the reverse: because it fights changes in current, its voltage leaps ahead and the current lags by 90 degrees. This timing offset is a phase shift, the very idea from the AC guide, now produced on purpose by a component.

The phasor picture

Tracking sine waves that are all the same frequency but shifted in time gets messy with trigonometry. The cure is the phasor: draw each sine wave as a little arrow whose length is the wave's amplitude and whose angle is its phase. Imagine every arrow spinning together at the frequency f; since they all spin at the same rate, we simply freeze the picture and read off the relative angles between them. A sine wave becomes a fixed arrow — and adding two sine waves of the same frequency becomes plain arrow (vector) addition, no trig identities required.

Now the 90-degree phase shifts from the last section become geometry. Point a resistor's voltage along the same direction as the current — they are in phase, so the resistor's contribution lies flat along the current arrow. A capacitor's or inductor's voltage sits at a right angle to the current, because of that quarter-cycle lead or lag. So resistance points one way and reactance points straight up or down, at 90 degrees to it. They are not just two numbers you can add; they live on perpendicular axes, and that is the key that unlocks the next idea.

Impedance: Ohm's law for AC

Combine the in-phase resistance R with the 90-degree reactance X and you get impedance, written Z — the total opposition a circuit shows to AC, complete with its timing. Because R and X sit at right angles, you add them like the two legs of a right triangle, not like ordinary numbers: the magnitude is |Z| = sqrt(R^2 + X^2), and the phase angle is theta = arctan(X / R). Impedance has both a size (how much it throttles the current) and an angle (how far it shifts the timing). It is the honest, complete generalization of resistance.

            o
           /|
          / |
  |Z|    /  |   X  = reactance   (90 deg, energy stored)
        /   |
       / θ  |
      o-----o
        R   = resistance   (in phase, energy burned)

   |Z| = sqrt( R^2 + X^2 )        tan(θ) = X / R

The payoff is enormous: with impedance in hand, Ohm's law works again, just dressed up as phasors — V = I times Z. The magnitude of the voltage is the magnitude of the current times |Z|, and the phase angle of Z is exactly how far the voltage and current end up out of step. Every DC tool you already own — series and parallel combinations, voltage dividers, Thevenin equivalents — carries straight over to AC the moment you replace plain resistances with impedances. Let us walk a real example through.

1. Take a resistor and capacitor in series, R = 1.6 kΩ and C = 100 nF, driven at f = 1 kHz. First find the capacitor's reactance: Xc = 1 / (2 times pi times 1000 times 100 times 10^-9) ≈ 1.6 kΩ.
2. Notice R and Xc happen to be equal at this frequency. On the triangle, R points along the current and Xc points straight down by 90 degrees (a capacitor, so the current leads).
3. Combine: |Z| = sqrt(R^2 + Xc^2) = sqrt(1600^2 + 1600^2) = 1600 times sqrt(2) ≈ 2.26 kΩ. The phase angle is theta = arctan(1600 / 1600) = 45 degrees.
4. Apply Ohm's law for AC: drive 1 V amplitude across the pair and the current is |I| = |V| / |Z| = 1 / 2260 ≈ 0.44 mA, with the current leading the applied voltage by 45 degrees.
5. Step back: the special frequency where Xc equals R is exactly the cutoff frequency f = 1 / (2 times pi times R times C) ≈ 1 kHz — the hinge of every RC filter, which is precisely where the next guide begins.