Inductive Reactance

Capacitors hate low frequency and love high frequency. Inductors do the reverse. An inductor's opposition to AC current — its inductive reactance $X_L$ — rises proportionally with frequency. Slow signals slip through untouched; fast ones are blocked.

The formula

For a pure inductor at frequency $f$:

Like $X_C$, inductive reactance is measured in ohms and plugs straight into a version of Ohm's Law:

The mirror-image of the capacitor case — and the mirror image of its phase relationship too. We'll get to that shortly.

Frequency dependence — the opposite of X_C

Double $f$ and $X_L$ doubles. A decade higher $f$ means a decade higher reactance.

• DC ($f = 0$): $X_L = 0$. The inductor looks like a short circuit (ignoring the copper winding's small ohmic resistance).
• High frequency: $X_L \to \infty$. The inductor looks like an open circuit.

The mirror image of capacitive reactance's DC-open, HF-short behaviour. Put the two together and you can build filters: an inductor in series with the signal path passes DC / blocks RF (a choke); a capacitor in series passes RF / blocks DC (a coupling cap). The low-pass filter in every power-supply output uses this.

Phase relationship — V leads I by 90°

An inductor's defining equation is $v = L \, dI/dt$. Feed it a sinusoidal current $i = I_p \sin(\omega t)$ and the voltage follows the derivative:

So the voltage across a pure inductor reaches its peaks a quarter-period before the current does. The mnemonic: “ELI” — in an L, E (voltage) leads I (current). Stack it with the capacitor mnemonic and you get the full memory aid “ELI the ICE man” — inductors are ELI, capacitors are ICE.

A detour into magnetism

Reactance is only half the story. To understand why inductance exists, and to make sense of the transformer later on, we need three scalar quantities every electrical engineer keeps in their head.

• Magnetic field intensity $H$ — the “driving force” produced by current around a coil. Measured in amp-turns per metre (A·t/m). For a coil of $N$ turns carrying current $I$ around a closed magnetic path of mean length $l$:
  $$H = \frac{N I}{l}$$
• Magnetic flux density $B$ — how densely field lines are packed. Measured in tesla (T) = Wb/m². Linked to $H$ by the permeability of the medium:
  $$B = \mu H = \mu_0 \mu_r H$$
  $\mu_0$ = 4π × 10⁻⁷ H/m is the permeability of free space; $\mu_r$ is the relative permeability of the core (air ≈ 1, iron ≈ 1 000 – 10 000, ferrites vary).
• Total flux $\Phi$ — how much field passes through a given cross-section $A$. Measured in webers (Wb):
  $$\Phi = B \cdot A$$

So a coil wound around a high-$\mu_r$ core generates a much bigger flux for the same current than the same coil in air — which is why every real inductor and transformer is wound on iron or ferrite.

The link back to inductance comes from Faraday's law: a changing flux induces an EMF, and the EMF opposes the change. For a coil of $N$ turns:

That's the chain: more turns → more flux per amp → higher inductance → higher reactance at a given frequency. Magnetism is the machinery underneath $X_L$.

The transformer principle

Wind two coils on the same core and you have a transformer. Primary $N_1$, secondary $N_2$. Since they share the same flux$\Phi$, each turn on either side sees the same $d\Phi/dt$, so the induced voltage per turn is identical — and the total voltages scale with the turn counts:

An ideal transformer conserves power ($P_1 = P_2$), which forces the currents to go the other way:

Step the voltage up 10× and the current steps down 10×. That's why national grids ship power at 400 kV instead of 240 V — the same watts flow with a thousandth of the resistive line loss ($P_{loss} = I^2 R$).

Why cores are laminated

The alternating flux that links primary and secondary also induces currents in the iron itself — the core is a conductor too. These eddy currents circulate inside the metal, dissipating energy as heat without doing any useful work. Left alone, a solid iron core would bleed 20–40% of the input power.

The fix is to slice the core into thin strips (typically 0.3 – 0.5 mm) laminated with an insulating varnish between them. Flux still runs freely along the length of each strip, but eddy currents can only swirl inside each thin slab instead of sweeping through the whole cross-section. Loss drops by one to two orders of magnitude and efficiency climbs to 95–98%.

The playground below skips all of that — it models an ideal transformer (η = 100 %) so the conservation rule $V_1 I_1 = V_2 I_2$ reads cleanly without a fudge factor. Real transformers sit a few per cent below; the gap is the core-loss story above.

What happens if V₂ halves — does I₂ double?

An ideal transformer always satisfies $P_{in} = P_{out}$ (and with a real one, $P_{in} = P_{out}/\eta$). The primary faithfully mirrors whatever the secondary draws. Whether halving $V_2$ makes $I_2$ double, halve, or do something else entirely depends on what the load is.

• Constant-resistance load. Heating elements, incandescent bulbs, simple resistor networks, printed-circuit shunts. Ohm's law runs the show: $I_2 = V_2 / R$. Halve $V_2$ → $I_2$ halves too → $P = V_2 I_2$ falls to a quarter of what it was. The load asks for less when the voltage drops. This is what the playground above models.
• Constant-power (regulated) load. A switch-mode power supply feeding a microcontroller, an LED driver with a constant-current stage, a variable-speed motor drive at fixed torque. The load has its own control loop and pulls whatever current it needs to hold $P$ fixed: $I_2 = P / V_2$. Halve $V_2$ → $I_2$ doubles → $P$ is unchanged. The load pulls harder when voltage drops. This is the scenario everyone remembers from textbook rated-power statements like “step down 10:1, so I jumps 10:1.”
• Constant-current load. A current source or a sensor driven by one. $I_2$ doesn't care about $V_2$ — power rises and falls with voltage. Rare as a primary load class but common inside electronics.

Real-world loads mix all three. A motor looks resistive at stall, roughly constant-power at rated speed, and a constant-current source during short-circuit protection. A distribution transformer feeding a neighbourhood sees a blend that shifts all day long. The transformer itself doesn't decide — it just passes through whatever the secondary circuit demands.

The playground below models a constant-power load — you pick the rated output power, and the load pulls whatever current is needed to hit it. Halve $V_2$ (raise $N_1$ or drop $V_1$) and watch $I_2$ double so $P_{out}$ stays locked. That's the textbook transformer rule in its cleanest form.

Common applications

• Power transformers step 11 kV or 33 kV distribution voltages down to 400 V / 230 V for homes and factories. Same physics as the playground above, just sized for kilowatts instead of watts.
• RF chokes sit in series with a DC supply feed. At DC they're a short — the rail works normally. At the operating frequency (say, 100 MHz) they're a high impedance, blocking the RF signal from leaking back into the power supply.
• Audio crossovers use an inductor in series with a woofer (blocks high frequencies), and a capacitor in series with a tweeter (blocks low frequencies). The band each speaker gets is set by picking L or C so its reactance equals the speaker's impedance at the crossover frequency.
• Power-supply output chokes smooth rectified DC. The inductor's high reactance at ripple frequencies (100 / 120 Hz for full-wave mains) blocks the AC component while passing the DC load current.
• Current transformers (CTs) flip the turn ratio around to measure current without breaking the line. A clamp-on ammeter is a CT with a split core.