Impedance & Phasors
Rotating phasor diagram. Builds on LRC Resonance.
You connect a pure inductor (no resistance) to an AC voltage source. The voltage across the inductor oscillates as V(t) = V₀ sin(ωt). You measure the current through the inductor.
What is the phase relationship between voltage and current?
Current lags voltage by 90° in a pure inductor! Remember ‘ELI the ICE man’: E (voltage) Leads I (current) in an inductor (L). The inductor resists changes in current — when voltage peaks, the current is still rising, reaching its peak a quarter-cycle later.
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Watch the phasor arrows rotate in the complex plane. For a pure inductor, the voltage phasor (yellow) is 90° ahead of the current phasor (cyan). The projections onto the real axis trace out the familiar sine waves — with the current lagging behind. Switch to a capacitor and watch the relationship flip!
Z = √(R² + (X_L − X_C)²)
Impedance is the AC generalization of resistance. The phase angle φ = arctan((X_L − X_C)/R) tells you the lag between current and voltage. At resonance, X_L = X_C and Z = R.
Speaker crossovers use impedance to split audio signals. The inductor’s impedance rises with frequency (blocking highs), while the capacitor’s drops (passing highs). This routes bass to woofers and treble to tweeters.
Industrial motors are inductive — current lags voltage, wasting power. Power companies add capacitors to cancel the inductive phase shift, bringing the current back in phase with voltage. This is power factor correction.
For maximum power transfer, a radio antenna’s impedance must match the transmitter’s. Impedance matching networks use combinations of L and C to cancel reactive components.
Your guitar’s tone knob is an RC low-pass filter. Turning it down increases the effective capacitance, lowering the cutoff frequency and creating a warmer, darker tone by filtering out highs.
“Phasors turn the time-domain chaos of AC circuits into elegant geometry. Voltages and currents become rotating arrows. Impedance becomes a triangle. And the dance between inductors and capacitors — one lagging, one leading — becomes a simple matter of adding vectors.”
Phasor Playground
Build AC circuits from R, L, and C components and watch the phasor diagram come alive. See how impedance, phase angle, and the impedance triangle change as you adjust component values and driving frequency.
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Start with just a resistor — the V and I phasors are perfectly aligned. Add an inductor and watch the current phasor rotate behind the voltage. Now add a capacitor too — it pulls the current forward. Sweep the frequency and watch the impedance triangle morph. Find the resonant frequency where X_L = X_C and the triangle collapses to just R. Compare φ at low, resonant, and high frequencies.
Phasors transform differential equations into vector addition. The impedance triangle — with R horizontal, net reactance vertical, and Z as the hypotenuse — captures all the AC behavior in a single geometric picture. At resonance, the triangle flattens, impedance is minimum, and current is maximum.