Impedance and Phasors in AC Circuits
Resistors, capacitors and inductors each set a different phase between current and voltage. Phasors add their voltages as vectors. Impedance magnitude Z describes the voltage-to-current amplitude ratio, while phase angle phi describes their phase difference.
What a learner can do afterwards
- States the reactance of a capacitor and an inductor and how each shifts the phase
- Adds voltages with a phasor diagram to get the impedance of a series RLC circuit
- Reads the phase angle between supply voltage and current off the diagram
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Use sinusoidal AC in steady state with ideal components. Angular frequency is omega = 2 pi f. A resistor's voltage and current are in phase. In a capacitor, current leads voltage by a quarter cycle; Xc = 1/(omega C). In an inductor, current lags voltage by a quarter cycle; XL = omega L. Reactance is the ratio of voltage amplitude to current amplitude, measured in ohms. It is not energy loss.
A phasor is an arrow whose length represents a sinusoid's amplitude and whose direction represents phase. In series, all elements share the same current. Take its direction as rightward: resistor voltage points right, inductor voltage up, and capacitor voltage down. The opposite vertical arrows subtract. Their sum with the resistor arrow gives supply voltage. The impedance magnitude is Z = sqrt(R^2 + (XL - Xc)^2), in ohms.
For an ideal series RLC circuit with R > 0, below resonance Xc exceeds XL and current leads supply voltage. Above resonance XL exceeds Xc and current lags. At resonance XL = Xc: the reactive voltage phasors cancel and Z = R. Supply voltage and current are then in phase. For a fixed supply voltage amplitude, this minimum impedance gives the greatest current amplitude.
The current amplitude is I0 = V0/Z, where V0 is the supply voltage amplitude. Do not mix peak and rms values. For V0 = 12 V and Z = 6 ohm, I0 = 12/6 = 2 A. First find the reactances, then Z, then divide the voltage amplitude by Z.
Instantaneous signed voltages add by the loop rule. Their positive peak amplitudes usually do not, because their peaks occur at different times. Inductor and capacitor voltages are half a cycle apart. For amplitudes VR = 6 V, VL = 10 V and VC = 2 V, the net reactive amplitude is 8 V. The supply amplitude is sqrt(6^2 + 8^2) = 10 V, not 18 V.
Define phi as supply voltage phase minus current phase. For R > 0, phi = arctan((XL - Xc)/R), between -90 and 90 degrees. Positive phi means current lags; negative phi means current leads. If XL - Xc equals R, tan phi = 1 and phi = 45 degrees. If XL = Xc, phi = 0. Z in this lesson is impedance magnitude; phi supplies its phase information.
Capacitor current leads voltage; inductor current lags. Phasor addition gives Z, I0 = V0/Z and the signed phase phi.
Original lesson, questions and study cards authored by Marble.
Rewritten readings were written for this dataset. Video references credit Khan Academy.
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