Applications: phasors & AC circuits
A sinusoid A cos(ωt + φ) is represented by the phasor Ae^(iφ), which strips out the time dependence common to every signal in the circuit. Differentiation then becomes multiplication by iω, so the differential equations of an AC circuit collapse into complex algebra: impedances are R, iωL and 1/(iωC), combined exactly like resistances.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Convert each source to a phasorAmplitude becomes the modulus and phase becomes the argument. The common e^(iωt) factor is dropped.
- Replace components by impedancesResistor R, inductor iωL, capacitor 1/(iωC). Engineers write j for i to avoid clashing with current.
- Combine as for resistorsSeries impedances add; parallel ones combine as reciprocals. Ohm’s law becomes V = IZ with complex quantities.
- Convert backThe modulus of the result is the amplitude and its argument is the phase shift.
Worked example
A resistor of 3 Ω is in series with an inductor of impedance 4i Ω. Find the total impedance and the phase angle.
- Series impedances add: Z = 3 + 4i.
- Magnitude: |Z| = √(9 + 16) = 5 Ω.
- Phase: arg Z = arctan(4/3) ≈ 0.927 rad ≈ 53.1°.
- A positive argument means the voltage leads the current.
Answer. Z = 3 + 4i Ω, with magnitude 5 Ω and phase 53.1° — the voltage leads the current by that angle.
Where marks get dropped
These are the specific errors that cost credit on applications: phasors & ac circuits questions — QED's rubric penalises each of them separately.
- Adding magnitudes instead of complex impedances. |3| + |4i| = 7, but the correct magnitude is 5.
- Getting the capacitor impedance sign wrong. 1/(iωC) = −i/(ωC), so capacitive reactance is NEGATIVE while inductive is positive.
- Confusing i with current. Electrical engineering writes j for the imaginary unit precisely to avoid this.
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Applications: phasors & AC circuits — frequently asked questions
Why do phasors work?
Because every quantity in a linear circuit at steady state oscillates at the same frequency ω. Factoring out e^(iωt) leaves a purely algebraic relation between amplitudes and phases.
What is resonance in this language?
The frequency where the inductive and capacitive reactances cancel, ωL = 1/(ωC), leaving a purely real impedance and maximum current. It gives ω = 1/√(LC).
Does this only apply to circuits?
No — the same phasor method handles any linear system driven sinusoidally: mechanical vibration, acoustics, optics and signal processing all use it.
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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