Questions and answers
Questions, with the board ready to try
Common questions on AC circuits, power, power factor correction and complex numbers. Each one has a short answer and a link that opens the lesson with the interactive board already set up: read it, try it, get it.
Alternating current 1 — Phasors and sinusoidal quantities: RMS value, phase, phasor diagram
A phasor is the fixed vector that represents the initial state (t = 0) of the vector rotating at angular speed ω, whose projection traces the sine wave instant by instant. Its magnitude equals the RMS value of the quantity and its angle with the real axis is the phase.
A sinusoidal quantity is written v(t) = VM sin(ωt + φ), with angular frequency ω = 2πf = 2π/T, amplitude VM and phase φ. The RMS value is V = VM/√2.
A vector of length VM rotating at constant speed ω has a vertical projection VM sin(ωt + φ): following its tip over time gives exactly the sine wave.
The RMS value is the DC voltage that would dissipate the same power in a resistor: the square root of the mean of the square. For a sine wave the mean of sin² is 1/2, so V = VM/√2 ≈ 0.707 VM.
Over a full period it is zero, because the positive and negative half-waves cancel out. Over half a period it is 2VM/π ≈ 0.637 VM.
It is the angle φ that the rotating vector makes with the reference axis at t = 0: in v(t) = VM sin(ωt + φ) it sets the point of its cycle where the sine wave starts. It is measured in radians or degrees.
Each sine wave is represented by a complex number: sums and derivatives of sine waves become sums and products of complex numbers, so AC circuits are solved with algebra, just like DC circuits.
Alternating current 2 — Impedance and series RLC circuits: reactance, impedance triangle, resonance
It is the ratio of the voltage and current phasors, Ż = V̇/İ = R + jX: the real part is the resistance, the imaginary part the reactance. Its magnitude |Z| gives the ratio of the RMS values, its angle φ the phase shift.
The two sine waves have the same frequency but reach their peaks at different times. In the phasor diagram the two vectors form an angle φ, the phase shift: in an ideal inductor, for instance, the current lags by 90°.
Because the voltage across an inductor is proportional to the rate of change of the current, v = L di/dt: it peaks when the current crosses zero, a quarter period before the current reaches its own peak.
Because the current is proportional to the rate of change of the voltage, i = C dv/dt: it peaks when the voltage crosses zero, a quarter period before the voltage reaches its own peak.
XL = ωL: the faster the current changes, the larger the voltage the inductor opposes. XC = 1/(ωC): at higher frequency the capacitor charges and discharges more often and lets more current through.
It is the right triangle with legs R and X and hypotenuse |Z| = √(R² + X²). The angle between R and Z is the phase shift φ between voltage and current, with cos φ = R/|Z|.
At the frequency f0 = 1/(2π√(LC)) the inductive and capacitive reactances cancel out: the impedance is minimum and equal to R, the current is maximum and in phase with the voltage, and L and C can show voltages much higher than the applied one.
Alternating current 3 — Admittance, susceptance and parallel RLC circuits: current resonance
It is the reciprocal of impedance, Ẏ = 1/Ż = G + jB, measured in siemens. In parallel connections admittances simply add up, like impedances in series, which makes the calculations much simpler.
They are the real part (G) and the imaginary part (B) of the admittance. The susceptance of an inductor has magnitude 1/(ωL), that of a capacitor ωC, and they have opposite signs.
All branches share the same voltage; the branch currents are out of phase with each other and add up as vectors (phasors), not as plain numbers: the total current is the hypotenuse of the current triangle.
At the frequency f0 = 1/(2π√(LC)) the currents in L and C are equal and opposite and cancel out: the line current is minimum and in phase with the voltage, while a much larger current can circulate between inductor and capacitor.
AC power — Active, reactive and apparent power: formulas and the power triangle
Active power P = V I cos φ (watts) is the part that can be turned into work or heat; reactive power Q = V I sin φ (var) is the share of power needed to build up the electric and magnetic fields of components (inductors, capacitors, motors, etc.) and is continuously exchanged between them and the sources; apparent power S = V I (VA) plays a role in sizing electrical lines and machines.
It is the average of the instantaneous power: the product of two sine waves shifted by φ contains a constant term, V I cos φ, and a term oscillating at twice the frequency whose average is zero.
p(t) = v(t)·i(t) is negative when voltage and current have opposite signs: at those moments the load gives back to the source the energy stored in its magnetic and electric fields.
With a 90° phase shift the instantaneous power swings symmetrically around zero: the energy taken in one quarter period is given back in the next. The average is zero (P = 0) and only reactive power is left.
It is the right triangle with legs P and Q and hypotenuse S = √(P² + Q²). The angle between P and S is the phase shift φ, and the power factor is cos φ = P/S.
In a circuit the total active power is the sum of the active powers of the single loads, and the total reactive power is the algebraic sum of the reactive powers (inductive positive, capacitive negative). Apparent powers, instead, do not add up.
Power factor correction — Power factor correction: capacitor sizing, single- and three-phase
To reduce the reactive power that inductive loads (motors, transformers) draw from the grid. For the same active power the line current drops, and with it the losses, the voltage drops and the utility penalties for a low power factor.
The reactive power to compensate is QC = P (tan φ1 − tan φ2), where φ1 and φ2 are the phase angles before and after correction. In single phase the capacitance is C = QC / (ω V²).
Correction shortens the Q leg: the active power P stays the same, while the apparent power S and the angle φ decrease and the power factor cos φ = P/S increases.
The reactive power to compensate is the same, but in delta each capacitor sees the line voltage, √3 times the phase voltage: the capacitance needed is one third of the star value (CΔ = CY/3).
It is the rise of the voltage at the receiving end of a line when the load is strongly capacitive, for instance with overcorrection or with a long line at light load. That is why capacitors must be sized carefully and switched out when the load drops.
Complex numbers 1 — Complex numbers: conjugate, modulus and argument, polar form, roots
It is a number of the form ż = a + ib, with a and b real and i the imaginary unit (i² = −1). a is the real part, b the imaginary part; in the Argand plane ż is the point (a, b).
The conjugate of ż = a + ib is ż* = a − ib: in the plane it is the mirror image across the real axis. The product ż·ż* = a² + b² is real and equals the squared modulus, which is the trick used for division.
The modulus is |ż| = √(a² + b²), the distance from the origin; the argument is the angle with the positive real axis. Mind the quadrants: when a < 0 the arctangent of b/a must be corrected by ±180°, and in the 3rd and 4th quadrants the angle is negative.
It is ż = r (cos α + i sin α), with r = |ż| the modulus and α the argument: the real part is r cos α and the imaginary part r sin α, the projections of the vector on the axes. From here you get the polar form r∠α.
In rectangular form (a + ib)(c + id) = (ac − bd) + i(ad + bc). In polar form it is simpler: the moduli multiply and the angles add, so the product rotates and stretches the vector.
Multiply numerator and denominator by the conjugate of the denominator, which then becomes real. In polar form the moduli divide and the angles subtract.
1/ż = ż*/|ż|²: multiply numerator and denominator by the conjugate. For ż = a + ib, 1/ż = (a − ib)/(a² + b²); in polar form the modulus becomes 1/|ż| and the angle changes sign.
Because i = 1∠90°: multiplying by i keeps the modulus and adds 90° to the angle. Two 90° rotations make 180°, a change of sign: that is why i² = −1.
All n roots have the same modulus, the n-th root of |ż|, and angles that differ by 360°/n: they are the vertices of a regular polygon inscribed in that circle.
They are the n complex numbers whose n-th power is 1: modulus 1 and angles 0, 360°/n, 2·360°/n, … They lie on the unit circle, at the vertices of a regular polygon with one vertex at 1.
Complex numbers 2 — Euler's formula, exponential form of complex numbers, complex logarithm
eiα = cos α + i sin α: the exponential with an imaginary exponent is the number with modulus 1 and angle α. It is the limit of (1 + iα/n)n as n → ∞, which the board builds step by step.
Just set α = π in Euler's formula: cos π + i sin π = −1. Rotating the number 1 by half a turn around the origin brings it to −1.
Adding and subtracting eiα = cos α + i sin α and e−iα = cos α − i sin α gives cos α = (eiα + e−iα)/2 and sin α = (eiα − e−iα)/(2i).
ż = r eiα, with r the modulus and α the argument. In this form multiplication, division and powers work on the exponents: r1eiα · r2eiβ = r1r2 ei(α+β).
(cos α + i sin α)n = cos nα + i sin nα (De Moivre's formula): the modulus stays 1 and the angle is multiplied by n, so the powers go round the unit circle.
Differentiating gives −sin x + i cos x = i (cos x + i sin x): the velocity vector is always perpendicular to the radius. It is the same property as the exponential eix, and leads to Euler's formula.
ln ż = ln|ż| + i(α + 2kπ): the real part is the logarithm of the modulus, the imaginary part is the argument, defined up to multiples of 2π. For example ln(−1) = iπ (principal value).