Alternating Current: common doubts, answered
The questions students ask most often about Alternating Current, each with a short answer. For the full chapter, read the Alternating Current notes.
About the chapter
When should I use peak values and when rms values in AC problems?
Use rms values for power, heating and meter readings, and peak values for the largest instantaneous value. Mains voltage, ammeter and voltmeter readings and P = VI cos φ are all rms. Expressions such as v = vm sin ωt and im = vm/Z use peaks. Because every peak value is √2 times its rms value, ratios such as Z = vm/im = V/I work with either kind, provided both are of the same kind.
Alternating voltage and a resistor
Read this section in the notes →If the average AC current over a cycle is zero, why does it still heat a resistor?
Because heating depends on i², which is never negative. Over one cycle the current is positive for half the time and negative for the other half, so its plain average is zero. The heat produced goes as i²R, and i² stays positive in both halves, averaging ½im². That is why the rms value, not the average value, is used for AC power and heating.
Are voltage and current in phase in a pure resistor in an AC circuit?
Yes. With v = vm sin ωt across it, the current is i = (vm/R) sin ωt, so the two reach their peaks and pass through zero at the same instants. A resistor stores no energy, so it responds immediately to the applied voltage, and Ohm's law holds at every moment. On a phasor diagram the voltage and current phasors point the same way.
RMS values and average power
Read this section in the notes →What does the rms value of an alternating current mean?
It is the steady direct current that would produce the same heating in a resistor: I = im/√2, about 0.707 times the peak. Likewise V = vm/√2. AC meters show rms values, and the mains supply is quoted that way. For a resistor, P = I²R, P = V²/R and P = VI keep their DC form when rms values are used; once L or C is present, the average power becomes P = VI cos φ.
Is 220 V household mains the peak voltage?
No, 220 V is the rms value; the peak is about 311 V. Since vm = √2 × V, the supply actually swings between +311 V and −311 V in each cycle. This matters for insulation and for components such as capacitors, which must survive the peak, and it is a classic trap when a question gives the mains voltage and asks for a peak current.
Phasors
Read this section in the notes →What is a phasor and why is it used in AC circuits?
A phasor is a vector that rotates anticlockwise at angular frequency ω, whose projection on the vertical axis gives the instantaneous value of a sinusoidal voltage or current; its length is the peak value. Phasors let you add quantities that are out of phase, such as the voltages across R, L and C, by simple vector addition instead of combining sine functions.
AC through an inductor
Read this section in the notes →Why does current lag the voltage by 90° in a pure inductor?
Because the voltage across an inductor depends on how fast the current changes, not on the current itself. With v = vm sin ωt, L di/dt = v gives i = im sin(ωt − π/2). The voltage is greatest when the current is changing fastest, which is when the current passes through zero, so the current reaches its own peak a quarter cycle later.
Why does inductive reactance increase with frequency?
Because at higher frequency the current must change faster, and the inductor's back emf grows with the rate of change. XL = ωL = 2πνL, so doubling the frequency doubles the opposition. For steady DC, at zero frequency, an ideal inductor has no reactance and acts like a plain wire. So an inductor lets low frequencies through easily and holds back high ones.
AC through a capacitor
Read this section in the notes →Why does a capacitor block DC but allow AC to flow?
Because charge never crosses the gap; the capacitor only charges and discharges. With DC it charges once and the current then stops, so in the steady state no current flows. With AC the plates are charged one way and then the other every half cycle, so current keeps flowing in the connecting wires. Since XC = 1/ωC, a higher frequency means a smaller reactance.
Why does current lead the voltage by 90° in a capacitor?
Because the current is largest when the capacitor's voltage changes fastest, a quarter cycle before the voltage peaks. With v = vm sin ωt, i = C dv/dt gives i = im sin(ωt + π/2). At the instant the voltage peaks the capacitor is fully charged and the current is momentarily zero. Remember: in a capacitor the current leads, in an inductor it lags.
Series LCR circuit and impedance
Read this section in the notes →Why can't I add VR, VL and VC directly in a series LCR circuit?
Because the three voltages are not in phase. VR is in phase with the current, VL leads it by 90° and VC lags it by 90°. They add as phasors: VL and VC point in opposite directions and partly cancel, and their difference is at right angles to VR. So V = √(VR² + (VL − VC)²), which can even be smaller than VL or VC alone.
What is the difference between reactance and impedance?
Reactance is the opposition offered by an inductor or a capacitor alone, XL = ωL or XC = 1/ωC, while impedance is the total opposition of the circuit, resistance included. For a series LCR circuit, Z = √(R² + (XC − XL)²) and the peak current is im = vm/Z. All three are measured in ohm, but only resistance turns electrical energy into heat.
How do I know whether current leads or lags in a series LCR circuit?
Compare XC with XL. If XC is larger, the circuit behaves like a capacitor and the current leads the voltage; if XL is larger, it behaves like an inductor and the current lags. The phase angle comes from tan φ = (XC − XL)/R. When the two reactances are equal, φ = 0 and the current and voltage are in phase, which is resonance.
Resonance
Read this section in the notes →What happens at resonance in a series LCR circuit?
The two reactances cancel, XL = XC, so the impedance falls to its smallest value, Z = R, and the current is largest. This happens at ω₀ = 1/√(LC). The voltages across L and C are equal and opposite, and each can be far larger than the source voltage. Current and voltage are in phase, so the power factor equals 1.
Can an RL or RC circuit show resonance?
No. Resonance needs XL = XC, so that the inductive and capacitive effects cancel each other. A circuit with only R and L, or only R and C, has a single reactance with nothing to cancel it, so its impedance changes smoothly with frequency and the current shows no sharp peak. Both an inductor and a capacitor must be present.
How does resonance help a radio pick one station?
The radio's tuning circuit is an LCR circuit whose resonant frequency is shifted by changing its capacitance. When that frequency matches a station's, the signal from that station drives the largest current, while other stations, off resonance, drive only small currents. The sharper the resonance, the more cleanly one station is separated from its neighbours.
Power factor
Read this section in the notes →Why is the average power zero in a pure inductor or capacitor even though current flows?
Because current and voltage are 90° out of phase, so cos φ = 0 in P = VI cos φ. During one quarter cycle the element takes energy from the source and stores it, and in the next quarter it gives all of it back. Averaged over a cycle nothing is used up. Such current is called wattless current, though it still heats the resistance of the wires.
What does the power factor tell us about a circuit?
It tells what fraction of the product VI is actually consumed as power: P = VI cos φ, where cos φ is the power factor. It is 1 for a pure resistor or a circuit at resonance and 0 for a pure inductor or capacitor. In a series LCR circuit cos φ = R/Z. A low power factor means large currents for little useful power, wasting energy in the supply lines.
Transformers
Read this section in the notes →How does a transformer change the voltage?
By mutual induction between two coils wound on the same soft iron core. Alternating current in the primary sets up a changing flux in the core, and that same flux threads the secondary, inducing the same emf in every turn. So the voltages are in the ratio of the turns, Vs/Vp = Ns/Np. More turns on the secondary step the voltage up; fewer step it down.
Does a step-up transformer increase power?
No. An ideal transformer delivers the same power it receives, so VpIp = VsIs. Raising the voltage lowers the current in the same ratio: Vs/Vp = Ns/Np = Ip/Is. A real transformer loses a little energy, so its output power is slightly less than its input. The gain is in voltage only, which is useful for cutting the current in long power lines.
Why doesn't a transformer work with a DC supply?
Because a steady current produces a steady flux, and a steady flux induces no emf in the secondary. Induction needs the flux to keep changing, which alternating current provides naturally. With DC, an emf would appear in the secondary only for a moment at switch-on or switch-off. The primary could also overheat: with no changing flux there is no back emf, so only the winding's small resistance would limit the current.
What are the energy losses in a real transformer and how are they reduced?
The main losses are flux leakage, heating of the windings, eddy currents and hysteresis. Some primary flux misses the secondary; the copper windings heat up as I²R; the changing flux drives eddy currents in the iron core; and repeatedly magnetising the core wastes energy. Winding the coils over one another, using thick wire, laminating the core and choosing a material with low hysteresis loss reduce these.
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