Alternating Current
Unlike the constant voltage of a battery, household electricity oscillates sinusoidally—it swaps direction 50 or 60 times per second depending on your…
Start with the simplest version: this lesson is about Alternating Current. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Unlike the constant voltage of a battery, household electricity oscillates sinusoidally—it swaps direction 50 or 60 times per second depending on your country. This alternating current (AC) seems chaotic compared to direct current (DC), yet it dominates modern power systems. Why? Because AC enables transformers, which electromagnetic-induction makes possible, allowing efficient long-distance power transmission. This chapter explores AC circuits and how our technology leverages the elegant mathematics of oscillating voltages and currents.
Advantages of AC Over DC
AC enables transformers: You can step voltage up for efficient transmission, then down for safe use. Transformers don't work with DC.
AC efficient to generate: Rotating a coil in a magnetic field naturally produces AC—it's mechanically simpler than mechanical commutators needed for DC.
AC safer for distribution: High voltage AC can be stepped down before reaching consumers.
These advantages explain why the grid uses AC despite the added complexity.
Sinusoidal AC Voltage and Current
AC voltage varies as:
v(t) = v_m sin(ωt)
Where:
- v_m is peak (amplitude) voltage
- ω = 2πf is angular frequency
- f is frequency (50 or 60 Hz)
The RMS (root mean square) voltage is what "110 volts" actually means:
V_RMS = v_m/√2 ≈ 0.707 v_m
RMS voltage gives the equivalent DC voltage that would produce the same average power. A 110V (RMS) AC supply has peak voltage around 155V.
Similarly for current: I_RMS = i_m/√2
AC Through Resistors: Ohm's Law Still Works
For pure resistance, Ohm's law applies to RMS values:
V_RMS = I_RMS × R
Voltage and current oscillate in phase—they reach peaks simultaneously. Power dissipated (as heat) is:
P_avg = V_RMS × I_RMS = I²_RMS × R
AC Through Inductors: Inductive Reactance
From electromagnetic-induction, inductors oppose changes in current. This opposition to AC flow is inductive reactance:
X_L = ωL = 2πfL
Measured in ohms. Higher frequency → greater reactance. An inductor acts like a frequency-selective resistor.
Current lags voltage by 90° in a pure inductor (voltage peaks before current peaks). No net energy is dissipated—energy oscillates between the magnetic field and the circuit.
AC Through Capacitors: Capacitive Reactance
From electrostatic-potential-and-capacitance, capacitors store charge. Their opposition to AC is capacitive reactance:
X_C = 1/(ωC)
Higher frequency → lower reactance. Capacitors pass high frequencies and block low frequencies (used for signal filtering).
Current leads voltage by 90° in a pure capacitor (opposite phase relationship to inductors).
LCR Circuit: Impedance and Resonance
A circuit with resistance, inductance, and capacitance together has impedance:
Z = √[R² + (X_L - X_C)²]
The voltage-current relationship becomes:
I_RMS = V_RMS/Z
Resonance occurs when X_L = X_C, making Z = R (minimum impedance). At resonance:
f_0 = 1/(2π√LC)
At resonance frequency, impedance is purely resistive (no reactive component), and impedance is minimum. Current is maximum for a given voltage. This is why radio tuning circuits work—they select the resonance frequency.
Power Factor
In AC circuits with reactive components, not all voltage contributes to real power. The power factor is:
PF = cos(φ) = R/Z
Where φ is the phase angle between voltage and current.
Real power: P = V_RMS × I_RMS × cos(φ)
If a circuit is purely reactive (φ = 90°), cos(φ) = 0 and no real power is transferred—just reactive power oscillates back and forth. Industrial facilities often have inductors from motors, creating poor power factors and requiring capacitors for correction.
Related Topics
current-electricity | electromagnetic-induction | electrostatic-potential-and-capacitance
Socratic Questions
- Why does AC transmission work better for long distances than DC, even though both can transmit power? What specifically about transformers and stepping voltage up makes the difference?
- In an ideal inductor (no resistance), no power is dissipated, yet you're applying voltage and current flows. Where does the energy go? Why don't we extract it?
- At resonance in an LCR circuit, reactances cancel (X_L = X_C). Is this a coincidence, or does it reveal something deeper about oscillating systems?
- A 60 Hz AC household outlet oscillates 120 times per second (both directions count). Why doesn't the flickering light from a light bulb at this frequency bother our eyes?
- Why do large electric motors reduce the power factor of a facility to below 1.0? What could a factory do to correct this?