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AC Circuits & EM Waves
Alternating current, impedance, resonance and the electromagnetic spectrum. 48 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Alternating current (AC) changes direction periodically, and it is the form of electricity delivered to homes. This area explains peak and RMS values, phasors, impedance of resistors, inductors and capacitors, resonance in RLC circuits, and the average power they consume. It finishes with the electromagnetic waves that AC circuits produce and the wider electromagnetic spectrum.
How the topics connect
AC circuits and electromagnetic waves extend the earlier electricity and magnetism to alternating quantities. Phasors and impedance handle circuits with resistors, inductors and capacitors, resonance explains tuning, and Maxwell's equations show that accelerating charges produce light and radio waves.
Real-world uses
Understand why the 230 V or 120 V quoted for household supply is an RMS value, and how this relates to the peak voltage.
Improve the efficiency of factories by correcting the phase difference between voltage and current, reducing losses in the supply.
Use resonance in an LC circuit to select one station out of many broadcast frequencies.
Relate the frequencies of radio, microwave and infrared waves to how they are used in broadcasting, Wi-Fi and remote controls.
Power networks use AC and transformers to transmit electricity efficiently, and power-factor correction keeps losses low in factories.
Radio, mobile phones, radar and fibre-optic networks all rely on electromagnetic waves and resonant circuits.
Alternating current
Current that reverses direction periodically.
i = I₀ sinωtAC current and voltage vary sinusoidally in time, reversing direction f times per second: i = I₀ sin ωt. Mains: 50 or 60 Hz. AC is easily transformed, so it's used for power distribution.
230 V RMS mains has a peak of 325 V.
AC is not 'weaker' than DC for the same RMS.
Peak and RMS values
RMS equals peak over root two.
I_rms = I₀/√2The RMS value is the DC equivalent in heating: I_rms = I₀/√2 ≈ 0.707 I₀. Meters and mains ratings quote RMS.
A 10 A peak current has 7.07 A RMS.
Average of a full sine cycle is zero.
Average power in AC
Average power over a cycle.
P = V_rms I_rms cosφP_avg = V_rms I_rms cosφ. For a purely resistive load P = I_rms²R.
230 V, 5 A with power factor 0.8: 920 W.
Don't use peak values for average power.
Phasors
Rotating vectors represent AC quantities.
Rotating vectors represent AC quantities; their lengths are amplitudes and angles are phases. Adding phasors gives total voltages.
Voltages across R, L, C add as vectors at 0°, +90°, −90°.
Phasors only apply for the same frequency.
Resistor in AC
Voltage and current are in phase.
Voltage and current are in phase: V = IR at all times and power oscillates at twice the frequency.
Heater elements.
No phase shift or reactance.
Inductor in AC
Current lags voltage by 90°.
X_L = ωLThe inductor's emf opposes the current change, so current lags voltage by 90°. Reactance X_L = ωL increases with frequency.
L = 0.1 H at 50 Hz: 31.4 Ω.
Passes DC but blocks high-frequency AC.
Capacitor in AC
Current leads voltage by 90°.
X_C = 1/ωCCurrent leads voltage by 90°, with reactance X_C = 1/ωC that decreases with frequency. It blocks DC.
C = 10 μF at 50 Hz: 318 Ω.
Opposite behaviour to an inductor.
LCR series circuit
Impedance combines R and net reactance.
Z = √(R²+(X_L−X_C)²)Impedance Z = √(R² + (X_L − X_C)²) and I = V/Z. The phase angle is tanφ = (X_L − X_C)/R.
R = 30 Ω, X_L = 80, X_C = 40: Z = 50 Ω.
Subtract the reactances before squaring.
Phase angle
Angle between voltage and current.
tanφ = (X_L−X_C)/Rφ is the angle between voltage and current: positive for inductive, negative for capacitive circuits, zero at resonance. It decides power transfer.
φ = 53° for R = 30, net X = 40.
Phase has a sign.
Series resonance
Impedance is minimum when X_L = X_C.
ω₀ = 1/√(LC)When X_L = X_C the impedance is minimum (= R) and the current maximum at ω₀ = 1/√(LC). Used to tune radios.
L = 1 mH, C = 100 pF: f₀ ≈ 503 kHz.
Voltages across L and C can exceed the supply.
Quality factor in LCR
Sharpness of resonance.
Q = ω₀L/RQ = ω₀L/R = 1/(ω₀RC) measures the sharpness of resonance; large Q means narrow bandwidth. It's the voltage magnification across L or C.
L = 1 mH, R = 10 Ω, ω₀ = 10⁶: Q = 100.
High Q needs low resistance.
Bandwidth
Frequency range around resonance.
Δω = R/LThe width of the resonance curve between half-power points: Δω = R/L = ω₀/Q. A narrow bandwidth gives selectivity.
Radio station selectivity.
Trade-off with response time.
Power factor
Cosine of the phase angle.
cosφ = R/Zcosφ = R/Z: the fraction of apparent power doing real work. Low power factor means large currents for the same power; industries add capacitors to correct it.
A motor at pf 0.7 draws 43% more current than at pf 1.
Utilities charge penalties.
Wattless current
Current with no average power at 90° phase.
Current with a 90° phase to voltage (in ideal L or C) transfers no average power; energy swaps back and forth with the source.
An ideal inductor draws current but no net power.
Still heats the wires.
LC oscillations
Energy sloshes between E and B fields.
ω = 1/√(LC)Charge in an LC circuit oscillates at ω = 1/√(LC), energy sloshing between E in the capacitor and B in the inductor; resistance causes damping.
Radio transmitters.
Analogous to a mass on a spring.
Parallel resonance
Impedance is maximum at resonance.
In a parallel LC circuit at resonance the impedance is maximum and the line current minimum (rejector circuit). Used in filters.
Tank circuit of oscillators.
Opposite of series resonance.
Filters
Low-pass, high-pass and band-pass circuits.
RC and RL combinations pass some frequencies and block others: low-pass (capacitor to ground), high-pass, band-pass. Cutoff f_c = 1/(2πRC).
R = 1 kΩ, C = 0.16 μF: f_c = 1 kHz.
Roll-off is 20 dB per decade for first order.
Transformers in power transmission
High voltage reduces line losses.
Stepping voltage up to 400 kV reduces the current and so I²R losses in lines; it's stepped down for use. For the same power, 10× voltage gives 100× less loss.
1 MW at 10 kV: 100 A, at 100 kV: 10 A.
Reason for AC grids.
Three-phase power
Three AC waves offset by 120° power industry.
Three AC voltages offset by 120° supply constant power to loads and give efficient motors. Line voltage = √3 × phase voltage.
400 V line in a factory = 230 V phase.
Used for industrial supply.
Rectification and smoothing
Convert AC to steady DC.
Diodes convert AC to pulsating DC; a capacitor smooths it and a regulator stabilises it. Ripple ∝ I/(fC).
Phone charger adaptors.
Larger C gives smoother output.
Displacement current
Allows current continuity through capacitors.
Between capacitor plates there's no charge flow but a changing electric field acts as a current: I_d = ε₀ dΦ_E/dt, completing the current loop and producing a magnetic field.
Equal to the conduction current in wires.
Introduced by Maxwell.
EM wave generation
Accelerating charges radiate EM waves.
Accelerating charges radiate; an oscillating current in an antenna produces self-propagating E and B oscillations. Hertz demonstrated them in 1887.
A radio mast radiates at its drive frequency.
Steady current doesn't radiate.
EM wave properties
Transverse waves with E and B perpendicular.
c = 1/√(μ₀ε₀)EM waves are transverse, E ⊥ B ⊥ direction, in phase, travelling at c = 1/√(μ₀ε₀) in vacuum, with E = cB. They carry energy and momentum and need no medium.
All colours of light travel at the same speed in vacuum.
They are not matter waves.
Speed of light
About 3×10⁸ m/s in vacuum.
c = 3×10⁸ m/sc = 299,792,458 m/s exactly (defines the metre). Light takes 8.3 minutes from the Sun, 1.3 s from the Moon. In media v = c/n.
Light travels 7.5 times around Earth in a second.
Nothing with mass reaches it.
EM wave equation
Derived from Maxwell's equations.
∇²E = μ₀ε₀ ∂²E/∂t²Maxwell's equations imply ∇²E = μ₀ε₀ ∂²E/∂t², a wave equation with speed 1/√(μ₀ε₀), matching the measured speed of light and revealing that light is electromagnetic.
This discovery united optics with electromagnetism.
Works for B as well.
Relation between E and B
Magnitudes related by c.
E = cBIn a plane EM wave E₀ = cB₀ and the fields are in phase. B is tiny: E = 300 V/m gives B = 1 μT.
Sunlight has E₀ ≈ 1000 V/m.
Both fields carry equal energy.
Energy of EM waves
Energy divided equally between E and B.
Energy density is shared equally by E and B: u = ε₀E² = B²/μ₀. Intensity I = cu_avg.
Sunlight gives 1361 W/m² above the atmosphere.
The energy is in the fields.
Poynting vector
Energy flux of an EM wave.
S = E×B/μ₀S = E × B/μ₀ gives the energy flow per unit area per unit time (W/m²), in the direction of propagation.
Flows into a resistor from the surrounding fields.
Time-average of S gives intensity.
Intensity of EM wave
Average power per area.
I = ½ε₀cE₀²I = ½ε₀cE₀² = cB₀²/2μ₀. It decreases as 1/r² from a point source.
E₀ = 100 V/m: 13 W/m².
Intensity ∝ amplitude squared.
Radiation pressure
Light carries momentum.
P = I/cAbsorbed light exerts P = I/c; reflected light 2I/c. Comet tails point away from the Sun partly due to this; solar sails use it.
Sunlight on a perfect absorber: 4.6 μPa.
Tiny at everyday intensities.
EM spectrum
Radio, microwave, infrared, visible, UV, X-ray, gamma.
Ordered by wavelength: radio (>1 m), microwave, infrared, visible (400–700 nm), ultraviolet, X-ray, gamma (<10 pm). Same nature, different frequency and photon energy.
A visible photon has about 2 eV.
Wavelength decreases as energy rises.
Radio waves
Used for broadcasting and communication.
Wavelengths from mm to km; used for broadcasting, mobile phones and Wi-Fi. AM waves can bounce off the ionosphere.
FM at about 100 MHz has λ = 3 m.
Antenna length ~λ/2 or λ/4.
Microwaves
Used in radar and ovens.
Wavelengths of 1 mm to 1 m; used for radar, satellite links and ovens (2.45 GHz excites water molecules).
Cosmic microwave background peaks at 160 GHz.
The oven's door mesh blocks microwaves.
Infrared
Thermal radiation sensed as heat.
Wavelengths 700 nm–1 mm emitted by warm bodies. Used in remote controls, thermal imaging and heating. Humans emit at ~10 μm.
Night-vision cameras.
Glass blocks far infrared.
Visible light
The narrow band our eyes see, 400–700 nm.
The small band 400–700 nm detected by the eye; violet shortest, red longest. White light is a mix of all colours.
Peak sensitivity at 555 nm (green).
Different from the range of most radiation.
Ultraviolet
Causes sunburn and sterilises.
Wavelengths 10–400 nm; causes tanning, sunburn, fluorescence and sterilises surfaces. Ozone absorbs UV-C and most UV-B.
UV lamps disinfect water.
Can damage DNA.
X-rays
Penetrating radiation for imaging.
Wavelengths 0.01–10 nm produced when fast electrons hit a metal target. They penetrate soft tissue but not bone; used for imaging and crystallography.
Medical 50 keV X-rays.
Ionising radiation.
Gamma rays
Highest-energy photons from nuclear processes.
The highest-energy EM radiation (>100 keV), from nuclear transitions and cosmic events. Penetrating and ionising, used in cancer therapy and sterilisation.
Cobalt-60 emits 1.17 and 1.33 MeV.
Shielded by lead and thick concrete.
Polarisation of EM waves
The E field oscillates in a particular direction.
Polarisation is the orientation of E: linear, circular or elliptical. Polarising sunglasses block horizontally polarised glare.
Radio antennas need matching polarisation.
Transverse waves only.
Antennas
Dipole antennas radiate and receive EM waves.
A half-wave dipole is resonant at length λ/2 and radiates mainly sideways. Larger or directional antennas (Yagi, dishes) focus the power.
FM 100 MHz: dipole length 1.5 m.
Gain describes directionality.
Waveguides and transmission lines
Guide EM waves along structures.
Coaxial cables and waveguides carry EM energy with low loss; impedance matching avoids reflections. Characteristic impedance Z₀ = √(L/C) per length.
TV cable 75 Ω.
Mismatches cause standing waves.
Modulation (AM and FM)
Encoding information onto carrier waves.
Information is carried by varying a carrier's amplitude (AM) or frequency (FM). FM is more resistant to noise and AM has narrower bandwidth.
Digital schemes (QAM, OFDM) power Wi-Fi.
Demodulation recovers the signal.
Optical fibre communication
Total internal reflection guides light pulses.
Light pulses travel along glass fibres by total internal reflection with losses of 0.2 dB/km. A single fibre carries terabits per second.
Undersea cables connect continents.
Dispersion limits speed over long distances.
Radar
Uses reflected radio waves to measure range and speed.
Radar sends radio pulses and times the echo: distance = ct/2. Doppler shift gives velocity. Used for air traffic, weather and speed cameras.
An echo delay of 100 μs means 15 km.
Stealth aircraft reduce the echo.
Lasers in communication
Coherent light carries data at high rates.
Lasers produce coherent narrow-band light that can be modulated at GHz rates for fibre links and satellite-to-ground links.
Laser links to the Moon transmit video.
Atmospheric turbulence is an issue.
Electromagnetic shielding
Metal enclosures block fields.
Conductive enclosures block EM waves by reflection and absorption (skin depth). Needed in sensitive electronics and MRI rooms.
Cables with braided shields.
Holes must be smaller than the wavelength.
Skin effect
AC concentrates near the conductor surface.
At high frequency AC concentrates near the surface of a conductor; skin depth δ = √(2ρ/ωμ). For copper at 50 Hz it's 9 mm; at 1 MHz 66 μm.
High-frequency cables use hollow or stranded conductors.
Raises effective resistance.
Resonant cavities
Metal boxes support standing EM waves.
Metal boxes support EM standing-wave modes at discrete frequencies; used in microwave ovens, lasers and particle accelerators. Resonant frequency f = (c/2)√((l/a)² + ...).
Magnetron cavities.
Modes depend on the cavity shape.
Frequently asked questions
Why is the RMS value used for AC voltage?
RMS, or root mean square, is the steady DC value that would deliver the same average power to a resistor. For a sine wave, the RMS value is the peak value divided by the square root of 2.
Why does an inductor oppose AC but pass DC?
A changing current induces a back emf in an inductor, which opposes the change. Steady DC does not change, so the inductor offers almost no opposition once the current is established.
What is the difference between impedance and resistance?
Resistance is the opposition to current in a resistor only. Impedance is the total opposition in an AC circuit, including the effects of inductors and capacitors, and it depends on frequency.
Why is AC used for homes instead of DC?
AC can be stepped up and down with transformers, which makes long-distance transmission efficient. Many devices then convert it to DC internally.
Do electromagnetic waves need a medium?
No. Changing electric and magnetic fields sustain each other, so light and radio waves travel through vacuum at the speed of light.
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