Physics · Electricity and magnetism · Grade 11

Alternating current and RLC

The socket says 220 V, yet the voltage reaches 311 V — how?

The situation

On the socket at home it says “220 V, 50 Hz”. But if you connect an oscilloscope, you see the voltage swinging from +311 V to −311 V 50 times every second. This current is produced by a generator in a hydroelectric or thermal power station: a coil rotating in a magnetic field. A kettle accepts this current without complaint, a capacitor seems to let it pass, while a coil “chokes” it; and at a certain frequency the voltages in the circuit can exceed the source voltage several times over. What does 220 V mean, and what resistance do a coil and a capacitor offer to alternating current?

Topic

Alternating current and the RLC circuit. Hydroelectric generator, 220 V at home and voltage resonance

Key ideas

Effective (r.m.s.) value
Reactance
Series resonance

Formulas

e = N B S ω \sin ω t, U = \frac{U_{m}}{\sqrt{2}}

Generator EMF and the effective (RMS) value: $311 / \sqrt{2} \approx 220$ V.

X_{L} = ω L, X_{C} = \frac{1}{ω C}

Inductive and capacitive reactance; $ω = 2 π ν$ .

I = \frac{U}{Z}, Z = \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}, tg φ = \frac{X_{L} - X_{C}}{R}

Ohm's law for a series RLC circuit and the total resistance (impedance).

ν_{0} = \frac{1}{2 π \sqrt{L C}}

Resonant frequency: $X_{L} = X_{C}$ , $Z = R$ , the current is greatest (series — voltage resonance, parallel — current resonance).

P = U I \cos φ, \cos φ = \frac{R}{Z}

Average power of alternating current and the power factor.

Theory

How alternating current is produced

The flux through a loop rotating at angular velocity $ω$ in a uniform magnetic field, $Φ = B S \cos ω t$ , varies periodically, and by Faraday's law an EMF is induced: $e = - N d Φ / d t = N B S ω \sin ω t$ . The EMF is greatest at the moment when the flux is zero — when the plane of the loop is parallel to the field lines. The ends of the loop are connected to the external circuit through slip rings and brushes. If, instead of the rings, a split ring — a commutator — is fitted, the connection is reversed every half turn and the current has one direction (pulsating) — a DC generator. In power stations, on the other hand, the magnet rotates and the windings are stationary; for 50 Hz a two-pole rotor turns 3 000 times a minute. Another way of obtaining energy is the MHD generator: an EMF is induced in hot plasma flowing through a magnetic field (details — in the “Plasma — the fourth state” experiment).

Amplitude and effective value

Voltage and current change from instant to instant, so the effective (RMS) value is used: the value of the direct current that releases in a resistor the same amount of heat as the alternating current does. Because $⟨ \sin^{2} ⟩ = 1 / 2$ , $U = U_{m} / \sqrt{2}$ , $I = I_{m} / \sqrt{2}$ . Voltmeters and ammeters read exactly this value: the 220 V in the mains is the effective value, and the amplitude is $220 \sqrt{2} \approx 311$ V. A lamp is at its brightest 100 times in 1 s (every half-period).

R, L and C in an alternating-current circuit

In a resistance $R$ the current and voltage are in phase, and energy is converted to heat. In a coil the self-induction EMF opposes the change in current: inductive reactance $X_{L} = ω L$ , and the current lags the voltage by $90 °$ (in a real coil slightly less than $90 °$ because of the winding resistance). No charge passes through a capacitor, but it is recharged every half-period: capacitive reactance $X_{C} = 1 / (ω C)$ , and the current leads the voltage by $90 °$ . In direct current ( $ν = 0$ ) a coil is only the resistance of its wire, and a capacitor is a broken circuit. In $X_{L}$ and $X_{C}$ no energy is used up on average — the energy goes back and forth into the field, so they are reactances. Lab work: measure a coreless coil with direct current to find $R_{L} = U / I$ , and with alternating current find $Z = U / I$ ; $X_{L} = \sqrt{Z^{2} - R_{L}^{2}}$ , $L = X_{L} / (2 π ν)$ .

Phasor diagram and Ohm's law

We represent each sinusoidal quantity by a vector (a phasor) rotating at angular velocity $ω$ : its projection on the vertical axis is the instantaneous value. In a series circuit the current is common, so ${\vec{U}}_{R}$ is drawn along the current, ${\vec{U}}_{L}$ $90 °$ ahead of it, and ${\vec{U}}_{C}$ $90 °$ behind; the source voltage is their vector sum: $U = \sqrt{U_{R}^{2} + {(U_{L} - U_{C})}^{2}}$ . From this follow Ohm's law for alternating current $I = U / Z$ and the total resistance $Z = \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}$ . In complex notation $Z = R + i (X_{L} - X_{C})$ is the impedance, and its argument is the phase shift $φ$ .

Resonance

When $X_{L} = X_{C}$ , that is at $ν_{0} = 1 / (2 π \sqrt{L C})$ , the total resistance of the series circuit is smallest ( $Z = R$ ), and the current is greatest and in phase with the voltage. The voltages across the coil and the capacitor cancel each other, but each can be $Q = X_{L} / R$ times greater than the source voltage — voltage resonance (this is how radio circuits select a station — “Oscillating circuit (LC)”). If L and C are connected in parallel, it is the other way round: the currents in the branches are large and opposite, while the total current is smallest — current resonance. Lab work: vary the frequency to obtain the $I (ν)$ curve and compare its peak with the theoretical $ν_{0}$ .

Power, three-phase current, rectification

The instantaneous power $p = u i$ also varies; its mean, $P = U I \cos φ$ , is the rate of doing work (W). $S = U I$ is the apparent power (V·A), and $\cos φ = P / S$ is the power factor. Motors are an inductive load: if $\cos φ$ is small, a larger current flows from the mains for the same useful power and the wires heat up more. A parallel capacitor compensates the reactive current and brings $\cos φ$ close to 1. In a three-phase system the three EMFs are shifted by $120 °$ from one another, and their sum at every instant is zero; the phase voltage is 220 V, and the voltage between phases (line voltage) is $220 \sqrt{3} \approx 380$ V. For long-distance transmission the voltage is raised with transformers — in the “Power transmission line” experiment. Rectification: a diode lets current through in one direction; a single diode cuts off a half-period (mean $U_{m} / π$ ), a diode bridge turns both half-periods the same way ( $2 U_{m} / π$ ), and a capacitor smooths the ripple. The cyclotron too uses an alternating voltage: the particle passes through the accelerating gap every half-turn — in the “Charged-particle accelerator” experiment.

Did you know?

In Uzbekistan, and in Europe and many countries of Asia, the mains frequency is 50 Hz, while in the USA, Canada and parts of Japan it is 60 Hz.
The three-phase current system was developed in 1888–1891 by the Russian engineer Mikhail Dolivo-Dobrovolsky; in 1891 three-phase current was transmitted over the Lauffen–Frankfurt line for about 175 km.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Plasma — the fourth state — Grade 10
Power transmission line — Grade 11
Microwaves and Wi-Fi — Grade 11
Radiation from a dipole antenna — Grade 11
Oscillating circuit (LC) — Grade 11
Radio communication and radar — Grade 11
Charged-particle accelerator — Grade 11
Thermoelectric phenomena — Year 2
Vector fields: div and curl — Year 3