Physics · Electricity and magnetism · Grade 11

Oscillating circuit (LC)

When you turn the radio dial, how does the circuit pick out a single station?

The situation

If you slowly turn the knob of an old radio, first one station and then another emerges from the noise. The knob rotates the plates of a variable capacitor inside, and the capacitor together with a coil forms an oscillating circuit. In this circuit electrical energy passes from the capacitor to the coil and back about a million times a second. Waves from dozens of stations are arriving at the aerial — how does the circuit “choose” just one of them, why do the oscillations die away, and how does a transmitter make them undamped?

Topic

Oscillating circuit. Radio receiver circuit: energy exchange between capacitor and coil

Key ideas

Thomson formula
Energy exchange
Resonance and damping

Formulas

T = 2 π \sqrt{L C}, ν_{0} = \frac{1}{2 π \sqrt{L C}}

Thomson’s formula: the natural period and frequency of the circuit.

W = \frac{q^{2}}{2 C} + \frac{L i^{2}}{2} = \frac{q_{m}^{2}}{2 C} = \frac{L I_{m}^{2}}{2}

The sum of the energies of the electric and magnetic fields (constant in an ideal circuit).

q = q_{m} e^{- β t} \cos ω t, β = \frac{R}{2 L}, ω = \sqrt{ω_{0}^{2} - β^{2}}

The solution of the damped oscillation equation $L \ddot{q} + R \dot{q} + q / C = 0$ .

δ = \ln \frac{q_{n}}{q_{n + 1}} = β T, Q = \frac{π}{δ} \approx \frac{1}{R} \sqrt{\frac{L}{C}}

The logarithmic decrement and the quality factor of the circuit.

Z = \sqrt{R^{2} + {(ω L - \frac{1}{ω C})}^{2}}, U_{C}^{r e s} = Q ℰ

The total impedance of the circuit; at resonance $Z = R$ and the voltage across the capacitor is $Q$ times the emf of the aerial.

Theory

What happens in the circuit?

A capacitor charged from a battery is connected to a coil. The capacitor starts to discharge, but the self-induction in the coil does not let the current grow instantly. When the capacitor is fully discharged ( $q = 0$ ) the current is at its greatest, and now self-induction does not let it stop instantly — the current recharges the capacitor with the opposite polarity. In this way energy keeps passing back and forth between the electric field (the violet rays between the plates, $W_{E} = q^{2} / 2 C$ ) and the magnetic field (the yellow lines around the coil, $W_{B} = L i^{2} / 2$ ). The electrons (blue dots) slosh back and forth in the wire.

Equation, amplitude, frequency, period

Kirchhoff’s law for the circuit: $L d i / d t + q / C = 0$ , that is $\ddot{q} + ω_{0}^{2} q = 0$ , $ω_{0} = 1 / \sqrt{L C}$ . The solution is a harmonic oscillation: $q = q_{m} \cos (ω_{0} t + φ_{0})$ , $i = \dot{q} = - q_{m} ω_{0} \sin (ω_{0} t + φ_{0})$ . $q_{m}$ is the charge amplitude, $I_{m} = q_{m} ω_{0}$ is the current amplitude. On the graph the current leads the voltage by a quarter of a period: when $u_{C} = 0$ , $| i |$ is greatest. Thomson’s formula $T = 2 π \sqrt{L C}$ : if you increase the capacitance by a factor of 4, the period becomes 2 times longer. With 200 µH and 150 pF, $T \approx 1.09$ µs and $ν \approx 919$ kHz — the medium-wave radio band. Time in the scene is slowed down a million times: 1 s on the screen is really 1 µs.

Mechanical analogy

A spring pendulum obeys the same equation: $m \ddot{x} + b \dot{x} + k x = 0$ . The correspondence: displacement $x \leftrightarrow q$ , velocity $v \leftrightarrow i$ , mass $m \leftrightarrow L$ (inertia — the coil resists a change in current), stiffness $k \leftrightarrow 1 / C$ , friction $b \leftrightarrow R$ ; potential energy $k x^{2} / 2 \leftrightarrow q^{2} / 2 C$ , kinetic energy $m v^{2} / 2 \leftrightarrow L i^{2} / 2$ , period $2 π \sqrt{m / k} \leftrightarrow 2 π \sqrt{L C}$ . The mass in the scene oscillates in the same way as the charge in the circuit.

Damped oscillations

In the resistance $R$ of the wires and the coil, energy is turned into heat ( $i^{2} R$ ), so the amplitude decreases like $e^{- β t}$ , $β = R / 2 L$ . The ratio of two successive peaks is constant: the logarithmic decrement $δ = \ln (q_{n} / q_{n + 1}) = β T$ . The quality factor $Q = π / δ$ — the oscillation dies away by a factor of $e$ in about $Q / π$ periods. In radio circuits $Q$ is about 50–200. If $R \geq 2 \sqrt{L / C}$ , there are no oscillations at all — the capacitor discharges slowly in one direction (an aperiodic process).

Forced oscillations: how a radio is tuned

The waves of all the stations induce small emfs in the aerial and produce forced oscillations in the circuit. The total impedance of a series circuit, $Z = \sqrt{R^{2} + {(ω L - 1 / ω C)}^{2}}$ , is smallest only at $ω = ω_{0}$ ( $Z = R$ ) — for that station the current is greatest, and the voltage across the capacitor is $Q$ times the emf (voltage resonance). For the other stations $Z$ is hundreds of times larger. The width of the resonance curve, $Δ ν \approx ν_{0} / Q$ , is the selectivity of the circuit. The parallel circuit and current resonance, and alternating current generators, are covered in the “Alternating current and RLC” experiment.

Transistor oscillator

For undamped oscillations, energy has to be added to the circuit at the right moment in every period — like pushing a swing at just the right time. A small feedback coil next to the main coil “senses” the direction of the current and opens the transistor in exactly the right half-period: the battery gives energy to the circuit. If the coupling is strong enough (in the model $k = R_{f b} / R > 1$ ), a small random oscillation grows, and when the transistor approaches saturation the growth stops: the loss and the energy supplied become equal — this is self-sustained oscillation. Oscillators like this work in radio transmitters and clocks. The model: the transistor acts like a negative resistance $R_{f b} (1 - i^{2} / I_{s}^{2})$ ; the steady amplitude is $I_{m} = I_{s} \sqrt{\frac{4}{3} (1 - 1 / k)}$ .

Did you know?

The formula for the oscillation period in a circuit was derived in 1853 by William Thomson (later Lord Kelvin) — which is why it is called Thomson’s formula.
In Europe, Africa and most of Asia (ITU Region 1) the medium-wave broadcasting band is 531–1 602 kHz, and the spacing between station frequencies is 9 kHz.

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

Magnetic properties of matter — Grade 10
Plasma — the fourth state — Grade 10
Power transmission line — Grade 11
Microwaves and Wi-Fi — Grade 11
Radiation from a dipole antenna — Grade 11
Alternating current and RLC — Grade 11
Radio communication and radar — Grade 11
Charged-particle accelerator — Grade 11
Thermoelectric phenomena — Year 2
Vector fields: div and curl — Year 3