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Analogue Signal Processing: LC Resonance Filters and the Ideal Op-Amp

An analogue signal changes smoothly. We process it in two big ways: an LC circuit picks out one frequency (a filter), and an operational amplifier (op-amp) makes a small difference in voltage much bigger. The LC circuit resonates at f₀ = 1/(2π√LC). The quality factor Q = f₀/Δf tells how sharp the peak is. An ideal op-amp has infinite open-loop gain, infinite input resistance, zero output resistance and infinite bandwidth.

🎬 Step-by-step story

  1. A capacitor (C) is charged. All the energy sits in its electric field, shown by the yellow block between the plates.
  2. The capacitor empties through the coil (L). Energy moves into the coil's magnetic field (green), then back again. It swings like a pendulum.
  3. Feed signals of many frequencies in. The bars show the output. The tallest bar is at the resonant frequency f₀ = 1/(2π√LC). This is a filter: it lets one band through.
  4. Raise Q (less energy loss). The peak gets taller and narrower. The orange bars are inside the bandwidth Δf, where output ≥ 0.707 of the peak.
  5. An ideal op-amp (purple triangle) has two inputs, V₊ and V₋. Its output is Vout = A(V₊ − V₋), with a huge gain A. Without feedback it acts as a comparator.
  6. Free play: change L, C and Q. Bigger L or C lowers f₀. Watch the energy swap and the peak move.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why does the energy keep swinging between C and L?

When C empties, current is flowing and the coil's magnetic field opposes any change, so current keeps going and charges C the other way. Then it repeats.

Why does the LC circuit pick one frequency?

At f₀ the coil's and capacitor's effects cancel, so the circuit responds most strongly. Away from f₀ one of them blocks the signal.

What does 0.707 have to do with bandwidth?

0.707 = 1/√2. At that voltage the power is half the peak power. The bandwidth is the frequency range above that line (orange bars).

Why does the op-amp output stick at +12 V or −12 V?

The gain is so huge that any small difference asks for a huge voltage, but the chip cannot give more than its supply. So it saturates.

Does a bigger capacitor raise or lower the tuning frequency?

It lowers it, because f₀ = 1/(2π√LC). Try it on the C slider.

What is analogue signal processing?

An analogue signal is a voltage that can take any value and changes smoothly, like the voltage from a microphone. Processing means changing it in a useful way: choosing some frequencies (filtering) or making it bigger (amplifying).

Two key tools: the LC resonance filter and the operational amplifier.

LC resonance filters

An inductor (coil, L) and a capacitor (C) joined together make an LC circuit. Energy moves back and forth: electric field in C → magnetic field in L → back to C. This is called oscillation.

It oscillates naturally at the resonant frequency:

f₀ = 1 / (2π√(LC))

If signals of many frequencies arrive, the circuit gives a big output only near f₀. So it works as a band-pass filter (a tuner).

Energy loss and damping

Resistance in the wire turns some energy into heat each cycle. The oscillation dies away (damping). More resistance = flatter, wider peak.

Q factor and bandwidth

The bandwidth Δf is the width of the peak where the output voltage is at least 1/√2 ≈ 0.707 of the maximum (the half-power points). The quality factor is Q = f₀ / Δf. A high Q means a sharp, narrow peak: the filter is very selective.

The ideal operational amplifier

An op-amp is a chip with two inputs: non-inverting (V₊) and inverting (V₋). Its output is

Vout = A₀ (V₊ − V₋)

where A₀ is the open-loop gain (gain with no feedback).

Properties of an ideal op-amp

The op-amp as a comparator

With no feedback, even a tiny difference is multiplied hugely, so the output jumps to the positive or negative supply (it saturates). If V₊ > V₋ the output is +Vs; if V₊ < V₋ it is −Vs. This compares two voltages, for example a light sensor turning on a street lamp.

Try it: feel resonance on a swing

Push a child on a swing. Push at the swing's own rhythm and it goes higher and higher; push at a random rhythm and it hardly moves. That is resonance. In the 3D, set L = 5 and C = 5, then L = 20 and C = 20, and see f₀ drop.

Key formulas and definitions

Worked examples

1. Find f₀ for L = 10 mH and C = 10 nF.

LC = 0.010 × 10×10⁻⁹ = 1×10⁻¹⁰. √LC = 1×10⁻⁵. f₀ = 1/(2π × 10⁻⁵) ≈ 15 900 Hz ≈ 15.9 kHz.

2. A tuner has f₀ = 1.0 MHz and bandwidth 20 kHz. Find Q.

Q = f₀/Δf = 1 000 000 / 20 000 = 50.

3. You need f₀ = 1.0 kHz with C = 100 nF. What L do you need?

L = 1/(4π²f₀²C) = 1/(4π² × 10⁶ × 10⁻⁷) = 1/3.95 ≈ 0.25 H.

4. An op-amp comparator has supply ±12 V and A₀ = 10⁵. V₊ = 2.001 V, V₋ = 2.000 V. Find Vout.

A₀(V₊ − V₋) = 10⁵ × 0.001 = 100 V. That is more than 12 V, so the output saturates at +12 V.

5. If C is made 4 times bigger, what happens to f₀?

f₀ ∝ 1/√C, so f₀ becomes 1/√4 = 1/2 of the old value.

6. An op-amp has A₀ = 2×10⁵ and the output is 6 V (not saturated). What is V₊ − V₋?

V₊ − V₋ = Vout/A₀ = 6 / (2×10⁵) = 3×10⁻⁵ V = 30 μV. Almost zero, which is why we say V₊ ≈ V₋.

Common mistakes

Practice quiz

1. Resonant frequency of an LC circuit is:
2. If Q increases, the bandwidth:
3. An ideal op-amp has input resistance:
4. In an LC circuit, when the capacitor is fully charged, the current is:
5. An op-amp without feedback with V₊ > V₋ gives:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What does an LC filter do?

It passes signals near its resonant frequency f₀ = 1/(2π√LC) and weakens others. Radios use it to pick one station.

What is the Q factor?

Q = f₀/Δf. It tells how sharp the resonance peak is. A high Q filter selects a very narrow band of frequencies.

Why is an ideal op-amp's input current zero?

Its input resistance is infinite, so no current can flow into the inputs. Real op-amps take only picoamps to nanoamps.

Where this is taught

England (GCSE, A level)Year 133.13 Electronics

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