Signals & Wireless (RF)
How an LC band-pass filter picks one station
A series LC band-pass filter is a coil and a capacitor placed one after the other in a signal's path. At one frequency their reactances are equal and opposite and cancel, so the pair acts like a plain wire and that frequency passes. Below it the capacitor blocks; above it the coil does. That's how a simple radio picks one station out of everything an antenna hears.
Opposite reactances
Each part holds back AC by its reactance, in ohms, and it changes with frequency. A coil's rises; a capacitor's falls:
They also act in opposite directions, so they're written with opposite signs: +j for the coil, −j for the capacitor. At 10 MHz a 2.5 µH coil is +j157 Ω and a 100 pF capacitor −j159 Ω. In series they add to about 2 Ω: next to nothing.
Find the capacitor
They cancel where XL = XC, at the resonant frequency:
For 10 MHz and 2.5 µH, C = 1 ÷ (4π² × 10¹⁴ × 2.5 × 10⁻⁶) ≈ 100 pF.
What gets through
With a 50 Ω receiver, the fraction of the signal that reaches it is R ÷ √(R² + X²), where X is what's left of the two reactances:
| Station | Net reactance | Reaches the receiver |
|---|---|---|
| 1 MHz | about −j1.6 kΩ | about 3 % |
| 10 MHz | about 0 | nearly all |
| 30 MHz | about +j420 Ω | about 12 % |
Live experiment
Slide the peak along the band
A 2.5 µH coil and a capacitor sit in series between the generator and a 50 Ω receiver, with stations on 1, 3, 10 and 30 MHz. Turn the capacitor from 1 nF down to 10 pF and the pass band slides from 3 MHz up past 30. At about 100 pF the 10 MHz station comes through whole.
“Select a band” is lesson 1 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. It’s open as soon as you sign up.
Tuning
Frequency goes as 1 ÷ √C, so the capacitor is the tuning knob. With 1 nF the same coil peaks near 3.2 MHz; to move up to 10 MHz, about 3.2 times higher, you need about a tenth of the capacitance. Old radios did this with a variable capacitor on the dial.
How sharp the peak is depends on its Q. Here Q = √(L ÷ C) ÷ R = 158 ÷ 50, about 3, so the pass band is a few megahertz wide: enough to separate stations at 1, 10 and 30 MHz, not neighbours 100 kHz apart.
Free lesson
Now build it yourself
Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.
“Select a band” is lesson 1 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. It’s open as soon as you sign up.
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Practice questions
Three quick questions from the lessons. Pick an answer to check it.
Show the answer
Answer: About 0 Ω: the pair is a straight wire. Inductive and capacitive reactances have opposite signs. Equal and opposite, they cancel, and the only thing left in the loop is the load.
- Why not “316 Ω”? Adding the sizes ignores their signs. A coil's +j and a capacitor's −j pull in opposite directions.
- Why not “−159 Ω”? That's the capacitor alone. The coil's +j157 Ω cancels nearly all of it.
- Why not “157 Ω”? That's the coil alone. The capacitor's −j159 Ω cancels it.
Show the answer
Answer: Smaller: about a tenth. f goes as 1 ÷ √C. About 3.2 times the frequency needs about a tenth of the capacitance (3.2² ≈ 10): 1 nF down to about 100 pF.
- Why not “Bigger”? More capacitance lowers the resonant frequency, moving the peak down, not up.
- Why not “The same”? With the same coil and capacitor the peak stays where it is, near 3 MHz.
- Why not “Removed”? Without the capacitor nothing cancels the coil, so there's no peak at all, just a coil blocking the high frequencies.
Show the answer
Answer: About 100 pF. C = 1 ÷ (4π² × (10⁷)² × 2.5 × 10⁻⁶) ≈ 1.0 × 10⁻¹⁰ F, which is 100 pF.
- Why not “10 nF”? More capacitance moves the peak down, to about 1 MHz. 10 MHz needs less.
- Why not “1 µF”? 1 µF with 2.5 µH resonates near 100 kHz, far below 10 MHz.
- Why not “Leave it”? 1 nF with 2.5 µH resonates near 3 MHz, not 10.
“Select a band” is lesson 1 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. It’s open as soon as you sign up.
Questions people ask
What is the difference between a series and a parallel LC filter?
A series LC placed in the signal's path is nearly a short at resonance, so that frequency passes and others are blocked. A parallel LC, a tank, is nearly an open circuit at resonance; placed from the signal to ground, it lets that frequency through and shorts the others away. Both pick out one frequency.
How do I make the pass band narrower?
Raise the Q. For a series LC driving a load R, Q = √(L ÷ C) ÷ R, and the bandwidth is roughly the centre frequency divided by Q. A bigger coil and a smaller capacitor for the same frequency, or a lower-resistance load, make it sharper.
How do you tune an LC filter to a different station?
Change L or C. Old radios turned a variable capacitor. Because the frequency goes as 1 ÷ √C, moving up by about 3.2 times needs a tenth of the capacitance.
Related topics
- LC resonant frequency explainedA coil and a capacitor swapping energy at one frequency: f₀ = 1 ÷ (2π√(LC)), tanks and tuning.
- Capacitive and inductive reactance explainedWhy a capacitor passes treble and a coil passes bass: X_C, X_L and Ohm's law for AC.
- What is impedance?Resistance and reactance combined: why 3 kΩ and 4 kΩ make 5 kΩ, and what phase means.
- Amplitude modulation (AM) explainedTurn a carrier's height up and down with a voice: modulation depth, sidebands, and why past 100% it tears.