Series and Parallel Resonance

Every antenna in the world is delivering a signal into your receiver at once. The reason you hear one station and not all of them is resonance: an inductor and a capacitor, chosen so that their opposite reactances cancel at exactly one frequency, and at no other. That single idea is the tuning dial, the IF filter, the crystal in the oscillator, the tank in the power amplifier and the antenna itself. If you understand one page of Module 2 properly, make it this one.

The resonance condition

The previous lesson established the two reactances and the fact that they pull in opposite directions:

• XL = 2πfL — rises as frequency rises
• Xc = 1 ÷ (2πfC) — falls as frequency rises

One line climbing and one line falling must cross somewhere, and they cross at exactly one frequency. That crossing is resonance:

Note carefully what is equal. It is the two reactances, in ohms — never the henries and the farads. Being equal in size and opposite in phase, they cancel completely, and the circuit stops behaving reactively at all: what is left is pure resistance, voltage and current come back into phase, and the power factor becomes unity.

The resonant frequency

Set the two reactances equal and solve for f:

2πfL = 1 ÷ (2πfC)  →  4π²f²LC = 1  →  f² = 1 ÷ (4π²LC)

Three consequences follow directly from where L and C sit in that expression, and the exam tests all three:

• Only L and C set the frequency. The resistance in the circuit does not appear. Resistance changes how sharp the response is, not where it is centred.
• Increasing either L or C lowers f₀. Both are in the denominator. Meshing a variable capacitor further in adds capacitance and tunes the receiver down the band; screwing a slug further into a coil adds inductance and does the same.
• The square root softens everything. Doubling C alone divides the frequency by √2, about 1.41. Doubling L and C makes the product LC four times bigger, so f₀ falls by √4 — it halves.

Worked example 1 — finding f₀

A tank circuit uses a 2.2 µH coil and a 235 pF capacitor. Where does it resonate?

1. L × C = 0.0000022 × 0.000000000235 = 5.17 × 10⁻¹⁶
2. √(LC) = 2.274 × 10⁻⁸
3. 2π × 2.274 × 10⁻⁸ = 1.429 × 10⁻⁷
4. f₀ = 1 ÷ 1.429 × 10⁻⁷ = 7.0 × 10⁶ Hz = 7.0 MHz

That is the middle of the 40 metre band, and those are realistic values for a home-built tank.

Series resonance: the acceptor

Put R, L and C in a single loop and feed the loop from a signal source. Off resonance, one reactance dominates and the impedance is high. At resonance the reactances cancel and:

• Impedance is at a minimum, and equals R alone. Not zero — a real coil always has some resistance, and it is that resistance which is left.
• Current is at a maximum, because I = V ÷ Z and Z is at its smallest.
• That current is in phase with the applied voltage.
• Large voltages appear across the coil and across the capacitor individually — Q times the applied voltage — even though they cancel when added.

Because it takes maximum current at its resonant frequency, a series tuned circuit is called an acceptor circuit. Wired in series with a signal path it lets the wanted frequency through; wired across a signal path it shorts one unwanted frequency to earth, which is how a trap works.

Parallel resonance: the tank

Put the coil and the capacitor side by side across the source instead and everything inverts. The two branches draw currents that are very nearly equal and almost exactly opposite in phase, so from the supply's point of view they cancel:

• Impedance is at a maximum at resonance.
• Line current — the current drawn from the source — is at a minimum.
• The current inside the L-C loop is large, Q times the line current. Energy sloshes from the capacitor's electric field into the coil's magnetic field and back, once per cycle, and the source only has to top up the losses.

Because it rejects current at its resonant frequency, a parallel tuned circuit is called a rejector circuit. Because energy sloshes about inside it, it is also called a tank circuit — the name you will hear for the output circuit of a transmitter's final stage.

The lab below keeps L, C and R fixed and lets you flip between the two circuits. Press 7 MHz tank, look at the series curve, then press Parallel (tank) and watch the same components produce the opposite shape. Then load 7 MHz, lossy coil and see what a higher resistance does to the width of the peak.

Q factor

Q, the quality factor, is a pure number that measures how little energy a tuned circuit wastes. Physically it is the ratio of energy stored to energy lost per cycle. Practically, in a series circuit, it is the ratio of the reactance to the resistance that is dissipating the energy:

Almost all of the loss lives in the coil: the resistance of its wire, losses in its core and in nearby metal. A capacitor of reasonable quality contributes little. So raising Q means thicker wire, better core material, a coil kept away from the chassis — and it is why a homemade air-cored coil wound on a plastic former can outperform a cramped one squeezed into a corner. A typical amateur tuned circuit reaches a Q of 100 to 250; a quartz crystal reaches tens of thousands, which is exactly why crystals are used where stability and selectivity matter.

Bandwidth and the half-power points

No tuned circuit responds to one frequency and nothing else — that would need infinite Q. It responds over a band, and the band is conventionally measured between the two frequencies at which the response has fallen to 0.707 of its peak value. Because power goes as the square of voltage, 0.707² = 0.5, so those are the points where the power has halved. In decibels that is a fall of 3 dB, so they are called the half-power points or the −3 dB points, and the span between them is the bandwidth.

Worked example 2 — bandwidth from Q

The 7 MHz circuit above has a Q of 100. What is its bandwidth?

BW = f₀ ÷ Q = 7 000 000 ÷ 100 = 70 000 Hz = 70 kHz.

Raise the Q to 700 and the same circuit passes only 10 kHz. Working backwards from Q also tells you the loss: at 7 MHz a 2.2 µH coil has XL = 2π × 7 000 000 × 0.0000022 = 96.8 Ω, so a Q of 100 means the circuit resistance is 96.8 ÷ 100 = 0.97 Ω. Under one ohm of stray resistance is all it takes to set the bandwidth of a 40 metre tank.

Selectivity, and what it costs

Selectivity is the ability to separate the wanted signal from its neighbours, and it is simply the other side of bandwidth: high Q, narrow bandwidth, sharp response, good selectivity. It is tempting to conclude that more Q is always better, and it is not.

The band you pass has to be wide enough for the signal you want. A CW signal needs only a few hundred hertz, so a very sharp filter suits it. An SSB signal occupies about 2.7 kHz and a filter narrower than that chops the top off the voice and makes it unintelligible. An AM broadcast signal needs perhaps 9 kHz. A tuned circuit sharp enough to reject the adjacent channel but wide enough to pass the wanted one is the whole design problem of a receiver front end, and the reason a good receiver has switchable filters.

Where resonance appears in a radio

That last row is worth pausing on. A dipole cut to length behaves exactly like a series resonant circuit: at its design frequency the reactance disappears, the impedance drops to a low resistance and the current is at a maximum. The antenna calculator is really a resonance calculator with a different set of units.

Coupled circuits

One tuned circuit gives a rounded peak. Two of them, coupled together, can be made to give a much better shape — flatter across the wanted band and steeper on the skirts. How much better depends entirely on how tightly they are coupled, which in a transformer-coupled pair means how close the two coils are and how much of one's flux links the other.

Critical coupling is defined as the coupling that transfers the greatest energy from the first circuit to the second at resonance. Couple more tightly than that and the peak does not grow further — it splits. The double-humped response of an over-coupled pair is wider overall and has steeper sides, which is often exactly what an IF filter wants, but it dips in the middle, so the designer chooses deliberately rather than by accident.