EE101 – Tuned Circuits & Resonance

In AC & Impedance, we saw that:

• An inductor has impedance that increases with frequency: Z_L = j ω L • A capacitor has impedance that decreases with frequency: Z_C = 1 / (j ω C)
That means:
• At low frequency: the capacitor dominates (big impedance), the inductor is small.
• At high frequency: the inductor dominates (big impedance), the capacitor is small.

If you put an inductor (L) and capacitor (C) together, there is a special frequency where their reactances (their “frequency behavior”) perfectly balance. That is called resonance, and the resulting circuits are called tuned circuits.

At the resonant frequency, the magnitudes of the inductor and capacitor impedances are equal:

Solving for ω (the angular frequency at resonance): In terms of ordinary frequency f (in hertz): This frequency f₀ is the natural frequency or resonant frequency of an ideal LC combination.

At this frequency:
• The inductor and capacitor reactances are equal and opposite.
• Energy swaps back and forth between the electric field of the capacitor and the magnetic field of the inductor.

A simple way to picture resonance:

• The capacitor stores energy as an electric field when it is charged.
• The inductor stores energy as a magnetic field when current flows through it.

If you connect an ideal charged capacitor to an ideal inductor:
• The capacitor begins to discharge, causing current to flow into the inductor.
• The inductor’s magnetic field builds up as the capacitor’s voltage falls.
• Then the process reverses: the magnetic field collapses and recharges the capacitor with opposite polarity.

The energy “sloshes” back and forth between the two, like water moving from one side of a tank to the other. In an ideal world (no resistance), this oscillation would continue forever at the resonant frequency f₀.

In real circuits, resistance and losses cause the oscillations to die down over time, but for many cycles the behavior still looks like a nice, ringing sine wave.

In a series LC circuit, the inductor and capacitor are in series with each other, and that series combination is driven by a source.

At resonance:
• The inductor and capacitor reactances cancel each other.
• The total impedance can become very small (limited mainly by resistance in the circuit).
• The circuit can draw a relatively large current at f₀.

Practical consequences:
• A series resonant circuit behaves almost like a short at its resonant frequency.
• The voltage across L or C individually can become quite large (even if the source voltage is modest), due to the big circulating current.

Uses:
• Series-tuned circuits can be used to pick out or pass a narrow band of frequencies.
• They can also appear in matching networks and filters where a low impedance at a specific frequency is useful.

In a parallel LC circuit, the inductor and capacitor are connected in parallel, and that combination is connected to the source.

At resonance:
• The currents through L and C are equal and opposite in phase.
• The net current drawn from the source can be very small.
• The overall impedance can become very large (again limited by real-world resistance).

Practical consequences:
• A parallel resonant circuit behaves almost like an open circuit at its resonant frequency.
• It tends to block current at f₀ while allowing other frequencies to pass more easily through other paths.

Uses:
• Parallel resonant circuits are widely used as band-stop or “notch” elements (blocking a single frequency).
• They also appear in oscillator designs and in RF front ends to help define which frequencies a circuit prefers.

Because L and C react differently with frequency, an LC circuit becomes naturally frequency-selective.

• At frequencies far from f₀: the LC combination has significant reactance and doesn’t respond strongly.
• Near f₀: the LC combination either becomes low-impedance (series) or high-impedance (parallel), depending on configuration.

This is why tuned circuits are the backbone of:
• Radio receivers (tuning to one station while rejecting others).
• Analog filters (band-pass, band-stop).
• Oscillators that need a stable, well-defined frequency.

In a radio, for example:
• The antenna picks up many frequencies at once.
• A tuned circuit is adjusted (by changing L or C) so that its resonant frequency matches the desired station.
• The tuned circuit responds strongly at that frequency while suppressing others, feeding mostly the station you want into the next stage of the receiver.

In a perfect world, LC resonance would be infinitely sharp and never lose energy. In reality:

• Wires and components have resistance.
• Inductors have core losses.
• Capacitors have leakage and other non-idealities.

These losses:
• Broaden the resonance curve (the circuit responds over a wider band of frequencies).
• Cause oscillations to die out over time if not driven.

Engineers often describe how “sharp” or “selective” a resonant circuit is using a parameter called Q (quality factor). Higher Q:
• Narrower bandwidth around f₀.
• Stronger peak response at resonance.
• Lower relative losses.

Resonance in an LC circuit is similar to many everyday systems:

• A swing in a playground has a natural frequency — push it at just the right rhythm and the motion grows.
• A tuning fork rings at its own pitch when struck.
• A bridge or building can vibrate at specific frequencies under wind or earthquakes.

In each case, energy prefers to move back and forth at a particular rate. In electronics, LC circuits give us a way to choose that rate and build systems that:
• Select certain frequencies.
• Reject others.
• Generate oscillations at a stable, repeatable frequency.

From simple Ohm’s Law through AC impedance and now LC resonance, we’ve built a picture where circuits don’t just carry current — they shape, select, and create signals over frequency.

Next, we’ll zoom out slightly and connect this to the idea of waves, showing how frequency, wavelength, and the speed of light all tie together:

Waves & Frequency – f, λ, and c