Resonance in LC Circuits

An LC circuit contains an inductor (L) and a capacitor (C) connected together. These two components can repeatedly exchange stored energy.

• A capacitor stores energy in an electric field.
• An inductor stores energy in a magnetic field.

Under suitable conditions, energy moves back and forth between these two fields, producing electrical oscillations.

At the circuit's resonant frequency, this exchange occurs naturally and efficiently. LC resonance is extremely important in radio, television, wireless communication, filters, oscillators, and many other electronic systems.

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The LC Circuit

The two main components are:

Capacitor (C)

Stores energy in an electric field.

Inductor (L)

Stores energy in a magnetic field.

The capacitor's stored energy is:

E_C = ½CV²

where:

• C = capacitance
• V = voltage across the capacitor

The inductor's stored energy is:

E_L = ½LI²

where:

• L = inductance
• I = current through the inductor

These two forms of stored energy can continually change into one another.

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Beginning with a Charged Capacitor

Imagine that the capacitor is initially fully charged.

At this moment:

• capacitor voltage is maximum
• charge on the capacitor is maximum
• current is zero
• magnetic field around the inductor is zero

Most or all of the circuit's stored energy is in the capacitor's electric field.

Therefore:

electric energy = maximum

magnetic energy = zero

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The Capacitor Begins to Discharge

When the capacitor is connected to the inductor, it begins to discharge.

Current starts flowing through the inductor.

As current increases:

• capacitor charge decreases
• capacitor voltage decreases
• electric-field energy decreases
• magnetic field around the inductor becomes stronger
• magnetic-field energy increases

Energy is being transferred:

electric field → magnetic field

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Maximum Current

Eventually, the capacitor becomes momentarily uncharged.

At this instant:

V = 0

so:

E_C = ½CV² = 0

However, current through the inductor is now at its maximum.

Therefore:

E_L = ½LI²

is maximum.

At this stage:

• electric-field energy is minimum
• magnetic-field energy is maximum
• current is maximum

The circuit's stored energy has moved from the capacitor into the inductor.

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Why the Current Does Not Immediately Stop

When the capacitor reaches zero charge, current does not simply stop.

The inductor opposes changes in current.

Its collapsing magnetic field produces an induced EMF that keeps current flowing.

As a result, the capacitor begins charging again — but with the opposite polarity.

Energy now moves:

magnetic field → electric field

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The Capacitor Charges in Reverse

As the inductor's magnetic field decreases:

• current decreases
• magnetic energy decreases
• charge builds on the capacitor
• capacitor voltage increases in the opposite direction
• electric-field energy increases

Eventually:

• current becomes zero
• magnetic field becomes zero
• capacitor charge reaches maximum again

But the capacitor's polarity is now opposite to its original polarity.

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The Cycle Repeats

The capacitor then begins discharging again in the opposite direction.

The complete process is:

charged capacitor

↓

current increases

↓

magnetic field strengthens

↓

capacitor becomes uncharged

↓

current reaches maximum

↓

magnetic field collapses

↓

capacitor charges with opposite polarity

↓

current reaches zero

↓

cycle repeats

This produces an electrical oscillation.

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Energy During One Complete Cycle

A useful way to visualize an ideal LC cycle is:

The process then repeats in the opposite direction.

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Conservation of Energy

In an ideal LC circuit, there is no resistance.

Therefore, no energy is lost.

Total energy remains constant:

E_total = E_C + E_L

so:

E_total = ½CV² + ½LI²

At different moments, the energy is divided differently between the electric and magnetic fields.

But the total remains constant.

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Maximum Electric Energy

When the capacitor is fully charged:

I = 0

Therefore:

E_L = 0

and:

E_total = ½CV_max²

All stored energy is electric-field energy.

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Maximum Magnetic Energy

When the capacitor is momentarily uncharged:

V = 0

Therefore:

E_C = 0

and:

E_total = ½LI_max²

All stored energy is magnetic-field energy.

Because total energy is conserved:

½CV_max² = ½LI_max²

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Worked Example: Energy Transfer

A capacitor has:

C = 20 μF

and is initially charged to:

V = 10 V

Convert:

20 μF = 20 × 10⁻⁶ F

Initial energy:

E = ½CV²

E = ½(20 × 10⁻⁶)(10²)

E = 1.0 × 10⁻³ J

E = 0.0010 J

In an ideal LC circuit, when the capacitor becomes completely discharged, this same energy is stored in the inductor's magnetic field.

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Worked Example: Maximum Current

Suppose the circuit from the previous example has:

L = 0.050 H

At maximum current:

½LI_max² = 0.0010 J

So:

I_max² = 2(0.0010)/0.050

I_max² = 0.040

I_max = √0.040

I_max = 0.20 A

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Oscillation and Simple Harmonic Motion

The behaviour of an ideal LC circuit is mathematically similar to simple harmonic motion.

Mechanical systems can exchange energy between:

• kinetic energy
• potential energy

An LC circuit exchanges energy between:

• magnetic-field energy
• electric-field energy

A mass-spring system and an LC circuit therefore have important mathematical similarities.

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Mechanical Analogy

Consider a mass attached to a spring.

At maximum displacement:

• speed = 0
• elastic potential energy = maximum
• kinetic energy = zero

At equilibrium:

• speed = maximum
• kinetic energy = maximum
• elastic potential energy = minimum

Similarly, in an LC circuit:

maximum capacitor charge → current = 0

and:

zero capacitor charge → current = maximum

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Resonance

Every ideal LC circuit has a natural frequency at which it tends to oscillate.

This is called its resonant frequency.

The resonant frequency is:

f₀ = 1/(2π√(LC))

where:

• f₀ = resonant frequency in hertz
• L = inductance in henries
• C = capacitance in farads

The values of L and C determine the frequency.

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Effect of Inductance

From:

f₀ = 1/(2π√(LC))

increasing L increases the denominator.

Therefore:

larger L → lower resonant frequency

Similarly:

smaller L → higher resonant frequency

The relationship is:

f₀ ∝ 1/√L

when C remains constant.

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Effect of Capacitance

Increasing C also increases the denominator.

Therefore:

larger C → lower resonant frequency

and:

smaller C → higher resonant frequency

The relationship is:

f₀ ∝ 1/√C

when L remains constant.

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Worked Example: Resonant Frequency

An LC circuit contains:

L = 0.020 H

C = 5.0 μF

Convert:

C = 5.0 × 10⁻⁶ F

Use:

f₀ = 1/(2π√(LC))

f₀ = 1/[2π√((0.020)(5.0 × 10⁻⁶))]

LC = 1.0 × 10⁻⁷

√LC ≈ 3.16 × 10⁻⁴

Therefore:

f₀ ≈ 1/[2π(3.16 × 10⁻⁴)]

f₀ ≈ 503 Hz

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Worked Example: Changing Capacitance

Suppose the capacitance in an LC circuit increases by a factor of 4 while L remains unchanged.

Since:

f₀ ∝ 1/√C

then:

f_new/f_old = 1/√4

f_new/f_old = 1/2

Therefore:

f_new = ½f_old

Quadrupling the capacitance halves the resonant frequency.

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Worked Example: Changing Inductance

Suppose inductance increases by a factor of 9.

Since:

f₀ ∝ 1/√L

then:

f_new/f_old = 1/√9

f_new/f_old = 1/3

Therefore:

f_new = ⅓f_old

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Finding Capacitance from Resonant Frequency

Starting with:

f₀ = 1/(2π√(LC))

the equation can be rearranged:

C = 1/(4π²f₀²L)

Suppose:

f₀ = 1000 Hz

L = 0.010 H

Then:

C = 1/[4π²(1000²)(0.010)]

C ≈ 2.53 × 10⁻⁶ F

or:

C ≈ 2.53 μF

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Finding Inductance

The resonant-frequency equation can also be rearranged:

L = 1/(4π²f₀²C)

This allows engineers to select an inductor that produces a desired resonant frequency with a particular capacitor.

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Resonance in a Driven Circuit

An LC circuit can also be driven by an external alternating voltage.

If the frequency of the external source is close to the circuit's natural frequency, resonance occurs.

At resonance, energy transfer between the source and circuit can become particularly effective.

The circuit can respond much more strongly near its resonant frequency than at frequencies far from resonance.

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Resonance and Frequency Selection

This frequency-dependent response allows an LC circuit to distinguish between signals of different frequencies.

Suppose many frequencies are present simultaneously.

A tuned LC circuit can be designed so that:

desired signal frequency ≈ resonant frequency

The desired frequency then produces a much stronger response than many unwanted frequencies.

This is the basis of tuning.

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Tuned Circuits

A tuned circuit is designed to respond strongly to a selected frequency or narrow range of frequencies.

LC circuits are especially useful for tuning because their resonant frequency can be controlled by changing:

• L
• C

The relationship is:

f₀ = 1/(2π√(LC))

Changing either component changes the frequency selected by the circuit.

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Variable Capacitors

Some traditional radio tuning systems use a variable capacitor.

Turning the tuning control changes C.

Since:

f₀ = 1/(2π√(LC))

changing C changes the resonant frequency.

The receiver can therefore be tuned to different stations.

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Example: Selecting a Radio Station

Suppose a receiver's antenna picks up signals at several frequencies.

The signals might contain frequencies such as:

• f₁
• f₂
• f₃
• f₄

The tuning circuit is adjusted so:

f₀ = f₃

The circuit responds strongly to f₃ while responding less strongly to the other frequencies.

This allows the receiver to select the desired signal.

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Resonance in Radio Receivers

Radio waves induce electrical signals in an antenna.

The antenna may receive signals from many transmitters simultaneously.

A tuned circuit helps select the desired carrier frequency.

The process can be represented as:

many radio signals reach antenna

↓

electrical signals are produced

↓

LC circuit selects desired frequency

↓

receiver processes selected signal

↓

information is recovered

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Resonance in Communication Systems

Modern communication systems must separate many signals sharing the electromagnetic environment.

Different signals can be assigned different frequencies.

Resonant and frequency-selective circuits help electronic systems distinguish among them.

Applications include:

• radio
• television
• wireless communication
• radar
• satellite communication
• some sensor systems

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Transmitters

Resonant circuits are also useful in transmitters.

A transmitter needs to generate electrical oscillations at a controlled frequency.

LC circuits can form part of an oscillator that produces or stabilizes high-frequency electrical signals.

These electrical oscillations can then be used to produce electromagnetic waves.

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Carrier Waves

Many communication systems use a high-frequency carrier wave.

Information is added to or encoded onto this carrier.

The carrier must have a carefully controlled frequency.

Resonant circuits can help generate or select particular frequencies.

The receiver can then tune to the appropriate frequency to recover the desired signal.

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Filters

LC circuits can also be used as electrical filters.

Filters can:

• pass certain frequencies
• reduce unwanted frequencies
• separate frequency ranges
• suppress noise
• select communication channels

Different arrangements of inductors and capacitors can produce different filtering behaviour.

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Band-Pass Behaviour

A band-pass filter allows a selected range of frequencies to pass more effectively than frequencies outside that range.

Resonant circuits can be used in band-pass systems.

This is useful when a receiver needs to accept signals near a desired frequency while rejecting others.

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Rejecting Unwanted Frequencies

A circuit can also be designed to strongly reduce a particular unwanted frequency.

Such frequency-selective behaviour is useful for:

• reducing interference
• removing noise
• separating channels
• improving signal quality

The exact behaviour depends on how the inductor, capacitor, resistor, source, and load are arranged.

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Real LC Circuits

A perfectly ideal LC circuit does not exist.

Real wires and components have resistance.

Therefore, some electrical energy is continually converted into thermal energy.

Instead of oscillating forever, an undriven real circuit's oscillations gradually decrease.

This is called damping.

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Damped Oscillations

In a real circuit:

electric energy ↔ magnetic energy

still occurs.

However, during each cycle some energy becomes thermal energy.

Therefore:

total oscillation energy decreases

and the amplitude gradually becomes smaller.

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Resistance and Resonance

Resistance affects how sharply a circuit responds around resonance.

Lower resistance generally allows:

• stronger oscillations
• less energy loss per cycle
• sharper frequency selection

Greater resistance generally causes:

• more damping
• greater energy loss
• broader and weaker resonance

This becomes particularly important in RLC circuits, which contain resistance, inductance, and capacitance.

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Quality Factor

The sharpness of resonance is often described by the quality factor, or Q factor.

A high-Q resonant circuit generally has:

• relatively low energy loss
• narrow resonance
• strong frequency selectivity

A lower-Q circuit has:

• greater damping
• broader resonance
• less precise frequency selection

High selectivity can be valuable in communication receivers.

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Energy at Different Points in the Cycle

Consider an ideal LC circuit beginning with a fully charged capacitor.

At t = 0:

• capacitor charge = maximum
• voltage = maximum
• current = zero
• electric energy = maximum
• magnetic energy = zero

One-quarter cycle later:

• capacitor charge = zero
• voltage = zero
• current = maximum
• electric energy = zero
• magnetic energy = maximum

One-half cycle later:

• capacitor charge = maximum in opposite direction
• voltage = maximum with opposite polarity
• current = zero
• electric energy = maximum
• magnetic energy = zero

Three-quarters of a cycle later:

• capacitor charge = zero
• current = maximum in opposite direction
• magnetic energy = maximum

One complete cycle later:

The circuit returns to its original state.

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Charge and Current Are Out of Phase

In an ideal LC circuit, capacitor charge and current do not reach their maximum values at the same time.

When charge is maximum:

I = 0

When current is maximum:

Q = 0

They are separated by one-quarter of a cycle.

This is a 90° phase difference.

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Voltage and Current

Capacitor voltage is proportional to capacitor charge:

V_C = Q/C

Therefore, capacitor voltage is also maximum when capacitor charge is maximum.

At those moments, current is zero.

When current reaches maximum magnitude, capacitor voltage passes through zero.

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Frequency and Angular Frequency

Resonance can also be described using angular frequency.

Angular frequency is:

ω₀ = 2πf₀

For an ideal LC circuit:

ω₀ = 1/√(LC)

Therefore:

f₀ = 1/(2π√(LC))

Both equations describe the same resonant behaviour.

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Worked Example: Angular Frequency

Suppose:

L = 0.025 H

C = 4.0 μF

Convert:

C = 4.0 × 10⁻⁶ F

Calculate:

ω₀ = 1/√(LC)

ω₀ = 1/√[(0.025)(4.0 × 10⁻⁶)]

ω₀ = 1/√(1.0 × 10⁻⁷)

ω₀ ≈ 3162 rad/s

Then:

f₀ = ω₀/(2π)

f₀ ≈ 503 Hz

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A Complete Resonance Problem

A tuned circuit contains:

L = 250 μH

C = 100 pF

Convert:

L = 250 × 10⁻⁶ H

C = 100 × 10⁻¹² F

Use:

f₀ = 1/(2π√(LC))

LC = (250 × 10⁻⁶)(100 × 10⁻¹²)

LC = 2.50 × 10⁻¹⁴

√LC ≈ 1.58 × 10⁻⁷

Therefore:

f₀ ≈ 1/[2π(1.58 × 10⁻⁷)]

f₀ ≈ 1.01 × 10⁶ Hz

or approximately:

f₀ ≈ 1.01 MHz

This frequency lies in the radio-frequency range, demonstrating why LC circuits are useful for radio tuning.

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Why LC Circuits Are Useful

LC circuits have several important advantages.

They can:

• produce electrical oscillations
• select particular frequencies
• reject unwanted frequencies
• store and exchange electrical energy
• form part of oscillators
• form part of filters
• tune communication equipment

Their resonant frequency can be designed by choosing suitable values of L and C.

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Applications of Tuned Circuits

Tuned circuits are used in many technologies, including:

• radio receivers
• radio transmitters
• television systems
• wireless communication equipment
• signal generators
• filters
• oscillators
• measurement equipment
• impedance-matching networks

Modern systems may also use digital and semiconductor technologies for many frequency-selection tasks, but LC resonance remains a fundamental principle in electronics.

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Resonance and Wireless Systems

Wireless systems depend on selecting and processing electromagnetic signals at particular frequencies.

Resonant circuits can help:

• select desired signals
• reject interference
• generate carrier frequencies
• filter unwanted frequencies
• improve energy transfer in some wireless systems

Resonance is therefore important far beyond traditional radio receivers.

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Resonant Wireless Power Transfer

Resonance can also improve some forms of wireless energy transfer.

If transmitting and receiving circuits are tuned to compatible resonant frequencies, energy transfer can become more effective.

This combines:

• mutual inductance
• electromagnetic induction
• LC resonance

Applications can include wireless charging and specialized contactless power systems.

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Comparing LC Energy Storage

The continuous exchange between these two energy stores produces LC oscillation.

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A Reliable Strategy for LC Resonance Problems

First identify:

• L
• C
• required resonant frequency

Convert all quantities into SI units:

• H
• F
• Hz

Common conversions include:

1 mH = 10⁻³ H

1 μH = 10⁻⁶ H

1 μF = 10⁻⁶ F

1 nF = 10⁻⁹ F

1 pF = 10⁻¹² F

Then use:

f₀ = 1/(2π√(LC))

If necessary, rearrange:

L = 1/(4π²f₀²C)

or:

C = 1/(4π²f₀²L)

Finally, check whether the answer is physically reasonable.

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Common Mistakes

Thinking the capacitor and inductor store the same type of energy

The capacitor stores electric-field energy. The inductor stores magnetic-field energy.

Thinking energy disappears when the capacitor becomes uncharged

At that moment, the energy can be stored in the inductor's magnetic field.

Thinking current is maximum when capacitor charge is maximum

Current is zero when capacitor charge reaches maximum magnitude.

Forgetting the square root in the resonance equation

The correct relationship is:

f₀ = 1/(2π√(LC))

Forgetting to convert units

Microfarads, nanofarads, picofarads, millihenries, and microhenries must be converted carefully.

Thinking larger L produces higher frequency

Increasing L decreases resonant frequency.

Thinking larger C produces higher frequency

Increasing C also decreases resonant frequency.

Assuming real LC circuits oscillate forever

Resistance causes energy loss and damping.

Thinking resonance amplifies every frequency equally

Resonant circuits respond particularly strongly near specific frequencies.

Confusing frequency with angular frequency

Frequency f is measured in hertz.

Angular frequency ω is measured in radians per second.

They are related by:

ω = 2πf

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Key Terms

LC circuit — circuit containing an inductor and capacitor that can exchange magnetic and electric field energy.

Resonance — strong response that occurs when a system is driven near its natural frequency.

Resonant frequency — natural oscillation frequency of an LC circuit.

Capacitance — ability of a capacitor to store separated electric charge and electric-field energy.

Inductance — property associated with magnetic-field energy storage and opposition to changes in current.

Electric-field energy — energy stored by a capacitor, given by E_C = ½CV².

Magnetic-field energy — energy stored by an inductor, given by E_L = ½LI².

Oscillation — repeated variation of a quantity about an equilibrium condition.

Tuned circuit — circuit designed to respond strongly to a selected frequency.

Natural frequency — frequency at which a system tends to oscillate when disturbed.

Angular frequency — frequency measured in radians per second, with ω = 2πf.

Damping — gradual reduction in oscillation amplitude caused by energy loss.

Quality factor (Q) — measure associated with the sharpness of resonance and energy loss in a resonant system.

Filter — circuit that selectively passes or reduces different frequencies.

Carrier wave — high-frequency wave used to carry information in a communication system.

Frequency selectivity — ability of a system to distinguish between signals of different frequencies.

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Key Takeaways

• An LC circuit contains an inductor and a capacitor.
• Capacitors store energy in electric fields.
• Inductors store energy in magnetic fields.
• LC oscillation involves repeated energy transfer between electric and magnetic fields.
• When capacitor charge is maximum, current is zero.
• When current is maximum, capacitor charge is zero.
• In an ideal LC circuit, total stored energy remains constant.
• Total energy can be written as E_total = ½CV² + ½LI².
• An ideal LC circuit has a natural or resonant frequency.
• Resonant frequency is f₀ = 1/(2π√(LC)).
• Increasing inductance decreases resonant frequency.
• Increasing capacitance decreases resonant frequency.
• At resonance, a driven circuit can respond strongly to an external alternating signal.
• Tuned circuits use resonance to select particular frequencies.
• LC resonance is important in radio receivers and transmitters.
• Changing L or C allows the resonant frequency to be adjusted.
• Resonant circuits can help separate desired communication signals from unwanted signals.
• Resistance causes damping in real circuits.
• High-Q circuits provide sharper frequency selection than strongly damped circuits.
• Resonance is important in communication, filtering, oscillators, measurement systems, and some wireless power-transfer systems.

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Check Your Understanding

1. What two components make up an LC circuit, and what type of energy does each store?
2. Describe what happens when a fully charged capacitor is connected to an inductor.
3. Why is the current zero when the capacitor reaches maximum charge?
4. Explain where the circuit's energy is stored when the capacitor is momentarily uncharged.
5. Write the equation for the resonant frequency of an ideal LC circuit and identify each variable.
6. Calculate the resonant frequency of a circuit with L = 0.050 H and C = 20 μF.
7. An LC circuit has a resonant frequency of 2000 Hz and an inductance of 0.010 H. Calculate the required capacitance.
8. If the capacitance of an LC circuit is increased by a factor of 4, what happens to its resonant frequency? Explain mathematically.
9. If both L and C are doubled, by what factor does the resonant frequency change?
10. A capacitor stores 0.020 J when fully charged. How much magnetic energy is stored in the inductor when the capacitor becomes completely discharged in an ideal LC circuit?
11. Explain why a real LC circuit will not continue oscillating forever without an external energy source.
12. What is meant by a tuned circuit, and how can its resonant frequency be adjusted?
13. Explain how an LC circuit can help a radio receiver select one station from many signals received by its antenna.
14. Explain the relationship between resonance, frequency selection, and communication systems.
15. A communication engineer wants a circuit that responds strongly to one particular frequency while rejecting nearby unwanted signals. Explain how an LC resonant circuit could accomplish this and discuss how resistance and Q factor would affect its performance.