8.2 RLC circuits

RLC circuits combine resistors, inductors, and capacitors into a single circuit that responds differently depending on the frequency of the AC signal driving it. This frequency-dependent behavior is what makes RLC circuits so useful for filtering, tuning, and signal processing. The core ideas here are impedance, resonance, and transient response.

Fundamentals of RLC circuits

An RLC circuit contains three types of passive components, each doing something different with energy. How these components interact determines the circuit's overall behavior at any given frequency.

Components of RLC circuits

Each component opposes current in its own way:

• Resistors dissipate electrical energy as heat. They follow Ohm's law: $V = IR$. Resistance doesn't depend on frequency.
• Inductors store energy in magnetic fields and resist changes in current. The voltage across an inductor is $V = L \frac{di}{dt}$. The faster the current changes (higher frequency), the more an inductor opposes it.
• Capacitors store energy in electric fields and resist changes in voltage. The current through a capacitor is $I = C \frac{dv}{dt}$. Capacitors oppose low-frequency signals more and let high-frequency signals pass more easily.

Because inductors and capacitors respond oppositely to frequency changes, combining them with a resistor creates a circuit whose behavior shifts dramatically across different frequencies.

Series vs parallel configurations

• In a series RLC circuit, all three components are connected end-to-end. The same current flows through each one, but the voltage drops across each component differ.
• In a parallel RLC circuit, all three components share the same voltage across them, but the current through each branch differs.

For series configurations, impedances add directly:

$Z_{total} = Z_1 + Z_2 + Z_3$

For parallel configurations, impedances combine reciprocally:

$\frac{1}{Z_{total}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \frac{1}{Z_3}$

The choice of configuration affects the resonant frequency, bandwidth, and whether the circuit reaches maximum or minimum impedance at resonance.

Impedance in RLC circuits

Impedance ($Z$) is the AC equivalent of resistance. It accounts for both energy dissipation (resistance) and energy storage (reactance). It's expressed as a complex number:

$Z = R + jX$

where $R$ is resistance and $X$ is the net reactance.

The two types of reactance behave oppositely with frequency:

• Inductive reactance: $X_L = \omega L$ (increases with frequency)
• Capacitive reactance: $X_C = \frac{1}{\omega C}$ (decreases with frequency)

The magnitude of the total impedance, $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$, determines how much current flows. The phase angle between voltage and current depends on whether $X_L$ or $X_C$ dominates.

Resonance in RLC circuits

Resonance is the frequency at which the inductive and capacitive reactances exactly cancel each other. At this point, the circuit's behavior simplifies dramatically, and some quantities (current or voltage, depending on configuration) reach extreme values.

Resonant frequency

Resonance occurs when $X_L = X_C$, meaning $\omega L = \frac{1}{\omega C}$. Solving for the frequency gives:

$f_0 = \frac{1}{2\pi\sqrt{LC}}$

At resonance in a series RLC circuit, the impedance drops to just $R$ (purely resistive), so current is at its maximum. In a parallel RLC circuit, impedance reaches its maximum at resonance, so current from the source is minimized.

This frequency is also the natural oscillation frequency of the circuit, which matters for transient behavior.

Quality factor

The quality factor ($Q$) measures how "sharp" the resonance is. It compares energy stored per cycle to energy lost per cycle. For a series RLC circuit:

$Q = \frac{\omega_0 L}{R}$

A high $Q$ (say, 50-100) means the circuit responds strongly at resonance but very weakly at nearby frequencies. A low $Q$ (around 1-5) means a broader, gentler response. Think of it this way: a high-$Q$ circuit is picky about frequency, while a low-$Q$ circuit is more forgiving.

Bandwidth

Bandwidth is the range of frequencies over which the circuit response stays within 3 dB of its peak value (that's the range where the power is at least half of the peak power). It's related to $Q$ by:

$BW = \frac{f_0}{Q}$

There's a direct trade-off: higher $Q$ means narrower bandwidth and more selectivity. If you're designing a radio tuner, you want a narrow bandwidth to isolate one station. If you're building an audio system, you might want a wider bandwidth to pass a range of frequencies.

AC analysis of RLC circuits

Analyzing RLC circuits with sinusoidal sources is much easier using phasors and complex numbers than trying to solve differential equations directly. These tools convert calculus problems into algebra.

Phasor diagrams

A phasor is a rotating arrow that represents a sinusoidal quantity. The arrow's length equals the amplitude, and its angle represents the phase. On a phasor diagram:

• The voltage across a resistor is in phase with the current.
• The voltage across an inductor leads the current by 90°.
• The voltage across a capacitor lags the current by 90°.

Adding phasors graphically (tip-to-tail) lets you find the total voltage or current in a circuit without solving trig equations. This is especially helpful for visualizing how the phase relationships shift as frequency changes.

Complex impedance

Complex impedance ($Z = R + jX$) lets you treat AC circuit analysis almost like DC Ohm's law. You can use $V = IZ$ where $V$, $I$, and $Z$ are all complex numbers.

• The magnitude $|Z|$ tells you the ratio of voltage amplitude to current amplitude.
• The angle of $Z$ tells you the phase difference between voltage and current.

For a series RLC circuit: $Z = R + j(\omega L - \frac{1}{\omega C})$. When $\omega L > \frac{1}{\omega C}$, the circuit is inductive (current lags voltage). When $\omega L < \frac{1}{\omega C}$, it's capacitive (current leads voltage).

Power factor

The power factor equals $\cos\phi$, where $\phi$ is the phase angle between voltage and current. It tells you what fraction of the apparent power actually does useful work.

• A power factor of 1 (purely resistive circuit) means all power is dissipated as real power.
• A power factor near 0 means most of the power is reactive, sloshing back and forth between the source and the reactive components without doing useful work.

Power companies care about this a lot. Low power factor means they have to supply more current for the same real power delivery. Correction is done by adding capacitors (to an inductive load) or inductors (to a capacitive load) to bring the phase angle closer to zero.

Transient response

Transient response describes what happens right after a sudden change, like closing a switch or applying a voltage step. The circuit doesn't instantly reach its steady state; instead, it goes through a transition period governed by the component values.

Overdamped vs underdamped systems

The relationship between $R$, $L$, and $C$ determines how the circuit settles to steady state:

• Overdamped ($R > 2\sqrt{\frac{L}{C}}$): The circuit returns to steady state slowly, without any oscillation. The resistance is large enough to prevent ringing.
• Underdamped ($R < 2\sqrt{\frac{L}{C}}$): The circuit oscillates around the steady-state value, with each swing getting smaller due to resistive losses. This is the "ringing" you sometimes see on oscilloscope traces.

The damping factor $\alpha = \frac{R}{2L}$ controls how quickly the oscillations die out in the underdamped case.

Critical damping

Critical damping is the boundary condition where $R = 2\sqrt{\frac{L}{C}}$. The circuit returns to steady state as fast as possible without oscillating. This is the sweet spot for applications like measurement instruments and control systems, where you want a quick response but can't tolerate overshoot.

Time constants

Time constants tell you how fast the transient dies away:

• For an RC circuit: $\tau = RC$
• For an RL circuit: $\tau = \frac{L}{R}$

After about $5\tau$, the transient has decayed to less than 1% of its initial value, and the circuit is effectively at steady state. For a full RLC circuit, the transient behavior involves multiple time constants that depend on the damping condition.

Applications of RLC circuits

Filters and tuning

RLC circuits selectively pass or block frequency ranges, which is the basis of signal filtering:

• Low-pass filters pass frequencies below a cutoff and attenuate higher ones (used in audio systems to isolate bass frequencies).
• High-pass filters pass frequencies above a cutoff (used in audio crossovers to send treble to tweeters).
• Band-pass filters pass a narrow range of frequencies around the resonant frequency (this is how AM/FM radio tuners select a single station).
• Notch (band-reject) filters block a specific narrow frequency range (used to eliminate 60 Hz power line hum from audio signals).

Oscillators

RLC circuits can generate sinusoidal signals when combined with an amplifier to compensate for resistive losses. The oscillation frequency is set by the LC values. Common types include:

• Colpitts oscillator: Uses a capacitive voltage divider in the LC tank circuit.
• Hartley oscillator: Uses a tapped inductor in the LC tank.
• Crystal oscillators: Replace the LC tank with a piezoelectric crystal for extremely stable frequency output.
• Voltage-controlled oscillators (VCOs): Allow the resonant frequency to be tuned by varying an applied voltage.

Power supplies

In power supply design, RLC circuits smooth the pulsating DC that comes out of a rectifier:

• LC filters reduce high-frequency ripple in the output.
• Pi filters (capacitor-inductor-capacitor) provide even better smoothing.
• Resonant converters use LC circuits to switch power efficiently at specific frequencies, reducing energy loss.
• Power factor correction circuits use RLC components to make the load appear more resistive to the power grid.

Energy storage in RLC circuits

Energy in an RLC circuit continuously moves between two forms: magnetic energy in the inductor and electric energy in the capacitor. The resistor steadily drains energy from the system.

Magnetic vs electric energy

• Energy stored in an inductor: $E_L = \frac{1}{2}LI^2$
• Energy stored in a capacitor: $E_C = \frac{1}{2}CV^2$

At resonance, the average energy stored in the inductor equals the average energy stored in the capacitor. Away from resonance, one form dominates depending on whether the circuit is behaving more inductively or capacitively.

Energy oscillations

In an underdamped RLC circuit, energy swings back and forth between the inductor and capacitor at the resonant frequency. Each cycle, the resistor converts some of that energy to heat, so the oscillations gradually shrink. This is exactly analogous to a pendulum swinging with friction: the energy transfers between kinetic and potential forms while friction slowly brings it to rest.

Power dissipation

Only the resistor dissipates power. Ideal inductors and capacitors store and return energy without loss. The average power dissipated is:

$P = I_{rms}^2 R$

The power factor determines how much of the total power flowing in the circuit is actually dissipated versus merely exchanged between reactive components. Efficient circuit design aims to minimize unnecessary resistive losses.

Frequency response

The frequency response shows how a circuit's output amplitude and phase change across a range of input frequencies. This is one of the most practical ways to characterize an RLC circuit's behavior.

Bode plots

Bode plots are the standard way to visualize frequency response. They consist of two graphs:

• Magnitude plot: Shows gain (output/input ratio) in decibels vs. frequency on a logarithmic scale.
• Phase plot: Shows the phase shift between output and input vs. frequency.

The logarithmic frequency axis lets you see behavior across many decades of frequency (e.g., 1 Hz to 1 MHz) on a single plot. Straight-line approximations make it easy to sketch Bode plots by hand for simple circuits.

Cutoff frequencies

The cutoff frequency is where the circuit's response drops to 3 dB below the passband level (which corresponds to half-power, or about 70.7% of the peak voltage).

• For a simple RC low-pass or high-pass filter: $f_c = \frac{1}{2\pi RC}$
• An RLC bandpass filter has two cutoff frequencies (one lower, one upper) that define the edges of the passband. The bandwidth is the difference between them.

Gain and phase shift

• Gain is the ratio of output amplitude to input amplitude. At resonance in a series RLC circuit, gain is at its maximum.
• Phase shift tells you how much the output is delayed relative to the input. For a simple RC low-pass filter, the phase shift is 0° at low frequencies and approaches -90° at high frequencies, passing through -45° at the cutoff frequency.

Both gain and phase vary continuously with frequency, and together they fully describe the circuit's linear behavior.

Measurement techniques

Impedance measurement

• LCR meters measure impedance magnitude and phase at a set frequency. These are the most common bench instruments for component characterization.
• Vector network analyzers sweep across a frequency range, giving detailed impedance data at every point.
• Impedance bridges compare an unknown impedance against a known standard for high-accuracy measurements.
• Four-wire (Kelvin) connections eliminate lead resistance errors, which matters for low-impedance measurements.

Q-factor determination

The most common method is the bandwidth method: measure the resonant frequency and the two -3 dB frequencies, then calculate:

$Q = \frac{f_0}{BW}$

You can also determine $Q$ by measuring the voltage magnification at resonance in a series circuit (the voltage across the inductor or capacitor can be $Q$ times larger than the source voltage).

Frequency response analysis

• Spectrum analyzers measure amplitude across a frequency range.
• Network analyzers provide both magnitude and phase data.
• Swept-sine techniques apply one frequency at a time for detailed, high-resolution measurements.
• Impulse response methods hit the circuit with a broadband pulse and use Fourier analysis to extract the frequency response quickly.