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5.8.6. RLC Circuits

Table of Contents

Combining Resistance, Inductance, and Capacitance in AC

An RLC circuit contains a resistor, an inductor, and a capacitor connected to an alternating voltage source. It is one of the most important AC circuits because it shows how opposition to current can come from resistance, magnetic effects, and electric charge storage at the same time.

In a resistor, voltage and current are in phase. In an inductor, current lags behind voltage. In a capacitor, current leads voltage. When all three are present together, the circuit behavior depends on how these effects combine.

The resistor dissipates energy as heat. The inductor stores energy in a magnetic field. The capacitor stores energy in an electric field. In an AC circuit, the inductor and capacitor can exchange energy back and forth, while the resistor removes energy from the system.

Series RLC Circuit

The simplest and most common case is the series RLC circuit, where the resistor, inductor, and capacitor are connected one after another in a single loop. Because they are in series, the same current flows through all three components.

If the applied voltage is sinusoidal, the voltage across each component is:

$$V_R = IR$$

$$V_L = I X_L$$

$$V_C = I X_C$$

where $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance.

The total opposition to current is not found by simply adding $R$, $X_L$, and $X_C$ as ordinary numbers, because the voltages are not all in phase. Instead, the net reactance is

$$X = X_L - X_C$$

and the total impedance is

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

The current amplitude is then

$$I = \frac{V}{Z}$$

where $V$ is the source voltage amplitude or RMS voltage, as long as the same type of quantity is used consistently.

For a series RLC circuit, the key formula is
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
and the current is
$$I = \frac{V}{Z}$$
This is the central rule for analyzing series RLC AC circuits.

Reactance Balance

The inductor and capacitor affect current in opposite ways. The inductor tends to reduce changes in current, while the capacitor tends to reduce changes in voltage. Their reactances are

$$X_L = \omega L$$

$$X_C = \frac{1}{\omega C}$$

where $\omega = 2\pi f$ is the angular frequency.

At low frequency, $X_C$ is large and $X_L$ is small, so the capacitor has the stronger effect. At high frequency, $X_L$ is large and $X_C$ is small, so the inductor has the stronger effect.

This means the behavior of an RLC circuit changes with frequency.

Phase Angle

Because the resistor and the net reactance affect current differently, the current and source voltage are generally not in phase. The phase angle $\phi$ is given by

$$\tan \phi = \frac{X_L - X_C}{R}$$

If $X_L > X_C$, the circuit behaves more like an inductor, and current lags voltage.

If $X_C > X_L$, the circuit behaves more like a capacitor, and current leads voltage.

If $X_L = X_C$, the circuit is purely resistive in its overall behavior, and current is in phase with voltage.

The phase angle in a series RLC circuit is
$$\tan \phi = \frac{X_L - X_C}{R}$$
If $\phi > 0$, the circuit is effectively inductive.
If $\phi < 0$, the circuit is effectively capacitive.
If $\phi = 0$, voltage and current are in phase.

Resonance

A very special condition occurs when the inductive reactance equals the capacitive reactance:

$$X_L = X_C$$

Using the formulas for reactance,

$$\omega L = \frac{1}{\omega C}$$

which gives the resonant angular frequency

$$\omega_0 = \frac{1}{\sqrt{LC}}$$

and the resonant frequency

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

At resonance, the inductor and capacitor cancel each other’s reactance. Then the impedance becomes

$$Z = R$$

so the current is maximum:

$$I_{\max} = \frac{V}{R}$$

At this condition, the circuit behaves like a pure resistor as seen by the source, even though the inductor and capacitor are still present.

At resonance in a series RLC circuit,
$$X_L = X_C$$
so
$$Z = R$$
and
$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$
Current is maximum at resonance.

Physical Meaning of Resonance

Resonance happens because the inductor and capacitor exchange energy most effectively at a particular frequency. The capacitor stores energy in its electric field, then releases it. The inductor stores energy in its magnetic field, then releases it. This repeated exchange can produce a large current if the driving frequency matches the natural frequency of the circuit.

The resistor prevents the current from becoming infinitely large by dissipating energy. A smaller resistance gives a sharper and stronger resonance. A larger resistance makes the resonance broader and less pronounced.

Voltage Across Individual Components

Even when the source voltage is moderate, the voltages across the inductor and capacitor can each become large near resonance. This happens because the current can be large, and

$$V_L = I X_L, \qquad V_C = I X_C$$

At resonance, $X_L = X_C$, so these two voltages can be equal in magnitude and opposite in phase. They cancel in the total loop voltage, but each can still be large individually.

This is an important feature of RLC circuits.

Frequency Behavior

The table below summarizes how a series RLC circuit behaves at different frequencies.

Frequency conditionRelative sizesCircuit behaviorCurrent
Low frequency$X_C > X_L$CapacitiveSmall
Resonance$X_C = X_L$Purely resistive overallMaximum
High frequency$X_L > X_C$InductiveSmall

Phasor Picture

A phasor diagram helps visualize the voltages in a series RLC circuit. The resistor voltage is in phase with the current, the inductor voltage leads current by $90^\circ$, and the capacitor voltage lags current by $90^\circ$.

The net reactive voltage is the difference between the inductor and capacitor voltages.

Phasor diagram for a series RLC circuit

In this drawing, the horizontal direction represents the part of voltage in phase with the current, and the vertical direction represents the net reactive part. The total source voltage is the vector sum.

Circuit Layout

Series RLC circuit connected to an AC source

This diagram represents an AC source driving a resistor, inductor, and capacitor in one loop.

Resonance Curve

If we graph current versus frequency, the current reaches a peak at the resonant frequency.

Current versus frequency in a series RLC circuit

The exact width and height of the peak depend on the resistance. Smaller $R$ gives a taller, narrower peak.

Summary Relations

The main formulas for a series RLC circuit are collected here.

QuantityFormula
Inductive reactance$X_L = \omega L$
Capacitive reactance$X_C = \frac{1}{\omega C}$
Net reactance$X = X_L - X_C$
Impedance$Z = \sqrt{R^2 + (X_L - X_C)^2}$
Current$I = \frac{V}{Z}$
Phase angle$\tan\phi = \frac{X_L - X_C}{R}$
Resonant angular frequency$\omega_0 = \frac{1}{\sqrt{LC}}$
Resonant frequency$f_0 = \frac{1}{2\pi\sqrt{LC}}$

In a series RLC circuit, resonance occurs when the inductive and capacitive reactances are equal. At that frequency, the impedance is smallest and equal to the resistance alone, so the current is largest.

Final Insight

An RLC circuit is a frequency-sensitive AC circuit. Its most important feature is resonance, where the inductor and capacitor balance each other. Because of this, RLC circuits are widely used in tuning, filtering, signal selection, and oscillatory electrical systems.

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