8.1 RLC Circuits: Series and Parallel

RLC Circuit Behavior

RLC circuits combine resistors (R), inductors (L), and capacitors (C) into a single loop or network. Because they contain two energy-storage elements (L and C), they produce second-order differential equations, which is why they're the focus of this unit. Their behavior includes oscillations, damping, and resonance, all of which show up in radio receivers, power systems, and signal processing.

Series and Parallel Configurations

In a series RLC circuit, every component shares the same current. The voltages across R, L, and C are generally different, and they add up (as phasors) to equal the source voltage.

In a parallel RLC circuit, every component shares the same voltage. The currents through R, L, and C are generally different, and they add up (as phasors) to equal the total source current.

A few key ideas apply to both configurations:

• Complex impedance is the tool you use to combine the effects of R, L, and C into a single quantity. It captures both magnitude and phase information.
• The phase relationship between voltage and current depends on whether the circuit is predominantly inductive (current lags voltage), capacitive (current leads voltage), or resistive (in phase).
• A sudden change in voltage or current triggers a transient response, which typically shows up as damped oscillations before the circuit settles to steady state.
• For sinusoidal (AC) inputs, steady-state analysis uses phasors and complex algebra to find voltages and currents without solving differential equations directly.

Current and Voltage Characteristics

Series RLC: The same current flows through every element. The individual voltage drops across R, L, and C add as phasors to give the total source voltage. For example, a series RLC driven by a 10 V source might have phasor voltage magnitudes of 3 V across R, 8 V across L, and 4 V across C. Notice these magnitudes don't simply add to 10 V as scalars; you need to account for phase. The resistor voltage is in phase with the current, the inductor voltage leads the current by 90°, and the capacitor voltage lags by 90°. The phasor sum still equals 10 V.

Parallel RLC: The same voltage appears across every element. The individual branch currents add as phasors to give the total source current. For instance, with 1 A total current, the branch currents through R, L, and C will generally have different magnitudes and phases that combine to 1 A as phasors.

Transient behavior: When you suddenly connect or disconnect a source, the circuit doesn't jump instantly to steady state. Instead, the natural response is an exponentially decaying sinusoid. The rate of decay depends on R (more resistance means faster decay), while the oscillation frequency depends on L and C.

Impedance and Resonant Frequency

Impedance Calculations

Impedance ($Z$) is the AC equivalent of resistance. It's a complex number that accounts for both energy dissipation (real part) and energy storage (imaginary part).

Series RLC impedance:

$Z = R + j(\omega L - \frac{1}{\omega C})$

Here $\omega L = X_L$ is the inductive reactance and $\frac{1}{\omega C} = X_C$ is the capacitive reactance. The $j$ indicates the reactive part is 90° out of phase with the resistive part.

Parallel RLC impedance is easier to compute using admittance ($Y = 1/Z$):

$Y = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C$

You find each branch's admittance, sum them, then take the reciprocal to get $Z$.

At resonance, the reactive parts cancel, leaving impedance purely resistive:

• Series RLC → impedance drops to its minimum value of $R$
• Parallel RLC → impedance rises to its maximum value

This contrast is worth remembering: series resonance means minimum $Z$, parallel resonance means maximum $Z$.

Resonant Frequency

Resonance occurs when the inductive reactance equals the capacitive reactance in magnitude:

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

Solving for the resonant frequency gives:

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

This formula is the same for both series and parallel RLC circuits.

At resonance, energy continuously swaps between the inductor's magnetic field and the capacitor's electric field. The resistor is the only element dissipating power. You can tune the resonant frequency by changing L or C, which is exactly how old radio dials worked: a variable capacitor shifted $f_0$ to select different stations.

Quality Factor and Bandwidth

Quality Factor (Q)

The quality factor ($Q$) is a dimensionless number that tells you how "sharp" the resonance peak is. A high-$Q$ circuit responds strongly at $f_0$ and weakly at nearby frequencies; a low-$Q$ circuit has a broader, flatter response.

Series RLC:

$Q = \frac{1}{R}\sqrt{\frac{L}{C}}$

Notice that smaller $R$ gives higher $Q$. That makes sense: less resistance means less energy lost per cycle, so the resonance is sharper.

Parallel RLC:

$Q = R\sqrt{\frac{C}{L}}$

Here, larger $R$ gives higher $Q$. In a parallel circuit, a large resistance means less current is "wasted" through the resistive branch, so the resonance stays sharp.

Physically, $Q$ represents the ratio of energy stored in the circuit to energy dissipated per cycle (scaled by $2\pi$). High-$Q$ circuits (like those in radio receivers) are very frequency-selective. Low-$Q$ circuits (like those in audio equalizers) pass a wider range of frequencies.

Bandwidth and Selectivity

Bandwidth ($BW$) is the range of frequencies over which the circuit's response stays within 3 dB of its peak value. The 3 dB points are also called half-power frequencies ($f_1$ and $f_2$), because at those frequencies the power delivered is half the maximum.

$BW = f_2 - f_1 = \frac{f_0}{Q}$

This relationship is important: if you know any two of $BW$, $f_0$, and $Q$, you can find the third.

• Narrow bandwidth (high $Q$): useful when you need to pick out one signal from many, such as selecting a single radio channel.
• Wide bandwidth (low $Q$): useful when you need to pass a broad range of frequencies, such as in broadband amplifiers.

Resonance in RLC Circuits

Resonance Characteristics

At resonance ($X_L = X_C$), the circuit behaves as if only the resistor is present:

• Series RLC at resonance: Impedance is at its minimum (just $R$), current is at its maximum, and the power factor equals 1 (unity). This is the condition for maximum power transfer from source to load.
• Parallel RLC at resonance: Impedance is at its maximum, and the current drawn from the source is at its minimum. Large circulating currents can exist between L and C internally, even though the external current is small.

The sharpness of the resonance peak is set by $Q$. A high-$Q$ circuit has a tall, narrow peak; a low-$Q$ circuit has a short, wide one.

Applications and Considerations

Where resonance helps:

• Radio tuning circuits use a variable capacitor in a series or parallel RLC to select a desired broadcast frequency.
• Wireless power transfer systems exploit resonance to maximize energy coupling between coils.
• Bandpass and notch filters in communication systems use resonant circuits to pass or reject specific frequency bands.
• Impedance matching networks use series resonance to ensure maximum power transfer between stages.
• Oscillators and frequency-selective amplifiers rely on parallel resonant circuits to sustain oscillations at a precise frequency.

Where resonance causes problems:

• In power systems, resonance can amplify harmonics and cause voltage spikes or equipment damage.
• Mechanical systems coupled to electrical ones can develop unwanted oscillations if resonant frequencies overlap.

Practical design concerns:

• Component tolerances (the actual L and C values may differ from their rated values) shift the resonant frequency. Tight-tolerance components matter in high-$Q$ designs.
• Temperature changes alter component values, which can detune a resonant circuit over time.
• When resonance is unwanted, damping (adding resistance or using lossy components) reduces the peak response and prevents instability.