LC Circuit: Parallel And Series Circuits, Equations & Transfer Function

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Key learnings:

LC Circuit Definition: An LC circuit consists of an inductor and a capacitor, oscillating energy without consuming it in its ideal state.
Series Configuration: In series LC circuits, the components share the same current but have different voltages across each, showing voltage summation.
Parallel Configuration: Parallel LC circuits maintain the same voltage across components while allowing different currents to flow through each.
Resonance: At resonance, series circuits minimize impedance and maximize current, while parallel circuits do the opposite, showcasing their filtering capabilities.
Transfer Function Analysis: The transfer function of an LC circuit describes how input voltages are converted to output voltages across capacitors and inductors, critical in signal processing.

What is an LC Circuit?

An LC circuit is an electrical circuit containing two passive circuit elements: an inductor (L) and a capacitor (C). Resonant applications also call it a tank or tuned circuit. Its connection to a source and load determines whether it behaves as a resonator, filter or impedance network.

An ideal LC circuit has no resistor and therefore no resistive loss. Energy moves between the capacitor’s electric field and the inductor’s magnetic field. By contrast, ideal models of RC circuits, RL circuits and RLC circuits include resistance that dissipates energy.

A practical LC circuit loses energy through inductor winding and core loss, capacitor equivalent series resistance, dielectric loss, radiation and connected loads. These losses damp free oscillation and limit the quality factor.

Why an LC Circuit is Called a Tuned Circuit or Tank Circuit?

As charge moves between the capacitor plates through the inductor, energy alternates between electric and magnetic fields. In an ideal circuit this continues indefinitely. In a practical unpowered circuit, losses make the oscillation decay.

The ideal circuit follows the same second-order equation as an undamped harmonic oscillator. The term tuned circuit refers to selecting a resonant frequency; tank circuit refers to the exchange of stored energy.

The circuit can act as an electrical resonator whose natural frequency is set by its inductance and capacitance.

Series LC Circuit

In the series LC circuit shown, the inductor and capacitor form one current path.

The same instantaneous current flows through both series elements.

$\begin{align*} i = i_L = i_C \end{align*}$

With the displayed reference polarities, the terminal voltage is the algebraic sum of the inductor and capacitor voltages.

$\begin{align*} V = V_L + V_C \end{align*}$

Resonance in Series LC Circuit

As frequency increases, the magnitude of inductive reactance increases:

$\begin{align*} X_L = \omega L = 2 \pi fL \end{align*}$

and the magnitude of the capacitive reactance decreases.

$\begin{align*} X_C = \frac{1}{\omega C} = \frac{1}{2 \pi f C} \end{align*}$

At ideal resonance, the inductive and capacitive reactance magnitudes are equal.

$\begin{align*} \begin{split} & X_L = X_C\\ & \omega L = \frac{1}{\omega C}\\ & \omega^2 = \frac{1}{LC}\\ & \omega = \omega_0 = \frac{1}{\sqrt {LC}}(where, \omega = angular frequency)\\ & 2 \pi f =\omega_0 = \frac{1}{\sqrt {LC}}\\ & f_0 =\frac{\omega_0}{2\pi} = \frac{1}{2 \pi \sqrt {LC}}\\ \end{split} \end{align*}$

Where, $\omega_0$ is a resonant angular frequency (radians per second).

$f_0$ is a resonant frequency (Hertz).

The impedance of a series LC circuit is calculated as follows:

$\begin{align*} \begin{split} & Z_L_C_(_s_e_r_i_e_s_) = Z_L + Z_C\ &= j \omega L + \frac{1}{j \omega C}\ &= j \omega L + \frac{j}{j^2 \omega C}\ &= j \omega L - \frac{j}{\omega C}\ &= j (\frac{\omega^2 LC - 1}{\omega C}) (where, j^2 = -1)\ \end{split} \end{align*}$

Now the angular resonant frequency is $\omega_0 = \frac{1}{\sqrt{LC}}$ , then impedance becomes

(1) $\begin{equation*} Z_L_C(\omega)_(_s_e_r_i_e_s_) = j L (\frac {\omega^2 - \omega_0^2} {\omega}) \end{equation*}$

At resonance, when $\omega = \omega_0$ , X_L and X_C cancel. The ideal model therefore has zero terminal impedance, so ( $I = \frac {V} {Z}$ ) predicts unbounded current from an ideal voltage source. Real resistance and source impedance make the current finite.

A series LC branch has minimum impedance at resonance. Placed in a suitable signal path with source and load resistance, it can form a band-pass filter . The full topology and output point determine the transfer response.

Below resonance, $f < f_0$ , $X_C >> X_L$ . The circuit is capacitive. The double-greater-than symbol in the protected expression is only accurate well below resonance; the required relation is simply that capacitive reactance is greater.
Above resonance, $f>f_0$ , $X_L >> X_C$ . The circuit is inductive. The double-greater-than symbol applies only well above resonance; the required relation is simply that inductive reactance is greater.
At resonance, $f = f_0$ , $X_L = X_C$ . Terminal impedance is minimum and source current is maximum for a finite-loss series circuit.

Parallel LC Circuit

In the parallel LC circuit shown, the inductor and capacitor share the same two nodes.

The voltage across each branch is the terminal voltage, subject to the same reference polarity.

$\begin{align*} V = V_L = V_C \end{align*}$

The terminal current is the algebraic or phasor sum of the inductor and capacitor branch currents.

$\begin{align*} i = i_L + i_C \end{align*}$

Resonance in Parallel LC Circuit

At ideal resonance, inductive reactance ( $X_L$ ) equals capacitive reactance ( $X_C$ ). The branch currents are equal and opposite at the terminals, giving zero source current and infinite input impedance in the ideal model. Real loss makes the source current non-zero and the impedance finite.

The resonant frequency is given by

$\begin{align*} f_0 = \frac {\omega_0} {2 \pi} = \frac {1} {2 \pi \sqrt{LC}} \end{align*}$

Now an Impedance of the Parallel LC circuit is given by

$\begin{align*} \begin{split} Z_L_C_(_P_a_r_a_l_l_e_l_) = \frac {Z_L Z_C} {Z_L + Z_C}\ &= \frac {j \omega L \frac{1}{j \omega C}} {j \omega L + \frac{1}{j \omega C}}\ &= \frac{\frac{L}{C}} { \frac{- \omega^2 LC + 1}{j \omega C}}\ &= \frac {j \omega L} {1 - \omega^2 LC} \ \end{split} \end{align*}$

Now the angular resonant frequency is $\omega_0 = \frac{1}{\sqrt{LC}}$ , then impedance becomes

(2) $\begin{equation*} Z_L_C(\omega)_(_p_a_r_a_l_l_e_l_) = - j (\frac {1}{C}) (\frac {\omega}{\omega^2 - \omega_0^2}) \end{equation*}$

At resonance, when $\omega = \omega_0$ , the ideal parallel impedance is infinite and the source current is zero, as indicated by ( $I = \frac {V} {Z}$ ). Practical losses limit the impedance peak.

A parallel LC branch has maximum impedance at resonance. In series with a signal path it can contribute to a band-stop response; as a shunt element it can contribute to a band-pass response. Source impedance, load impedance and the output point determine the complete filter response.

Below resonance, f<f₀, X_L < X_C. The parallel network is inductive.
Above resonance, f>f₀, X_C < X_L. The parallel network is capacitive.
At resonance, f = f₀, X_L = X_C. Terminal current is minimum and impedance is maximum for a finite-loss parallel circuit.

LC Circuit Equations

Current and voltage equation

The two displayed initial-value equations use an inconsistent phase convention for voltage:

$\begin{align*} I(0) = I_0 sin\phi \end{align*}$

$\begin{align*} V(0) = -\omega_0 L I_0 sin\phi \end{align*}$

The displayed oscillation equations place current and voltage in phase, although ideal LC current and capacitor voltage are one-quarter cycle apart:

$\begin{align*} I(t) = I_0 sin (\omega_0 t + \phi) \end{align*}$

$\begin{align*} V(t) =\sqrt {\frac{L}{C}} I_0 sin (\omega_0 t + \phi) \end{align*}$

LC circuit differential equation

$\begin{align*} \frac {d^2 i(t)}{dt^2} + \frac{1}{LC} i(t) = 0 \end{align*}$

$\begin{align*} S^2 i(t) + \frac{1}{LC} i(t) = 0 \end{align*}$

$\begin{align*} S^2 + \omega_0^2 = 0 \,\, (where, \omega = \omega_0 = \frac{1}{\sqrt{LC}}) \end{align*}$

Impedance of the Series LC circuit

$\begin{align*} Z_L_C(\omega)_(_s_e_r_i_e_s_) = j L (\frac {\omega^2 - \omega_0^2} {\omega}) \end{align*}$

Impedance of the Parallel LC circuit

$\begin{align*} Z_L_C(\omega)_(_p_a_r_a_l_l_e_l_) = - j (\frac {1}{C}) (\frac {\omega}{\omega^2 - \omega_0^2}) \end{align*}$

Settling Time

An ideal LC circuit is undamped, so its natural oscillation does not settle to a constant final value. A practical circuit has loss and is better represented by an RLC model; its transient may decay towards a steady state.

Settling time is a control-system measure defined by a selected tolerance band, often plus or minus 2% around a non-zero final value. The sustained natural oscillation of an ideal LC circuit has no settling time.

LC Circuit Analysis: Series And Parallel Circuits, Equations And Transfer Function