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 Current

Assume $I(t)$ is the instantaneous loop current. The inductor voltage is $V = L \frac{dI(t)} {dt}$ and the capacitor voltage magnitude is $V = \frac {Q}{C}$ , where Q is charge on the referenced capacitor plate. Signs depend on the selected current and voltage reference directions.

Now according to Kirchhoff’s voltage law, the sum of potential drops across the various components of a closed-loop is equal to zero.

(3) $\begin{equation*} L \frac {dI(t)}{dt} + \frac {Q}{C} = V \end{equation*}$

Differentiating the constant-source equation with respect to time gives the homogeneous current equation below. The protected intermediate line incorrectly substitutes Q = It; for time-varying current, current is the time derivative of charge.

$\begin{align*} \frac{d^2 I(t)}{dt^2} + \frac{d}{dt} \frac{Q}{LC} = \frac{dV}{dt} \end{align*}$

$\begin{align*} \frac{d^2 I(t)}{dt^2} + \frac{1}{LC} \frac{d}{dt} (It) = 0 (where, Q = It) \end{align*}$

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

(4) $\begin{equation*} \frac{d^2 I(t)}{dt^2} = - \frac{1}{LC} I(t) \end{equation*}$

Now the current in a simple harmonic oscillations form is given by:

(5) $\begin{equation*} I (t) = I_0 sin (\omega t + \phi) ( I = I_m sin \omega t ) \end{equation*}$

Where $I_0 > 0$ and $\phi$ are constants.

Put the value of equation (5) into (4) we get,

$\begin{align*} \frac{d^2}{dt^2}I_0 sin(\omega t+\phi) = - \frac{1}{LC}I_0 sin(\omega t+\phi) \end{align*}$

$\begin{align*} \frac{d}{dt} [\frac{d}{dt}I_0 sin(\omega t+\phi)] = - \frac{1}{LC}I_0 sin(\omega t+\phi) \end{align*}$

$\begin{align*} \frac{d}{dt} [\omega I_0 cos(\omega t+\phi)] = - \frac{1}{LC}I_0 sin(\omega t+\phi) [\frac{d}{dx} sinax = acosax] \end{align*}$

$\begin{align*} -\omega^2 I_0 sin(\omega t+\phi) = - \frac{1}{LC}I_0 sin(\omega t+\phi) [\frac{d}{dx} cos ax = -asinax] \end{align*}$

$\begin{align*} - \omega^2 = - \frac{1}{LC} \end{align*}$

(6) $\begin{equation*} \omega = \frac{1}{\sqrt{LC}} \end{equation*}$

The ideal homogeneous LC equation therefore supports undamped sinusoidal oscillation at its natural angular frequency.

LC Circuit Voltage

Now according to equation (3), the induced voltage across an inductor is minus the voltage across the capacitor.

$\begin{align*} V = -L \frac {dI(t)}{dt} \end{align*}$

Put the equation of current from equation (5), we get

$\begin{align*} \begin{split} V(t) = - L \frac{d}{dt} [I_0 cos (\omega t + \phi)] \ &= - L I_0 \frac{d}{dt} [cos (\omega t + \phi)] \ &= - L I_0 [-\omega sin (\omega t + \phi)] \ &= \omega L I_0 [sin (\omega t + \phi)] \ &= \frac{1}{\sqrt{LC}} L I_0 [sin (\omega t + \phi)] (where,\omega = \frac{1}{\sqrt{LC}}) \\ V(t) = \sqrt\frac{L}{C} I_0 [sin (\omega t + \phi)] \ \end{split} \end{align*}$

Voltage magnitude reaches a maximum when current is zero, and current magnitude reaches a maximum when voltage is zero. The ratio of voltage amplitude to current amplitude is $\sqrt\frac{L}{C}$ , the LC characteristic impedance. The protected voltage derivation changes the assumed current from sine to cosine without stating the phase change.

Transfer Function of LC Circuit

With zero initial conditions, the transfer function from input voltage to capacitor voltage is given below. The displayed result is written as a sinusoidal frequency response in omega even though it is labelled as a function of s.

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

The corresponding inductor-voltage expression follows. Both ideal responses have an unbounded value at resonance because the model contains no loss; every practical circuit has finite Q and finite voltage magnification.

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

LC Step Response and Natural Frequency

The illustrated circuit starts with zero capacitor voltage and zero inductor current. Switch K remains open for a long time, then closes at t=0 and applies a voltage source. This is a zero-state step-response setup, not a source-free natural response.

At t=0^–switch K is open

Continuity of ideal inductor current and capacitor voltage gives the displayed zero initial conditions.

$\begin{align*} I_L(0^-) = 0 = I_L(0^+) \end{align*}$

$\begin{align*} V_C(0^-) = 0 = V_C(0^+) \end{align*}$

Finite voltage cannot change ideal inductor current instantaneously, and finite current cannot change ideal capacitor voltage instantaneously.

For all t>=0⁺ switch K is closed

Now the voltage source is introduced in the circuit. Hence applying KVL to the circuit, we get,

$\begin{align*} \begin{split} - V_L(t) - V_C(t) + V_S = 0 \\ V_L(t) + V_C(t) = V_S \\ L \frac{di(t)}{dt} + \frac{1}{C} \int i(t) dt = V_S \\ \end{split} \end{align*}$

Here voltage across the capacitor is expressed in terms of current.

The above equation is called the integro-differential equation. Differentiating both sides of the above equation with respect to t, we get,

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

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

Equation (7) indicates a second-order differential equation of an LC circuit.

Replace $\frac{d^2}{dt^2}$ with s², we get,

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

The characteristic roots should be a complex-conjugate pair: positive and negative imaginary natural angular frequency. The protected derivation below omits both the imaginary unit and the negative root.

$\begin{align*} S_1,_2 = \frac {\sqrt{\frac{4}{LC}}} {{2}} = \frac {\frac{2}{\sqrt{LC}}} {2} = \frac{1}{\sqrt{LC}} \end{align*}$

Here, $\frac{1}{\sqrt{LC}}$ is the natural angular frequency in radians per second, not frequency in hertz.

LC Circuit Frequency Response

Using the impedance method, frequency response is the output phasor divided by the input phasor:

$\begin{align*} H(\omega) = \frac{Y(\omega)}{X(\omega)} = \frac{V_o_u_t}{V_i_n} \end{align*}$

Assume that the output voltage occurs across the capacitor terminals, apply potential divider rule to the above circuit

(9) $\begin{equation*} V_o_u_t = V_i_n \frac {Z_C}{Z_C + Z_L} \end{equation*}$

Where, $Z_C =$ Impedance of the capacitor $= \frac{1}{j \omega C}$

$Z_L =$ Impedance of the inductor $= {j \omega L}$

Substitute it in equation (9), we get

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

(10) $\begin{equation*} H(\omega) = \frac{V_o_u_t}{V_i_n} = \frac{1}{1 - \omega^2 LC} \end{equation*}$

Assume that the output voltage occurs across the inductor, apply potential divider rule to the above circuit

(11) $\begin{equation*} V_o_u_t = V_i_n \frac {Z_L}{Z_C + Z_L} \end{equation*}$

Substitute value of $Z_C$ and $Z_L$ in above equation, we get

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

(12) $\begin{equation*} H(\omega) = \frac{V_o_u_t}{V_i_n} = -\frac{\omega^2 LC}{1 - \omega^2 LC} \end{equation*}$

Equations (10) and (12) give the capacitor-output and inductor-output sinusoidal responses of the ideal series LC divider. They are real away from resonance because the two reactances share the same imaginary factor; the ideal expressions are undefined at resonance.

LC Circuit Differential Equation

$\begin{align*} L \frac{di(t)}{dt} + \frac{1}{C} \int i(t) dt = V \end{align*}$

This is an integro-differential equation in which capacitor voltage is expressed as the time integral of current. Any initial capacitor voltage must also be included when it is non-zero.

Now, differentiating above equation both sides with respect to t, we get,

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

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

This is the second-order homogeneous differential equation for ideal LC loop current after differentiating a constant applied voltage.

Replace $\frac{d^2}{dt^2}$ with s², we get,

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

Now, $\omega_0 = \frac{1}{\sqrt{LC}}$ therefore, $\omega_0^2 = \frac{1}{LC}$ , put it in above equation we get,

$\begin{align*} S^2i(t) + \omega_0^2 i(t) = 0 \end{align*}$

$\begin{align*} S^2 + \omega_0^2 = 0 \end{align*}$

LC Circuit Charging and Discharging

Both LC elements store energy. The inductor stores energy in its magnetic field (B) as a function of current. The capacitor stores energy in the electric field (E) between its plates as a function of voltage.

Assume the capacitor initially holds charge q and inductor current is zero. All initial energy is then in the capacitor’s electric field. In the protected derivation below, the final denominator should be C rather than C squared.

$\begin{align*} \begin{split} E_C =\frac{1}{2} CV^2 \ &= \frac{1}{2} C \frac{q^2}{C^2} \ &= \frac{1}{2} \frac{q^2}{C^2} (V = \frac{q}{C}) \ \end{split} \end{align*}$

Charging And Discharging Of LC Circuit — Charging and Discharging of LC CIrcuit

When an inductor is connected across a charged capacitor, capacitor voltage drives current and builds a magnetic field around the inductor. Capacitor voltage falls to zero as charge moves. The protected expression ( $I = \frac{q}{t}$ ) gives only average current over an interval; instantaneous current is the rate of change of charge.

At the first zero crossing of capacitor voltage in the ideal circuit, all energy is in the inductor and current magnitude is maximum. The intended inductor-energy expression is shown as ( $E_L = \frac{1}{2} LI^2)$ ; its closing parenthesis is misplaced inside the protected formula.

In the ideal lossless model, total stored energy remains constant, so maximum capacitor energy equals maximum inductor energy. In a practical circuit, resistance and other losses reduce the energy on each cycle.

At this instant stored energy in the magnetic field around an inductor induces a voltage across the coil according to the faraday’s law of electromagnetic induction ( $e = N \frac{d\phi}{dt}$ ). This induced voltage causes a current to flow through the capacitor and the capacitor begins to recharge with a voltage of opposite polarity.

The capacitor then discharges again, producing current through the inductor in the opposite direction.

Energy therefore alternates between capacitor and inductor. The ideal response continues at constant amplitude; a practical source-free response decays because of internal and external losses.

The figure shows the charging and discharging voltage and current waveform.

Charging and Discharging Lc Circuit Waveform — Charging and Discharging Voltage and Current Waveform

LC Circuit Applications

Common LC circuit applications include:

Resonant networks in radio transmitters, receivers, televisions, amplifiers, oscillators, filters, tuners and frequency mixers.
Oscillator frequency selection and filtering a selected band from a signal containing several frequencies.
High-Q resonators, where designers reduce loss to obtain a narrow bandwidth or low oscillator phase noise. Other filters deliberately use loading to set a wider bandwidth.
A driven series resonant circuit can provide voltage magnification across either reactive element, limited by Q and component ratings.
A driven parallel resonant circuit can support large circulating branch current while drawing a smaller terminal current, again limited by loss.

What is Damping?

Damping is the decay of free-oscillation amplitude caused by energy loss. Resonance is a larger steady-state response of a driven system near a natural frequency. Lower damping usually makes the resonance peak higher and narrower, but source and load conditions still determine its amplitude.