---
title: RLC Resonance
module: Alternating Current
moduleNumber: 9
lessonNumber: 3
order: 903
summary: >
  Put a resistor, inductor, and capacitor in one loop and their reactances work against
  each other: inductive reactance grows with frequency while capacitive reactance
  shrinks, and at one frequency they cancel exactly. There the branch looks purely
  resistive, the current peaks, and the inductor and capacitor voltages can swing far
  above the source. We locate that resonance at $\omega_0=1/\sqrt{LC}$, measure how
  sharp the peak is with the quality factor $Q=\omega_0L/R$, tie its half-power
  bandwidth $R/L$ to the ringdown of the unforced circuit, and read the same poles off
  as bandpass and peaked filters at the R, L, or C terminals.
topics: [Alternating Current]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 29 — Alternating-Current Circuits; §29-6 Driven RLC Circuits"
---

## The series RLC loop

A series RLC circuit contains one branch current. The same instantaneous current
passes through the resistance, inductance, and capacitance; the three terminal
voltages need not have the same phase. Let $q(t)$ be the capacitor charge on the
plate selected as positive and let $i(t)=\d q/\d t$ be the branch current. With passive
voltage references across all three components, Kirchhoff's loop law gives

$$
v_s(t)=Ri(t)+L\frac{\d i}{\d t}+\frac{q(t)}{C}.
$$

Substitution of $i=\d q/\d t$ gives the charge equation

$$
L\frac{\d^2q}{\d t^2}+R\frac{\d q}{\d t}+\frac{q}{C}=v_s(t).
$$

The $L$ term sets the response to a current slope, the $R$ term removes energy from
the branch, and the $q/C$ term records capacitor voltage. The equation applies to
the terminal values of an ideal series model. The resistance may represent an
intentional resistor plus stated winding and source resistance, provided those losses
remain approximately linear over the chosen amplitude and frequency range.

$$
% caption: Series RLC loop with one branch current. The same current crosses R, L, and C in series, and the source voltage equals the sum of the three terminal voltages.
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The state variables have distinct continuity rules. Capacitor voltage cannot jump
without an impulse of current, because $i=C\,\d v_C/\d t$. Inductor current cannot jump
without an impulse of voltage, because $v_L=L\,\d i/\d t$. A source step therefore starts
the circuit from its pre-step capacitor voltage and branch current. The resistor has
no stored state: its voltage changes immediately when current changes. These facts
are enough to set the two initial conditions required by the second-order equation.

A sinusoidal source $v_s(t)=V_0\cos(\omega t)$ produces a differential equation with
two parts. A transient solves the homogeneous equation and retains information about
the initial capacitor charge and inductor current. A forced steady-state term has the
source frequency and remains after the transient has decayed. The separation matters
in a short measurement. A frequency generator can be set to its nominal final value
while the circuit current still contains a decaying contribution from the previous
frequency or source phase.

The homogeneous equation is

$$
L\frac{\d^2q_h}{\d t^2}+R\frac{\d q_h}{\d t}+\frac{q_h}{C}=0.
$$

Define the undamped angular frequency and damping rate by

$$
\omega_0=\frac{1}{\sqrt{LC}},
\qquad
\gamma=\frac{R}{2L}.
$$

The characteristic roots are $s=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}$. Their
form divides free response into three cases. The comparison uses $R$ against
$2\sqrt{L/C}$, the resistance for which the roots coincide.

When $R<2\sqrt{L/C}$, the transient has the form

$$
q_h(t)=A e^{-\gamma t}\cos(\omega_d t+\alpha),
\qquad
\omega_d=\sqrt{\omega_0^2-\gamma^2}.
$$

The current is the derivative of this charge, so its phase and envelope differ from
the charge trace. The envelope $e^{-\gamma t}$ is set by the total series resistance,
whereas the ringing frequency is slightly below $\omega_0$. Small resistance yields
many visible cycles; resistance close to the critical value suppresses repeated zero
crossings. The resistance must include the source impedance present after switching.
A generator's stated output resistance can therefore change an observed ringdown even
when the circuit board has no added resistor.

Critical damping occurs at $R=2\sqrt{L/C}$. The transient form

$$
q_h(t)=(A+Bt)e^{-\omega_0t}
$$

returns to equilibrium without ringing. Above this resistance, the two exponential
rates are real and distinct. A large resistance can make one rate very slow because
the capacitor has little current path through which to discharge. The phrase
"fastest return" applies only to the ideal unforced model with fixed $L$, $C$, and
series $R$; source loading and measurement input impedance can add paths that change
the observed decay.

$$
% caption: Free response of the unforced series RLC branch from the same initial capacitor charge. Underdamping reverses sign, critical damping returns without a reversal, and overdamping decays along a slower real root.
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$$

## Sinusoidal steady state and impedance

After the transient has become small relative to the forced response, represent each
sinusoidal voltage and current by an RMS phasor at the drive frequency. The series
impedance is

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

The reactive term is negative below the natural frequency and positive above it. Its
sign records the current-voltage phase relation. The magnitude and phase are

$$
|Z|=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2},
\qquad
\phi_Z=\atanTwo\!\left(\omega L-\frac{1}{\omega C},R\right).
$$

The branch current for input phasor $\widetilde V_s$ is

$$
\widetilde I=\frac{\widetilde V_s}{Z},
\qquad
I_{\rm rms}=\frac{V_{s,\rm rms}}{|Z|},
\qquad
\phi_I=\phi_s-\phi_Z.
$$

The current leads a source voltage below resonance because the net impedance is
capacitive. It lags above resonance because the net impedance is inductive. At the
frequency where the reactive term vanishes, source voltage and current have the same
phase. The sign convention belongs in the experimental record: reversing a current
sense resistor or one voltage probe shifts a reported phase by $180^\circ$.

$$
% caption: Phasor sum above resonance. The resistor voltage lies along the branch current; the inductor voltage leads and the capacitor voltage lags, and since the branch is inductive above resonance their net is upward, tilting the source phasor ahead of the current.
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$$

The resistor, inductor, and capacitor voltage phasors are

$$
\widetilde V_R=\widetilde I R,
\qquad
\widetilde V_L=j\omega L\widetilde I,
\qquad
\widetilde V_C=-\frac{j}{\omega C}\widetilde I.
$$

Adding these phasors enforces the loop law without mixing amplitudes that occur at
different times. The magnitudes $V_L=I\omega L$ and $V_C=I/(\omega C)$ can each
exceed the source magnitude near resonance. Their phasors then have opposite vertical
directions and nearly cancel in the source sum. A meter placed across one component
measures that component voltage, not the much smaller residual difference.

$$
% caption: Series impedance magnitude versus frequency. It falls to its minimum value R where the reactances cancel, and rises on either side as the net reactance grows.
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$$

A time record gives the same phase information. If the source voltage is selected as
zero-phase reference, the current peaks before the source peak below resonance and
after it above resonance. At resonance, their peaks and zero crossings align after
instrument-delay correction. A single trace cannot establish a phase relation. Record
both waveforms on a common time base or calculate current from a measured sense
resistor voltage with a stated polarity.

## Resonance as reactive cancellation

Series resonance occurs when the inductive and capacitive reactances have equal
magnitudes:

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

The resonance angular frequency and ordinary frequency are therefore

$$
\omega_0=\frac{1}{\sqrt{LC}},
\qquad
f_0=\frac{1}{2\pi\sqrt{LC}}.
$$

At this frequency the ideal series impedance reduces to $Z=R$. The current amplitude
is limited by the total series resistance, and the current phase equals the source
phase. Resonance does not remove the inductor or capacitor from the circuit. Their
individual terminal voltages remain present, their stored energies continue to vary,
and a small change in frequency restores a nonzero net reactance. The cancellation
belongs to the vector sum of terminal voltages, not to the disappearance of either
component's physical field.

The balance condition also gives a practical consistency test. Calculate $\omega_0$
from the selected $L$ and $C$, then calculate $X_L$ and $X_C$ at that frequency.
They should agree within the parameter tolerance and model range. A measured current
maximum far from that predicted frequency can arise from an incorrect capacitor value,
additional series inductance, source resistance, a load connected across the circuit,
or a component value measured at a different bias or amplitude. Start with terminal
measurements before assigning the shift to a vague "resonance error."

$$
% caption: Reactive cancellation at series resonance. The inductor and capacitor voltage phasors are equal and opposite, so they sum to zero and the source phasor reduces to the resistor voltage along the branch current.
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$$

The current-amplitude curve follows directly from impedance magnitude:

$$
I_{\rm rms}(\omega)=
\frac{V_{s,\rm rms}}
{\sqrt{R^2+\left(\omega L-1/(\omega C)\right)^2}}.
$$

Its largest value is $V_{s,\rm rms}/R$ at $\omega_0$ in the ideal model. The curve
need not be symmetric on an ordinary linear frequency axis. The reactive expression
contains both $\omega$ and $1/\omega$, so equal additive offsets above and below
$f_0$ do not generally give equal current. On a logarithmic frequency axis, the
low-frequency capacitive side and high-frequency inductive side have a more balanced
appearance. A plot should state its frequency scale before a claimed symmetry is
interpreted physically.

$$
% caption: Series branch current versus drive frequency. The current maximizes where inductive and capacitive reactance cancel; below that the branch is capacitive and above it inductive.
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$$

Near resonance, the individual reactive voltage magnitudes can be much larger than
the source voltage. At exact resonance,

$$
V_{L,\rm rms}=I_{\rm rms}\omega_0L,
\qquad
V_{C,\rm rms}=\frac{I_{\rm rms}}{\omega_0C},
\qquad
V_{L,\rm rms}=V_{C,\rm rms}.
$$

Since $I_{\rm rms}=V_{s,\rm rms}/R$, both reactive voltage magnitudes equal
$Q V_{s,\rm rms}$ when $Q=\omega_0L/R$. The result is called voltage
magnification. It follows from nearly opposite inductor and capacitor phasors, with
no active voltage-gain element. The inductor and capacitor voltages are nearly opposite
in phase, so their vector sum can remain small while each terminal voltage is large.
Voltage ratings, probe limits, and insulation spacing must be checked against the
individual component voltage. Reading only the source setting can miss the stressed
quantity by a large factor.

The stored energies explain the sharp current response without invoking any active
amplification. The capacitor energy is $U_C=q^2/(2C)$ and the inductor energy is
$U_L=Li^2/2$. In an ideal free oscillation, their sum remains constant and moves from
electric storage to magnetic storage and back. With a source at resonance, the source
replaces energy removed by the resistance over repeated cycles. A small series
resistance removes only a small fraction of stored energy per cycle, so the steady
current can build to a large value before loss balances the source input.

The resonance frequency of a driven circuit and the damped free-oscillation frequency
are distinct, although related. The ideal drive-current maximum occurs
at $1/\sqrt{LC}$. The underdamped free transient oscillates at
$\sqrt{\omega_0^2-\gamma^2}$. Their difference is small at high $Q$ and becomes
measurable when damping is substantial. A ringdown measurement and a swept-source
measurement therefore answer related but distinct questions. The first identifies the
natural transient poles of the loaded circuit; the second identifies the forced
response at the selected source amplitude and termination.

Source coupling alters measured resonance. A large source resistance adds to $R$ and
broadens the current curve. A load across the capacitor alters the effective branch
network and shifts both frequency and damping. A high-impedance voltage probe can add
capacitance that matters when the selected capacitor is small. Report the source
resistance, the load, and the probe topology with any resonance frequency or quality
factor. Those conditions form part of the circuit definition.

## Bandwidth, quality factor, and selectivity

Bandwidth describes the frequency interval over which a resonant response remains
near its maximum. A series RLC branch uses the two frequencies where current
amplitude has fallen to $1/\sqrt2$ of its resonance value. Since the resonance
current magnitude is $V_{s,\rm rms}/R$, that condition gives

$$
\frac{V_{s,\rm rms}}{|Z|}
=\frac{1}{\sqrt2}\frac{V_{s,\rm rms}}{R},
\qquad
|Z|=\sqrt2R.
$$

The reactive magnitude at each boundary is therefore $R$:

$$
\left|\omega L-\frac{1}{\omega C}\right|=R.
$$

The term half-power is commonly used for these points because resistor heating is
proportional to current squared. Here the points mark a current-amplitude ratio for the
series branch. They do not replace a full AC-power analysis for a source, a load, or a
non-sinusoidal waveform.

$$
% caption: Half-power bandwidth of the series current. The two edge frequencies where the current falls to one over root two of its peak bound the passband, and their separation is the bandwidth R over L.
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$$

Solving the two quadratic equations gives the positive edge frequencies

$$
\omega_-=
\frac{\sqrt{R^2+4L/C}-R}{2L},
\qquad
\omega_+=
\frac{\sqrt{R^2+4L/C}+R}{2L}.
$$

Their difference and product are especially informative:

$$
\Delta\omega=\omega_+-\omega_-=\frac{R}{L},
\qquad
\omega_+\omega_-=\omega_0^2.
$$

The ideal series model has angular-frequency bandwidth $R/L$ under the stated
current-ratio definition. The two edge frequencies are not, in general,
located equal additive distances from $\omega_0$. Their geometric mean equals
$\omega_0$. A logarithmic frequency axis displays this multiplicative symmetry
directly; a broad linear axis does not. In hertz, $\Delta f=R/(2\pi L)$. Doubling the series
resistance doubles bandwidth; doubling inductance halves it while also changing the
chosen capacitance required for a fixed center frequency.

The quality factor compresses center frequency and bandwidth into a dimensionless
number:

$$
Q=\frac{\omega_0L}{R}
=\frac{1}{\omega_0CR}
=\frac{\omega_0}{\Delta\omega}
=\frac{f_0}{\Delta f}.
$$

Large $Q$ means a narrow current peak relative to its center frequency. It does not
mean that every component in the circuit has negligible loss. It means that the
resistance used in the stated series model is small compared with the reactance at
resonance. A circuit can have a narrow measured current peak and still have a source
whose available voltage falls under load, a capacitor whose value drifts with bias,
or an inductor whose resistance rises with temperature. Those effects change the
fitted $R$, $L$, or $C$ and therefore change the measured quality factor.

The energy definition gives a complementary interpretation for a lightly damped
branch. Let $U$ be the energy stored near a current or voltage maximum and let
$\Delta U_{\rm cycle}$ be the energy removed by the series resistance in one cycle.
Then, for small fractional energy loss,

$$
Q\approx2\pi\frac{U}{|\Delta U_{\rm cycle}|}.
$$

The approximation is a cycle-to-cycle statement. It becomes less accurate when the
energy falls appreciably within one oscillation or when the element values depend on
current. It nonetheless provides a direct bridge between a ringdown envelope and a
frequency sweep: slow energy loss gives a long ringdown and a narrow driven response.

$$
% caption: Two quality factors at one natural frequency. The smaller-resistance branch, with large Q, gives a taller, narrower current peak; both peak at the same center frequency.
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$$

The phase curve provides a second selectivity measurement. Since

$$
\tan\phi_Z=
\frac{\omega L-1/(\omega C)}{R},
$$

the impedance phase crosses zero at resonance. Close to that point,

$$
\omega L-\frac{1}{\omega C}
\approx2L(\omega-\omega_0),
\qquad
\left.\frac{\d\phi_Z}{\d\omega}\right|_{\omega_0}
\approx\frac{2L}{R}=\frac{2Q}{\omega_0}.
$$

High-$Q$ circuits change phase rapidly across a narrow interval. That phase slope can
be more sensitive than amplitude when a measurement chain has stable channel delay
and adequate signal-to-noise ratio. It can also be less reliable when a fixed timing
skew has not been calibrated, because a constant delay creates a phase error that
increases with frequency. Record the source and current references before using a
zero-phase crossing as a resonance estimate.

$$
% caption: Impedance phase through resonance. Both curves cross zero phase at the natural frequency; the large-Q branch swings from lagging to leading over a narrower band, giving a steeper crossing.
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$$

Selectivity comes with a time-domain consequence. A narrow current response takes
many source cycles to settle after a frequency change because the stored energy must
adjust to the new balance between source input and series loss. A rapid sweep can
therefore distort a narrow measured peak: the generator reaches a new frequency
before the branch current reaches its corresponding steady amplitude. Reduce sweep
speed, dwell at each frequency, or fit the transient explicitly. The requirement
becomes stronger as $Q$ rises and as the source changes phase discontinuously between
sweep points.

Component tolerance also limits selectivity. Differentiating the center-frequency
relation gives, for small independent fractional changes,

$$
\frac{\Delta f_0}{f_0}
\approx-\frac12\frac{\Delta L}{L}
-\frac12\frac{\Delta C}{C}.
$$

A narrow bandwidth cannot compensate for a center frequency that moves with
capacitance tolerance, temperature, or bias. The design task has two independent
parts: select a bandwidth through the total series resistance, then establish a
center-frequency tolerance through $L$ and $C$. A report that gives only $Q$ omits
the frequency stability needed to determine whether a desired signal remains inside the
band.

## Measuring a resonance curve

A resonance measurement needs a defined source, a defined series resistance, and a
defined measurement loading. Drive the series branch with a sine source and place a
known sense resistor $R_s$ in series with the RLC elements. Record the phasor across
the sense resistor and the phasor across the RLC branch. The current and branch
impedance follow from

$$
\widetilde I=\frac{\widetilde V_sense}{R_s},
\qquad
\widetilde Z_{\rm branch}=
R_s\frac{\widetilde V_{\rm branch}}{\widetilde V_{\rm sense}}.
$$

These relations use measured terminal voltages rather than a generator-panel setting.
They remain valid when the source has a finite output resistance, as long as the sense
resistor and branch form the only current path between the two measured nodes. The
source amplitude can then change with frequency without corrupting the extracted
branch impedance. A source with current limiting or waveform clipping still changes
the result, so retain both voltage records and inspect their sine-fit residuals.

$$
% caption: Series-sense measurement. A known sense resistor in series gives the branch current from its own voltage, and the voltage measured across the RLC branch divided by that current gives the branch impedance at each frequency.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[thick] (.7,1.5)--(1.15,1.5);
\draw[thick] (1.15,1.5) circle (.36);
\draw[thick] plot[domain=.93:1.37,samples=26] ({\x},{1.5+.11*sin(360*(\x-.93)/.44)});
\node[below] at (1.15,1.02) {source};
\draw[thick] (1.51,1.5)--(2.0,1.5);
\draw[thick] (2.0,1.5)--(2.1,1.62)--(2.28,1.38)--(2.46,1.62)--(2.64,1.38)--(2.82,1.62)--(3.0,1.38)--(3.1,1.5);
\draw[thick] (3.1,1.5)--(3.5,1.5);
\node[above] at (2.55,1.74) {sense R};
\draw[fill=acc!10,draw=acc,thick] (3.5,1.04) rectangle (5.0,1.96);
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\draw[<->,acc,thick] (3.5,.9)--(5.0,.9) node[midway,below] {branch V};
\draw[->,black,thick] (2.1,2.2)--(3.0,2.2) node[midway,above] {current};
\end{tikzpicture}
$$

Set the sense resistance from the expected impedance scale. If $R_s$ is far smaller
than the branch impedance, the current-sense voltage can approach the instrument noise
floor and its phase becomes unstable. If $R_s$ is far larger, it adds substantial
series resistance, lowers $Q$, and changes the very bandwidth being measured. A
value comparable to the expected branch impedance near the sweep edges produces
measurable voltage on both channels, but its own frequency behavior must be known.
A noninductive resistor is preferable when the target frequency makes a wound resistor
reactive. Its resistance tolerance and temperature must be included in the fitted
series $R$ uncertainty.

The source and the sense resistor dissipate energy during the sweep. Estimate the
peak expected current from the resonance model, then check the source current limit,
the resistor power rating, the inductor current rating, and the capacitor voltage
rating before raising amplitude. At a high quality factor, the source voltage may be
small while the capacitor and inductor each carry a much larger voltage. Start at a
low amplitude, verify linear scaling of current with source voltage, and raise the
drive only after the measured resonance frequency and phase behavior agree with the
small-signal model.

Each frequency point should retain enough information to rebuild a phasor. Fit a
recorded voltage to

$$
v[n]=a\cos(\omega t_n)+b\sin(\omega t_n)+d,
\qquad
A=\sqrt{a^2+b^2},
\qquad
\phi=\atanTwo(-b,a).
$$

The offset $d$ checks for DC coupling or amplifier offset. The amplitude $A$ is a
peak value, so convert both channels together if the report uses RMS. Sine and cosine
fitting uses the full sampled record and separates amplitude from arbitrary trigger
placement. It also exposes harmonics through the residual trace. A zero-crossing
comparison provides an independent check when waveforms are clean, but it becomes
noisy near a shallow slope or when the source contains distortion.

Use a logarithmic frequency grid for an initial search spanning decades. The grid
places similar fractional spacing at the low and high sides of the expected center.
Once a peak has been located, add closely spaced points around the peak and both
half-height crossings. A data set with one point at the maximum and no points on the
slopes cannot determine bandwidth reliably. A data set with amplitude alone cannot
separate a shifted center from a phase-reference mistake. Plot current magnitude,
impedance phase, and the source amplitude on aligned frequency axes.

The sweep must settle. After a step in source frequency, branch energy from the
previous point persists for a time of order several damping times. The amplitude
envelope of a lightly damped free component falls approximately as $e^{-\gamma t}$.
Choose a dwell time that reduces this remnant below the requested amplitude tolerance.
For example, five damping times leave an envelope factor of $e^{-5}$, about seven
thousandths of its starting value. The requirement assumes a stationary sine source;
automatic generators can introduce phase resets or amplitude changes that need their
own verification.

Instrument references require explicit treatment. Two oscilloscope channels can have
different propagation delays from probes, attenuator settings, digital filters, or
current probes. A fixed delay $\Delta t$ creates a phase term proportional to
frequency. Send the same sine source through both complete measurement paths, measure
the residual phase over the sweep, and apply the signed correction to the branch-sense
phase difference. Do not correct a phase plot by a constant number of degrees across
a wide frequency interval unless the calibration establishes that behavior.

The probe return path is part of the circuit at high frequency. Long ground leads add
loop inductance, and two earth-referenced probe clips may join circuit nodes through
the oscilloscope. Use a common reference node, a differential probe, or an isolated
measurement input appropriate to the topology. Keep voltage-sense points near the
component terminals. A breadboard rail several centimeters from the capacitor can
include enough lead inductance to shift a narrow resonance measurement.

Fit the complete complex data rather than extracting $L$, $C$, and $R$ from three
unrelated points. A least-squares model can compare measured
$\widetilde Z(\omega)$ against $R+j(\omega L-1/(\omega C))$ across the selected
band. Plot residual real and reactive parts after the fit. Random residuals near the
instrument uncertainty support the model over that band. A smooth residual trend with
frequency indicates missing lead inductance, loading, or frequency-dependent loss.
An amplitude-dependent residual indicates a nonlinear or heated element. Exclude that
range from the quoted ideal-series fit; a single averaged parameter would conceal the
departure.

Repeat a few points after remounting the circuit and after the sweep returns to its
starting frequency. The comparison checks contact repeatability and thermal drift.
Report the sweep direction, dwell time, source amplitude convention, sense resistance,
source impedance, probe settings, sample rate, fitting method, and calibration record.
Those details allow a later reader to distinguish a property of the RLC branch from a
property of the measurement path.

## Output selection and narrow-band filtering

The series branch has one current but three different component voltages. Selecting
one of those voltages as an output gives a frequency-dependent transfer function.
With an ideal source driving the complete series combination, define

$$
H_R(\omega)=\frac{\widetilde V_R}{\widetilde V_s},
\qquad
H_L(\omega)=\frac{\widetilde V_L}{\widetilde V_s},
\qquad
H_C(\omega)=\frac{\widetilde V_C}{\widetilde V_s}.
$$

The phasor voltage laws give

$$
H_R=\frac{R}{Z},
\qquad
H_L=\frac{j\omega L}{Z},
\qquad
H_C=\frac{-j/(\omega C)}{Z}.
$$

These functions describe a specified output port under a specified load. Connecting
an amplifier, a meter, a cable, or a second circuit to that port adds an impedance
that changes the result. A claimed bandpass shape is incomplete without the output
node, return node, source impedance, and load impedance. The resistor output is often
convenient for a current measurement because $V_R=IR$, but its magnitude also depends
on the full series resistance used in the model.

Across the resistor, the magnitude is

$$
|H_R|=
\frac{R}{\sqrt{R^2+\left(\omega L-1/(\omega C)\right)^2}}.
$$

It approaches zero at very low and very high frequency and reaches one at resonance
for an ideal source and a resistor that represents all series loss. The resistor
output therefore has a bandpass response. Its current-based half-height edges are the same frequencies used
for the branch bandwidth. A finite source resistance reduces the on-resonance output
unless that resistance is included in $R$ and the output is defined across the full
resistive series path. A resistor located at only one part of the total loss reports
only its own voltage fraction.

The capacitor and inductor outputs have different limiting forms. At low frequency,
capacitor reactance dominates the series sum and $H_C$ approaches one in magnitude,
while $H_L$ approaches zero. At high frequency, inductor reactance dominates and
$H_L$ approaches one in magnitude, while $H_C$ approaches zero. Near resonance,
both can have magnitudes near $Q$ and can exceed the input magnitude. The C output
therefore resembles a low-pass response with resonant peaking; the L output resembles
a high-pass response with resonant peaking. The resemblance identifies the chosen
port and its limiting behavior, not a universal replacement for a dedicated
low-pass or high-pass network.

$$
% caption: Capacitor and inductor output magnitudes from one series branch. The capacitor output dominates below resonance and the inductor output above it; both peak near the resonant frequency, rising to about Q times the input.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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\node[acc] at (1.35,2.05) {C out};
\node[black] at (4.75,2.05) {L out};
\node[above,fill=white,inner sep=1pt] at (3.0,2.72) {peak};
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$$

All three transfer functions share the same poles but have different zeros. With
$s$ as the complex frequency variable,

$$
H_R(s)=\frac{RCs}{LCs^2+RCs+1},
\qquad
H_C(s)=\frac{1}{LCs^2+RCs+1},
\qquad
H_L(s)=\frac{LCs^2}{LCs^2+RCs+1}.
$$

All three outputs share the same denominator because the same branch stores energy.
Their numerators specify how the output port suppresses low frequency, high frequency,
or both. The resistor output has one zero at $s=0$, giving low-frequency rejection,
and falls at high frequency because the denominator has higher order. The capacitor
output has nonzero DC transfer. The inductor output has a double zero at DC. These
statements assume an unloaded output; a load changes the numerator and can change the
denominator as well.

Pulse response follows the same poles. A high-$Q$ resistor-output network rings after
a short burst because the burst excites the stored-energy mode. The envelope then
decays with the series damping rate. A high-$Q$ bandpass circuit separates nearby
continuous frequencies well, but it responds slowly to a sudden change in amplitude
or frequency. The time-bandwidth tradeoff appears directly in the differential
equation: lowering $R$ narrows the frequency response and reduces damping at the same
time.

Output loading can invalidate the displayed filter curve. A detector connected across
the capacitor adds a parallel path; an amplifier connected across the resistor can
change the effective output resistance; a cable connected across the inductor adds
capacitance. These changes can lower $Q$, shift the center frequency, and alter the
edge frequencies. Include the intended load in the circuit model, or buffer the port
with a known high-input-impedance stage. The buffer still has input capacitance and
must be included when the selected capacitor is small or the required bandwidth is
narrow.

Signal selection also needs a rejection requirement. A narrow response centered at
$f_0$ rejects an interferer at $f_i$ according to the ratio
$|H(f_i)|/|H(f_0)|$. Numerical frequency difference $|f_i-f_0|$ alone does not set
that ratio. A nearby
interferer may lie within the half-height bandwidth and remain substantial. A distant
interferer can be strongly reduced even when its absolute frequency difference looks
small on a broad scale. State input amplitudes, desired output ratio, source and load
impedances, and the frequency grid used for verification.

## Design calculation, model limits, and reporting

> **Worked example (resonance, bandwidth, and voltage magnification).** A series branch
> has $R=20.0\ \Omega$, $L=10.0\ \mathrm{mH}$, and $C=2.53\ \mu\mathrm F$, driven by a
> $1.00\ \mathrm V$ RMS sinusoid. Find the resonant frequency, quality factor,
> bandwidth, half-power edges, resonant current, and the reactive-component voltages.
>
> The natural frequency follows from $L$ and $C$:
> $$
> \omega_0=\frac{1}{\sqrt{LC}}=6.29\times10^3\ \mathrm{rad\,s^{-1}},
> \qquad
> f_0=\frac{\omega_0}{2\pi}=1.00\ \mathrm{kHz}.
> $$
> The quality factor and bandwidth are
> $$
> Q=\frac{\omega_0L}{R}=3.14,
> \qquad
> \Delta\omega=\frac{R}{L}=2.00\times10^3\ \mathrm{rad\,s^{-1}},
> \qquad
> \Delta f=\frac{\Delta\omega}{2\pi}=318\ \mathrm{Hz}.
> $$
> Solving the two half-power conditions gives edge frequencies of $853\ \mathrm{Hz}$
> and $1.17\ \mathrm{kHz}$; their difference reproduces $\Delta f$ and their geometric
> mean is $f_0$. At resonance the impedance is $Z=R$, so
> $$
> I_{\rm rms}=\frac{V_{s,\rm rms}}{R}=\frac{1.00\ \mathrm V}{20.0\ \Omega}
> =50.0\ \mathrm{mA},
> $$
> and each reactive component carries $QV_{s,\rm rms}=3.14\ \mathrm V$ RMS — over three
> times the source voltage, even though no element supplies gain. Check these figures
> against the source-current, capacitor-voltage, and inductor-current limits before
> choosing a laboratory drive.

$$
% caption: Design curve for R equals 20 ohm, L equals 10 millihenry, C equals 2.53 microfarad. The half-power current edges fall at about 853 hertz and 1.17 kilohertz around the 1.00 kilohertz center.
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\node[below] at (2.12,.5) {853 Hz};
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\node[left] at (.7,2.105) {half power};
\end{tikzpicture}
$$

The calculation describes a parameter set, not a guarantee about a physical assembly.
The total series resistance includes the explicit resistor, source resistance referred
to the branch, inductor winding resistance, connection resistance, and any intended
load in the series path. Leaving out a $10\ \Omega$ source resistance in this example
changes the predicted $Q$ from $3.14$ to $2.09$ and increases the nominal bandwidth
from $318\ \mathrm{Hz}$ to $477\ \mathrm{Hz}$. The center frequency can remain nearly
unchanged while the response becomes much less selective. A source data sheet value
is only one part of the check; measure source voltage under the actual branch load.

The inductance and capacitance used in a design equation must match the operating
condition. A value obtained with a small test signal at one frequency and temperature
may differ from the value present during a larger sweep. The series RLC model does
not identify which component caused a change. It reports the change as a shifted
effective $L$, $C$, or $R$. Component-level characterization belongs to the separate
capacitor and inductor lessons. In an RLC experiment, use controlled amplitude,
temperature, and lead geometry, then state the resulting fitted effective values.

Lead inductance and stray capacitance can matter even when their values are small
compared with the selected component values. A few centimeters of loop area adds a
series contribution; a cable or probe can add a shunt contribution. Their effects are
most visible where the ideal residual reactance is near zero. A broad, low-$Q$ circuit
may tolerate such additions. A narrow circuit can show a measurable center shift or
an asymmetric residual with the same fixture. Keep the source-to-branch loop compact,
bring sense leads to the selected terminals, and repeat the measurement after a
deliberate change in lead placement. The difference estimates fixture sensitivity.

An amplitude sweep tests linearity. At several source levels, divide measured branch
current by measured source voltage and compare the normalized curves. A linear series
model gives the same normalized resonance curve at every level, within noise and
heating effects. A peak that shifts with amplitude points to a changing effective
reactance. A peak that broadens only after a long dwell may indicate temperature-driven
resistance. A distorted sine record can also bias phasor fitting, so inspect the raw
waveform and its residuals before attributing every departure to the branch itself.

The transient provides another diagnostic. Disconnect or sharply change the source
only when the experiment and equipment permit it, then record the branch response
with a high-impedance measurement path. Fit the ringdown envelope and oscillation
frequency. Compare the inferred damping rate $\gamma$ with $R/(2L)$ from the swept
fit. Agreement supports one common series model. Disagreement can arise from source
loading during the sweep, a load removed during ringdown, or a frequency-dependent
loss that a constant-$R$ fit averages differently in the two experiments.

Uncertainty must be attached to the reported quantity. For a center frequency
measured from a fitted current curve, include frequency-reference accuracy, point
spacing, source settling, fitting uncertainty, and repeatability after remounting.
For bandwidth, include uncertainty in both edge frequencies and the rule used to
identify the current-ratio level. For quality factor, propagate the uncertainty in
both center and bandwidth. A narrow bandwidth can make an absolute frequency error
small while making its fractional effect on $Q$ large.

If independent standard uncertainties are available for $f_0$ and $\Delta f$, a
first-order estimate is

$$
\left(\frac{u(Q)}{Q}\right)^2
\approx
\left(\frac{u(f_0)}{f_0}\right)^2+
\left(\frac{u(\Delta f)}{\Delta f}\right)^2.
$$

Shared fitting parameters can correlate the two estimates. In that case, retain the
fit covariance or use repeated complete sweeps to estimate the uncertainty in $Q$
directly. Reporting many digits from a numerical curve fit does not create physical
precision. The number of justified digits follows the calibration and repeatability,
not the display resolution of the fitting program.

A complete series-RLC result records the circuit and the experiment together:

- **Branch definition.** List the measured or nominal R, L, and C, their connection
  order, source resistance, intended load, lead arrangement, and temperature.
- **Drive condition.** State the source waveform, peak or RMS convention, amplitude,
  frequency grid, sweep direction, dwell time, and any amplitude-linearity check.
- **Measurement path.** Give the sense resistor, voltage-reference polarities, probe
  type, sample rate, record length, phasor-fit method, and channel-delay calibration.
- **Extracted response.** Report center frequency, bandwidth definition, quality
  factor, current peak, phase crossing, uncertainty method, and fit residuals.
- **Model range.** State the frequency and amplitude interval for which the ideal
  series RLC fit represents the data, plus the observed condition where it fails.

A reproducible resonance result includes the circuit, sweep, measurement path, fitted
response, and residuals. Its fitted center frequency and bandwidth apply to the stated
source, cable, load, temperature, and drive amplitude; changing any of those conditions
requires a new model check.

## Analytical checks across the response

Several limiting calculations catch sign and scale errors before a sweep begins. Far
below the natural frequency, the capacitor term dominates the series impedance:

$$
Z\approx R-\frac{j}{\omega C},
\qquad
|Z|\approx\frac{1}{\omega C},
\qquad
I_{\rm rms}\approx\omega C V_{s,\rm rms}.
$$

The current should rise in direct proportion to frequency on this low-frequency side,
and it should lead the source voltage. Far above the natural frequency, the inductor
term dominates:

$$
Z\approx R+j\omega L,
\qquad
|Z|\approx\omega L,
\qquad
I_{\rm rms}\approx\frac{V_{s,\rm rms}}{\omega L}.
$$

The current should fall inversely with frequency on this high-frequency side and lag
the source voltage. A measured curve that rises on both far sides often indicates that
the plotted quantity is a component voltage rather than branch current, that the
source level changes across the sweep, or that the selected frequency range has
already reached an unmodeled parasitic limit.

Near the natural frequency, introduce a fractional detuning

$$
\epsilon=\frac{\omega-\omega_0}{\omega_0}.
$$

For $|\epsilon|\ll1$, the net reactance has the local form

$$
\omega L-\frac{1}{\omega C}\approx2\omega_0L\epsilon.
$$

The normalized current becomes

$$
\frac{I(\omega)}{I(\omega_0)}
\approx\frac{1}{\sqrt{1+\left(2Q\epsilon\right)^2}}.
$$

The factor $2Q\epsilon$ controls the local response. A fractional detuning of
$1/(2Q)$ lowers current to approximately $1/\sqrt2$ of its peak value.
A high-$Q$ circuit needs a finer frequency grid because the characteristic fractional
scale decreases as $Q$ rises. Use the approximation only near the peak; the full
impedance expression remains the correct model across a broad sweep.

The component-voltage check is equally important. Compute

$$
\widetilde V_R=\widetilde I R,
\qquad
\widetilde V_L=j\omega L\widetilde I,
\qquad
\widetilde V_C=-\frac{j}{\omega C}\widetilde I,
$$

then add the three complex values and compare the result with the measured source
phasor. Adding their magnitudes is invalid because the inductor and capacitor
voltages have opposite quadrature signs. A calculation that reports a source voltage
smaller than both reactive voltages can be fully consistent near resonance. The
complex sum, not a scalar sum, enforces the loop law.

The same calculation separates an instrument error from a component error. Suppose
the measured branch current agrees with the predicted magnitude but its inferred
phase has a nearly linear frequency offset. A channel-delay calibration is the first
test. Suppose the phase crosses zero near the predicted center while the current peak
is lower and wider than predicted. The total series resistance is the first parameter
to remeasure. Suppose both peak position and phase crossing shift together. The
effective product $LC$ has changed, so inspect component values, lead geometry, and
the output load. This sequence uses measured patterns rather than a single adjusted
number.

Settling time can be specified from the damping rate. If a remnant transient envelope
must fall below a fraction $\eta$ of its initial value, choose a dwell satisfying

$$
t_{\rm dwell}\gtrsim\frac{\ln(1/\eta)}{\gamma}
=\frac{2L}{R}\ln\!\left(\frac{1}{\eta}\right).
$$

At a target of one percent, the logarithm is about $4.61$. The dwell estimate uses the
free-response envelope of the stated series model. A generator whose frequency changes
gradually can have a different settling history, and a source whose phase resets at
each point can inject a larger transient. Inspect several records around the peak to
verify that the chosen dwell produces stable fitted amplitude and phase.

Dimensional checks remain simple and effective. $LC$ has units of seconds squared,
so $1/\sqrt{LC}$ has units of radians per second. $R/L$ has the same units and is
therefore a valid bandwidth in angular frequency. $\omega_0L/R$ is dimensionless,
as required for $Q$. A unit mismatch usually identifies a misplaced factor of
$2\pi$, a capacitance entered in microfarads rather than farads, or an inductance
entered in millihenries rather than henries. Write numerical prefixes explicitly in
the calculation record; a silent prefix error can move a predicted center frequency
by three orders of magnitude.

The checks keep the model local and testable. An ideal series RLC equation describes
the stated branch and connection state. Within the stated drive, source, load,
temperature, and frequency interval, the measured branch either agrees with that
model to its uncertainty or yields a specific residual for a broader model.

For data reduction, preserve the raw phasors as well as derived magnitudes. A later
change in source calibration, current-reference polarity, or output loading can be
applied to complex records without recreating the experiment. Preserve the actual
frequency values rather than rounded axis labels. Store the fitted sine coefficients,
the time stamps, and the circuit photograph or wiring drawing with the sweep. These
records allow the calculated branch voltage, current, phase, bandwidth edges, and
quality factor to be recalculated under a revised uncertainty model. They also expose
whether a visually narrow peak came from the branch, a sparse frequency grid, or a
plotting scale that hid the response tails.