RLC Resonance · study.

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) = R i (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 ^{2} q}{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.

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.

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 $v_{s} (t) = V_{0} cos (ω t)$ 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 ^{2} q _{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

ω_{0} = \frac{1}{L C}, γ = \frac{R}{2 L} .

The characteristic roots are $s = - γ \pm γ^{2} - ω_{0}^{2}$ . Their form divides free response into three cases. The comparison uses $R$ against $2 L / C$ , the resistance for which the roots coincide.

When $R < 2 L / C$ , the transient has the form

q_{h} (t) = A e^{- γ t} cos (ω_{d} t + α), ω_{d} = ω_{0}^{2} - γ^{2} .

The current is the derivative of this charge, so its phase and envelope differ from the charge trace. The envelope $e^{- γ t}$ is set by the total series resistance, whereas the ringing frequency is slightly below $ω_{0}$ . 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 $R = 2 L / C$ . The transient form

q_{h} (t) = (A + B t) e^{- ω_{0} t}

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.

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.

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 (ω) = R + j (ω L - \frac{1}{ω C}) .

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 ∣ = R^{2} + (ω L - \frac{1}{ω C})^{2}, ϕ_{Z} = atan2 (ω L - \frac{1}{ω C}, R) .

The branch current for input phasor $V_{s}$ is

I = \frac{V _{s}}{Z}, I_{rms} = \frac{V _{s, rms}}{∣ Z ∣}, ϕ_{I} = ϕ_{s} - ϕ_{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 $18 0^{\circ}$ .

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.

The resistor, inductor, and capacitor voltage phasors are

V_{R} = I R, V_{L} = j ω L I, V_{C} = - \frac{j}{ω C} I .

Adding these phasors enforces the loop law without mixing amplitudes that occur at different times. The magnitudes $V_{L} = I ω L$ and $V_{C} = I / (ω C)$ 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.

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.

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:

ω L = \frac{1}{ω C} .

The resonance angular frequency and ordinary frequency are therefore

ω_{0} = \frac{1}{L C}, f_{0} = \frac{1}{2 π L C} .

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 $ω_{0}$ 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.

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.

The current-amplitude curve follows directly from impedance magnitude:

I_{rms} (ω) = \frac{V _{s, rms}}{R ^{2} + ( ω L - 1/ ( ω C ) ) ^{2}} .

Its largest value is $V_{s, rms} / R$ at $ω_{0}$ in the ideal model. The curve need not be symmetric on an ordinary linear frequency axis. The reactive expression contains both $ω$ and $1/ ω$ , 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.

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.

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

V_{L, rms} = I_{rms} ω_{0} L, V_{C, rms} = \frac{I _{rms}}{ω _{0} C}, V_{L, rms} = V_{C, rms} .

Since $I_{rms} = V_{s, rms} / R$ , both reactive voltage magnitudes equal $Q V_{s, rms}$ when $Q = ω_{0} L / R$ . 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 $U_{C} = q^{2} / (2 C)$ and the inductor energy is $U_{L} = L i^{2} /2$ . 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 $1/ L C$ . The underdamped free transient oscillates at $ω_{0}^{2} - γ^{2}$ . 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 $1/ 2$ of its resonance value. Since the resonance current magnitude is $V_{s, rms} / R$ , that condition gives

\frac{V _{s, rms}}{∣ Z ∣} = \frac{1}{2} \frac{V _{s, rms}}{R}, ∣ Z ∣ = 2 R .

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

ω L - \frac{1}{ω C} = 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.

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.

Solving the two quadratic equations gives the positive edge frequencies

ω_{-} = \frac{R ^{2} + 4 L / C - R}{2 L}, ω_{+} = \frac{R ^{2} + 4 L / C + R}{2 L} .

Their difference and product are especially informative:

Δ ω = ω_{+} - ω_{-} = \frac{R}{L}, ω_{+} ω_{-} = ω_{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 $ω_{0}$ . Their geometric mean equals $ω_{0}$ . A logarithmic frequency axis displays this multiplicative symmetry directly; a broad linear axis does not. In hertz, $Δ f = R / (2 π L)$ . 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{ω _{0} L}{R} = \frac{1}{ω _{0} C R} = \frac{ω _{0}}{Δ ω} = \frac{f _{0}}{Δ 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 $Δ U_{cycle}$ be the energy removed by the series resistance in one cycle. Then, for small fractional energy loss,

Q \approx 2 π \frac{U}{∣Δ U _{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.

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.

The phase curve provides a second selectivity measurement. Since

tan ϕ_{Z} = \frac{ω L - 1/ ( ω C )}{R},

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

ω L - \frac{1}{ω C} \approx 2 L (ω - ω_{0}), \frac{d ϕ _{Z}}{d ω}_{ω_{0}} \approx \frac{2 L}{R} = \frac{2 Q}{ω _{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.

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.

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{Δ f _{0}}{f _{0}} \approx - \frac{1}{2} \frac{Δ L}{L} - \frac{1}{2} \frac{Δ 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

I = \frac{V _{s} e n se}{R _{s}}, Z_{branch} = R_{s} \frac{V _{branch}}{V _{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.

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.

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 (ω t_{n}) + b sin (ω t_{n}) + d, A = a^{2} + b^{2}, ϕ = atan2 (- 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 $e^{- γ t}$ . 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 $Δ t$ 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 $Z (ω)$ against $R + j (ω L - 1/ (ω C))$ 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} (ω) = \frac{V _{R}}{V _{s}}, H_{L} (ω) = \frac{V _{L}}{V _{s}}, H_{C} (ω) = \frac{V _{C}}{V _{s}} .

The phasor voltage laws give

H_{R} = \frac{R}{Z}, H_{L} = \frac{j ω L}{Z}, H_{C} = \frac{- j / ( ω 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 $V_{R} = I R$ , but its magnitude also depends on the full series resistance used in the model.

Across the resistor, the magnitude is

∣ H_{R} ∣ = \frac{R}{R ^{2} + ( ω L - 1/ ( ω C ) ) ^{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.

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.

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

H_{R} (s) = \frac{R C s}{L C s ^{2} + R C s + 1}, H_{C} (s) = \frac{1}{L C s ^{2} + R C s + 1}, H_{L} (s) = \frac{L C s ^{2}}{L C s ^{2} + R C s + 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 $R = 20.0 Ω$ , $L = 10.0 mH$ , and $C = 2.53 μ F$ , driven by a $1.00 V$ 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 $C$ :

ω_{0} = \frac{1}{L C} = 6.29 \times 1 0^{3} rad s^{- 1}, f_{0} = \frac{ω _{0}}{2 π} = 1.00 kHz .

The quality factor and bandwidth are

Q = \frac{ω _{0} L}{R} = 3.14, Δ ω = \frac{R}{L} = 2.00 \times 1 0^{3} rad s^{- 1}, Δ f = \frac{Δ ω}{2 π} = 318 Hz .

Solving the two half-power conditions gives edge frequencies of $853 Hz$ and $1.17 kHz$ ; their difference reproduces $Δ f$ and their geometric mean is $f_{0}$ . At resonance the impedance is $Z = R$ , so

I_{rms} = \frac{V _{s, rms}}{R} = \frac{1.00 V}{20.0 Ω} = 50.0 mA,

and each reactive component carries $Q V_{s, rms} = 3.14 V$ 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.

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.

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 $10 Ω$ source resistance in this example changes the predicted $Q$ from $3.14$ to $2.09$ and increases the nominal bandwidth from $318 Hz$ to $477 Hz$ . 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 $γ$ with $R / (2 L)$ 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 $Δ f$ , a first-order estimate is

(\frac{u ( Q )}{Q})^{2} \approx (\frac{u ( f _{0} )}{f _{0}})^{2} + (\frac{u ( Δ f )}{Δ f})^{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}{ω C}, ∣ Z ∣ \approx \frac{1}{ω C}, I_{rms} \approx ω C V_{s, 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 ω L, ∣ Z ∣ \approx ω L, I_{rms} \approx \frac{V _{s, rms}}{ω 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

ϵ = \frac{ω - ω _{0}}{ω _{0}} .

For $∣ ϵ ∣ ≪ 1$ , the net reactance has the local form

ω L - \frac{1}{ω C} \approx 2 ω_{0} L ϵ .

The normalized current becomes

\frac{I ( ω )}{I ( ω _{0} )} \approx \frac{1}{1 + ( 2 Q ϵ ) ^{2}} .

The factor $2 Q ϵ$ controls the local response. A fractional detuning of $1/ (2 Q)$ lowers current to approximately $1/ 2$ 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

V_{R} = I R, V_{L} = j ω L I, V_{C} = - \frac{j}{ω C} 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 $L C$ 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 $η$ of its initial value, choose a dwell satisfying

t_{dwell} ≳ \frac{ln ( 1/ η )}{γ} = \frac{2 L}{R} ln (\frac{1}{η}) .

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. $L C$ has units of seconds squared, so $1/ L C$ has units of radians per second. $R / L$ has the same units and is therefore a valid bandwidth in angular frequency. $ω_{0} L / R$ is dimensionless, as required for $Q$ . A unit mismatch usually identifies a misplaced factor of $2 π$ , 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.