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A band-pass filter preferentially transmits signals within a range of frequencies and attenuates sufficiently lower and higher frequencies. Taking the output voltage across the resistor in a series RLC circuit gives one example: the resonant series RLC response selects a band around $1/\sqrt{LC}$. ^[ref].

A critically damped RLC response is the boundary between underdamped RLC response and overdamped RLC response. The free series RLC circuit has a repeated characteristic root $-{\omega }_0$, with general solutionFor initial conditions $q(0)=q_0$, $i(0)=0$,In the plotted normalization, $q(1)=2/e\approx 0.735759$ and $i(1)=-1/e\approx -0.367879$. The electric charge returns without ringing; the polynomial factor $1+{\omega }_{0}t$ matters, so a single exponential function does not give the whole response.

For fixed $L,C$ and these initial conditions, critical damping avoids ringing while giving a faster asymptotic return than any larger $R$: the overdamped RLC response has a root closer to zero. This makes critical damping a useful tuning target for settling after switching. An MIT laboratory exercise makes the distinction observable by asking students to compare all three cases with $L=47\mathrm{mH}$ and $C=47\mathrm{nF}$. Those values give $R_{\mathrm{crit}}=2\sqrt{L/C}=2000\Omega$ and ${\omega }_0\approx 21276.6{\mathrm{s}}^{-1}$. ^[ref].

Write ${\omega }_0^2=1/(LC)$ and $\gamma =R/L$ for a series RLC circuit. The squared charge response has denominator $D=({\omega }_0^2-{\omega }^2{)}^2+{\gamma }^2{\omega }^2$. Its nonzero resonance peak occurs at ${\omega }_r^2={\omega }_0^2-{\gamma }^2/2$. The two squared half-height frequencies, when both exist, areTheir difference in angular frequencies is only approximately $\gamma$ at large quality factor. The electric current instead has $|I{|}^2=|V{|}^2/[R^2+(\omega L-1/(\omega C){)}^2]$. Its exact half-power frequencies satisfy $|\omega L-1/(\omega C)|=R$, whose positive roots differ by $R/L$, independently of capacitance. Thus an exact capacitance-independent bandwidth applies to current or resistor-voltage output; capacitor-voltage output is a low-pass response with a possible resonant peak.

Electrical engineering designs systems that generate, transmit, measure and control electrical energy and information. An electrical circuit makes this concrete: resistors, capacitors and inductors turn electric potential differences into predictable electric currents and time-dependent responses. Signal processing then asks which information those responses preserve, suppress or reveal.

The connection is useful in both directions. A series RLC circuit selects a band of frequencies through its physical dynamics; a moving average selects frequencies by arithmetic on recorded samples. Both can be studied through a frequency response. The models differ, but the question is shared: what happens to each component of the input, and what does that do to the output we care about?

Electrical impedance is the ratio of terminal electric potential difference and electric current phasors at one specified positive angular frequency $\omega$. For an electrical circuit modeled as a linear time-invariant system, use the sinusoidal steady state after any decaying transient has settled. With the passive sign convention, positive electric current enters the terminal assigned positive electric potential difference. Define peak complex amplitudes byThe physical $v(t)$ and $i(t)$ are real; the complex numbers encode their amplitudes and wave phases. The unit of electrical impedance is the ohm, $1\Omega =1\mathrm{V}/\mathrm{A}$. Both phasors may instead use root mean square magnitudes, with the same ratio $Z$. A nonlinear diode can generate additional frequencies; a single $Z(\omega )$ then does not describe the whole response.

Write the electrical impedance in Cartesian and polar forms asHere $R$ is the resistive real part, $X$ the signed electrical reactance, and $\varphi$ the voltage-minus-current phase difference, defined modulo a full turn. The Pythagorean theorem gives the modulus from a right triangle in the complex plane. The complex argument gives the rotation from the current phasor to the voltage phasor. Thus $V_{\mathrm{rms}}=|Z|I_{\mathrm{rms}}$, while negative $\varphi$ means electric current leads terminal electric potential difference. For $R>0$, $\varphi =\arctan (X/R)$ lies between $-\pi /2$ and $\pi /2$; outside that case, use the quadrant of the complex number rather than a bare inverse tangent. The geometric magnitude and phase difference are the two pieces of the AC relation.^[ref]

Multiplication by $Z$ acts on phasors, not on instantaneous physical signals. For example, when $i(t)=I_0\cos \omega t$,The second term is in phase quadrature with electric current; it cannot be represented by pointwise multiplication $v(t)=Zi(t)$ when $X\ne 0$. This is the useful compression: one complex multiplication replaces separate amplitude and phase difference calculations, while the real part restores the physical signal.

An ideal resistor, inductor and capacitor have $Z_R=R$, $Z_L=\mathrm{i}\omega L$ and $Z_C=1/(\mathrm{i}\omega C)$. Their electrical impedances add in a series RLC circuit, giving $Z=R+\mathrm{i}(\omega L-1/(\omega C))$. The current amplitude is $V_0/|Z|$, so cancellation of the imaginary parts explains the resonant series RLC response. The component-law derivation and opposite complex exponential convention are in reactance phase convention.

Electrical reactance $X$ is the signed imaginary part of electrical impedance $Z=R+\mathrm{i}X$, measured in ohms. It describes the part of the terminal electric potential difference in phase quadrature with the electric current in a sinusoidal steady state. In the $e^{\mathrm{i}\omega t}$ convention, an ideal inductor has $X_L=\omega L>0$ and an ideal capacitor has $X_C=-1/(\omega C)<0$ for $\omega >0$. A large $|X|$ reduces electric current at fixed applied voltage, but an ideal purely reactive component returns the energy it receives over each cycle. This is why electrical reactance and electrical resistance have the same units yet describe different physical behavior. ^[ref].

In a series RLC circuit, electrical impedances add because the same electric current passes through every component. For ideal positive inductance $L$, capacitance $C$, and total electrical resistance $R_{\Sigma }>0$,Here $\varphi$ is voltage wave phase minus current wave phase. Below ${\omega }_0=1/\sqrt{LC}$, $X<0$ and electric current leads terminal voltage; above it, $X>0$ and electric current lags. At ${\omega }_0$, the two signed electrical reactances cancel, leaving $Z=R_{\Sigma }$ and the maximum electric current at fixed terminal voltage. This is the resonant series RLC response. Electrical resistance does not cancel: the toy circuit gives $f_0={\omega }_0/(2\pi )=1591.55\mathrm{Hz}$ and $I_{\mathrm{rms}}=9.091\mathrm{mA}$ there. These numbers follow directly from the stipulated component values. ^[ref].

Cancellation concerns terminal voltage, not the disappearance of stored energy. In the toy circuit at resonance, the individual inductor and capacitor voltages are each $0.9091\mathrm{V}$ in root mean square magnitude, with opposite wave phases, while their sum is zero. Their peak stored magnetic energy and capacitor energy are each $0.8264\mu \mathrm{J}$, attained at different times. With smaller positive $R_{\Sigma }$, either component voltage can exceed the applied voltage. This ability to select a frequency while sustaining internal energy exchange connects electrical reactance to tuning and frequency response.

For an ideal inductor or capacitor, the absorbed instantaneous power equals the rate of change of stored energy:Positive power stores energy; negative power returns it. Over a periodic steady state, each stored energy returns to its initial value, so its cycle-averaged absorbed power is zero. An ideal resistor instead has $p_R=R{i}^2\ge 0$, transferring energy into Joule heating. More generally, the average terminal power of this series RLC circuit isZero average power in an ideal reactive component does not imply zero instantaneous power, zero electric current, or zero stored energy. Real coils and capacitors have losses, so zero ideal reactive loss is a model assumption rather than a claim about every component. ^[ref].

The motivating observation is a pair of sinusoids on an oscilloscope: changing frequency changes both their relative amplitudes and their phase difference. Gary Tuttle's EE 201 laboratory uses resistor voltage as the current-phase reference, compares predictions with readings at several frequencies, and explicitly asks students to measure component values and account for coil electrical resistance. The following bench arrangement applies that method to a series RLC circuit; its component values and illustrated traces are our own toy example. ^[ref].

Connect a low-voltage sine generator to a series chain ending in a known sensing resistor $R_s$ at the common return. Use a resistor whose unwanted inductance is negligible over the chosen frequency range. An oscilloscope displays voltage against time: channel A reads the loaded voltage across the whole chain, and channel B reads the voltage $v_s$ across $R_s$. Both channel ground connections and the generator return meet at that same node. Ordinary bench oscilloscope grounds are electrically common; attaching one to an intermediate circuit node would change the circuit. A separate floating component voltage needs a suitable differential measurement, rather than moving a ground clip. ^[ref].

After the free transient decays, record frequency $f$, period $T=1/f$, both voltage amplitudes, and matched rising zero-crossing times $t_A,t_B$. Check that the traces are sinusoidal. Define $\Delta t=t_B-t_A$ on the branch $|\Delta t|<T/2$. With negligible probe loading and negligible sensing-resistor electrical reactance, $v_s=R_{s}i$, so channel B is the electric current reference, and in the reactance phase conventionA later current crossing gives $\Delta t>0$, hence positive electrical reactance; an earlier crossing gives negative electrical reactance. Use the same amplitude convention for both channels. An amplitude-only ratio recovers $|Z|$, so it cannot distinguish the two signs: timing supplies the missing information. The formulas follow from $Z=|Z|e^{\mathrm{i}\varphi }$ and the chosen terminal orientation.

Begin with a known resistor replacing the reactive chain: its expected phase difference is approximately zero, testing channel polarity and relative delay. Measure the same signal with both channels to establish the instrumental delay baseline, then exchange probes to check that a phase offset does not belong to one channel. Next test an inductor and a capacitor separately with $R_s$: after subtracting $R_s$ from the inferred real part, expect approximately $X/f=2\pi L$ and $fX=-1/(2\pi C)$, respectively. Finally combine $L,C,R,R_s$ and sweep through the predicted resonance. These are proposed controls and consequences of the model, not reported observations.

For the toy series RLC circuit, the prediction at $500\mathrm{Hz}$ is $X=-286.894\Omega$, $|Z|=307.259\Omega$, $\varphi =-69.02^{\circ }$, and $V_{s,\mathrm{rms}}=32.55\mathrm{mV}$. At $5000\mathrm{Hz}$, $X=+282.328\Omega$ and $\varphi =+68.71^{\circ }$. Near $1591.55\mathrm{Hz}$ the model predicts zero phase difference and $V_{s,\mathrm{rms}}=90.91\mathrm{mV}$. Keep actual loaded $V_{A,\mathrm{rms}}$ fixed or normalize by it at each frequency; the generator's display alone does not establish that voltage. The $10\Omega$ sensing resistor is part of the $110\Omega$ total and therefore changes the predicted peak electric current.

Real components add unintended electrical impedance: coil wire has electrical resistance, nearby turns add capacitance, and capacitor leads add inductance. These effects can produce a component's own resonance, beyond which the ideal $\omega L$ or $-1/(\omega C)$ approximation fails. Work in a range where that approximation is adequate, measure coil wire electrical resistance, and compare the inferred $R_{\mathrm{chain}}$ with the sensing resistor, intended resistor, and component losses. A measured nonzero real part is evidence of loss in the assembly, not negative or positive electrical reactance becoming heat. ^[ref]; ^[ref].

The probes also add electrical resistance and capacitance to the measured nodes; their loading becomes more important as frequency increases. Use compensated probes with known input electrical impedance and short ground connections, and repeat with a smaller drive voltage to test whether the inferred $X$ depends on amplitude. ^[ref]. Record actual voltage and phase readings separately from calculated predictions, with component tolerances and timing uncertainty. A phase error $\delta \varphi$ alone produces approximately $\delta X=|Z|\cos \varphi \delta \varphi$, so near zero electrical reactance, a small delay can imitate a small residual $X$. Agreement within stated uncertainties tests the model; the smooth toy curves establish no experimental accuracy.

An ideal electrical circuit exchanges energy between an inductor and a capacitor. With no electrical resistance or applied drive, the electric charge obeys $L{q}^{\prime \prime }+q/C=0$, giving angular frequency ${\omega }_0=1/\sqrt{LC}$. A series RLC circuit adds dissipation and can include an external drive.

A linear filter obeys the superposition principle: filtering $\alpha x+\beta z$ gives $\alpha T[x]+\beta T[z]$. Many familiar examples are also linear time-invariant systems, represented by an impulse response and convolution. Linearity alone does not imply time invariance: multiplying $x[n]$ by the index $n$ is linear but changes its behavior when the same input is shifted.

The moving average is a finite impulse response example implemented by arithmetic; a band-pass filter can instead be built from a series RLC circuit. Their frequency responses explain which oscillations pass through. Choosing a filter means choosing how much amplitude, timing and detail may change, not simply removing every visible fluctuation.

An overdamped RLC response is a free series RLC circuit response with $R>2\sqrt{L/C}$, equivalently $\alpha >{\omega }_0$. Put $\beta =\sqrt{{\alpha }^2-{\omega }_0^2}$. The distinct negative characteristic roots are $r_{\pm }=-\alpha \pm \beta$, so the general electric charge isFor initial conditions $q(0)=q_0$, $i(0)=0$,For $q_0>0$, this electric charge decreases to zero without reversing sign. In the dimensionless example ${\omega }_0=1$, $\alpha =2$, the characteristic roots are $-2\pm \sqrt{3}\approx -0.268,-3.732$. The slow term dominates late times. Stronger damping can therefore make the final return slower: for $\alpha \gg {\omega }_0$, $r_{+}\sim -{\omega }_0^2/(2\alpha )=-1/(RC)$. Nonoscillation alone does not imply a fast response, and monotonicity here uses the specified initial conditions.

The PDF gives $V=H(t)$, the Heaviside step function; the TeX incorrectly changes this to $V=I(t)$. A unit voltage step has derivative the Dirac delta function, so the current is the causal Green function for $L{d}^2/d{t}^2+Rd/dt+1/C$.

The current must be continuous at zero: a jump would create a delta derivative through $L{I}^{\prime \prime }$, absent on the right-hand side. Integrating across zero therefore gives $L[I^{\prime }]+R[I]=1$, so $I(0^{+})=0$ and $I^{\prime }(0^{+})=1/L$. PutThese distinct negative characteristic roots give the overdamped RLC responseTogether with $I(t)=0$ for $t<0$, this solves the step response. The eventual current is zero: the capacitor charges and blocks steady current in the series RLC circuit.

Let $Q(t)$ denote the electric charge on the initially positive capacitor plate and define the discharge electric current by $I=-\dot{Q}$. Kirchhoff's voltage law for the series RLC circuit gives the damped harmonic oscillator equationThe initial conditions are $Q(0)=Q_0$ and $\dot{Q}(0)=0$: the open switch initially prevents electric current, and finite inductance prevents an instantaneous jump in it. Put $\gamma =R/(2L)$ and ${\omega }_0^2=1/(LC)$. The two characteristic roots are $s=-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}$. An oscillatory electric current requires a nonzero imaginary part, henceFor positive $L,C$ and nonnegative $R$, this is the underdamped regime. The initial conditions giveAt equality there is critical damping, and above it there is an overdamped response, with no repeated changes of sign of the discharge electric current. A nonzero initial electric charge is understood; if $Q_0=0$, the stated initial conditions give no electric current at all.

Put $x=\omega /{\omega }_0$. Maximizing the output wave amplitude is equivalent to minimizingIts derivative is $D^{\prime }(x)=2x[2x^2-2+Q^{-2}]$. An interior minimum at positive frequency therefore requires $Q>1/\sqrt{2}$, and occurs atIndeed $D^{\prime \prime }(x_{\mathrm{res}})=8x_{\mathrm{res}}^2>0$, and the peak value is $|H{|}_{\max }=Q/\sqrt{1-1/(4Q^2)}$. For $0<Q\le 1/\sqrt{2}$, $D$ increases for $x>0$ and the largest output is the DC value $|H(0)|=1$. At equality the putative peak has reached zero frequency; there is no distinct nonzero-frequency resonance. This threshold concerns capacitor voltage peaking: the series RLC circuit's electric current still has its maximum at ${\omega }_0$ for every positive $R$.

For $m\ddot{x}+b\dot{x}+kx=F_0\cos \omega t$, define ${\omega }_0=\sqrt{k/m}$ and $\beta =b/(2m)$. The parameter relating quality factor and viscous damping regimes isAt weak damping it also equals $2\pi$ times stored energy divided by energy lost per cycle. The stored-energy convention and frequency matter: this cycle-based interpretation should not be extended to free motions that never complete a cycle.

For $\beta <{\omega }_0$ the free motion is $e^{-\beta t}[A\cos {\omega }_{d}t+B\sin {\omega }_{d}t]$, where ${\omega }_d=\sqrt{{\omega }_0^2-{\beta }^2}$. Thus underdamping means $Q>1/2$. High $Q$ gives many oscillations per amplitude-decay time, narrow resonance and strong frequency selectivity; a lightly loaded tuning fork, optical cavity or low-loss electrical resonator provides examples. The amplitude decay time is ${\tau }_A=1/\beta =2Q/{\omega }_0$, while the slowly averaged energy decays with ${\tau }_E=1/(2\beta )=Q/{\omega }_0$. These times differ by a factor of two.

At critical damping, $\beta ={\omega }_0$ and $Q=1/2$, with $x=(A+Bt)e^{-{\omega }_{0}t}$. There is no ringing; a simple critically damped position indicator can settle without oscillation. For $\beta >{\omega }_0$ and $Q<1/2$, there are two real negative decay rates $-\beta \pm \sqrt{{\beta }^2-{\omega }_0^2}$. Increasing very strong drag slows the long-time return, whose rate approaches ${\omega }_0^2/(2\beta )$. A strongly viscously damped mechanical indicator illustrates this regime. These classifications concern the linear homogeneous equation, not every possible nonlinear damping law.

The driven displacement amplitude isThe displacement-resonance threshold of a viscous oscillator requires $\beta <{\omega }_0/\sqrt{2}$, or $Q>1/\sqrt{2}$; it lies at $\sqrt{{\omega }_0^2-2{\beta }^2}$. Some underdamped systems therefore have no such maximum. Average absorbed power instead peaks at ${\omega }_0$ for this viscous model. Its half-power frequencies are $\sqrt{{\omega }_0^2+{\beta }^2}\pm \beta$, so their separation is $2\beta$ and $Q={\omega }_0/\Delta \omega$. For broad response this ratio remains a useful parameter but not a claim of many free oscillations. The familiar displacement-linewidth version is a high-$Q$ approximation.

A series RLC circuit has ${\omega }_0=1/\sqrt{LC}$ and $Q={\omega }_{0}L/R$ for the charge oscillation. Resistance dissipates energy, inductance provides the inertial analogue and capacitance the restoring analogue. Ringdown and a calibrated half-power scan independently estimate loss; parasitic resistance, loading and radiation change the measured value. A large $Q$ increases frequency selectivity but also lengthens settling time, so it is not universally the best instrumental choice.

The original diagram is a series RLC circuit, so the same electric current $i=\dot{q}$ passes through all three elements. Choose capacitor electric charge $q$ positive on the plate into which positive $i$ enters. Kirchhoff's voltage law givesDividing by inductance $L$ givesThe correspondence to the forced harmonic oscillator is $m\leftrightarrow L$, $b\leftrightarrow R$, $k\leftrightarrow 1/C$, and $x\leftrightarrow q$. The stored energy is $L{i}^2/2+q^2/(2C)$; its balance is $\dot{E}=Vi-R{i}^2$, confirming the damping sign.

Choose positive electric current in the initial discharge direction, since the source diagram does not label the initial plate polarity or a current arrow. Let $q$ be the charge remaining on the initially positive plate, so $I=-\dot{q}$. Kirchhoff's voltage law for the series RLC circuit givesThe natural angular frequency, damping coefficient and quality factor areSince $\alpha <{\omega }_0$ (equivalently $Q>1/2$), this is an underdamped RLC response. With ${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}=815.82\mathrm{rad}{\mathrm{s}}^{-1}$, satisfying the initial conditions givesand differentiating givesThe electric current starts at zero with $\dot{I}(0)=V_0/L=100\mathrm{A}{\mathrm{s}}^{-1}$. Its wave period is $T_d=2\pi /{\omega }_d=7.702\mathrm{ms}$, its zeros occur every $3.851\mathrm{ms}$, and its positive and negative wave envelopes decay with time constant $30.0\mathrm{ms}$. The first maximum obeys $\tan {\omega }_{d}t={\omega }_d/\alpha$ and is $0.1151\mathrm{A}$ at $1.875\mathrm{ms}$. The electric current tends to zero as the initial capacitor energy is dissipated in the resistor.

A mass $M$ on a spring of stiffness $K$, subject to linear viscous damping and a sinusoidal drive, obeys the forced harmonic oscillator equationHere $x$ is displacement, $2M\gamma$ has units $\mathrm{kg}{\mathrm{s}}^{-1}$, and $F_0$ is a force in newtons. A series RLC circuit is an electrical analogue, with $M\leftrightarrow L$, $2M\gamma \leftrightarrow R$, $K\leftrightarrow 1/C$ and $x\leftrightarrow q$.

For free motion set $F_0=0$ and solve the characteristic equation $s^2+2\gamma s+{\omega }_0^2=0$. The roots $s=-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}$ distinguish three cases:

The constants are set by initial position and velocity. The mechanical energy and its rate of change in free motion areFor weak damping, the cycle-averaged energy decays approximately as $e^{-2\gamma t}$. The quality factor is $Q={\omega }_0/(2\gamma )$; for $Q\gg 1$ it equals $2\pi$ times stored energy divided by the loss per cycle. Logarithmic decrement of successive free-decay peaks determines $\gamma$ experimentally.

For driven motion the complete answer is the decaying free transient plus a steady response at the driving angular frequency:At low frequency, $X\to F_0/K$ and the displacement is nearly in phase with the drive. At $\Omega ={\omega }_0$, the lag is $\pi /2$ and $X=F_0/(2M\gamma {\omega }_0)$. At high frequency, $X\simeq F_0/(M{\Omega }^2)$ and the lag tends to $\pi$. The displacement resonance maximum lies atonly when $\gamma <{\omega }_0/\sqrt{2}$. Thus an underdamped oscillator need not have a nonzero-frequency displacement resonance peak.

In the steady state the mean input power equals the mean dissipated power:This absorption peaks exactly at $\Omega ={\omega }_0$, rather than at ${\Omega }_r$. Its half-power points have bandwidth $\Delta \Omega =2\gamma$, so $Q={\omega }_0/\Delta \Omega$ for this linear model. For weak damping, a high quality factor therefore means a narrow strong resonance and a long free-decay time. Exactly zero damping at resonant driving has no finite steady response: a secular amplitude grows with time.

For sinusoidal steady state with convention $e^{i\omega t}$, complex impedance is the voltage phasor divided by the current phasor. Its modulus is the amplitude ratio and its argument is the voltage-current phase difference. Since $V_L=LdI/dt$,The original circuit is a series RLC circuit with output across the inductor. Its transfer function isFor the original capacitance, the outputs are $10.15\mathrm{mV}$ at $9\mathrm{MHz}$ and $4.65\mathrm{mV}$ at $10\mathrm{MHz}$, a ratio $0.458$. The current resonance is $f_0=1/(2\pi \sqrt{LC})=8.967\mathrm{MHz}$, so the circuit favors the first transmitter.

The resistor voltage is in phase with the current; the inductor voltage leads it by $\pi /2$ and the capacitor voltage lags by $\pi /2$. Large reactive voltages therefore nearly cancel in their phasor sum. A large inductor voltage is consistent with a small supply voltage: it is inductor voltage magnification, not power gain in a passive circuit.

Reducing the capacitance raises $f_0$ to $9.452\mathrm{MHz}$, between the two transmitters. Direct evaluation gives $7.02\mathrm{mV}$ and $7.22\mathrm{mV}$, respectively. Their similar detunings give approximately equal amplitudes; they are not exactly equal, partly because the numerator of $H$ also depends on frequency.

The two-tone output of a linear resonant circuit, for coherent monochromatic sources of equal input amplitude and arbitrary fixed input phases, isWith $A\simeq B=V_o$, this is approximatelyThus a $9.5\mathrm{MHz}$ rapid oscillation has a beat envelope repeating in magnitude at $1\mathrm{MHz}$. The signed envelope cosine has frequency $0.5\mathrm{MHz}$. Unequal amplitudes leave a small nonzero envelope minimum; independently drifting transmitter phases make the beating drift. No new frequency is generated by this ideal linear circuit.

Use $e^{i\omega t}$ phasors. The relation $V=L\dot{I}$ gives $Z_L=i\omega L$. The three branches form a parallel RLC circuit and share the source voltage. The source-current phasor and its real time dependence areConsequently its amplitude is $V_0\sqrt{R^{-2}+(\omega C-1/(\omega L){)}^2}$ and its phase relative to the voltage is ${\tan }^{-1}[R(\omega C-1/(\omega L))]$. At low frequency it is inductive and lags toward $-\pi /2$; at high frequency it is capacitive and leads toward $+\pi /2$. The ideal amplitude diverges at either extreme and has a finite minimum between them; real component losses and source electrical impedance limit those asymptotes.

Zero phase requires cancellation of electrical susceptances:The inductor and capacitor currents are nonzero, equal in magnitude and opposite in phase, while the resistor supplies the in-phase source current. energy oscillates between the reactive elements; dissipation occurs in the shunt resistor. A source-current maximum would describe a different, series-resonant circuit.

The specified ratio isIt exceeds $1/\sqrt{2}$ when $|\omega C-1/(\omega L)|<1/R$. Define $b=1/(RC)$ and ${\omega }_0=1/\sqrt{LC}$. The two positive endpoints are ${\omega }_{\pm }=[\sqrt{b^2+4{\omega }_0^2}\pm b]/2$, soThe endpoints have geometric mean ${\omega }_0$. Increasing the parallel resistance reduces dissipative loading and narrows this band, in contrast with increasing the resistance of a series RLC circuit. Equivalently, $Q={\omega }_0{U}_{\max }/P_{\mathrm{loss}}$, with $U_{\max }=C{V}_0^2/2$ and $P_{\mathrm{loss}}=V_0^2/(2R)$. The parallel RLC bandwidth here belongs to the explicitly defined current ratio, not a half-maximum rule applied to a source-current peak that does not exist.

After switching, Kirchhoff's voltage law gives the free series RLC circuit equationIts characteristic roots are $-\gamma \pm i{\omega }_d$, whereA real, nonzero oscillation frequency requires $R^2<4L/C$. For a passive resistor,The endpoint $R=0$ is the undamped limit; strictly decaying oscillations have $0<R<894\Omega$. The critical endpoint itself does not have a nonzero ${\omega }_d$.

Let $Q=C{V}_0$ denote the initial electric charge, so $Q=30\mu \mathrm{C}$ and $\dot{q}(0)=0$. Distinguish $Q$ from the amplitude coefficient $q_0$ in an exponential-times-cosine representation: imposing zero initial electric current generally requires a nonzero phase.