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An electrical circuit assembled from separately made components, such as individual transistors, resistors and capacitors. Several functions can share a package without forming a monolithic integrated circuit; packaging alone does not establish how the components were fabricated.

The time derivative of the electric displacement field enters the macroscopic Ampère-Maxwell equation alongside free electric current density. In vacuum, ${\mathbf{J}}_D={ϵ}_0{\partial }_t\mathbf{E}$. It is not transport of free charges through the vacuum gap of a capacitor; its surface integral nevertheless supplies the term needed for compatibility with the continuity equation.

George Elwood Smith reports fabricating MOS capacitors, thermally generating the first packet, and clocking it to an output that pushed it into the substrate for electric current detection. A later integrated eight-bit device demonstrated imaging. Dark or reversed-clock control traces are not specified; feasibility alone does not calibrate charge-transfer efficiency.^[ref]

A discriminating repeat would compare identical cycles with and without initial carrier injection, then suppress or reverse the transfer sequence while retaining output switching. An electric current transient synchronized with gate switching can include capacitor feedthrough; only the difference tied to initial electric charge and the expected transfer delay tests packet transport. These are proposed controls, not controls documented in the lecture. For an optical version, a blocked exposure estimates dark current and read noise; a spatially isolated illuminated optical detector pixel exposes any deferred charge in CCD readout trail. Photon shot noise persists even with ideal transfer.

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.

An actual teaching experiment turns the phasor definition into readings. Leipzig University's laboratory handout Measurements with an oscilloscope, experiment E 3e, uses a function generator, a two-channel oscilloscope, a known resistor and a capacitor. Figure 5 places the capacitor above the resistor, with a common return below the resistor. Channel 1 measures the applied terminal electric potential difference $v_1$; channel 2 measures $v_R$. The handout asks for peak-to-peak values and relative timing, then comparison of measured and calculated electrical impedances with $R=3\mathrm{k\Omega }$, $C=0.47\mu \mathrm{F}$ at $100$ and $300\mathrm{Hz}$. It also asks students to check the oscilloscope time base against a digitally counted generator frequency. These are apparatus and procedural specifications, not reported measured results.^[ref]

For this arrangement, Ohm's law gives $i(t)=v_R(t)/R$ if probe loading and the sensing resistor's electrical reactance are negligible. Record both voltage magnitudes with the same convention: peak, peak-to-peak, or root mean square. After the transient settles, compare matched rising zero crossings $t_1$ of $v_1$ and $t_R$ of $v_R$, with period $T=1/f$. ThenThe recovered electrical impedance includes the known resistor. For a separate unknown component in series with that resistor, subtract $R$ from the inferred total electrical impedance. A current crossing earlier than the applied voltage crossing gives $\Delta t<0$, hence negative $\varphi$. For the stipulated series RC impedance example at $1000\mathrm{Hz}$, this predicts $\Delta t=-0.160717\mathrm{ms}$; it is a calculated timing, not a lab observation. Timing supplies the complex argument that an amplitude ratio alone cannot recover.

As additional measurement controls, first feed the same sinusoid to both channels to check relative delay and polarity; then replace the reactive chain with a known resistor, expecting nearly zero phase difference. Keep both probe returns at the common return shown in figure 5. A reversed voltage polarity adds half a turn to its inferred wave phase. Match crossings with the same slope and choose $-T/2<\Delta t\le T/2$; adjacent cycles otherwise change $\varphi$ by a full turn. A sweep continuous in frequency helps track that branch. These controls test the signed inference rather than assuming that the scope's delay readout has the desired sign.

Measure the loaded terminal electric potential difference on channel 1 at every frequency. The generator's output impedance can change the applied magnitude; its front-panel setting is not the measured terminal value. The generator's internal electrical impedance is outside the terminal ratio above, while the known sensing resistor is inside it. Probe loading, finite timing resolution, component tolerances and distorted waveforms limit the inference. A repeat at lower drive amplitude tests whether one linear electrical impedance still fits. Electrical reactance bench measurement develops these loading and reference-direction issues further.

An electrical insulator has sufficiently small transport of electric charge for its operating conditions. Insulation is a practical requirement involving electric potential difference, temperature, geometry and duration, rather than a universal claim of zero electrical conductivity. Electric polarization can be substantial in an electrical insulator; dielectric breakdown can end the insulating regime.

For an idealized uniform slab with small constant electrical conductivity $\sigma$, the Ohmic conductor model predicts leakage electrical resistance $R=d/(\sigma A)$ in parallel with capacitance $C=ϵA/d$. An isolated capacitor then hasThis is a derived model of slow discharge, not an assertion that every leakage mechanism obeys Ohm's law. Its time scale matches charge relaxation. The fixed-charge comparison in dielectric capacitor response needs an observation time short compared with $RC$, but long enough for electric polarization to settle. During a maintained voltage, leakage consumes power $V^2/R$ as Joule heating; delayed electric polarization can add dielectric loss under changing electric fields.^[ref]

Large relative permittivity alone does not rule out either mechanism.^[ref]

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.

In the $e^{i\omega t}$ phasor convention, electrical admittance is $Y=G+iB$, with conductance $G$ and electrical susceptance $B$. An ideal capacitor has $B=\omega C$ and an ideal inductor has $B=-1/(\omega L)$. electrical susceptances add for parallel branches; their cancellation gives a purely resistive input admittance.

An electric battery converts the available chemical Gibbs free energy of one or more galvanic cells into electrical work delivered to an electrical load. The reacting materials are a finite reservoir: a battery can sustain a terminal electric potential difference while discharging, but cannot supply energy indefinitely. Unlike a capacitor, whose capacitor energy resides in an electric field, a battery chiefly stores its available energy in the chemical state of its materials. The detailed half-reactions and apparatus of a Daniell cell explain this conversion at the electrode level.

During discharge, electrons move through metallic wires from the negative anode, through the electrical load, to the positive cathode; external conventional current points the other way. Inside the battery, ions carry electric current through the electrolyte, and redox chemical reactions connect ionic transport to electronic transport at the electrodes. The same electron does not make a lap through both wire and electrolyte. Electric charge conservation closes the circuit; changing chemical composition supplies the energy. See galvanic cell for the two transport paths and electrolyte for how oppositely moving ions contribute to one net electric current.

The source's electromotive force $\mathcal{E}$ measures its driving work per unit electric charge, in volts; the loaded terminal electric potential difference $V_L$ measures the electrical work per unit electric charge delivered outside the battery. They agree in an ideal lossless source, or approximately near open circuit, but generally differ during discharge.^[ref]

In the ideal fixed-voltage model used in fixed-voltage capacitor energy balance, a battery maintains a prescribed terminal electric potential difference while electric charge flows to or from the capacitor. This is a constraint for that calculation, not an inexhaustible reservoir. The fixed-voltage capacitor energy balance also accounts for electrical energy returned to the source; an actual cell can accept sustained charging only if its chemistry and operating conditions permit it.

An electrostatic accelerator gives a charged particle kinetic energy by taking it through a sustained electric potential difference. Its ion source, accelerating tube and target form a particularly direct chain: prepare a charge, let an electric field do work, and measure what arrives or what the target emits. A vacuum reduces collisions with gas before the beam reaches its target. The Cockcroft-Walton generator supplies the potential by accumulating electric charge on capacitors.

Electrostatics studies time-independent electric charge distributions and their electric fields. In vacuum, its local equations in SI units areHere $\rho$ is charge density, ${ϵ}_0$ is vacuum permittivity, and $\Phi$ is electric potential. Gauss's law supplies the divergence equation; the static form of Faraday's law of induction supplies the zero curl. The Poisson equation becomes Laplace's equation where there is no electric charge. Prescribed electric charges give electric fields through Coulomb's law and the superposition principle; mobile electric charges on an electrical conductor instead require solving for their distribution together with the electric potential. ^[ref].

The useful distinction is between prescribed sources and prescribed boundary conditions. On a grounded electrical conductor, an external connection fixes its electric potential relative to an electrical reference potential and allows electric charge to enter or leave. On an isolated electrical conductor, total electric charge is fixed and its constant electric potential is an unknown. A neutral isolated electrical conductor can therefore have a nonzero local surface charge density through electrostatic induction. Equal electric potentials also do not imply equal electric charges: charge sharing between distant connected spheres gives a simple geometric example. ^[ref].

Electrostatics is useful because a spatial boundary value problem replaces the need to follow every moving charge carrier during charge relaxation. It describes the settled state of a capacitor and explains electrostatic shielding of sensitive equipment. Its assumptions must match the apparatus: stationary boundaries and sources, negligible time variation of the magnetic field, and enough time for charge relaxation. A metal carrying a maintained electric current through finite electrical resistance generally needs a nonzero internal electric field by Ohm's law; it is not a conductor in electrostatic equilibrium. Static macroscopic electric charge also does not mean that every electron is motionless. ^[ref].

An envelope detector extracts the slowly varying nonnegative wave envelope of a rapid carrier wave. In a simple electrical version a diode charges a capacitor near positive carrier peaks and an attached resistor discharges it between peaks. When $0\le m<1$, ideal envelope extraction gives $E(t)=A[1+m\cos (\Omega t)]$; removing the constant $A$ leaves $Am\cos (\Omega t)$. This permits recovery without a synchronized carrier reference. Carrier injection thus trades transmitted power for simplicity.^[ref]

A working peak detector needs margins as well as $m<1$: $E_{\min }=A(1-m)$ should comfortably exceed the diode's forward-voltage scale and noise, and the time constant $\tau =RC$ must suppress carrier ripple while following a falling wave envelope. The coarse scale window isA time constant that is too short leaves carrier ripple; one that is too long makes the discharging capacitor fall more slowly than the envelope. In the ideal zero-drop, continuous-tracking approximation, following a falling $E$ requires $E/\tau \ge -E^{\prime }$. For $0<m<1$,Differentiating the ratio locates its maximum at $\cos \theta =-m$. This original slope check explains why operation near $m=1$ is demanding even before overmodulation: the upper bound tends to zero while the need to smooth carrier cycles persists. It is an approximate circuit criterion, not a guarantee in the presence of diode drops, loading or noise. At $m=0$ there is no varying message to track; at $m=1$ ideal envelope extraction remains algebraically possible, but the simple detector has no positive trough margin.

For $m>1$, even an ideal noiseless envelope detector returns $A|q(t)|$. Removing its mean or its original offset cannot restore the discarded sign. A more elaborate receiver can use the retained wave phase through coherent demodulation.

The farad, symbol $\mathrm{F}$, is the SI unit of capacitance:A capacitor of capacitance $1\mathrm{F}$ carries separated electric charge magnitude $1\mathrm{C}$ at terminal electric potential difference $1\mathrm{V}$. A picofarad is $10^{-12}\mathrm{F}$, so the worked parallel-plate capacitor's capacitance $88.54\mathrm{pF}$ is $8.854\times 10^{-11}\mathrm{F}$. ^[ref].

For a quasistatic change of capacitance at fixed terminal electric potential difference, the capacitor energy is $U=C{V}^2/2$ and plate electric charge is $Q=CV$. Electrical work supplied to the electric field by the battery is $VdQ$, so mechanical work done on the capacitor is $d{W}_{\mathrm{mech}}=dU-VdQ=-V^{2}dC/2$. Increasing the separation of a parallel-plate capacitor makes $dC<0$: mechanical work is positive even though electrostatic energy decreases, because electrical energy returns to the source. Conversely, when $dC>0$, the battery supplies $V^{2}dC$: half increases capacitor energy and half is available as mechanical work done by the electric field. Differentiating capacitor energy alone at fixed terminal electric potential difference omits this exchange with the battery.

Southern Methodist University's teaching lab, Introduction to the Parallel Plate Capacitor Lab, makes the response tangible with a capacitor assembled from aluminium foil, paper and taped wire contacts. Its instructions specify approximately $10\mathrm{cm}\times 10\mathrm{cm}$ foil rectangles, aligned with the paper completely separating them. A multimeter in capacitance mode supplies the readout; a ruler measures plate dimensions and a micrometer measures specimen thickness. The lab replaces the paper with other sheets, including cardstock, plastic wrap and wax paper. It asks students to keep foil area and alignment consistent, measure thickness at several positions, estimate thickness uncertainty, and wait for the capacitance reading to stabilize. These are published preparation and measurement instructions, not measured results reported here.^[ref]

The inference uses each specimen's own thickness $t$, not just a comparison of raw meter readings. With corrected capacitance $C_s$ and overlapping area $A$, the ideal estimate is ${ϵ}_r=C_{s}t/({ϵ}_{0}A)$. A thicker sheet can give a smaller capacitance despite larger relative permittivity. This is why changing material while letting the plate gap change does not isolate electric susceptibility by itself.

Useful additional controls are to keep leads in the same positions, use capacitance measurement calibration, repeat one sheet after the others to check drift, and repeat the assembly to assess contact reproducibility. An air-gap reference must have known spacers maintaining the same separation; removing a supporting sheet and letting the foil touch is not such a reference. Record the instrument's test frequency and measurement mode when available. A capacitance display alone does not establish negligible dielectric loss or a strictly static permittivity. These are proposed controls extending the published teaching instructions.

Air gaps produce a particularly revealing systematic error. Suppose a uniform sheet of thickness $t$ and relative permittivity ${ϵ}_r$ is separated from the electrodes by air layers of total thickness $g$. Approximate air permittivity by vacuum permittivity, neglect the fringing field of a capacitor and use the same area in each layer. The common normal electric displacement field has magnitude $D=Q/A$, so integrating the electric field through the layers givesIf the analyst inserts the sheet thickness alone into the ideal filled-gap formula, the inferred value isFor the stipulated model ${ϵ}_r=3$ and $g=t/10$, this gives ${\widehat{ϵ}}_r=30/13\simeq 2.308$, not three. These values are calculated examples, not measured sheet properties. If total plate separation $d=t+g$ is used instead, the effective estimate is $(t+g)/(g+t/{ϵ}_r)$; it still mixes the sheet and the air. Thus measuring a thickness accurately cannot compensate for an unrecognized layer.

The fringing field of a capacitor and additive lead capacitance further limit a foil measurement. Keep lateral plate dimensions large relative to separation, and use a guard electrode of a capacitor in a suitable fixture for more demanding work.^[ref] A family of thickness measurements can help distinguish material response from an additive offset, but air gaps can vary with the assembly. A large, repeatable reading supports a capacitance change; extracting a material electric susceptibility requires the geometry, gaps, instrument response and dielectric loss assumptions to survive these checks. No measured relative permittivity is claimed for any of the lab's sheets.

A half-wave rectifier with one diode and a resistor load passes one half of each input cycle.^[ref]

For an original ideal example, let an ideal voltage source supply the sinusoid $v_S(t)=V_p\sin (\omega t)$, with $V_p=5\mathrm{V}$ and angular frequency $\omega >0$. Connect its positive reference terminal to the diode anode, the diode cathode to the upper terminal of a load resistor with electrical resistance $R_L=1\mathrm{k\Omega }$, and the lower terminal to the source return. Let $v_L$ be the upper-minus-lower load electric potential difference, and $i_L$ the downward load electric current. Assume the ideal diode model, zero source electrical resistance, and no filter capacitor, inductor or switching delay.

The Kirchhoff cycle law gives $v_S=V_D+v_L$, and Ohm's law gives $v_L=R_L{i}_L$. If $v_S>0$, the blocking assumption would give $V_D=v_S>0$, contradicting the ideal diode model, so the diode conducts with $V_D=0$. If $v_S<0$, conduction would give $i_L=v_S/R_L<0$, also forbidden, so the diode blocks. At the crossings both values vanish. ThusThe peak load electric current is $5\mathrm{mA}$. The output is unidirectional but still varies in time, reaching zero throughout every negative half-cycle. Over a period of a function $2\pi /\omega$, write the wave phase as $\theta =\omega t$. Its average value of a function and root mean square follow from two integrals:Consequently the average load power is $v_{L,\mathrm{rms}}^2/R_L=6.25\mathrm{mW}$, not ${\overline{v}}_L^2/R_L$. The ideal diode has zero loss; the resistor converts supplied energy into Joule heating. The same switching argument predicts $v_L=\min (0,v_S)$ when the diode is reversed. Directional selection changes the waveform and its average value of a function; it does not by itself produce a constant output.

Inductance relates electric currents to the magnetic flux linkage they produce. For a single linear coil with fixed geometry and magnetic response, its self-inductance $L$ is the constant inHere $\Lambda$ is magnetic flux linkage from the coil's own electric current $I$. The SI unit of inductance is the henry. Knowing inductance lets us predict the induced electromotive force when electric current changes, and the magnetic energy stored by an inductor. ^[ref].

For example, a coil of self-inductance $0.5\mathrm{H}$ carrying electric current $3\mathrm{A}$ has magnetic flux linkage $1.5\mathrm{Wb}$. If its $100$ turns link the same magnetic flux, each turn has magnetic flux $0.015\mathrm{Wb}$. Keeping the geometry and magnetic response fixed while doubling electric current doubles magnetic flux linkage and quadruples the inductor's magnetic energy.

The comparison with capacitance is useful: a linear capacitor has $Q=CV$ and stores capacitor energy $C{V}^2/2$ in its electric field, whereas a linear inductor has $\Lambda =LI$ and stores magnetic energy $L{I}^2/2$ in its magnetic field. The RLC circuit brings both forms of energy storage together.

For series resistance $R$, inductance $L$ and capacitance $C$, with output measured across the inductor, the $e^{i\omega t}$ transfer function is $H_L=i\omega L/[R+i(\omega L-1/(\omega C))]$. At current resonance ${\omega }_0=1/\sqrt{LC}$ its modulus is ${\omega }_{0}L/R=Q$. Large and oppositely phased inductor and capacitor voltages nearly cancel in the loop sum. This is voltage magnification in a passive circuit, not power gain. The maximum of $|H_L|$ need not occur exactly at the current-resonance frequency.