Electrical engineering

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?

Digital electronics represents information by discrete signal levels. Combinational logic computes from current inputs; sequential logic also uses stored state.

A digital circuit implementing binary multiplication. A combinational logic implementation forms shifted partial products with AND gates and combines them with adders. A sequential circuit can instead reuse one adder, storing a partial sum, a shifted multiplicand, and the remaining multiplier in digital registers. If those quantities are $P,M,Q$, the update $P\leftarrow P+(Q_0?M:0)$, $M\leftarrow 2M$, $Q\leftarrow \lfloor Q/2\rfloor$ preserves $P+MQ$. Starting with $(0,X,Y)$ and stopping when $Q=0$ therefore produces $P=XY$, provided widths prevent overflow.

A temporary excursion of an output logic signal to its active level. A clocked controller may specify the duration in complete clock periods, rather than relying on an analogue delay.

A display whose seven separately controlled bars can form digit shapes. A combinational logic decoder maps an encoded digit to seven output bits. Active-high versus active-low control and the required behaviour for unused input codes belong to the decoder's specification.

Represent a decimal number by coding each decimal digit separately as four bits of weights $8,4,2,1$. The codes 10–15 are unused for one digit; a Boolean circuit must specify whether these combinations produce a defined result or are don't-care conditions.

An arbiter selects at most one asserted request, searching cyclically after the most recent successful recipient. After a grant it advances that remembered position. This implements circular priority among bits representing competing requests; it is distinct from assigning a fixed priority forever.

Store both the last recipient index $L$ and a current-grant valid bit $V$. With $k$ possible recipients, an idle period clears $V$ while preserving $L$. The Moore machine grants requester $j$ precisely when $V\wedge [L=j]$ holds. For $k>1$, its $k$ idle states are distinguishable by a subsequent pair of competing requests with different positions in the cyclic order; active states have distinct outputs. Thus all $2k$ states are distinguishable and $\lceil {\log }_2(2k)\rceil$ state bits suffice and are necessary for this output convention. A continuously asserted request receives a grant within $k$ decisions, because before that grant the cyclic scan cannot pass it to choose a later asserted request.

A recurring digital signal whose designated clock edge triggers a state update. A rising clock edge is the transition from the low level to the high level; a falling clock edge is the reverse.

One complete repetition of a periodic clock signal, for example from one rising clock edge to the next. Its duration is the clock period. A synchronous sequential circuit can update its stored state once per designated clock edge; a one-cycle output pulse remains asserted for one such interval.

The time between equivalent clock edges in consecutive cycles of a periodic clock signal. It is the reciprocal of the clock's frequency.

The reciprocal of the processor clock period. Reducing a combinational critical path can permit a higher frequency, subject to the remaining timing requirements.

The interval between successive designated clock edges of a processor's clock signal. In a synchronous sequential circuit it must be long enough for the required register-to-register propagation and setup timing.

A circuit producing a periodic output at an integer fraction of the input frequency. Each toggle stage divides the triggering frequency by two, so stage $n-1$ in an $n$-stage binary ripple counter has ideal output frequency $f_{\mathrm{in}}/2^n$.

A sequential circuit has stored state. Its future output can therefore depend on earlier inputs as well as the present inputs.

An electronic display cycling through the six die faces while a switch is pressed. A modulo-six counter and combinational logic can drive seven lights arranged as die pips. The apparent unpredictability comes from uncertainty in release timing relative to a rapid clock signal; deterministic cycling alone does not establish equal outcome probabilities.

A fixed-width collection of flip-flops storing a binary representation. A synchronous load-enabled digital register captures all its input bits on the active clock edge when load is asserted and otherwise retains its old value. One multiplexer before each D flip-flop can select new data or feedback from its old output. A processor register is a particular use inside a processor; temporary state in a standalone binary multiplier is another.

A chain of digital registers that transfers each stored value to the next stage at a clock edge. A serial-in parallel-out shift register exposes the stored history simultaneously. With the current input and $k-1$ delayed samples, combinational logic can evaluate a function of the latest $k$ samples.

A shift register accepting one sample at each clock edge and exposing every stored stage as a parallel output.

A directed graph with one vertex for each state and labelled edges for the permitted transitions. A finite counter has a directed cycle when every state has one successor.

A counter stores a binary representation of a state, usually advancing by one increment or decrement modulo $2^n$ when it has $n$ bits.

A digital counter cycling through six stored values. With ordinary binary representation, the valid three-bit states are $000,001,010,011,100,101$. The unused encodings require either an initialization assumption or specified recovery transitions.

A cascade in which an output transition of one flip-flop clocks the next. With positive-edge-triggered toggle stages and uncomplemented $Q$ connections, it decrements modulo $2^n$: a zero bit rises to one and propagates a borrow, while a one bit falls to zero and stops propagation. Ripple propagation takes time, so intermediate transient states are not the settled counter state.

A flip-flop stores one bit and updates it at a specified clock edge; an asynchronous clear can override that clocked update. Signal polarity is part of the specification, not something inferred from the name alone.

The time from a triggering clock edge to the corresponding settled output of a flip-flop, under specified loading. The time available between successive active clock edges must accommodate this delay, the combinational critical path, the receiving flip-flop's setup time, and any clock uncertainty.

The data input must remain stable for a specified interval before the sampling clock edge and a specified interval after it. Violations can prevent a well-defined sampled bit.

The minimum interval after an active clock edge during which the data input of a flip-flop must remain stable for specified operation. It constrains how quickly a newly launched value may reach the receiving flip-flop.

The minimum interval before an active clock edge during which the data input of a flip-flop must remain stable for specified operation. It constrains the latest arrival of new data at the receiving flip-flop.

A control that resets stored state independently of the clock edge. An active-low clear is asserted by a low signal; clocked sampling resumes after the clear is released.

At the active clock edge, $Q^{+}=D$. An asynchronous clear, when provided and asserted, takes precedence over this data-sampling rule.

At the active clock edge, the next state is $Q^{+}=J\neg Q\vee \neg KQ$: inputs $00$ hold, $01$ reset, $10$ set, and $11$ toggle. In particular, $K=0$ gives $Q^{+}=J\vee Q$, whereas $J=K=1$ gives $Q^{+}=\neg Q$. The standard function table is illustrated by a positive-edge device in the ^[ref]. Its asynchronous reset is active high; another circuit may specify an active-low clear.

An electric lamp whose current heats a filament until it emits visible thermal radiation. Joule heating converts electrical input into thermal energy and radiation. Its electrical resistance changes with temperature, so one fixed resistor value need not describe different operating voltages.

Unwanted stochastic fluctuations of voltage or electric current. Contributions include thermal noise, shot noise, and resistance fluctuations under bias; their physical origins and dependence on frequency need not be identical. A low-pass filter attenuates the high-frequency part of such fluctuations along with any high-frequency signal.

Detection and inference about remote objects using transmitted radio waves and their returned electromagnetic radiation. A received signal depends on propagation, directional antenna gain, target scattering and receiver sensitivity.

For a far-field free-space link with one reciprocal antenna transmitting and receiving at range $D$, the outward intensity is $P_{T}G/(4\pi D^2)$. The backward scattered intensity at the receiver is this times ${\sigma }_{\mathrm{back}}/(4\pi D^2)$. Multiplying by the matched antenna effective area $G{\lambda }^2/(4\pi )$ gives the displayed received power. Attenuation, misalignment and polarization mismatch require additional factors. The threshold for a root mean square voltage $V_m$ across a real matched load $R$ is $V_m^2/R$.

The monostatic radar cross section is $4\pi$ times the backward differential scattering cross-section. Equivalently, ${\sigma }_{\mathrm{back}}={\lim }_{r\to \infty }4\pi r^2{S}_{\mathrm{back}}/S_{\mathrm{inc}}$. It is a directional scattering measure with dimensions of area, not generally the geometric area or the total scattering cross-section. The definition may additionally specify the transmitted and received polarizations.

Retaining only the electric dipole induced in a small sphere gives $\mathbf{p}=4\pi {ϵ}_0{a}^3{\mathbf{E}}_{\mathrm{inc}}$. In the backward direction its differential scattering cross-section is $k^4{a}^6$, so the radar cross section is $4\pi k^4{a}^6$. This response model omits the sphere's induced magnetic dipole moment; it must be distinguished from perfect-conductor sphere dipole interference.

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An analog-to-digital converter produces a discrete numerical code from an input voltage. Its specifications include input range, resolution, conversion rate, input impedance, offset, gain error and nonlinearity. A uniform $b$-bit converter covering voltage span $V_{\mathrm{span}}$ has nominal spacing $q\simeq V_{\mathrm{span}}/2^b$. The uniform-quantization error bound applies to rounding inside the range, while clipping is a different error. Sampling and quantization are separate operations: neither sufficient sampling rate nor a large bit count alone guarantees accurate reconstruction.

oversampling takes samples faster than the minimum rate required by the desired signal bandwidth. With suitable uncorrelated conversion noise, digital filtering removes noise outside that bandwidth before decimation. Independent-error averaging improves standard deviation as $N^{-1/2}$; fourfold averaging ideally gains one bit of amplitude resolution. Repetition of an unchanged biased code does not produce the same benefit.

dither is an intentionally introduced fluctuation that makes a quantizer's output vary between codes, reducing signal-correlated quantization artifacts. A suitable distribution can make the averaged code carry information about a constant voltage smaller than one conversion step. The improvement requires filtering or averaging; extra noise does not improve an individual conversion or repair arbitrary calibration errors.

Let a constant input lie a fraction $f$ of the way between two consecutive quantizer levels separated by $\Delta$. With independent uniform additive dither spanning one step and nearest-level rounding, the two output codes occur with probabilities $1-f$ and $f$. Their mean equals the input and their sample-average variance is the displayed expression. This simple model shows both the sub-step information and the remaining noise cost.

rounding an unclipped voltage to equally spaced levels of separation $\Delta$ leaves error in $[-\Delta /2,\Delta /2]$. If the fractional position within a conversion interval is uniformly distributed, direct integration gives zero mean and expectation value $⟨{ϵ}^2⟩={\Delta }^2/12$. This is a statistical model for the error, not a claim that the converter adds independent white noise for every possible input.

A smooth input probability density varying slowly on the scale of one conversion interval makes the fractional uniform quantization error approximately uniform. Large overall variance is not sufficient: discrete inputs aligned with conversion levels provide a counterexample. Clipping, coarse sampling and correlations between error and signal require separate analysis.

A full-scale range $V_{\mathrm{FS}}$ and allowed quantization step $\Delta$ require $b\ge \lceil {\log }_2(V_{\mathrm{FS}}/\Delta )\rceil$ bits. The ideal uniform-quantizer rms error is approximately $\Delta /\sqrt{12}$. Bandwidth determines the sampling-rate requirement, but does not by itself determine a bit depth.

A transmission line guides travelling voltage and current waves. Its distributed electrical properties determine its characteristic impedance. A mismatch at a terminal produces reflected waves; available power, delivered power and returned power are distinct quantities.

A length $\ell$ with impedance ratio $\alpha$ between identical matched lossless lines has multiple internal reflections. Summing their amplitudes gives the displayed power transmission when the propagation speed is the same in all sections. It is unity at half-wave resonances even though either individual interface reflects. Matched remote terminations remove external echoes but not this frequency dependence.

A periodic lossless line can consist of equal-length sections with impedances $Z_1,Z_2$. Its semi-infinite input is a fixed point of the two-section impedance transformation. The physical solution selects a forward-energy or decaying Bloch wave. At special frequencies where the cell matrix is the identity, the fixed-point equation alone loses uniqueness; limiting absorption or a termination specifies the response.

For a real-frequency lossless reciprocal periodic line, a cell transfer matrix has determinant one. If half its trace lies outside $[-1,1]$, the Bloch multipliers are real reciprocal numbers rather than unit-modulus propagation factors. The physical interior field decays and carries no net power to infinity. A real reference line is then totally reflected by a semi-infinite structure; a finite structure can still tunnel.

The voltage-to-current ratio of a selected Bloch eigenmode at a specified position within a periodic cell. It depends on the chosen cell boundary, unlike the cell's propagation eigenvalues. In a pass band a physical semi-infinite mode carries positive real power; in an ideal stop band its input impedance is reactive and the mode decays into the line.

A shunt electrical resistance $R$ inserted between the conductors of a uniform lossless transmission line of characteristic impedance $Z$ has voltage reflection coefficient $r=-Z/(2R+Z)$ and transmission amplitude $t=1+r$. Equally spaced shunts produce coherent reflected waves. The round-trip phase between neighbours is $2kL$, so interference repeats in frequency with spacing $v_p/(2L)$ for nondispersive phase speed $v_p$.

Summing only paths that reflect at one shunt, with all previous shunts crossed once in each direction, gives a geometric series with ratio $t^2{e}^{2ikL}$. This is not the full multiple-reflection solution: it excludes paths that bounce between distinct shunts. For a semi-infinite array the exact amplitude instead obeys $R=r+t^2{e}^{2ikL}R/(1-r{e}^{2ikL}R)$. Near coherent reflection peaks the omitted paths need not be small, even if each shunt is weak.

On a uniform lossless transmission line, incident and reflected sinusoidal waves give voltage maxima $|V^{+}|(1+|\Gamma |)$ and minima $|V^{+}|(1-|\Gamma |)$. Their ratio is the voltage standing wave ratio, $(1+|\Gamma |)/(1-|\Gamma |)$, where $\Gamma$ is the voltage-wave reflection coefficient. A matched termination gives one, while perfect reflection gives an infinite ratio. These expressions assume a common real characteristic impedance and no propagation attenuation between the compared extrema.

Two perfectly conducting wide strips of width $a$ and separation $d\ll a$, filled with a homogeneous nonmagnetic dielectric of absolute permittivity $\epsilon$, support a transverse electromagnetic mode. Ignoring edge fields, their distributed capacitance is $C^{\prime }=\epsilon a/d$ and distributed external inductance is $L^{\prime }={\mu }_{0}d/a$. The magnetic-energy integral ${\mu }_0(I/a{)}^{2}ad/2=L^{\prime }{I}^2/2$ gives the latter. Their characteristic impedance is $Z_0=(d/a)\sqrt{{\mu }_0/\epsilon }$ and propagation speed is $1/\sqrt{{\mu }_0\epsilon }$. These parameters are per unit length, distinct from a lumped capacitor and inductor.

A distributed transmission line with per-unit-length series resistance $R^{\prime }$, series inductance $L^{\prime }$, shunt conductance $G^{\prime }$ and shunt capacitance $C^{\prime }$ obeys the displayed coupled first-order equations by applying Kirchhoff's voltage law and Kirchhoff's current law to a short element. With $R^{\prime }=G^{\prime }=0$, both voltage and current satisfy a wave equation with speed $1/\sqrt{L^{\prime }{C}^{\prime }}$ and characteristic impedance $\sqrt{L^{\prime }/C^{\prime }}$. The quantities $R^{\prime },L^{\prime },G^{\prime },C^{\prime }$ are distributed parameters, not a single lumped resistor or capacitor.

For per-length line parameters $R^{\prime },L^{\prime },G^{\prime },C^{\prime }$ and $e^{-i\omega t}$ convention, a forward factor $e^{-\Gamma z}$ has ${\Gamma }^2=(R^{\prime }-i\omega L^{\prime })(G^{\prime }-i\omega C^{\prime })$. Expanding about the lossless limit with $R^{\prime }\ll \omega L^{\prime }$ and $G^{\prime }\ll \omega C^{\prime }$ gives $\Gamma \simeq \alpha -i\omega \sqrt{L^{\prime }{C}^{\prime }}$, with $Z_0=\sqrt{L^{\prime }/C^{\prime }}$ and the displayed amplitude attenuation. Power decays twice as fast in the exponent. Terminal reflection is a separate power-return mechanism and need not be dissipative.

The characteristic impedance is the voltage-to-current ratio of a single forward travelling wave on a uniform transmission line. For real $Z_0$ and a real load resistance $R_L$, the voltage-wave reflection coefficient is $(R_L-Z_0)/(R_L+Z_0)$ and the power accepted by the load is the incident power times $4R_L{Z}_0/(R_L+Z_0{)}^2$.

A lossless uniform transmission line has a forward travelling-wave voltage and current related by $V_0=Z_0{I}_0$ with $Z_0=\sqrt{L/C}$. Arbitrary independent DC voltage and current fields cannot simply acquire the same forward wave factor while retaining Maxwell consistency.

An electrical transformer transfers energy between windings through electromagnetic induction. Its operation depends on changing magnetic flux, rather than a static magnetic field alone.

The reflected impedance is the contribution of a coupled secondary load to a primary winding's input impedance. For two ideal mutually coupled inductors, choose winding signs so that positive electric currents reinforce their magnetic fluxes. If the secondary loop has complex impedance $Z_s$, its Kirchhoff's voltage law gives ${\tilde{I}}_s=-i\omega M{\tilde{I}}_p/Z_s$. Substitution in ${\tilde{V}}_p=i\omega L_p{\tilde{I}}_p+i\omega M{\tilde{I}}_s$ gives $Z_{\mathrm{in}}=i\omega L_p+{\omega }^2{M}^2/Z_s$. For $Z_s=R+i\omega L_s$, the reflected real part is ${\omega }^2{M}^{2}R/[R^2+(\omega L_s{)}^2]$, and primary mean input power equals secondary mean Joule heating.

For an ideal electrical transformer with turns ratio $n=N_p/N_s$, the secondary load $R_L$ appears at the primary as $n^2{R}_L$, while a source resistance $R_g$ appears at the secondary as $R_g/n^2$. A finite magnetizing inductance $L_m$ contributes a parallel branch $i\omega L_m$ at the primary; its magnitude must greatly exceed the reflected load for the loaded input to be approximately resistive. The corresponding unloaded secondary self-reactance is $\omega L_m/n^2$. That reactance is distinct from the source's transformed output resistance. A specification that simultaneously calls the loaded input matched to a small resistance and calls the primary impedance very large must distinguish these two branches.

For a coaxial cable with inner and outer radii $a,b$, a radial interface $c$, and permittivities $ϵ,{ϵ}^{\prime }$, define $T=\ln (c/a)/ϵ+\ln (b/c)/{ϵ}^{\prime }$. With terminal electric potential difference $V$, the electric displacement field is $D_{\rho }=V/(T\rho )$. The electric field is $E_{\rho }=V/(T{ϵ}_i\rho )$ in each layer $i$. If the outward and return electric currents have magnitudes $I$, the magnetic field strength in the gap is $H_{\phi }=I/(2\pi \rho )$ and the magnetic field is $B_{\phi }={\mu }_{i}I/(2\pi \rho )$.

The proof uses Gauss's law without free interface electric charge, the electric potential difference integral, and the magnetostatic Ampère law. The axial Poynting vector is $VI/(2\pi T{ϵ}_i{\rho }^2)$, so its cross-sectional surface integral is $VI$, independently of the radial material layering and magnetic permeabilities. The empty inner and exterior regions have zero electric field and magnetic field for the regular solution with bounded exterior electric potential and no applied magnetic field.

A small longitudinal electric field drives conductor current in a lossy coaxial line. Its cross product with the azimuthal magnetic field sends energy radially into both conductors, where it becomes heat; forward power decreases with distance.

For ideal coaxial conductors of radii $a<b$ filled with a homogeneous permittivity $ϵ$ and magnetic permeability $\mu$, Gauss's law gives $E_r=\lambda /(2\pi ϵr)$ and hence $C^{\prime }=2\pi ϵ/\ln (b/a)$. The magnetic field $H_{\phi }=I/(2\pi r)$ has energy per unit length $L^{\prime }{I}^2/2$, giving $L^{\prime }=\mu \ln (b/a)/(2\pi )$. These are the external-field parameters when internal magnetic penetration into the conductors is neglected. Their product is $L^{\prime }{C}^{\prime }=\mu ϵ$, and the characteristic impedance is $\sqrt{L^{\prime }/C^{\prime }}=\sqrt{\mu /ϵ}\ln (b/a)/(2\pi )$.

A chart recorder records a changing signal against time on a moving chart.

An antenna couples electrical circuits to propagating electromagnetic radiation for transmission or reception.

The antenna directivity compares angular radiated power $U=d{P}_{\mathrm{rad}}/d\Omega$ with its average over directions. Its integral is $4\pi$. Unlike antenna gain, it uses radiated power rather than accepted input power, so it omits dissipative losses. A narrow beam can have high directivity without a larger total radiated power.

A phased array combines radiation from spatially separated antennas with controlled relative phases. Changing those phases steers the direction of constructive interference without moving the elements. For an equally spaced linear array with spacing $d$, free-space wavenumber $k$ and phase increment $\psi$, a principal beam at angle ${\theta }_0$ from the array normal satisfies $\psi =kd\sin {\theta }_0$ when the time dependence is $e^{-i\omega t}$ and the element drives are $e^{ij\psi }$.

A grating lobe is an additional principal maximum caused by spatial sampling in a phased array. A regularly spaced linear array steered towards ${\theta }_0$ also has maxima wherever $\sin \theta =\sin {\theta }_0+m\lambda /d$ lies in $[-1,1]$. Taking $d<\lambda /2$ excludes such maxima for all steering angles; $d=\lambda /2$ is the usual limiting choice, with an endfire degeneracy at the extreme scan directions.

The array factor describes the interference between the elements of a phased array, separately from the radiation pattern of each element. For $N$ identical equally spaced elements,At $\chi =2\pi m$ the limiting value has magnitude $N$. The total intensity is proportional to $|F{|}^2$ times the single-element antenna beam pattern.

Radiation resistance represents emitted electromagnetic power as a real input resistance of an antenna. It does not describe conversion to heat inside the conductor. If peak current amplitude is used, $P_{\mathrm{rad}}=|I_0{|}^2{R}_{\mathrm{rad}}/2$. Real ohmic loss adds a separate resistance and lowers radiation efficiency. For an ideal short uniform-current element of length $d$, the Hertzian dipole gives $R_{\mathrm{rad}}=80{\pi }^2(d/\lambda {)}^2$ in ohms, using the usual vacuum impedance approximation $120\pi \Omega$.

A fixed electrically small loop of area A with peak sinusoidal current $I_0\cos \omega t$ has magnetic moment $m=AI$. The magnetic dipole radiation law gives mean power ${\mu }_0{A}^2{I}_0^2{\omega }^4/(12\pi c^3)$. Defining it as $I_0^2{R}_{\mathrm{rad}}/2$ yields $R_{\mathrm{rad}}={\mu }_0{A}^2{\omega }^4/(6\pi c^3)$. This equivalent resistance accounts for radiation, not heating in the conductor. Rotating the loop changes its vector moment and cannot in general be modelled by the same scalar current alone.

The Friis transmission equation describes a far-field free-space antenna link with aligned directions, matched polarizations and negligible additional loss. It follows by multiplying the transmitter's axial energy flux $P_t{G}_t/(4\pi d^2)$ by the receiver's antenna effective area ${\lambda }^2{G}_r/(4\pi )$. A receiving impedance mismatch multiplies the available received power by $1-|{\Gamma }_L{|}^2$. Wavelength scaling must state which gains, apertures and impedances are held fixed.

Two aligned lossless Hertzian dipoles in one another's broadside radiation zone have gain $G=3/2$ and one-way received-power fraction $\eta =G^2[\lambda /(4\pi r){]}^2$. If the second remains conjugately matched, its matched-load scattering of a Hertzian dipole reradiates the same power as its load absorbs. Applying the reciprocal far-field link on the return gives $P_{\mathrm{echo}}=P_t{\eta }^2$, proportional to $r^{-4}$. The statement concerns physical dipole reradiation; a pure antenna load-mismatch echo model omits that scattering and is not interchangeable with it.

In an ideal matched-port two-port network for an antenna pair, the incident amplitude traverses the link, reflects from the receiver's load with amplitude coefficient ${\Gamma }_L$, then traverses the reciprocal link again. If each one-way power transfer factor is $\mathcal{F}$, the returned power fraction is $|{\Gamma }_L{|}^2{\mathcal{F}}^2$. With fixed gains, $\mathcal{F}\propto {\lambda }^2$ and this echo scales as ${\lambda }^4$. Actual antenna structural scattering or an existing input mismatch can add a coherent return not specified by the Friis transmission equation alone.

An antenna aperture describes the collecting area associated with a receiving antenna. The antenna effective area need not equal its geometric projected area.

The antenna effective area converts incident energy flux into available received power for a specified direction and matched polarization of an electromagnetic wave. In a lossless free-space link, a transmitter of antenna gain $G$ at range $r$ and a receiver of effective area $A_{\mathrm{eff}}$ have $P_r/P_t=G{A}_{\mathrm{eff}}/(4\pi r^2)$. Impedance or polarization mismatch reduces the delivered power.

For a reciprocal antenna with matched polarization and load, the available antenna effective area is proportional to the directional antenna gain. For a lossless antenna, a thermal equilibrium argument fixes the constant: one polarization of Rayleigh-Jeans law radiance gives incident power $(k_{B}T/{\lambda }^2)\int A_{\mathrm{eff}}d\Omega d\nu$, and matched thermal electrical noise gives $k_{B}Td\nu$. Thus $\int A_{\mathrm{eff}}d\Omega ={\lambda }^2$. Since $\int Gd\Omega =4\pi$, the displayed relation follows. Reciprocal losses multiply both area and gain by the same efficiency.

Antennas with effective areas $A_{e,1}$ and $A_{e,2}$ receiving the same polarization-matched plane wave have available powers in that ratio. If their matched-load resistances are equal, terminal voltage magnitudes have ratio $\sqrt{A_{e,1}/A_{e,2}}$. With unequal matched resistances the ratio becomes $\sqrt{A_{e,1}{R}_1/(A_{e,2}{R}_2)}$. Open-circuit voltages additionally depend on each antenna's source impedance. Geometry alone therefore fixes available-power estimates more directly than unspecified terminal voltages.

Antenna gain compares the far-field energy flux in a specified direction with the flux that an isotropic radiator would produce using the same input power. Thus $S_r=G{P}_{\mathrm{in}}/(4\pi r^2)$. Gain includes radiation efficiency; it must be distinguished from a voltage or current amplifier gain and from directivity when losses matter.

The fraction of accepted input power radiated by an antenna. With consistent feed definitions, antenna gain is radiation efficiency times antenna directivity. Ohmic losses reduce the efficiency, while feed mismatch is usually tracked separately.

The antenna temperature expresses the noise power delivered by a receiving antenna as an equivalent input noise temperature. It depends on the sky and surroundings weighted by the antenna beam pattern, as well as losses in the antenna; it need not be the physical temperature of either the antenna or a single astronomical source.

For a uniformly illuminated circular aperture of diameter D, normalized received power is $[2J_1(x)/x{]}^2$ with $x=\pi D\sin \theta /\lambda$. The first null has $\sin \theta =1.21967\lambda /D$ and the half-power point has $\sin \theta \simeq 0.5145\lambda /D$. A pointing tolerance must name the tolerated power loss; the first-null angle is not a good low-loss tolerance. These estimates require an ideal aperture pattern and are only approximate for a dish a few wavelengths across.

For a narrow angular power pattern $\exp [-(u^2+w^2)/(2{\alpha }^2)]$ about an equatorial beam centre, the solid angle measure is locally $dudw$. Extending the small lobe to the plane gives its beam solid angle ${\Omega }_A\simeq 2\pi {\alpha }^2$. Both sides of the azimuthal lobe must be included. A lossless antenna then has peak antenna gain $G_0\simeq 2/{\alpha }^2$ and peak antenna effective area $A_0\simeq {\lambda }^2/(2\pi {\alpha }^2)$.

A Hertzian dipole is an ideal electrically short oscillating electric dipole used to model radiation from a localized current element. In vacuum its far-field electric dipole radiation has ${\sin }^2\theta$ angular power distribution and total power $P={\mu }_0|\ddot{\mathbf{p}}{|}^2/(6\pi c)$, evaluated at the retarded time. The electric-dipole approximation for radiation requires a source much smaller than its radiation wavelength.