Transmission line

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.