A lossless ladder line (series inductance per unit length

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Plasma Physics · Waves in plasmas

Modes, Packets and Media, Ch Transmission lines and electromagnetic waves in matter, plasmas and guides

Problem

A lossless

LC

ladder line (series inductance

L'

per unit length, shunt capacitance

C'

per unit length) is modified by connecting each section of length

\mathrm{d}x

to earth through an additional shunt inductor of value

L_{\rm p}'/\mathrm{d}x

(a leakage-inductance path in parallel with the shunt capacitor), so that

L_{\rm p}'

is an inductance times a length, in

\mathrm{H\,m}

. ((a)) Write Kirchhoff's voltage and current laws for a section of length

\mathrm{d}x

, take the continuum limit, and eliminate

I

to obtain a single wave equation for

V(x,t)

. ((b)) Substitute a travelling-wave solution

\tilde V=\tilde V_0e^{i(\omega t-kx)}

and show that

\omega^2=\omega_{\rm c}^2+v^2k^2

, giving

v

and

\omega_{\rm c}

in terms of

L'

C'

and

L_{\rm p}'

. ((c)) For

L'=1.0\,\mu\mathrm{H\,m^{-1}}

C'=100\,\mathrm{pF\,m^{-1}}

and

L_{\rm p}'=4.0\,\mathrm{mH\,m}

, evaluate

v

and the cutoff frequency

f_{\rm c}=\omega_{\rm c}/2\pi

. ((d)) Below

f_{\rm c}

this line reflects an incoming signal without dissipating it. State which quantity in a plasma plays the role of the shunt inductor

L_{\rm p}'

, and why.