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TotalTemp Technologies (Thermal Platforms) - RF Cafe

Transmission Line Equations

Transmission lines take on many forms in order to accommodate particular applications. All rely on the same basic components - two or more conductors separated by a dielectric (insulator). The physical configuration and properties of all the components determines the characteristic impedance, distortion, transmission speed, and loss.

See a discussion on transmission lines and coaxial connectors.

The following formulas are presented in a compact text format that can be copied and pasted into a spreadsheet or other application.

For the following equations, ε is the dielectric constant (ε = 1 for air)

Two Conductors in Parallel (Unbalanced)

Above Ground Plane

For D << d, h

Z0= (69/ε½) log10{(4h/d)[1+(2h/D)2]}

Parallel conductors above a ground plane - RF Cafe

Single Conductor Above Ground Plane

 

For d << h

Z0= (138/ε½) log10(4h/d)

Single conductor above a ground plane - RF Cafe

Two Conductors in Parallel (Balanced)

Above Ground Plane

For D << d, h1, h2

Z0= (276/ε½) log10{(2D/d)[1+(D/2h)2]}

Parallel conductors above a ground plane - RF Cafe

Two Conductors in Parallel (Balanced)

Different Heights Above Ground Plane

For D << d, h1, h2

Z0= (276/ε½)log10{(2D/d)[1+(D2/4h1h2)]}

Parallel conductors different heights above a ground plane - RF Cafe

Single Conductor Between

Parallel Ground Planes

For d/h << 0.75

Z0= (138/ε½) log10(4h/πd)

Single conductor between parallel ground planes - RF Cafe

Two Conductors in Parallel (Balanced)

Between Parallel Ground Planes

For d << D, h

Z0= (276/ε½) log10{[4h tanh(πD/2h)]/πd}

Two conductors between parallel ground planes - RF Cafe

Balanced Conductors Between

Parallel Ground Planes

For d << h

Z0= (276/ε½) log10(2h/πd)

Balanced conductors between parallel ground planes - RF Cafe

Two Conductors in Parallel (Balanced)

of Unequal Diameters

Z0= (60/ε½) cosh-1 (N)

N = ½[(4D2/d1d2) - (d1/d2) - (d2/d1)]

Parallel conductors - unequal diameters - RF Cafe

Balanced 4-Wire Array

For d << D1, D2

Z0= (138/ε½) log10{(2D2/d)[1+(D2/D1)2]}

Balanced 4-wire array - RF Cafe

Two Conductors

in Open Air

Z0= 276 log10(2D/d)

Two conductors in open air - RF Cafe

5-Wire Array

For d << D

Z0= (173/ε½) log10(D/0.933d)

5-wire array - RF Cafe

Single Conductor in

Square Conductive Enclosure

For d << D

Z0≈ [138 log10(ρ) +6.48-2.34A-0.48B-0.12C]/ε½

A = (1+0.405ρ-4)/(1-0.405ρ-4)

B = (1+0.163ρ-8)/(1-0.163ρ-8)

C = (1+0.067ρ-12)/(1-0.067ρ-12)

ρ= D/d

Single conductor in square conducting enclosure - RF Cafe

Air Coaxial Cable with

Dielectric Supporting Wedge

For d << D

Z0≈ [138 log10(D/d)]/[1+(ε-1)(θ/360)]½)

ε = wedge dielectric constant

θ= wedge angle in degrees

Air coaxial cable with dielectric supporting wedge - RF Cafe

Two Conductors Inside Shield

(sheath return)

For d << D, h

Z0= (69/ε½) log10[(ν/2σ2)(1-σ4)]


ν = h/d       σ = h/D

Twin conductors inside shield - RF Cafe

Balanced Shielded Line

For D>>d, h>>d

Z0= (276/ε½) log10{2ν[(1-σ2)/(1+σ2)]}


ν = h/d       σ = h/D

Balanced shielded line equation - RF Cafe

 
Two Conductors in Parallel (Unbalanced)

Inside Rectangular Enclosure

For d << D, h, w

                          ∞

Z0= (276/ε½) {log10[(4h tanh(πD/2h)/πd)- ∑ log10[(1+μm2)/(1-νm2)]}

                            m=1

μm=sinh(πD/2h)/cosh(mπw/2h)

νm=sinh(πD/2h)/sinh(mπw/2h)

Balanced 2-conductor line inside rectangular enclosure - RF Cafe

 

 

Equations appear in "Reference Data for Engineers," Sams Publishing 1993

RF Cascade Workbook 2018 - RF Cafe
Triad RF Systems Amplifiers - RF Cafe
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Copper Mountain Technologies (VNA) - RF Cafe

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About RF Cafe

Kirt Blattenberger - RF Cafe Webmaster

Copyright: 1996 - 2024

Webmaster:

    Kirt Blattenberger,

    BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The World Wide Web (Internet) was largely an unknown entity at the time and bandwidth was a scarce commodity. Dial-up modems blazed along at 14.4 kbps while tying up your telephone line, and a nice lady's voice announced "You've Got Mail" when a new message arrived...

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