Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are...

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Transmission lines

Transcript of Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are...

Page 1: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

Transmission lines

Page 2: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

Fig 22: induction in a transmission line

𝐼

Page 3: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

From Eqs. 269 and 270 one gets:

(𝐶𝑑𝑉 = 𝑑𝑞)

(𝐿𝑑 ሶ𝐼 = 𝑑𝑉𝑒𝑚𝑓)

Page 4: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

Differentiating Eq.271 with respect to t and of Eq. 272 in respect to x yield correspondingly:

Page 5: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

𝑍 is the characteristic impedance of the transmission line and measured in ohms Ω .

271

272

271 272

Page 6: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

𝑰Fig 24: Termination of transmission line with impedance R.

Page 7: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

(271)

(272)

𝑉 = 𝑉0𝑒−𝑖𝜔𝑡 1 + 𝐾 286 𝑎𝑛𝑑

𝑉0

𝑍𝑒−𝑖𝜔𝑡 1 − 𝐾 (287)

Solving the equation for 𝐾, which gives the relative amplitude and phase of the “reflected” wave one gets:

If the impedance of the load is 𝑍𝐿 then:

Note: when 𝑍𝐿 = 𝑍,⟹ no reflected wave i.e. the termination is matched to the characteristic transmission line impedance.

Page 8: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

(291)

(290)

Page 9: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

(271)

(272)

Page 10: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable
Page 11: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable
Page 12: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

Note that since:

Page 13: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

The center wire and surrounding sheath are

separated by a dielectric of permittivity 𝜖 and

permeability 𝜇.

Fig. 26: Coaxial cable transmission line

Page 14: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

+𝝀 −𝝀

Page 15: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

𝒍

Page 16: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

𝒍

Page 17: Transmission lines · 2020-02-27 · 11 Transmission lines Transmission lines are used (as are waveguides) to guide electromagnetic waves from one place to an- other. A coaxial cable

With the expression in Eq. 307 ; for the capacitance per unit length one gets:

𝐶 =𝜆

𝑉=

2𝜋𝜀

ln( Τ𝑑 𝑎)