A transmission line of length L connects a load to a sinusoidal voltage source with an oscillation frequency f. Assuming that the velocity of wave propagation on the line is c, for which of the following situations is it reasonable to ignore the presence of the transmission line in the solution of the circuit?

(a) l = 20 cm, f = 20 kHz

(b) l = 50 km, f = 60 Hz

(c) l = 20 cm, f = 600 MHz

(d) l = 1 mm, f = 100 GHz

a. negligible

b. Border

c. Non negligible

d. Non negligible

Explanation:

On a transmission line, the velocity is expressed as

V=fλ

Hence we can write the wavelength as

λ=v/f

experimentally, a transmission line is negligible when

l/λ ≤ 0.01

hence we can write the above equation as

fl/v ≤0.01

a. for when l = 20 cm = 0.2m, f = 20 kHz=20000Hz, and v=speed of light=3*10⁸m/s

we have

(0.2*20000) / 3*10⁸=1.33*10⁻⁵·

since 1.33*10⁻⁵ < 0.01 it is negligible

b. for when l = 50km = 50000m, f = 60Hz and v=speed of light=3*10⁸m/s

we have

(50000*60) / 3*10⁸=0.01·

since 0.01 = 0.01 it is On the border.

c. for when l = 20 cm = 0.2m, f = 600 MHz=6*10⁸Hz, and v=speed of light=3*10⁸m/s

we have

(0.2*6*10⁸) / 3*10⁸=0.40

since 0.40 > 0.01 it is nonnegligible.

d. for when l = 1mm = 0.001m, f = 100GHz and v=speed of light=3*10⁸m/s

we have

(0.001*100*10⁹) / 3*10⁸=0.83

since 0.83 > 0.01 it is nonnegligible