Question

A 75 ohm coaxial transmission line has a length of 2.0 cm and is terminated with a load impedance of 37.5 + j75 Ohm. If the dielectric constant of the line is 2.56 and the frequency is 3.0 GHz, find the input impedance to the line, the reflection coefficient at the load, the reflection coefficient at the input, and the SWR on the line.

Answer 1

The load reflection coefficient,
`\Gamma =0.62\angle 82.875^(\circ) \Omega`

Reflection coefficient at input,
`\Gamma = 0.62\angle - 147.518^(\circ) \Omega`

SWR = 4.26

Given:

Characteristic impedance of the co-axial cable,
`Z_(c) = 75 \Omega`

Length of the cable, L = 2.0 cm = 0.02 m

`Z_(Load) = 37.5 + j75 \Omega`

Dielectric constant, K = 2.56

frequency, f = 3.0 GHz =
`3.0 * 10^(9) Hz`

Step-by-step explanation:

In order to calculate the reflection coefficient at load, we first calculate these:

The line input impedance
`Z_(i)` is given by:

`Z_(i) = Z_(c)(Z_(Load) + jZ_(c) tan(\beta L))/(Z_(c) + jZ_(Load) tan (\beat L))` (1)

Now, we calculate the value of
`\beta`:

`\beta = (2\pi)/(\lambda') = \farc{2\pi f√(K)}{c}`

(since,
`\lambda' = \farc{c}{f√(K)}`)

`\beta = \farc{2\pi f√(2.56)}{3* 10^(8)} = 100.53`

Now, Substituting the value in eqn (1):

`Z_(i) = 75(37.5 + j75 + j75 tan(100.53* 0.02))/(75 + j(37.5 + j75) tan ( 100.53* 0.02)) = 18.99 - j20.55 \Omega = 27.98\angle - 47.257^(\circ) \Omega`

Now, the load reflection coefficient is given by:

`\Gamma = (Z_(Load) - Z_(c))/(Z_(c) + Z_(Load))}`

Thus

`\Gamma = (37.5 + j75 - 75)/(75 + 37.5 + j75)} = 0.077 + j0.615 = 0.62\angle 82.875^(\circ) \Omega`

Similarly,

Reflection coefficient at input:

`\Gamma' = (Z_(i) - Z_(c))/(Z_(c) + Z_(i))}`

`\Gamma' = (18.99 - j20.55 - 75)/(75 + 18.99 - j20.55)} = - 0.523 - j0.334 = 0.62\angle - 147.518^(\circ) \Omega`

Now, the SWR is given by:

SWR, Standing Wave Ratio =
`(1 +|\Gamma|)/(1 - |\Gamma|)`

SWR =
`(1 +|0.62|)/(1 - |0.62|) = 4.26`