Question

Describe how to calculate the reflection coefficient from S-parameters.

Answer 1

To calculate the reflection coefficient (
Γ
Γ) from S-parameters, you need to use the S11 parameter, which represents the reflection coefficient at Port 1 of a two-port network. The S-parameters describe the behavior of a linear network in terms of the incident and reflected waves at each port. The reflection coefficient is a complex quantity and is commonly represented in polar form as
Γ
=
∣
Γ
∣
⋅



Γ=∣Γ∣⋅e
jθ
, where
∣
Γ
∣
∣Γ∣ is the magnitude and

θ is the phase angle.

To calculate the reflection coefficient (
Γ
Γ) from S11, follow these steps:

Obtain the S11 parameter:
S11 is defined as the ratio of the reflected voltage wave (

reflected
V
reflected
    ​

) to the incident voltage wave (

incident
V
incident
    ​

) at Port 1:


11
=

reflected

incident
S
11
    ​

=
V
incident
    ​

V
reflected
    ​

    ​


Convert S11 to the reflection coefficient (
Γ
Γ):
The reflection coefficient is related to S11 as follows:

Γ
=

11
Γ=S
11
    ​


Analyze the magnitude and phase of
Γ
Γ:
The reflection coefficient in polar form can be expressed as:

Γ
=
∣
Γ
∣
⋅



Γ=∣Γ∣⋅e
jθ

The magnitude of
Γ
Γ (denoted as
∣
Γ
∣
∣Γ∣) represents the ratio of the amplitude of the reflected wave to the amplitude of the incident wave:

∣
Γ
∣
=
∣

11
∣
∣Γ∣=∣S
11
    ​

∣

The phase angle of
Γ
Γ (denoted as

θ) represents the phase difference between the reflected and incident waves:


=
arg
(

11
)
θ=arg(S
11
    ​

)

Note: The phase angle

θ is usually expressed in degrees or radians.

Keep in mind that the reflection coefficient
Γ
Γ can also be represented in rectangular form as
Γ
=
Γ
real
+

Γ
imag
Γ=Γ
real
    ​

+jΓ
imag
    ​

, where
Γ
real
Γ
real
    ​

 and
Γ
imag
Γ
imag
    ​

 are the real and imaginary parts of
Γ
Γ, respectively.

In summary, to calculate the reflection coefficient from S-parameters, you need to extract the S11 parameter and convert it to polar form to obtain its magnitude and phase angle.