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4-6
Section IV – Operating Instructions
Armed with this last result, and calling the Feed-through stand (F subscript) the “generator” (g subscript)
and terminating sensor (M subscript) the “load” (l subscript), and substituting in the definitions for cal
factor from earlier, we get the more general equation for transferring between a feed-through and a
terminating sensor:
𝑘
𝑀
=
𝑘
𝐹
𝑃
𝑆𝑆𝑆𝑀
𝑃
𝑆𝑆𝑆𝐹
|1
− Γ
𝑀
Γ
𝐹
|
2
Where:
k
M =
Calibration factor of the Terminating Mount
k
F =
Calibration factor of the Feed-through Mount
P
SubM
= Power measured terminating mount
P
SubF
= Power measured Feed-through mount
Γ
Μ
=
Gamma Correction full vector data Terminating Mount
Γ
Φ
=
Gamma Correction full vector data Feed-through Mount
Now in this general equation, the Gamma terms are the reflection scattering parameter of the respective
port noted in the subscript. Gamma is a complex vector with scalar values denoting the real and
imaginary magnitudes:
Γ
≡ 𝜌∠𝜙
=
𝜌
cos
𝜙
+ i
𝜌
sin
𝜙
In the general transfer equation, the term,
|1
−
Γ
𝑀
Γ
𝐹
|
2
is the scalar “gamma correction” or “port
match” term. Inside the absolute value brackets, however, is a vector subtraction. Expanding out to
make the angles explicit, this becomes:
|1
− 𝜌
𝑀
𝜌
𝐹
cos(
𝜙
𝑀
+
𝜙
𝐹
)
− 𝑖 𝜌
𝑀
𝜌
𝐹
sin(
𝜙
𝑀
+
𝜙
𝐹
)|
2
Where the i represents
√−
1
2
, or the “imaginary” component.
The absolute value, or length of a vector, is given by the Pythagorean formula, which is the square
root of the square of the magnitudes of the real and imaginary components. It’s convenient that we
are looking for the square of the magnitude, so we don’t have to worry about the square root part.
Our correction term becomes the scalar,
(1
− 𝜌
𝑀
𝜌
𝑀
cos(
𝜙
𝑀
+
𝜙
𝐹
))
2
+ (
𝜌
𝑀
𝜌
𝐹
sin(
𝜙
𝑀
+
𝜙
𝐹
))
2
When the squares are evaluated, this expands to:
1
−
2
𝜌
𝑀
𝜌
𝐹
cos(
𝜙
𝑀
+
𝜙
𝐹
) +
𝜌
𝑀
2
𝜌
𝐹
2
cos
2
(
𝜙
𝑀
+
𝜙
𝐹
) +
𝜌
𝑀
2
𝜌
𝐹
2
sin
2
(
𝜙
𝑀
+
𝜙
𝐹
)
